Chapter 1Introduction to Portfolio Management

WHAT IS INVESTMENT?

Man, it is said, lives on hope. But, hope is only a necessary condition for life, but not sufficient. There are many other materialistic things that he needs - food, clothing, shelter, etc. And, like his hope, his needs too keep changing through his life. To make things more uncertain, his ability to fulfill the needs too changes significantly. When his current ability (current income) to fulfill his needs exceeds his current needs (current expenditure), he saves the excess. The savings may be buried in the backyard, or hidden under a mattress. Or, he may feel that it is better to give up the current possession of these savings for a future larger amount of money that can be used for consumption in future.

In contrast to the above situation, if the amount available for current consumption is less than the current needs, he has to engage in negative saving, or borrowing. The funds thus borrowed may be used for current consumption. But, the lender of the money who foregoes his current possession and hence its consumption will ask for more than what he has lent. That is. the borrower should be willing to pay more than he has borrowed.

This trade off between the amount available for present consumption and future consumption is at the heart of all savings and investments. The ratio between the amount of current consumption that can be exchanged for a certain future consumption is called the pure or risk-free rate of interest or pure time value of money. This relationship is influenced by the forces of supply and demand and is determined in the capital markets. If foregoing Rs.100 of certain income today gives a certain income of Rs.104 after one year, the pure rate of exchange or pure rate of interest is said to be 4 percent.

The pure rate of interest referred to above is a real rate as it is based on two amounts of money that are fully certain. If the lender expects the purchasing power of money to fall during the time he lends money, he expects, in addition to the pure or risk-free rate, an amount to compensate him for the fall in the purchasing power of money. If the realization of the future amount is uncertain, he will expect much more and such excess is called the risk premium.

Keeping all the above in view, it can be said that an investment is an agreement for a current outflow of money for some period of time in anticipation of a future inflow that will compensate for the changes in the purchasing power of money, as well as the uncertainty relating to the inflow of the money in future. This understanding describes well all the possible investments, like stocks, bonds, commodities or real estate by all classes of investors like individuals, institutions, governments, etc. In all these investments, the trade off is between a known amount that is invested today, in return for an expected amount in future. While the amount being invested is certain, as it is now in our hands or rather is going out of our hands, the expected future inflow carries with it uncertainties regarding its realization and its real worth will be known only when it is due for realization.

Are all investments speculative?

We know that investment means sacrificing or committing some money today in anticipation of a financial return later. The investor indulges in a bit of speculation as to how much return he is likely to realize. There is an element of speculation involved in all investment decisions. It does not follow though that all investments are speculative by nature.

Genuine investments are carefully thought out decisions. They involve only calculated risks. The expected return is consistent with the underlying risk of the investment. A genuine investor is risk averse and usually has a long-term perspective in mind. The government officer's investment in the units of UTI (transaction 4), the college professor's Reliance stockholding (transaction 5), and the lady clerk's Post Office Savings Deposit (transaction 6), all may be regarded as genuine investments. Each person seems to have made carefully thought out decision and each has taken only a calculated risk.

Speculative investments on the other hand are not carefully thought out decisions. They are based on rumors, hot tips, inside dopes and often simply on hunches. The risk assumed is disproportionate to

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the return expected from speculation. The intention is to profit from short-term market fluctuations. In other words, a speculator is relatively less risk averse and has a short-term perspective for investment. Our friend John's decision to invest all his savings in the new issue of Fraternity Electronics based only on the rumors (transaction 7) may be labelled as speculative investment. John does not seem to have carefully thought out this decision. He is taking a high risk by putting all his savings in just one stock and that too in a new stock.

So, an investment can be distinguished from speculation by (a) the time horizon of the investor and (b) the risk-return characteristics of the investments. A genuine investor is interested in a good rate of return, earned on a rather consistent basis for a relatively long period of time. The speculator, on the other hand, seeks opportunities promising very large returns, earned rather quickly. In this process, he assumes a risk that is disproportionate to the anticipated return.

From the foregoing discussion, it cannot be however, inferred that there exists a clear-cut demarcation between investment stocks and speculative stocks. The same stock can be purchased as a speculation or as investment, depending on the motive of the purchaser. For example, the decision of the professor to invest in the stock of Reliance Industries is considered as a genuine investment because he seems to be interested in a regular dividend income and prospects of long-term capital appreciation. However, if another person buys the same stock with the anticipation that the share price is likely to raise to Rs.350 very quickly and gain from the rise, such decision will be characterized as speculation.

Are Investment and Gambling the Same?

Gambling is defined in Webster's Dictionary as 'An act of betting on an uncertain outcome'. Since the prospective return on investment is uncertain at the time investment is made, one may say that there is an element of gambling involved in every investment. This is particularly so in the case of those investments in respect of which little information exists at the time of investment decision. However, genuine investments cannot be labelled as gambling activities.

In gambling, the outcome is largely a matter of luck; no rational economic reason can be given for it. This is in contrast to what we can say about genuine investments. Unlike investors and speculators, the gamblers are risk lovers in the sense that the risk they assume is quite disproportionate to the expected reward. Though the pay-off, if won, is extraordinary, the chances of winning the bet are so slim that no risk averse individual would be willing to take the associated risk. The cricket fan's bet of Rs.100 on the outcome of test match in England (transaction 3) is an act of gambling; it is not a genuine investment.

It should, however, be noted that a clear demarcation between investment, speculation, and gambling is not always easy. Often it becomes a matter of degree and opinion. Aggressive investors are likely to decide on investments based, among other things, on their speculative and gambling instincts more than the defensive or conservative investors do.

Having understood what genuine financial investments are, let us consider the objectives sought to be fulfilled by investors seeking such investments.

Investment Objectives and Constraints

Investment Objectives

Rationally stating, all personal investing is designed in order to achieve a goal, which may be tangible (e.g., a car, a house, etc.) or intangible (eg., social status, security, etc.). Goals can be classified into various types based on the way investors approach them viz:

a. Near-Term High Priority Goals: These are goals which have a high emotional priority to the investor and he wishes to achieve these goals within a few years at the most. Eg: A new house. As a result, investment vehicles for these goals tend to be either in the forms equivalent to cash or as fixed-income instruments with maturity dates in correspondence with the goal dates. Because of the high emotional importance these goals have, investor, especially the one with moderate means will not go for any other form of investment which involves more risk especially where his goal is just in sight.

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b. Long-Term High Priority Goals: For most people, this goal is an indication of their need for financial independence at a point some years ahead in the future. Eg: Financial independence at the time of retirement or starting a fund for the higher education of a three-year old child. Normally, we find that either because of personal preference or because the discounted present value is large in relation to their resources, the time of realization for such goals is set around 60 years of age for people of moderate means. Because of the long-term nature of such goals, there is not a tendency to adopt more aggressive investment approaches except perhaps in the last 5 to 10 years before retirement. Even then, investors usually prefer a diversified approach using different classes of assets.

c. Low Priority Goals: These goals are much lower down in the scale of priority and are not particularly painful if not achieved. For people with moderate to substantial wealth, these could range from a world tour to donating funds for charity. As a result, investors often invest in speculative kinds of investments either for the fun of it or just to try out some particular aspect of the investment process.

d. Enterpreneurial or Money Making Goals: These goals pertain to individuals who want to maximize wealth and who are not satisfied by the conventional saving and investing approach. These investors usually put all the spare money they have into stocks preferably of the company in which they are working/owning and leave it there until it reaches some level which either the individual believes is enough or is scared of losing what has been built-up over the years. Even then, the process of diversification and building up a conventional portfolio usually takes him a long time involving a series of opportunities and sales spread over many years.

Investment Constraints

An investor seeking fulfillment of one of the above goals operates under certain constraints:

• Liquidity• Age• Need for Regular Income• Time Horizon• Risk Tolerance• Tax Liability

The challenge in investment management, therefore, lies in choosing the appropriate investments and designing a unit that will meet the investment objectives of the investor subject to his constraints. To take on this challenge the first step will be to get acquainted with the different types of investments that are available in our financial market.

Investment Classification

Broadly speaking investment can be categorized as follows:

This study will concentrate more on the financial investment part and so only financial instruments are elaborated with a brief introduction to real investments.

Figure 1.1

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Investment Motives or Goals

The desire to postpone current consumption for higher consumption in future manifests itself in many ways. To take a look at their investment motives, let us divide investors into two classes - individuals and institutions.The common investment goals of individuals are• Buying a new house• Financing child's education• Saving for independence in old age• Saving for a trip abroad• Saving now to start a venture later.

While the goals of individuals are determined by their physical, emotional and other needs, the goals of institutions generally stem from their source of funds and the promises they have made to the providers of funds. Their goals are frequently like

• To generate at least the promised return for the investors. (In the case of a mutual fund which promised a minimum return. This also applies to a defined benefit pension plan.)

• To generate the maximum possible return for all the subscribers. (In case of a defined contribution pension plan, all mutual funds in general and insurance companies in of particular.)

The risk in investment has been ignored completely in describing the investment goals above, We will discuss the investment goals in greater detail in a later chapter, as also the risk perceptions and constraints of the two types of investors.RISKS IN INVESTMENT

Before proceeding to know how risk arises in investments, let us first understand the two terms that are often used interchangeable - risk and uncertainty. In the context of an investment, a situation of certainty is one in which the return from the investment is known for sure. Let us say, an individual invests in government securities and holds them to maturity. The individual can be sure about the redemption of the amount invested on maturity and payment of interest. Therefore, his rate of return is known for sure.

The term risk, in the context of investments, refers to the variability of the expected returns. It is an attempt to quantify the probability of the actual return being different from the expected return. Though there is a subtle distinction between uncertainty and risk, it is common to find the use of both the terms interchangeably.

Types of Risk

The variability of the return or the risk can be segregated into many components, based on the factors that give rise to it. Broadly, risk is said to be made up of three components: business risk, financial risk and liquidity risk. Let us understand them briefly.

Business risk can be easily understood in the context of an investment in a business entity. This risk is

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the variability of returns introduced by the nature of business of the entity invested in. Changes in prices of raw materials and finished goods, changes in supply and demand for raw materials and finished goods, changes in wage rates, changes in fuel costs, changes in the economic lives of assets, changes in tax laws and changes in operating costs are some of the factors that cause business risk. These factors have a direct impact on the profitability of the investee and which in turn influences the share price and the dividend payment or the ability of the firm to repay its debt with interest. The share price at the time of sale and the dividend payments or the interest payments and redemption amount determine the return to an investor. Through this book we will mostly focus on investments in business entities. If we need to draw a parallel in the context of a consumer credit to an employee, it can be related to job security, career prospects. In the context of a government bond it may mean the ability of the government to generate adequate revenues. However, this becomes less relevant because of its ability to monetize a deficit.

Financial risk arises from the financing pattern of the investee company. In other words, it is the variability of the returns from investments made in the company brought about by the financing mix used by the company. If a company uses only equity, its financial risk will be relatively less, as there are no obligatory payments to be made. A company using debt will carry more risk, as the obligatory payments on account of interest and repayment of principal have to be met before any money is available for distribution to the equity investors. And, inability to meet the obligations may result in compulsory liquidation. These factors create variability in the profits of the firm and its share price.

Liquidity risk refers to the uncertainty of the ability of an investor to exit from an investment when she desires. The exit route primarily depends on the secondary market where the securities are traded. Though issuer may step in to provide liquidity in the form of buy back of shares, options on bonds, redemption of securities, all such provisions have a time dimension which is determined by the issuer. However, the term liquidity refers to the ability of investor to exit according to her requirements. When an investor approaches secondary market for liquidity, her concerns are two-fold.• Time taken for liquidation• Price realization.

If the security is illiquid, it may become necessary to sell at a price lower than the market price to reduce the time taken for liquidation. Such discount/reduction in price is called Price Concession. Hence price concession on a security and liquidity are inversely related.

The buyer too faces the same uncertainties - how long will it take to buy it and at what price can it be bought? The greater the uncertainty regarding these two, higher the liquidity risk. Investments like T-Bills can be sold or bought instantly while those like investments in real estate in remote areas take considerable amount of time and effort to buy or sell.

The risk premium mentioned earlier is, therefore, a function of these three types of risks. To sum up, the factors causing volatility are the business risk, financial risk and liquidity risk.

NEED FOR PORTFOLIO MANAGEMENT

A portfolio is a collection of assets. But the question is, why should we invest in a collection or group of assets, rather than a single asset? And, why does it become necessary to manage a group of assets? Let us consider these questions now.To help us understand the answers better, we take the help of the following data.

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Table 1.1 Weekly Returns during the last quarter of 1999

ACC HDFC Tata -0.73 1.05 -4.157.17 2.19 5.18 4.54 -0.10 -5.8010.53 1.22 1.59

-20.44 -8.17 -5.28-0.57 2.91 -3.173.39 0.16 -0.50

-8.78 -2.98 -1.075.74 9.06 -0.58

-3.17 43.18 5.3023.64 11.56 6.21-2.44 17.52 -3.7722.42 -13.94 -1.75

-12.66 0.62 -0.41

Looking at the returns from any of the above companies, we can say that they are fairly volatile. An investor investing in any one of the above companies will have to be ready to face the volatility. But, let us look at the returns from an investment in ACC and HDFC or HDFC and Tata Power in equal amounts.

ACC + HDFC HDFC + Tata Power

0.160 -1.550

4.680 3.685

2.220 -2.950

5.875 1.405

-14.305 -6.725

1.170 -0.130

1.775 -0.170

-5.880 -2.025

7.400 4.240

20.005 24.240

17.600 8.885

7.540 6.875

4.240 -7.845

-6.020 0.105

The standard deviation of the above two combinations (8.56 for ACC + HDFC and 7.62 for HDFC + Tata Power) are considerably less than the standard deviation of returns of ACC (11.66) or HDFC (13.03). The returns from the combination of the two shares, obviously, are less volatile than the returns from any one of the shares. But, what about the returns? The returns, as can be seen, are neither as high nor as low as the returns on the individual stocks. That is, the returns on the stocks have been evened out; reducing the variation, but the reduction in the variation has also reduced the returns. Thus, we can see that by investing in a combination of stocks instead of a single stock, we can alter the total return we get as well as the variability of the return. It means that we can create a new

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investment opportunity, whose risk-return profile is different from the existing investments. While there is a new opportunity, the question is whether it is in any way better than the existing ones. An investment can be considered as a better one if it offers higher returns than the existing ones at a given level of risk or offers a lower level of risk than the existing ones for a given level of return or both.

Figure 1.2

From the above figure we can quickly conclude the following:A is better than B C is better than B C is better than D E is better than D E is better than F G is better than F

By using a combination of say B and D if we are able to create C which is better than B and D, then it is worthwhile to create such a new investment opportunity.

In the earlier example, we have used an equal combination of the two stocks. But, we can combine them in different proportions to get different levels of return and standard deviation of return. The following is the output from a combination of different proportions of ACC and HDFC.

ACCADFC

WEEKS

10

0.80.2

0.60.4

0.40.6

0.20.8

01

1234567891011121314

STD

-0.737.174.5410.53-20.44-0.573.39-8.785.74-3.1723.64-2.4422.42-12.6611.66

-0.3746.1743.6128.668

-17.9860.1262.744-7.6206.4046.10021.2241.55215.148-10.0049.580

-0.0185.1782.6846.806

-15.5320.8222.098-6.4607.06815.37018.8085.5447.876-7.3488.550

0.3384.1821.7564.944

-13.0781.5181.452-5.3007.73224.6416.3929.5360.604-4.6928.940

0.6943.1860.8283.082

-10.6422.2140.806-4.1408.39633.9113.97613.528-6.668-2.03610.590

1.052.19-0.101.22-8.172.910.16-2.989.0643.1811.5617.52-13.940.6213.03

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As can be seen from the above table, the standard deviation of return decreases as the proportion of HDFC increases, and reaches minimum for a 60-40 combination of ACC and HDFC while increasing the proportion of HDFC any further leads to increase in the standard deviation. The increase in standard deviation after the 60-40 proportion suggests that there is an ideal mix of the two securities where the standard deviation is the lowest.

Having established the need for investing in portfolios rather than individual assets, we can now move on to the question of management of portfolios. Portfolios are (or rather, should be) built to suit the return expectations and/or the risk appetite of (he investor. That is, a combination of assets or securities is formulated which meets the level of return he expects provided he is willing to meet the associated risk, or the return possible at the level of risk he is willing to bear. This is because, often, building a portfolio which meets both the return expectations and the risk taking ability of the investor is not possible. After all, who does not like to own a portfolio of risk-free assets yielding (at least) 100 percent per annum?Designing portfolios to suit investor requirements often involves making several projections regarding the future, based on the current information. When the actual situation is at variance from the projections portfolio composition needs to be changed. One of the key inputs in portfolio building is the risk bearing ability of the investor. Change in it also calls for a change in the portfolio composition to match the current risk bearing ability. Investment presupposes some future needs; changes in the needs will demand changes in the portfolio composition. Lastly, the returns and the risk characteristics of the assets which make up the portfolio may undergo a change, warranting changes in the composition of the portfolio.

When so many factors are likely to influence the changes in the portfolio composition, is it necessary to keep a close watch on the portfolio and manage or respond to the changes carefully? The answer is yes, and hence the existence of a separate discipline of finance called Portfolio Management.

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THE PROCESS OF PORTFOLIO MANAGEMENT

Though portfolio management has been in existence for a very long time, its treatment in various works of literature on the subject has not been systematic. The focus has been on matching the characteristics of the assets with the needs of the investors on an ad hoc basis, ignoring the fact that portfolio management is a continuous process and not a set of" discrete events.

Figure 1.3

Figure 1.3: The Portfolio Construction, Monitoring and Revision Process

Portfolio management can be described, according to Maginn and Tuttle , as a systematic, continuous, dynamic and flexible process which involves: Identifying and specifying an investor's objectives, preferences and constraints to develop clear

investment policies Developing strategies by choosing optimal combinations of financial and real as sets available in

the market and implementing the strategies Monitoring the market conditions, relative asset values, and the investor's circumstances Making adjustments in the portfolio to reflect significant changes in one or more relevant

variables.

The above process applies, by and large, to the activities of all kinds of investment managers and investors.

Portfolio Management: An Evaluation

Portfolio managers, to be successful, have to work on any one or more of the following strategies: Timing the market. Selection of superior stocks or groups of stocks. Making changes in the portfolio structure and/or strategy. Having a long-term investment philosophy.While the above strategies do look impressive on paper, empirical studies made on the performance of the fund managers have proved that it is very unlikely that a fund manager consistently outperforms the market from these strategies in the long run and that it is therefore better to hold the market

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portfolio.

The reasons cited in the studies for the failure of market timing are as follows:

• The market performs just as well when the investor is in, as when he is out of it. That is, the returns from the market will be the same, whether the investor has invested into stocks or is holding cash. By being invested for part of the time and moving out of the market rest of the time, the investor is losing the gains during that period of time. Therefore, it is not possible to beat the market with market timing.

• If, from the gains obtained by portfolio managers (like pension funds) over a period of time of about eight years, the gains from those obtained during the ten best days during the period are reduced, the gains obtained from them reduced drastically. If the gains from twenty best days are reduced, the returns fell below those on T-Bills. This means, the trick lies in being invested in stocks during the twenty crucial days than the whole period. And, the proportion of those twenty days being a small fraction of the total period under consideration, it can be concluded that it is highly unlikely that anybody would succeed in timing the market.

Similarly, the studies have revealed that the stocks rejected from their investment or sold expecting a fall in prices have performed as well as the stocks selected by them. Portfolio strategies like investing in growth stocks and investing based on technical too have their share of bad reputation for giving exceptionally high returns during some periods and large negative returns during the other periods. And, very few investment philosophies have been formulated in the past few decades that have stood the test of time.

All the four strategies mentioned earlier have one thing in common - they all depend on the mistakes of the others for success. That is, the expectations of the investor adopting the strategies have to come true and those of the others should not. But, with there being so many investment managers competing with each other, it may be naive to expect others to make errors consistently. This will further strengthen the argument that it is not possible to beat the market.

But, these studies were based on the performance of the stock market over a very long time, up to about four decades. The argument for using such a long time period is that, if somebody starts investing at the age of twenty and continues investing till he is sixty, will he not be in the market for about four decades? If over -frame, investing in the stock market using any of the strategies does not provide returns above the market, then there is no point in investing because the investor will lose what he might gain over the market during a boom. There exists only a timing difference between the gains and losses.

Efficient Portfolios and Efficient Frontier

Let us look at the risk-return attributes of combinations of risky assets. In theory, we can plot all conceivable combinations of risky assets in the risk-return space. Assume that our exercise results in plot of points as shown in figure 1.4.

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Figure 1.4: Risk-Return Possibilities for Assets and Portfolios

It is reasonable to assume that investors would prefer more returns to less; and would prefer less risk to more. Thus, if we can find a set of portfolios which offer higher return for the same level of risk, or offered a lower risk for the same return, we would have identified the portfolios which an investor would like to hold; and other portfolios possibilities can be ignored. Given this background, let us examine figure 1.4. We find that portfolio A would be preferable to portfolio H because it carries less risk at the same level of return, In fact, portfolio A cannot be eliminated from our consideration because there is no other portfolio which has less risk for the same return; or more return for the same risk. Hence we term portfolio A as an efficient portfolio (portfolio A is referred as the global minimum variance portfolio because it is the portfolio that has the lowest risk among all feasible portfolios plotted in the figure).

A re-look at the figure reveals that portfolio B dominates portfolio C because it offers more return for the same level of risk. We also find that portfolio D dominates portfolio E because it offers a higher return for the same level of risk. But portfolio B dominates D because it offers same return for a lower risk. Since there is no portfolio which dominates B either along the risk dimension or along the return dimension, portfolio B can be termed as most dominant portfolio for that level of risk, in other terms it can be called as an efficient portfolio in that plane. From figure 1.4, we find that portfolio F dominates portfolio G because it offers more return for the same level of risk. We also find that portfolio F is an efficient portfolio in that plane and it can be viewed as maximum return portfolio among the portfolios plotted in figure 1.4. The boundary ABF is referred to as the efficient frontier. Therefore, we can formally define an efficient portfolio as follows:An efficient or dominant portfolio is that portfolio which has no alternative withi. the same E(RP) and lower σp

ii. the same σp and higher E(RP) iii. a higher E(RP) and a lower σp.

The curve enveloping all the portfolios that lie between the portfolio that has the highest variance and the portfolio that offer the highest return is the efficient frontier.

The above discussion is, however, neither intended to discourage the student from studying portfolio management nor to suggest that outperforming the market is impossible. The key issue here is a time horizon. While looking at the historical data one can easily identify the time when the returns are abnormally high. And such instances are not exceptions. Hence it depends on the ability of a fund manager to identify/forecast such instances that enable abnormal returns. The empirical studies also

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question the ability of consistently outperforming the market but they do not rule out the possibility of outperforming the market. Hence the need for portfolio management is all the more in practice.

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Chapter 2Capital Market Theory

INTRODUCTION

A risk-averse and rational investor would like to maximize the expected return for a given risk or would like to minimize the risk for a given expected return. Portfolio theory provides a normative approach for the analysis and identification of such risk-minimizing portfolios. Using Markowitz or Sharpe single-index models an investor can identify the set of portfolios that maximize expected return at each level of risk. The set of efficient portfolios thus obtained is efficient frontier. Now every investor, in analyzing the risk and return of individual securities, should choose a portfolio, which lies on the efficient frontier.

One important implication of the normative approach provided by the portfolio theory is pricing of financial assets. If all investors act in a manner that maximizes expected return at a given level of risk, what are the results of this aggregate behavior in terms of the relationship between risk and expected return? Capital market theory relates to the pricing of financial assets and the equilibrium relationship between risk and expected return that results from the aggregate behavior of investors seeking to maximize expected return.

MARKOWITZ MODEL AND EFFICIENCY FRONTIER

Markowitz model of portfolio analysis generates an efficient frontier, which is a set of efficient portfolios. A portfolio is said to be efficient if it offers maximum expected return for a given level of risk or it offers minimum risk for a given level of expected return.

The concept of efficient portfolio can be illustrated with an example. Suppose you have three portfolios A, B and C. Risk-return characteristics of these portfolios are

Portfolio

Expected Return E(ri) %

Standard Deviation of E(ri)

ABC

88 10

12 18 18

If you are to choose between portfolios A and B, you would choose portfolio A since it gives you the same return as B, but has a lower risk than B. That is, portfolio A dominates B and is considered to be superior or efficient. In the same way portfolio C dominates B and is considered to be efficient. If you can identify all such efficient portfolios and plot them, you will get what is called efficient frontier.

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Figure 2.1: Markowitz Efficient Frontier

In figure 2.1, EF is efficient frontier, which is a set of efficient portfolios. Portfolios lying above the efficient frontier are desirable but are not available. Portfolios below the efficient frontier are attainable but not desirable since they are dominated by efficient portfolios. Therefore, efficient frontier is also a frontier demarcating the possible and impossible portfolios.

Markowitz devised an ingenious computational model to trace out the efficient frontier and to identify the portfolios that comprise the efficient frontier. In the calculations he used the technique of quadratic programming. He assumed that one could deal with N securities or fewer. Using the expected return, variances and all pair-wise covariance’s among securities being considered for inclusion in the portfolio, he was able to calculate risk and return for any portfolio comprising of some or all of securities. In particular, for any specific value of expected return, using the programming calculations he determined the least-risk portfolio. With another value of expected return, a similar procedure again yields the minimum risk combination. By this tracing out process, the efficient frontier is derived. When only securities that have non-zero variances (or standard deviation) about their return are included in the analysis, the efficient frontier derived is given in figure 2.1.

Assumptions of Markowitz Model

As with any model building exercise, Markowitz portfolio theory is also based on few assumptions. They arei. Investors are risk-averse and thus have a preference for expected return and dislike for risk. This

is general behavior of a rational investor. An investor would like to get the highest return possible for a given risk or would like minimizing the risk for a given expected rate of return.

ii. Investors act as if they make investment decisions on the basis of the expected return and the variance (or standard deviation) about security return distributions. That is, investors measure their preferences and dislike for investments through the expected return and variances (or standard deviations) about security return.

CAPITAL ASSET PRICING MODEL (CAPM)

The CAPM was developed in mid-1960s. The model has generally been attributed to William Sharpe, but John Linter and Jan Mossin made similar independent derivations. Consequently, the model is often referred to as Sharpe-Linter-Mossin (SLM) Capital Asset Pricing Model. The CAPM explains the relationship that should exist between securities' expected returns and their risks in terms of the means and standard deviations about security returns. Because of this focus on the mean and standard deviation the CAPM is a direct extension of the portfolio models developed by Markowitz and Sharpe. Although the model has been extensively examined, modified and extended in the literature,

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the original SLM version of the CAPM still remains the central theme in the Capital Market Theory as well as in the current practices of investment management.

Using a set of simplifying assumptions, the CAPM is an equation that expresses the equilibrium relationship between security's (or portfolio's) expected return and its systematic risk. Because the CAPM is relatively a simple model, it has been employed in a wide variety of academic and institutional applications such as measuring portfolio performance, testing of market efficiency, identifying under and overvalued securities, determining consensus price of risk implicit in the current market prices and capital budgeting.

Assumptions of CAPM

As we have discussed, CAPM is an extension of Markowitz portfolio theory. The assumptions on which Markowitz portfolio theory is based on are applicable to CAPM also. Apart from the three assumptions listed under Markowitz model, the following additional assumptions are made in deriving the CAPM.

Table 2.1: Assumption for CAPM

Common to both the Markowitz model and CAPM

a. Investors are risk-averse, expected utility maximizers.

b. Investors chose portfolios on the basis of their expected mean and variance

of return.

Additional assumptions

a. Borrowing and lending at the risk-free rate are unrestricted.

b. All investments are perfectly divisible.

c. All investors have uniform, single period investment horizon and

expectations regarding means, variances and covariances of security returns

are homogeneous.

d. There are no imperfections in the market.

e. Capital markets are the equilibrium.

a. There is a riskless asset that earns a risk-free rate of return. Furthermore, investors can lend or invest at this rate and also borrow at this rate in any amount.

b. All investments are perfectly divisible. This means that every security and portfolio is equivalent to a mutual fund and that fractional shares for any investment can be purchased in any amount.

c. All investors have uniform investment horizons and have about homogeneous expectations with regard to investment horizons or holding periods to forecasted expected returns and risk levels of securities. This means that investors form their investment portfolios and revise them at the same intervals of time (say, every six months). Furthermore, there is complete agreement among the investors as to the return distribution for each security or portfolio.

d. There are no imperfections or frictions in the market to impede investor buying and selling. Specifically, there are no taxes or transaction costs involved with security transaction. Thus there are no costs involved in diversification and there is no differential tax treatment of capital gains and ordinary income. This assumption makes possible the arbitraging of ‘mispriced’ securities, thus forcing an equilibrium price.

e. All security prices fully reflect the changes in future inflation expectations. That is, there is no

15

uncertainty about expected inflation.f. Capital markets are in equilibrium. That is, all investment decisions have been made and there is

no further trading without new information.

Lending and Borrowing at the Riskless Rate

The consideration of a riskless asset alters the efficient frontier considerably. Figure 2.2 displays the efficient frontier, in terms of expected return E(r), and standard deviation (a), along with the riskless asset f and three risky portfolios, A, B and M. Since the riskless asset f has no risk, (i.e. σ f = 0), its E(r) and σ plot on the zero-risk, vertical axis at the point IT, which represent the expected rate of return on the riskless asset f.

Figure 2.2: Borrowing and Lending at the Riskless Rate rf and Investing in the Risky Portfolio M

With the riskless asset f and the ability to borrow or lend (invest) at risk-free rate r f, it is now possible to form portfolios that have risky assets as well as the risk-free asset within them. Furthermore, all combinations of any risky portfolio and the riskless asset will lie along a straight line connecting their E(r), σ plots. For example, portfolios containing f and the risky portfolio A will lie along the line segment rfA, as shown in figure 2.2. Similarly, combination of f with either portfolio B or portfolio M will lie along segments rfB and rfM, respectively. Therefore, combining any risky portfolio with a riskless asset produces a linear relationship between their respective E(r), σ points.

The portfolio expected return for any portfolio i that combines f and M is

E (ri) = WfRf + (1 - Wf) E(rM)

Where,

Wf = The percentage of the portfolio invested in the riskless security f

1 - Wf = The percentage of the portfolio invested in the risky portfolio M

The portfolio variance for portfolio i is

σ2i = Wf

2σf2 + (1 - Wf)2σM

2 + 2Wf (1 - Wf) σf,m

Where,

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σf2 = Variance of risk-free asset

σM2 = Variance of portfolio M

σf,m = Covariance between f and M. By definition, σf

2 =0. Thus,σi

2 = (1 - Wf)2 σM2 (or)

σi = (1 - Wf) σM

Combinations of a risky portfolio with the riskless asset are generally referred to as lending portfolios, since some of the investment is invested or lent at the riskless rate r f. That is, 1 > Wf > 0. Therefore, the above equations for E(ri) and σi

2 are lending portfolios.

Illustration 2.1

Suppose the risk-free rate, rf, is 8% and expected return on the risky portfolio, rM, is 20% with a standard deviation of 25%. If an investor would like to invest 20% of his portfolio in the risk-free asset, f, and the balance in the risky portfolio M, risk-return characteristics of the portfolio will be Expected return on the portfolio

E(ri)= Wfrf + (1 - Wf)rM

= (0.20 x 8) + (1 - 0.20)20= 17.6%

Standard deviation of expected return of the portfolioσi = (1 - Wf) σM

= 0.80 x 25= 20%The following table shows portfolio expected returns and standard deviations for various combinations of lending at rf and risky portfolio M.

Wf 0.00 0.25 0.50 0.75 1.00(1 - 1.00 0.75 0.50 0.25 0.00E(ri) 20.00 17.00 14.00 11,00 8.00σi 25,00 18.75 12.50 6.25 0.00

We have assumed that investors can not only lend or invest at the risk-free rate r f, but they can borrow unlimited amount at the same risk-free rate rf. That is, the proportion of funds invested in the risk-free asset, Wf, becomes negative.

When the percentage of portfolio invested in riskless security f is negative, that is Wf < 0, the resulting portfolio is referred to as borrowing portfolio. This is because, additional funds are borrowed at rf

and invested in the risky portfolio. This borrowing portfolio would be analogous to short sale of riskless security, f.

The portfolio expected return for a borrowing portfolio i is

E(ri)=-Wfrf+(l + Wf)E(rM).

The portfolio variance for a borrowing portfolio i is

σi2 = (1 + Wf)2 σM

2

Illustration 2.217

In illustration 2.1 we have seen how investment in f influences the risk-return characteristics of the portfolio. Now, we will see the risk-return characteristics of borrowing portfolio. We assume the same rf and E(rM) and σM as in illustration 2.1.

Suppose, the investor borrows 20% of his portfolio and invests in the market portfolio. That is, W f = - 0.20 and (1 - Wf) = 1.20. Expected return on this portfolio

E(ri) = (-0.20 x 8) + (1.20x20)= - 1.6 + 24.0= 22.4%

Standard deviation of expected return of the portfolio σi = (1 - Wf) = 25

= 1.2 x 25 = 30%

The following table shows portfolio expected returns and standard deviations for various combinations of borrowing at rf and portfolio M.

Wf 0 -0.25 -0.50 -0.70 -1.00

(1-Wf) 1.00 1.25 1.50 1.70 2.00

E(ri) 20.00 23 26 29 32

σi 25.00 31.25 37.50 43.75 50

An important implication of introducing riskless rate of lending and borrowing is the transformation of the efficient frontier. With the introduction of rf, the efficient frontier is transformed into a liner form. Furthermore, as long as E(rM) > rf, investors can continually increase expected return and risk by borrowing increasing amounts at rf and investing the borrowed proceeds in portfolio M. Figure 2.3 shows the efficient frontier with borrowing and lending portfolios.

THE DOMINANT PORTFOLIO M

By borrowing and lending at the riskless rate rf, investors can alter the risk/expected return profile of any efficient portfolio to meet personal preferences for risk and expected return. In figure 2.4, regardless of whether investors want to borrow or lend, portfolio M is the best efficient portfolio. This is because investors can invest in portfolio M and then borrow or lend at r f to suit their preference. That is, by borrowing and lending at rf, in conjunction with investing in portfolio M, they can create portfolio combinations along line rf M, in such a way that for a given level of risk it is possible to find a combination of M and risk-free borrowing/lending which offers a return that is higher than the one available for a portfolio on the efficient frontier. Figure 2.4 illustrates this.

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Figure 2.3: Borrowing and Lending at the Riskless Rate rf and Investing in the Risky Portfolio M

For example, portfolios A and B have risk-return parameters of, σArA and σBrB. Both of them represent portfolios that offer the highest return for the given level of risk. With the introduction of unlimited risk-free borrowing and lending, it is possible to construct a portfolio consisting of M and risk-free lending/borrowing which are represented by A' and B' that have risk levels of σA and σB but have returns higher than rA. and rB. Hence the portfolios on the line rfM always dominate the portfolios on the efficient frontier. Further, all the portfolios along the line rfM dominate portfolios along other two lines, rfB and rfA shown in figure 2.4.

Figure 2.4

Suppose that your acceptable level of risk is σ i’, which is a lending portfolio, with our risky portfolios A, M and B and the riskless asset f. You can opt for portfolios A, F or G. In this case, portfolio G dominates both A and F since for the same risk σi’. G offers greater returns. In the same way for a borrowing (or) leveraged portfolio with a risk of σi’', portfolio H dominates B and I. This way, all portfolios along the line rf GMH dominate portfolios beneath them in terms of either E(ri) or σi.

Because of this dominance, all investors should choose efficient portfolio M in conjunction with their preferences for lending or borrowing at the risk-free rate, rf. Graphically, portfolio M represents the

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tangency point between a ray drawn from the intercept, r f, to the efficient frontier. This line of tangency drawn from rf to M has the greatest slope for any line drawn from r f to the efficient set of risky portfolio. That is, point M is the efficient portfolio that maximizes the value of [E(r) - r f]/ σ, risk premium.

Thus portfolios along this line will maximize E(r) at their respective σ levels, when compared to portfolios along lower rays drawn from rf to any other portfolio along the efficient frontier.

Market Portfolio

Since every investor should choose to hold portfolio M, it follows that portfolio M must be a portfolio containing all securities in the market. Such a portfolio that contains all securities is called the Market Portfolio. Because all investors should choose the market portfolio, it should contain all available securities. If it did not, securities not included would not be demanded by any investor and prices of these securities, therefore, would fall and their expected return would rise. At some point the increased expected returns would be attractive to some investors.

Now let us see why this must be the case. At equilibrium, market value of the risky assets should be equal to the funds available for investing in the risky assets. Now, consider a security A, which constitutes one percent of market value of all risky securities. Assume that each investor places only 0.50 percent of his risky portfolio in security A. That is, 0.50 percent of total funds at risk are invested in security A. But, security A constitutes of one percent of the market value of the total risky assets. These two figures are not consistent since both indicate one and the same thing. This is because the market value of risky assets is nothing but the amount of funds invested in risky assets by all the investors. Therefore, under the assumptions made in this model, the optimal combination of risky securities is that existing in the market. Hence, M is the market portfolio where each risky asset i has the weight (Wi) equal to

Wi =

THE SEPARATION THEOREM

With the inclusion of the riskless asset in the investment opportunity set, all investors should choose the same portfolio M, because that risky portfolio, in conjunction with borrowing or lending at r f, will enable them to reach the highest level of expected return for their level of desired risk. This result is of critical importance to the development of the CAPM.

According to the portfolio theory, each investor should choose an appropriate portfolio along the efficient frontier. The particular portfolio chosen may or may not involve borrowing or the use of leveraged, or short, positions. The investment decision (which efficient portfolio to choose) and the financing decision (whether or not their portfolio involved borrowing, or short sales) were determined simultaneously in accordance with the risk level, identified by the investor at an acceptable level.

Therefore, the investment decision is given and is the same for all investors - everyone should choose to invest in portfolio M. The financing decision, or how much to borrow or lend, will vary from investor to investor according to individual preferences for risk and expected return. That is, the individual investor will invest in portfolio M and then borrow or lend at rf in an amount such that their utility function, as represented by their indifference curves, is just tangent to the line rfM. For example, figure 2.5 illustrates that investor A's optimal portfolio calls for lending; whereas investor B's optimal portfolio one of borrowing. This analysis suggests that both types of investors should hold

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identically risky portfolios. Desired risk levels are then achieved through combining portfolio M with lending or borrowing and the separation of investing and financing decisions is called the separation theorem and provides a fundamental result for the development of the CAPM.

Figure 2.5: Personal Preferences for Risk and Expected Return

Illustration 2.3

Suppose there are two investors A and B. The objective of investor 'A' is to earn a return of 25% and is willing to assume the relevant risk. On the other hand, the objective of investor B is to limit his risk to a variance of 400(%)2 . Assuming same rf E(rM) and σM as in illustrations 2.1 and 2.2 the financing decisions of A and B are determined as follows:

Investor A

E(ri) = Wfrf+(1 -Wf)rM

Targeted E(ri) for A is 25%

.'. 25 = (Wf x 8) + (1 - Wf) 20

25 = 8 Wf + 20 - 20 Wf

-12 Wf = 5

Wf = -5/12 = - 0.4167

That is, A should borrow 41.67% of his portfolio at the rate of 8% and invest in the market portfolio M to obtain the expected return of 25%. Standard deviation of this portfolio

σi = (1 - Wf) 25

.'. = 1.4167 x 25

= 35.42%

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Investor B

Targeted risk σi2 = 400

.'. σi = 20

20 (1 - Wf) 25

1-Wf =

Wf = 1 -

= 0.20

That is, B should invest 20% of this portfolio in risk-free asset f to limit his risk at σ i = 20%. Expected return will be E(ri) = Wfrf + (1 - Wf)rM

= (0.20 x 8) + (0.80 x 20)= 17.6%.

THE CAPITAL MARKET LINE (CML)

With the ability to borrow and lend at the risk-free rate rf, in conjunction with an investment in market portfolio M, the old curved efficient frontier is transformed into a new efficient frontier, which is a line passing from rf, through market portfolio M. This new linear efficient frontier is called the Capital Market Line, or simply the CML. This CML, together with the old efficient frontier, is illustrated in figure 2.6. An inspection of the figure indicates that all portfolios lying along the

Figure 2.6: The Capital Market Line (CML)

CML will dominate, in terms of E(r) and σ, the portfolios along the previous curved efficient frontier. The CML not only represents the new efficient frontier, but it also expresses the equilibrium pricing relationship between E(r) and σ for all efficient portfolios lying along the line. Since the equation for any line can be expressed as y = a + bx, where a represents the vertical intercept and b represents the slope of the line, the pricing relationship given by the CML can be easily determined. In figure 2.6, a = rf and b = [E(rM) - rf]/ σM.

Thus the CML relationship for any efficient portfolio i is provided in equation

22

CML: E(ri) = rf +{[E(rM) - rf]/ σM}σi

In words, the above equation states that the expected return on any efficient portfolio i, E(r i) is the sum of two components: (1) the return on the risk-free investment, rf, and (2) a risk premium, {[E(rM) - rf]/ σM}σi that is, proportional to the portfolio's σi. The slope of the CML (E(rM) - rf/ σM is called the market price of risk, and this component is the same for all portfolios lying along the CML. Thus the factor that distinguishes the expected returns among CML portfolios is the magnitude of the risk, σ i. The greater the σi, the greater the risk premium and the expected return on the portfolio.

Illustration 2.4

Assume that the expected return on the market portfolio M, rM, is 20%, with a standard deviation, σM, of 25%. If the risk-free rate, rf, is 8%, the slope of the CML would be

= 0.48%

The slope of the CML indicates the equilibrium price of risk in the market. In our illustration, a risk premium of 0.48% indicates that the market demands this amount of return for each percentage increase in the portfolio's risk.Now, the CML's intercept would be rf = 8%

Therefore, the CML equation isE(ri) = ri = 8 + 0.48 σi

It is important to recognize that the CML pricing holds only for efficient portfolio that lie along its line. That is, only the most efficient, in terms of risk-reducing potential, portfolios that are constructed of combinations of the risk-free asset f and market portfolio M lie along the CML. All individual securities and inefficient portfolios lie under the curve. For the efficient set of portfolios along the CML, their total risk, as measured by σi, represents their systematic risk, since all unsystematic risk has been diversified. That is, the efficient frontier not only produces the set of optimal portfolios in terms of risk and expected returns, securities expected returns and their covariances with the market portfolio is called the Security Market Line (SML), which is commonly referred to as the but it also represents most efficient set of diversified portfolios at different levels of expected returns. Thus the CML represents portfolios that are not only efficient in a risk/expected return sense, but it also represents zero unsystematic risk portfolios. Since total risk, σ i, is the sum of systematic and unsystematic risk, a portfolio that is well diversified has its total risk equal to its systematic risk. Therefore, for well diversified efficient portfolio that lie along CML, their risk, σ i, can be thought of as either total risk or systematic risk. Thus the CML states that the appropriate measure of risk that is to be priced for these efficient portfolios is the level of systematic risk present in these portfolios.

The Capital Asset Pricing Model (CAPM)

The CML is important in describing the equilibrium relationship between expected return and risk for efficient portfolios that contain no unsystematic risk. It is not, however, the appropriate equitation for explaining the theoretical relationship that should exist between expected return and risk for securities and portfolios in general. Two important questions then, are (1) What is the appropriate measure or risk that investors should use to evaluate the expected returns for all securities and portfolios, efficient or inefficient? And (2) What is the equilibrium relationship that should exist between the expected return on a security and its relevant measure of risk?

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The formula for the various of a portfolio of n securities:

σ n2 = wi Wj σij

When portfolios are equally weighted, that is, when Wi = 1/n, the expected level of portfolio risk can be expressed as:

E(σn2) = 1/n [E(σi

2) – E (σij)] + E(σij)

Where,

E(σi2) = Average vaiance of an individual security that is included in the portfolio

E(σij) = Average pair wise covariance between securities in the portfolio.

Expressing portfolio risk in this manner provides some insights into the effects an individual security has on portfolio risk.As the above equation indicates, whenever the investor adds a security to an existing portfolio, that new security affects expected portfolio risk, E(σn

2), in two ways.

First, it affects the average variance value, E(σ i2). Thus if the variance of new security is greater (less)

than the average variance across the other securities already in the portfolio, then the level of E(σ i2)

will increase (decrease) when the new security is added. However, even if the variance of the new security is large, as the portfolio size, n, increase, the impact of this effect becomes smaller. That is, the total impact of the average variance component on the total risk of the portfolio is very small and equals (1/n) E(σi

2). Thus if n is sufficiently large, the impact of a single security’s variance on the overall risk of the portfolio is negligible. Furthermore, since investor should hold the market portfolio M, n is very large and the impact of a single security’s variance on the total risk of the market portfolio is negligible.

The second, and more important, impact of a security on the expected risk of an investor’s portfolio is through the average covariance element, E(σij). Whenever a security is added to the portfolio, it affects the average covariance component through its relationship with all the other n – 1 securities in the portfolio. Furthermore, as the above equation indicates, a portion of this average covariance term is not affected by n. Thus if the covariance of the new security is greater (less) than the existing average covariance among the securities in the portfolio, the security can significantly raise (lower) the overall portfolio risk.

Therefore, effective diversification involves adding new securities whose returns have low levels of covariance or correlation with the returns of those securities already included in the portfolio. Thus securities whose returns have low or even negative levels of covariance with the returns of the other securities will be in great demand by investors who choose to diversify their holdings. The prices (returns) of these securities should rise (fall) in response to investor’s demand for their desirable diversification benefits. In other words, investors will require less, in terms of expected return, for a security that has a low covariance or correlation with other securities in the portfolio, because of the effect that this security can have on reducing portfolio risk. Conversely, securities whose returns will be required to offer more expected return in exchange for their potential in adding to the overall risk of the investors portfolio.

On the basis of these arguments, it can be concluded that a security’s expected return should be positively related to the level of covariance between that security’s return and the return on the

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investor’s personal portfolio. The greater the covariance, the higher the required return. As we previously stated, since all investors should hold the same portfolio, Market portfolio M, the required return should be a function of the covariance between the security’s return and the market portfolio. The equilibrium relationship between securities expected returns and their covariances with the market portfolio is called the Security Market Line (SML), which is commonly referred to as the Capital Asset Pricing Model (CAPM).

Figure 2.7: The Capital Asset Pricing Model (CAPM)

Figure 2.7 displays the CAPM relationship. We see that, as with the CML, the theoretical relationship is linear. The CAPM is the theoretical relationship that should hold for all securities and portfolios, both efficient and inefficient. In equilibrium, all securities and portfolios' [E(r i), σiM] plots should lie on the CAPM line.

Mathematically, the CAPM relationship is described by the equation of the line depicted in figure 2.7. This equation can be formulated by recognizing that Market portfolio M must also lie on the line. Using the relationship y = a + bx and recognizing that

(σMM) = (σM 2), the CAPM is given by

CAPM: E(ri) = rf + {[E(rM) - rf]/( σM 2)} σiM

The above equation states that the expected return on any security or portfolio i, E(r i), is the sum of two components: (1) the risk-free rate of return, rf, and (2) amarket risk premium, [E(rM) - rf]/( σM 2)/σiM which is proportional to how the security's rate of return co-varies with the market's return. The slope of the

CAPM is given by [E(rM) - rf]/ σM 2 and is same for all securities. The magnitude of the covariance, σ iM

is, what determines how much additional return, over and above rf, security or portfolio i must provide in order to compensate the investor for its covariance with market portfolio, M.

When analyzing these results, it is important to recognize that since all investors can and should diversify by holding market portfolio M, the relevant measure of risk in the pricing of security expected returns is the security's systematic risk, as measured by σ iM. Thus the CAPM says that unsystematic risk should not be priced, since investors can and should costlessly diversify or eliminate this portion. The CAPM relationship depicted in figure 2.7 can also be expressed in terms of a security's (or portfolio's) beta, βi.

As we know,25

βi = σiM/ σM 2

Substituting βi for σiM/ σM 2 IN the CAPM relation above beta version of CAPM is:

CAPM: E(ri) = rf + [E(rM) - rf] βi

This equation says that the risk premium for security or portfolio i equals the market price of risk, E(rM) - rf, times the security's systematic risk as measured by its beta. The greater the beta, the higher should be the required return, assuming, of course, E(rM) > rf.

This version of CAPM is depicted in figure 2.8.

Figure 2.8 The CAPM Relationship in terms of Beta

The CML vs. The CAPM

Before we turn to a discussion on the non-standard forms of the CAPM, let us review the similarities and differences between the CML and the CAPM. The CML sets forth the relationship between expected return and risk for efficient, well-diversified portfolios, whereas the CAPM is a pricing relationship that is applicable for all securities and portfolios, both efficient and inefficient. In both the CML and the CAPM the appropriate measure of risk is the systematic portion of total risk. However, since the CML assumes well-diversified portfolios, its measure of risk, σ i , which is the total risk for a portfolio, is the same as its systematic risk, since there is no unsystematic risk present in well-diversified portfolios. The CAPM utilizes the beta, βi, or covariance, σiM, as its measure of systematic risk.

Finally, it is interesting to note that CML relationship is a special case of the CAPM. To see this, consider equation for the CAPM:

E(r ) = r + [E(rM) - rf]

Recall that

Inserting this result into equation produces:

E(r ) = r + [E(rM) - r ]

For portfolios whose returns are perfectly positively correlated with the market and thus lie along the CML, = 1. Therefore, for these portfolios, the CAPM % relationship reduces to

26

E(r ) = r + {[E(r ) – rf]/ M} i

This is the CML relationship. Thus the CAPM is the. general risk/expected-return pricing relationship for all assets, whereas the CML is a special case of the CAPM and represents an equilibrium pricing relationship that holds only for widely diversified, efficient portfolios.

NON-STANDARD FORMS OF CAPM

Differential Borrowing and Lending Rates

One of the major assumptions of the CAPM is that investors can borrow and lend in any amount at the risk-free rate. It may seem reasonable that investors should be able to lend or invest at some risk-free rate, say the rate on a treasury bill, but nearly all investors have a borrowing rate that generally exceeds their lending rate. That is, most investors cannot borrow at the same rate as the government. Allowing for the existence of differential borrowing and lending rates complicates greatly the equilibrium results of the original CAPM m model.

Figures 2.9 and 2.10 display the implications of multiple interest rates for the CML and the CAPM. As shown in figure 2.9, the CML under the condition of differential borrowing and lending rates starts at rL, the lending rate, then moves along CMLL, the capital market line with lending, until it intersects the lending efficient risky market portfolio, ML. It, then, moves along the curved efficient frontier to the borrowing risky market portfolio MB, and then proceeds outward on CMLB, the CML borrowing line. The dashed portfolio of the CMLL, and CMLB straight lines are not relevant since they pertain to borrowing and lending segments, respectively, that are no longer feasible. As figure 2.9 indicates, the greater the differential between rB and rL the longer will be the curved section of the CML and more portfolios there will be along for which no precise linear pricing relationship exists.

Figure: 2.9: The CML with Differential Borrowing (rB) and Lending (rL) Rates

27

Figure: 2.10: The CAPM with Differential (rB) and Lending (rL) Rates

2.10 displays the effects of differential borrowing and rates for the CAPM pricing relationship. As the two figures indicate, since rB will differ from investor to investor, each investor will, in principle, be facing a different CML and CAPM. Thus there will be no unique equilibrium pricing relationship that exists for all securities across all investors.

Zero Beta CAPM

At the other extreme of having multiple riskless rates, we could assume that there is no riskless asset at which investors can lend or borrow. The absence of a riskless asset means that there is no available investment whose return is certain. Eliminating the existence of a riskless asset has been termed the zero beta versions of the CAPM. This version of the model is illustrated in figures 2.11 and 2.12.

Figure 2.11: The Efficient Frontier with No Riskless Asset and the Zero Beta Portfolio

Figure 2.11 displays the traditional Markowitz efficient frontier with the market portfolio MZ, the global minimum-variance portfolio, (MVP), and portfolio Z, which represents the minimum-variance zero beta portfolio. MZ is termed the market portfolio for the zero beta version of the CAPM since it is at the tangency point of a ray drawn from E(rz), the expected return on the minimum-variance zero beta portfolio Z and the efficient frontier. A zero beta portfolio is a portfolio with no systematic (i.e. (βz = 0) risk. However, unlike the riskless asset, it does have unsystematic risk. Portfolio Z does have some amount of risk and therefore does not lie along the vertical, zero risk, axis. Along line segment ZZ' in figure 2.11 lie the set of portfolios whose returns have zero correlation (and thus have zero betas) with the market portfolio's (Mz) return. Portfolio Z is that portfolio, among the set of portfolios along ZZ', that has the lowest variance. Alternatively, Z represents the portfolio, among all zero beta portfolios, that has the smallest unsystematic risk.

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When no riskless asset exists, investors choose portfolios along the efficient frontier on or above the minimum-variance portfolio, MVP. Portfolio lying between points MVP and Mz are formed from combining MZ and Z in positive proportions, that is, both WM and Wz are greater than zero. However, those portfolio positions above point MZ are constructed by purchasing Mz and selling Z short.

Figure 2.12 displays graphically the zero beta version of the CAPM. In the zero beta version of the CAPM, the pricing line intersects the expected return axis at the point

Figure 2.12: The CAPM with the Zero Beta Portfoliorepresenting the expected return for the minimum-variance zero beta portfolio, E(r z), and then passes through market portfolio Mz. That is, in the absence of a true riskless asset, the equilibrium pricing relationship can be established with Mz and portfolio Z, which has an expected return of E(rz) and the lowest variance among all zero beta portfolios. The pricing relationship for the zero beta CAPM graphed in figure 2.12 is

Zero beta CAPM: E(ri) = E(rz) + [E(rM) - E(rz)] βi

This equilibrium relationship states that the expected return on any security or portfolio i is a linear function of the expected return on the market and the minimum-variance zero beta portfolio, Z.

Tax-adjusted CAPM

The assumption that there are no taxes implies that an investor is indifferent between receiving income in the form of capital gains or dividends; and that all investors hold the same portfolio of risky assets. But given the fact that normally long-term capital gains are taxed at a concessional rate, while dividends are taxed at the marginal rate, we would expect investors under different tax brackets to hold different portfolios of risky assets. For example, individual investors in the upper tax brackets would prefer to hold low-yield-high-capital gains stocks in order to maximize their post-tax return while investors who are subject to lower tax rates or zero tax liability may prefer to hold high-yield stocks. Consequently, the equilibrium price of asset will depend on tax implications.

Michael Brennan was the first researcher to formalize the impact of taxes on the CAPM. Assuming a differential tax treatment for (long-term) capital gains and dividends, he developed the following version of CAPM which is referred to as the tax-adjusted CAPM:

E(ri) = rf (l - T) + bi[E(rM) - rf (1 - T) - TDm] + TDi

Where,

E(ri) = Expected return on stock i rf - Risk-free rate of interest

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βi = Beta coefficient of stock i

Dm = Dividend yield on the market portfolio

Di = Dividend yield on stock i

(Td-Tg)

T = (1_TD

} = tax factor

Td = Tax rate on dividends

Tg = Tax rate on (long-term) capital gains.

Examining equation, we find that if T = 0, the equation boils down to the original CAPM equation. If T is positive, we find that the expected pre-tax return is an i increasing function of the dividend yield. The following illustration clarifies this point.

Illustration 2.5

The risk-free rate of interest is 9% and the expected return on the market is 15%. The dividend yield on the market portfolio is 4%. The tax rate on dividend income is 40%, while the tax rate on long-term capital gains is 20%. Determine the expected return for a stock with a beta of 1.2 and a dividend yield of 6%. Will the expected return change if the dividend yield is taken as 8%?

Solution

The tax factor T =

=

= 0.25

The equation for the tax-adjusted CAPM will be

E(rj) = 9% (1 - 0.25)+ βi [15% - 9%

(1 - 0.25) - 0.25 x 4%] + 0.25 Di

= 6.75 + 7.25βi + 0.25Di

With βi = 1.2 and Di = 6%

E(ri) = 6.75 + 7.25(1.2) + 0.25(6)

= 16.95%

With βi = 1.2 and Di = 8%

E(ri) = 6.75 + 7.25(1.2) + 0.25(8)

= 17.45%.

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Thus, we find that stocks with higher dividend yields are expected to offer higher pre-tax returns than stocks with low dividend yields for the same level of systematic risk. This finding is intuitively appealing because an increase in dividend yield results in a larger tax outflow and as a result, the investor demands a higher pre-tax return.

A positive T has some interesting implications for the investors. The most important implication is that investors should tilt their portfolios towards or away from high-yield stocks depending on their tax status and tax bracket. For instance, investors, in the higher tax brackets, will prefer to hold a higher percentage of low-yield high capital-gains stocks in order to maximize their post-tax return. On the other hand, investors in tax brackets, where Td (marginal tax rate) is not significantly different from Tg (capital gains tax rate) may prefer to hold high-yield stocks.

The other implication is that investors have to contend with additional unsystematic risk when they have tilted portfolios. Put differently, tilted portfolios have more unsystematic risk than portfolios which are well diversified across all yield levels. Therefore, the investor opting for tilted portfolios must determine whether the incremental post-tax return resulting from the tilted portfolio is more than what is mandated by the additional unsystematic risk.

APPLICATION OF CIVIL AND CAPM

We have understood that the Capital Market Line (CML) depicts the linear relationship between expected return and total risk of all efficient portfolios; while the Security Market Line (SML) depicts the linear relationship between expected return and systematic risk of all individual securities and all portfolios. As we have already seen, the CAPM is a general risk/expected return pricing relationship for all assets whereas the CML is a special case of the CAPM. Hence understanding of applications of the CAPM also covers the application of the CML.

There are a number of applications of ex post SML in security analysis and portfolio management. Among these are (a) evaluating the performance of portfolio; (b) tests of asset-pricing theories; and (c) tests of market efficiency. Ex ante SML can be used in identifying mispriced securities. Ex ante SML represents the linear relationship between the expected rates of return for securities and their expected betas. For the discussion on ex ante and ex post SMLs. 'Risk and Return' of our textbook 'Security Analysis'.

Performance Evaluation of Portfolios

The performance of portfolio is frequently evaluated based on the security market line criterion - a large positive alpha being taken as an indicator of superior (above-normal) performance and a large negative alpha being taken as an indicator of inferior (below-normal) performance. The following illustration explains the mechanics involved.

Illustration 2.6

The equation of the ex post SML for the period under review is estimated to be r = 1 + 1.63 i. Evaluate the performance of the mutual fund given below using the SML approach.

Data on Monthly Returns (%)

Mutual Fund BSE-NATNovember, 1998 -3.85 1.62December 4.00 -1.06January, 1999 30.77 20.34February 58.82 24.02March 87.50 47.59

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April -23.46 -13.35May -29.03 -21.40June -4.55 -0.22July -12.38 -7.53August 26.09 8.17September 20.69 9.86October -25.00 -13.26November -8.57 -9.45December 13.54 2.64January, 2000 2.75 2.58February -16.07 -0.46March -25.53 -15.56April -5.00 -5.50May 6.77 5.61June -3.52 1.94July 10.22 4.67August 11.92 14.25September -1.78 5.07

The first step is to calculate the mean monthly returns on the BSE-NAT and the mutual fund; and the beta coefficient of the mutual fund by regressing the portfolio returns on the index returns. These statistics are tabulated below.

Mutual Fund

BSE-NAT

Average Monthly Return Standard Deviation of Monthly Returns Beta Coefficient

4.97%14.76%

1.77

2.63% 27.25%

Plugging in the beta coefficient in the ex post SML equation, we get

ri = 1 + (1.63 x 1.77) = 3.89%

= 4.97 - 3.89 = 1.08%

The positive alpha indicates that the mutual fund scheme has generated above-normal returns.

While the ex post SML approach presents a conceptually sound basis for evaluating the performance of a portfolio, it must be borne in mind that the relative performance of the portfolio, as measured by alpha, can vary depending upon which index is used to determine the beta of the portfolio. We will have more on performance evaluation in a later chapter.

Tests of Asset Pricing Theories

The CAPM pricing model is given by the equation E(ri) = rf + [E(rM) - rf]βi

According to the theory, the expected return on security i, E(r i), is related to the risk-free rate, rf, plus a risk premium, [E(rM) - rf] βi which includes the expected return on the market portfolio. Conceptually,

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all the variables in the above equation are ex ante expectations of what investors believe will be the values for E(ri), rf, E(rM) and βi over the coming relevant investment horizon. However, since large-scale data for individual security expectations do not exist, almost all empirical tests of the CAPM have been conducted using ex post, realized return data. That is, ex post data are assumed to be suitable proxies for expectations.

Generally, a two-step procedure is employed to test the CAPM. In the first step, betas are estimated using the holding period returns for securities and the market index. The Single Index Model (Sharpe Single Index Model) used for estimating the betas is

ri,t = αi,t +βi,t rM, t + εi,t

Where,ri,t = Return on security i in time period tRM,T = Return on the market index in time period tαi,t = A constant, the portion of return on stock i that is not related to the market returnεi,t = Error term, the portion of the security's return that is not captured by αi and βi.

The second step tests whether or not the betas are related to expected returns in the manner predicted by the CAPM. The step involves the estimation of a regression of the formri,t = to,t + t1,t βi + εi,t

Where,ri,t = Realized return (Holding Period Yield) on

portfolio i in period t βi = Beta of portfolio ito,t + t1,t = Regression parameters estimated in

period t εi,t = Error term from the regression.

By running the above regression over different periods, it can be determined whether or not To.t and Ti,t conform to the CAPM theory.

Tests of Market EfficiencySML can also be used for testing market efficiency. As we know, when markets are efficient the scope for abnormal returns will not be there and returns on all securities will be commensurate with the underlying risk. That is, all assets are correctly priced and provide a normal return for their level of risk and the difference between return earned on the asset and required rate of return on the asset should be statistically insignificant if markets are efficient. To test for efficiency we need to estimate the required rate of return along with the realized rate of return. SML comes handy in estimationof the required rate of return.

Identifying Mispriced Securities

The ex ante SML can be used for identifying mispriced (under and overvalued) securities. The following illustration explains the mechanics involved.

Illustration 2.7

The conditional returns on three stocks - Alpha, Beta and Gamma - and on the market index are as follows:

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Economic ScenarioProbabilit

yConditional Expected One-Period Returns

(%)Alpha Beta Gamma Market

Recession and high interest rates

0.20 -27 -26 -8 -15

Recession and low interest rates

025 24 32 -4 16

Boom and high interest rates

0.30 16 64 42 32

Boom and low interest rates

0.25 50 24 40 40

An analyst has estimated the equation of the ex ante SML as R(ri) =12 + 8 i, where R(r ) is the required return on security i. Based on the ex ante SML identify the under and overvalued stocks in the above table.

Solution

The following table provides the expected returns on the stocks and the market index; the covariances of the stock returns and market returns; the variance of market returns; and the beta coefficients of the stocks.

Alpha Beta Gamma MarketA. B.C.D.

Expected Return (%) Covariance of Returns with Market Index (%) Variance of Rectums on Market Index (%) Beta Coefficient (= B/C)

18.5 496

1.14

28516

1.18

20.00 416.00

0.95

20

436

Plugging in the beta coefficients in the equation for the ex ante SML, we get

R(r )= 12 + 8(1.14) = 21.12%

R(rB)= 12+ 8(1.18) = 21.44%

R(r ) = 12 + 8(0.95) = 19.60%

Comparing the required rates of return with the expected returns on the securities, we find that stock Alpha is overvalued; stock Beta is undervalued; and stock Gamma is more or less correctly valued. The plots of the three securities and the security market line are displayed in figure 2.13.

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Figure 2.13: Security Market Line and Security Valuation

We must note that the expected returns on the three securities can be expressed as

E(ri) =

Where,

E(Pi) = Expected price of security i at the end of the one-period time horizon

P0 = Current market price

E(Div) = Expected dividend per share during the given time horizon.

Therefore, if a security is classified as an undervalued security by the SML it implies that the market price P must increase from the current level to a higher level such that the expected return equals the required rate of return. Buying the security at this point of time and selling it off after the price has appreciated to the required level will produce an abnormal gain (return). Similarly, if a security is classified as an overvalued security, its market price must decline below the current level to that level at which the expected return will equal the required rate of return. Short selling this security will produce an abnormal gain.

Stocks plotting off the security market line provide evidence of mispricing in the market. There is always bound to be some degree of mispricing in the securities market; and there are three reasons for this phenomenon. The first is transactions costs that may reduce investors' incentive to correct minor deviations from the SML; the cost of adjustment may be greater than or at least equal to the potential opportunity presented by the mispricing. Second, investors subject to taxes might be reluctant to sell an overvalued security with a capital gain and incur the tax. Finally, imperfect information can affect the valuation of a security. Some investors are less informed than others and may not observe mispricing and hence not act on these opportunities.

Figure 2.14: Security Market Line in the presence of Market Imperfections

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Figure 2.14 is an illustration of how the SML would look when actual market conditions are as we just described. In this case, all securities would not be expected to lie exactly on the SML. Therefore, in practice, the SML is a band instead of a thin line. The width of this band varies directly with the imperfections in the market.

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Chapter 3Portfolio Analysis

In 1952, Markowitz proposed the modern portfolio theory approach to investing. He assumed that an investor has a fixed amount to invest at the present for a particular time horizon which is known as the investor's holding period. After the expiration of the holding period, the investor will sell the securities that were purchased at the starting of the holding period. Subsequently the investor can either consume or reinvest or partly consume and partly reinvest. This approach can be viewed as a single-period approach, where the starting of the period is denoted as t = t0, while the end of the period is denoted as t – t1. The investor is required to take a decision on the type of the securities he would like to purchase at t = to and hold until t – t1. Since a portfolio contains various securities, this decision can be viewed as equivalent to selecting an optimal portfolio from a set of probable portfolios. Hence it is referred as portfolio selection problem. In deciding about where to invest at t – t0, the investor should estimate the expected returns on various securities under consideration and then invest in the one with the highest expected returns. At the same time investor should also consider the uncertainty attached with the realization of the return.

Risk and Investor Preferences

The main substance in portfolio theory is to find an efficient frontier or locus of possible portfolio opportunities. After determining this locus, the next question that arises is, how should investors select the most suitable option?

Figure 3.1: Figure of Indifference Curves for Risk-Averse Investor

The most appropriate way to know about the best option on the efficient frontier is to assess the satisfaction an investor receives from the investment opportunities. The risk-return trade-off on any portfolio determines an investor's perception towards that portfolio. We will try to judge how the risk of a portfolio affects the investor's preference. Indifference curves or utility functions represent an investor's preference for risk and return and can be drawn on a two-dimensional figure, where the horizontal axis indicates risk as measured by standard deviation (denoted by p) and the vertical axis indicates benefit measured by expected return (denoted by rp). Let us draw a set of preferences or indifference curves for a hypothetical investor as shown in the figure.

Each curve represents equal satisfaction along its length. Higher curves indicate more desirable

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situation attached to it compared to the lower indifference curves. Each curved line indicates one indifference curve for the investor and represents all combinations of the portfolio that provide the investor with a given level of desirability. The figure depicted above shows the indifference curves in increasing order of desirability. The investor with the indifference curves in the above figure would find portfolios A and B equally desirable, even though they have different expected returns and standard deviations. This happens because both the portfolios lie on the same indifference curve I2. Portfolio B has a higher standard deviation (18%) than portfolio A (12%) and is, therefore, less desirable on that dimension. However, exactly offsetting this loss in desirability is the gain in desirability provided by the higher expected returns of B (11%) compared to A (8%). This example proves that all portfolios resting on a given individual indifference curve are equally desirable to the investor. This important feature of the indifference curves implies that two indifference curves cannot intersect. To verify this, assume two indifference curves L and L intersect at a point X. Since all the portfolios on I are equally desirable, therefore, X t which lies on L is also equally desirable. Similarly, all the portfolios resting on Iq are equally desirable. Because X also lies on L, it is equally desirable compared to all portfolios resting on L. Now, given that X is on both indifference curves, all the portfolios on L must be equally desirable to as those on I2. However, this situation (indicates a conflict) does not agree with the basic characteristics of L and L which states that two curves are supposed to represent different levels of desirability. Hence it can be inferred that these curves cannot intersect.

Since most investors would expect more return for additional risk consumed, indifference curve will be always positively sloped. In contrast to risk-averse investors, a set of indifference curves for risk-lover investors will have negative sloping and skewed towards the origin.

The degree of slope associated with indifference curves will indicate the degree of the risk aversion for any chosen investor. The comparison between an aggressive and a conservative investor has been shown below. The conservative investors will require substantial increase in return for assuming small increase in risk. On the other hand, aggressive investors may satisfy with small increase in returns for accepting the same increase in risk.

Although differences may occur in the slope of indifference curves, they are assumed to be positive sloping for most rational investors. Now coming to map of indifference curves for a risk-averse investor depicted in figure 3.1 he would find portfolios A and B equally desirable but he would find portfolio C with an expected return of 10% and standard deviation of 14%, preferable to both of them. This is because portfolio C rests on an indifference curve 13, that is located to the north of 12- Portfolio C has a sufficiently large expected return relative to A, which more than offsets its higher standard deviation, and on balance, makes it more useful than A. Similarly C has sufficiently smaller standard deviation than B which more than offsets its smaller expected return and, on balance, makes it more desirable than portfolio B. This characteristic leads to the second important feature of the indifference curves which implies that an investor will find any portfolio that is lying on an indifference curve that is further north to be more desirable than any portfolio lying on an indifference curve that is not as further north.

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Figure 3.2: Indifference Curves for Different Types of Risk-Averse Investors

Lastly, it is important to note that there can be an infinite number of indifference curves for an investor. Therefore, it will be always possible to plot an indifference curve between the two given indifference curves. As shown below in figure 3.3, we can plot a third indifference curve I* lying between indifference curves 1* and I2. Again it is quite possible to plot another indifference curve above I2 and yet another below Ij. Now the question arises: How does an investor determine what his or her indifference curves will look like? The indifference curve for each investor will be unique. One possible method to determine the indifference curve involves presenting the investor with a set of hypothetical portfolios, along with their expected returns and standard deviations. After this, he or she would be asked to choose the most desirable portfolios. Given the choice that is made, the shape and locations of investor's indifference curves can be estimated. Here it is expected that the investor would have acted as if he or she has indifference curves in making this choice, even though indifference curves would not have been explicitly used.

Figure 3.3: Plotting an Indifference Curve between Two Others

In short, we can say that each investor has a map of indifference curves representing his or her preferences for expected returns and standard deviations. This means that the investor should plot the curve for each possible return-risk combination as shown in figure 3.1. Then from these curves the

39

investor makes a choice of his portfolio which lies on the indifference curve which is further north-west.

The Efficient Set Theorem

As discussed earlier, an infinite number of portfolios can be formed from a set of N securities. Suppose, an investor is considering stocks of 4 firms namely Reliance, Tisco, Infosys and Ranbaxy, for his investment purposes. Therefore, N in this case is equal to 4. The investor could invest only in Infosys, or invest only in Reliance. Alternatively, the investor could purchase a combination of Infosys and Reliance stocks. The investor could invest in all the four firms. For example, the investor could put 50% of his or her money in Reliance and Infosys or 25% in Reliance and 75% in Infosys or 33% in Infosys and 67% in Reliance or any percent (between 0% and 100%) in Infosys with the rest going into Reliance. Therefore, without even considering the other two companies i.e. Ranbaxy and Tisco, there are infinite number of possible portfolios that could be purchased. Now the question arises, does the investor need to evaluate all these portfolios. Fortunately, the answer is "no". The key to why the investor needs to look at only a subset of the available portfolios lies in the efficient set theorem, which states that

a. All investors will choose their optimal portfolio from the set of portfolios that

• offer maximum expected returns for varying levels of risks,• offer minimum risk for varying levels of expected returns.

b. All the sets of the portfolios satisfying these two conditions are known as the efficient sets or efficient frontiers.

The Feasible Set

Figure 3.4 represents the location of the feasible set, also known as the opportunity set, from which the efficient set can be identified. The feasible set simply represents all portfolios that could be formed from a group of N securities. That is, all possible portfolios that could be constructed from the N securities lie either on or within the boundary of the feasible set. The points denoted as K, L, M, N and O in figure 6.4 are examples of such portfolios. Normally, this set will have an umbrella type shape similar to the one illustrated in figure 6.4. The exact shape of the feasible set depends on the particular securities involved, it may be more to the right or left, or higher or lower, or flatter or thinner than shown in the figure. However, the shape of the feasible set, except in the unique situations, looks almost similar to what appears in the figure 6.4.

The efficient set can now be traced by applying the efficient set theorem to this feasible set. First, the set of the portfolios that satisfy the first condition of the theorem should be located. From the above figure, it is quite clear that, there is no portfolio offering less risk than portfolio N, This is because if a vertical line was drawn through N, there would be no point in the feasible set that was to the left of the line. Again there is no portfolio offering more risk than that of portfolio L. This is because if a vertical line was drawn through L, there would be no point in the feasible set to the right of the line. Thus the set of portfolios giving maximum expected returns for different levels of risk is the set of portfolios lying on the western boundary of the feasible set between points M and L.

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Figure 3.4: Feasible and Efficient Sets

Now, coming to the second criterion of the efficient set theorem, there is no portfolio offering an expected return greater than portfolio M, because no point in the feasible set lies above a horizontal line going through M. Similarly, there is no portfolio offering a lower expected return than portfolio K, because no point in the feasible set lies below a horizontal going through K. Thus the set of the portfolios offering minimum risk for varying levels of the expected return is the set of portfolios lying on the western boundary of the feasible set between points K and M. As we know that both the conditions have to meet in order to identify the efficient set, it can be seen that only those portfolios lying on the boundary between points N and M do so. Therefore, these portfolios form the efficient set, and it is from this set of efficient portfolios that the investor will choose his or her optimal portfolio. Rest of the other feasible portfolios are inefficient portfolios and should be avoided.

Portfolio Effect in the Two-Security Case

We have shown the effect of diversification on reducing risk. The key was not that two stocks provided twice as much diversification as one, but that by investing in securities with negative or low covariance among themselves, we could reduce the risk. Markowitz's efficient diversification involves combining securities with less than positive correlation in order to reduce risk in the portfolio without sacrificing any of the portfolio's return. In general, the lower the correlation of securities in the portfolio, the less risky the portfolio will be. This is true regardless of how risky the stocks of the portfolio are when analyzed in isolation. It is not enough to invest in many securities; it is necessary to have the right securities.

Let us conclude our two-security example in order to make some valid generalizations. Then we can see what three-security and larger portfolios might be like. In considering a two-security portfolio, portfolio risk can be defined more formally now as:

σp = √X2xσ2

x + X2yσ2

y + 2XxXy(rxyσxσy) (i)

where:σp = portfolio standard deviationXx = percentage of total portfolio value in stock XXy = percentage of total portfolio value in stock Yσx= standard deviation of stock Xσy= standard deviation of stock Yrxy = correlation coefficient of X and Y

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Note= rxyσxσy = covxy

Thus, we now have the standard deviation of a portfolio of two securities. We are able to see that portfolio risk (σp is sensitive to (1) the proportions of funds devoted to each stock, (2) the standard deviation of each stock, and (3) the covariance between the two stocks. If the stocks are independent of each other, the correlation coefficient is zero (rxy = 0). In this case, the last term in Equation (i) is zero. Second, if rxy is greater than zero, the standard deviation of the portfolio is greater than if r xy = 0. Third, if rxy is less than zero, the covariance term is negative, and portfolio standard deviation is less than it would be if rxy were greater than or equal to zero. Risk can be totally eliminated only if the third term is equal to the sum of the first two terms. This occurs only if first, rxy = -1.0, and second, the per-centage of the portfolio in stock X is set equal to Xx = σy / (σx + σy).

To clarify these general statements, let us return to our earlier example of stocks X and Y. In our example, remember that:

STOCK X

STOCK Y

Expected return (%) Standard deviation (%)

92

94

We calculated the covariance between the two stocks and found it to be —8. The coefficient of correlation was —1.0. The two securities were perfectly negatively correlated.

CHANGING PROPORTIONS OF X AND Y

What happens to portfolio risk as we change the total portfolio value invested in X and Y? Using Equation (i), we get:

STOCK X (%)

STOCK Y (%) PORTFOLIO STANDARD DEVIATION

100 0 2.080 20 0.866 34 0.020 80 2.80 100 4.0

Notice that portfolio risk can be brought down to zero by the skillful balancing of the proportions of the portfolio to each security. The preconditions were rxy = -1.0 and X = σx / (σx + σy) or 4/(2 + 4) = .666.

Changing the Coefficient of Correlation

What would be the effect using x = 2/3 and y = 1/3 if the correlation coefficient between stocks X and Y had been other than -1.0? Using Equation 17.2 and various values for rxy, we have:

Y PORTFOLIO STANDARD -0.5 1.34* 0.0 1.9

+ 0.5 2.3 + 1.0 2.658

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*σp = √(.666)2 (2)2 + (.334)2 + (2) (.666)(.334)(-.5)(2)(4) = √1.777 + 1.777 – (.444)(4) = √1.777 = 1.34

If no diversification effect had occurred, then the total risk of the two securities would have been the weighted sum of their individual standard deviations:

Total undiversified risk = (.666)(2) + (.334)(4) = 2.658

Because the undiversified risk is equal to the portfolio risk of perfectly positively correlated securities (rxy = +1.0), we can see that favorable portfolio effects occur only when securities are not perfectly positively correlated. The risk in a portfolio is less than the sum of the risks of the individual securities taken separately whenever the returns of the individual securities are not perfectly positively correlated; also, the smaller the correlation between the securities, the greater the benefits of diversification. A negative correlation would be even better.

In general, some combination of two stocks (portfolios) will provide a smaller standard deviation of return than either security taken alone, as long as the correlation coefficient is less than the ratio of the smaller standard deviation to the larger standard deviation:

rxy < σx / σy

Using the two stocks in our example:

-1.00 < 2/4

-1.00 < +.50

If the two stocks had the same standard deviations as before but a coefficient of correlation of, for example, +.70, there would have been no portfolio effect because +.70 is not less than +.50.

Graphic Illustration of Portfolio Effects

The various cases where the correlation between two securities ranges from -1.0 to +1.0 are shown in Figure 3.5. Return is shown on the vertical axis and risk is measured on the horizontal axis. Points A and B represent pure holdings (100 percent) of securities A and B. The intermediate points along the line segment AB represent portfolios containing various combinations of the two securities. The line segment identified as rah = + 1.0 is a straight line. This line shows the inability of a portfolio of perfectly positively correlated securities to serve as a means to reduce variability or risk. Point A along this line segment has no points to its left; that is, there is no portfolio composed of a mix of our perfectly correlated securities A and B that has a lower standard deviation than the standard deviation of A. Neither A nor B can help offset the risk of the other. The wise investor who wished to minimize risk would put all his eggs into the safer basket, stock A.

The segment labeled rab = 0 is a hyperbola. Its leftmost point will not reach the vertical axis. There is no portfolio where σp = 0. There is, however, an inflection just above point A that we shall explain in a moment.

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Figure 3.5 Portfolios of Two Securities with Differing Correlation of Returns

The line segment labeled rab = — 1.0 is compatible with the numerical example we have been using. This line shows that with perfect inverse correlation, portfolio risk can be reduced to zero. Notice points L and M along the line segment AGB, or rab = -1.0. Point M provides a higher return than point L, while both have equal risk. Portfolio L is clearly inferior to portfolio M. All portfolios along the segment GLA are clearly inferior to portfolios along the segment GMB. Similarly, along the line segment APB, or rab = 0, segment BOP contains portfolios that are superior to those along segment PNA.

Markowitz would say that all portfolios along all line segments are ''feasible" but some are more "efficient" than others.

The Three-Security Case

In Figure 3.6 we depict the graphics surrounding a three-security portfolio problem. Points A, B, and C each represent 100 percent invested in each of the stocks A, B, and C. The locus AB represents all portfolios composed of some proportions of A and B, the locus A C represents all portfolios composed of A and C, and so on. The general shape of the lines AB, AC, and BC suggests that these security pairs have correlation coefficients less than + 1.0.

What about portfolios containing some proportions of all three securities? Point G can be considered some combination of A and B. The locus CG is then a three-security line. The number of such line segments representing three-security mixtures can be seen from Figure 3.10, where any point inside the shaded area will represent some three-security portfolio. Whereas the two-security locus is generally a curve, a three-security locus will normally be an entire region in the Rp, σp diagram.

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Figure 3.6 Three-Security Portfolios

Consider for a moment three portfolio points within the Rp,σp diagram in Figure 3.10. Call the portfolios P1, P2, and P3. If we stop to think for a moment, the number of three-security portfolios is enormous—much larger than the number of two-security portfolios. Faced with the order of magnitude of portfolio possibilities, we need some shortcut to cull out the bulk of possibilities that are clearly nonoptimal. Looking at portfolios P1 and P2, we might observe the fact that because P2 lies to the left of and below P1, P2 is probably more appealing to the conservative investor, and P1 appeals to those willing to gamble a bit more. Would a rational investor select P3? We think not because it involves a lower return than P2 but has the same risk. Thus, we say that a portfolio is inefficient, or dominated, if some other portfolio lies directly above it (or directly to the left of it) in the risk-return space.

In general, an efficient portfolio has either (1) more return than any other portfolio with the same risk or (2) less risk than any other portfolio with the same return. In Figure 3.7 the boundary of the region identified as the curve LC dominates all other portfolios in the region. Portfolios along the segment AL represent inefficient portfolios because they show increased risk for lower return. Each point

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Figure 3.7 Regions of Portfolio Points with Three Securities

The Sharpe Index Model

William Sharpe, who among others has tried to simplify the process of data inputs, data tabulation, and reaching a solution, has developed a simplified variant of the Markowitz model that reduces substantially its data and computational requirements.

First, simplified models assume that fluctuations in the value of a stock relative to that of another do not depend primarily upon the characteristics of those two securities alone. The two securities are more apt to reflect a broader influence that might be described as general business conditions. Relationship between securities occur only through their individual relationships with some index or indexes of business activity. The reduction in the number of covariance estimates needed eases considerably the job of security-analysis and portfolio-analysis computation.

Figure 3.8 Attainable Efficient Frontiers (Return less assumed 5 percent inflation rate.)46

Thus the covariance data requirement reduces from (N2 - N)/2 under the Markowitz technique to only N measures of each security as it relates to the index. In other words:

NUMBER OF

MARKOWITZ SHARPE INDEX SECURITI

ES COVARIANCES COEFFICIENT

S 10 45 10 50 1,225 50

100 4,950 100 1,000 499,500 1,000 2,000 1,999,000 2,000

However, some additional inputs are required using Sharpe's technique, too. Estimates are required of the expected return and variance of one or more indexes of economic activity. The indexes to which the returns of each security are correlated are likely to be some securities-market proxy, such as the Dow Jones Industrial Average (DJIA) or the Standard & Poor's 500 Stock Index. The use of economic indexes such as gross national product and the consumer price index was found by Smith to lead to poor estimates of covariances between securities. Overall, then, the Sharpe technique requires 3N + 2 separate bits of information, as opposed to the Markowitz requirement of [N(N + 3)]/2.

Sharpe's single-index model has been compared with multiple-index models for reliability in approximating the full covariance efficient frontier of Markowitz. The more indexes that are used, the closer one gets to the Markowitz model (where every security is, in effect, an index). The result of multiple-index models can be loss of simplicity and computational savings inherent in these shortcut procedures. The research evidence suggests that index models using stock price indexes are preferable to those using economic indexes in approximating the full covariance frontier. However, the relative superiority of single versus multiple index models is not clearly resolved in the literature.RISK-RETURN AND THE SHARPE MODEL

Sharpe suggested that a satisfactory simplification would be to abandon the covariances of each security with each other security and to substitute information on the relationship of each security to the market. In his terms, it is possible to consider the return for each security to be represented by the following equation:

Ri = αi + βiI + ei

where.

Ri = expected return on security iαi = intercept of a straight line or alpha coefficient, - βi = slope of straight line or beta coefficientI = expected return on index (market)ei = error term with a mean of zero and a standard deviation which is a constantIn other words, the return on any stock depends upon some constant (α), plus some coefficient (β), times the value of a stock index (I), plus a random component (e). Let us look at a hypothetical stock and examine the historical relationship between the stock's return and the returns of the market (index).

If we mathematically "fit" a line to the small number of observations, we get an equation for the line of the form y = a + βx. In this case the equation turns out to be y = 8.5 — .05x.

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Figure 3.9 Security Returns Correlated with DJIA

The equation y = a + βx. has two terms or coefficients that have become commonplace in the modern jargon of investment management. The α, or intercept, term is called by its Greek name alpha. The β, or slope, term is referred to as the beta coefficient. The alpha value is really the value of y in the equation when the value of x is zero. Thus, for our hypothetical stock, when the return on the DJIA is zero the stock has an expected return of 8.5 percent [y = 8.5 -.05(0)]. The beta coefficient is the slope of the regression line and as such it is a measure of the sensitivity of the stock's return to movements in the market's return. A beta of +1.0 suggests that, ignoring the alpha coefficient, a 1 percent return on the DJIA is matched by a 1 percent return on the stock. A beta of 2.5 would suggest great responsiveness on the part of the stock to changes in the DJIA. A 5 percent return on the index, ignoring the alpha coefficient, leads to an expected return on the stock of 12.5 percent (2.5 times 5 percent). While the alpha term is not to be ignored, we shall see a bit later the important role played by the beta term or beta coefficient.

The Sharpe index method permits us to estimate a security's return then by utilizing the values of α and β for the security and an estimate of the value of the index. Assume the return on the index (I) for the year ahead is expected to be 25 percent. Using our calculated values of α = 8.5 and β = -.05 and the estimate of the index of I = 25, the return for the stock is estimated as:

Ri = 8.5 - .05(25) = 8.5 - 1.25 = 7.25

The expected return on the security in question will be 7.25 percent if the return on the index is 25 percent, and if α and β are stable coefficients.

For portfolios, we need merely take the weighted average of the estimated returns for each security in the portfolio. The weights will be the proportions of the portfolio devoted to each security. For each security, we will require α and β estimates. One estimate of the index (I) is needed. Thus:

Rp = Xi (αi + βiI)where all terms are as explained earlier, except that Rp is the expected portfolio return, Xi is the proportion of the portfolio devoted to stock i, and N is the total number of stocks.

The notion of security and portfolio risk in the Sharpe model is a bit less clear on the surface than are return calculations. The plotted returns and some key statistical relationships are shown below.

YEAR

SECURITY

RETURN

INDEX RETURN

(%)

48

1 6 202 5 403 10 30Average = 7 30Variance from average = 4.7 66.7 Correlation coefficient = -.189Coefficient of determination = .0357

Notice that when the index return goes up (down), the security's return generally goes down (up). Note changes in return from years 1 to 2 and 2 to 3. This reverse behavior accounts for our negative correlation coefficient (r).

The coefficient of determination (r2) tells us the percentage of the variance of the security's return that is explained by the variation of return on the index (or market). Only about 3.5 percent of the variance of the security's return is explained by the index; some 96.5 percent is not. In other words, of the total variance in the return on the security (4.7), the following is true:Explained by index = 4.7 X .0357 = .17 Not explained by index = 4.7 X .9643 = 4.53Sharpe noted that the variance explained by the index could be referred to as the systematic risk. The unexplained variance is called the residual variance, or unsystematic risk.

Sharpe suggests that systematic risk for an individual security can be seen as:

Systematic risk = β2X (Variance of index) = β2σ2I

= (-.05)2(66.7)

= (.0025)(66.7)

= .17

Unsystematic risk = (Total variance of security return) - (Systematic risk)

= e2

= 4.7-.17 = 4.53

Then:

Total risk = β2σ2I + e2

And portfolio variance is

where all symbols are as before, plus:

σp2 = variance portfolio

σI2 = expected variance of index

ei2 = variation in security’s return not caused by its relationship to the index

49

Generating the Efficient Frontier

Using the required inputs, the Sharpe model and a computer, a series of "corner" portfolios was generated rather than an infinite number of points along the efficient frontier. The traceout of the efficient frontier connecting corner portfolios is shown in Figure 3.10. The Table given below shows the stocks and relative proportions invested at several corner portfolios.

Corner portfolios are portfolios calculated where a security either enters or leaves the portfolio. Corner portfolio 1 is a one-stock portfolio. It contains the stock with the greatest return (and risk) from the set—in this case, USAir. Notice in theTable that the return of 20.5 percent (.205) and the standard deviation or risk of 16.6 percent (.1659) for corner portfolio 1 (USAir) correspond to the earlier calculations shown to arrive at these figures. The computer program proceeds down the efficient frontier finding the corner portfolios. Corner portfolio 2 is introduced with the appearance of a second stock, High Voltage Engineering. Typically, the number of stocks increases as we move down the frontier until we reach the last corner portfolio—the one that provides the minimum attainable risk (variance) and the lowest return. To understand better what is happening between any two successive corner portfolios, examine numbers 8 and 9. Between these two, Pitney Bowes stock makes its initial appearance.

Figure 3.10 Efficient Frontier Connecting "Corner" Portfolios

The actual number of stocks entering into any given efficient portfolio is largely determined by boundaries, if any, set on the maximum and/or minimum percentage that can be devoted to any one security from the total portfolio. If these percentages (weights) are free to take on any values, the efficient frontier may contain one- or two-security portfolios at the low or high extremes. Setting maximum (upper-bound) constraints assures a certain minimum number of stocks held. The efficient frontier in Figure 3.10 had no constraints placed upon weights.

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Chapter 4Selection of the Optimal Portfolio

Selection of the Optimal Portfolio

The next question arises, how will the investor select an optimal portfolio? As shown in figure 4.1, the investor should first draw his efficient set of the portfolios and then he has to plot his indifference curves on this figure of the efficient set and then proceed to choose the portfolio that is on the indifference curves, which will have minimum risk for the required level of return. This portfolio will correspond to the point where an indifference curve is just tangent to the efficient set. This can be seen from the figure that this is portfolio O on indifference curve I2. Although the investor will prefer a portfolio on 13, but no such feasible asset exists as there is no contact of the indifference curve with the efficient set. With regard to Ij, there are several portfolios that the investor could choose (for example P). However, the figure shows that portfolio O dominates such portfolios because it is on the indifference curve that is further north. Point O is the only point which is on the efficient frontier and also on one of the indifference curves. In other words, at point O the indifference curve is tangent to the efficient set frontier.

Figure 4.1: Selecting an Optimal Portfolio

For highly risk-averse and slightly risk-averse investors, the position of the indifference or preference curves will change and accordingly they will choose their optimum portfolio. For example, as depicted in figure 4.2, a highly risk-averse investor will choose a portfolio close to N while a slightly risk-averse investor will select a portfolio close to M.

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Figure 4.2: Portfolio Selection for a Highly Risk-Averse Investor

Figure 4.3; Portfolio Selection for a Slightly Risk-Averse Investor

From our earlier discussion on indifference curves, it is evident that an investor will select the portfolio that put him or her on the indifference curve further north-west. This is rightly incorporated in the efficient set theorem which suggests that the investor need not be concerned with portfolios that do not lie on the north-west boundary of the feasible set. Indifference curve for the risk-averse investors will be always positively sloped and convex. It can be easily shown that the efficient set is generally positively sloped and concave, reflecting that if a straight line is drawn between two points on the efficient set, the straight line lies below the efficient set. This characteristic of the efficient set is important because it means that there will be only one tangency point between the investor's indifference curves and the efficient set and on this point the optimum portfolio should lie.

Optimal Portfolio Selection Using Lagrangian Multiplier

Before starting discussion on the optimal portfolio selection using Lagrangian multiplier, it is important to understand the basic steps followed in the constrained optimization with Lagrangian multipliers.

Constrained Optimization with Lagrangian Multiplier:

Differential calculus is used to maximize or minimize a function subject to given constraints. Let us

52

assume a function f(x,y) is subject to a constraint g(x,y) - k (a constant). A new function F can be formed by following the three steps.

1. Setting the constraint equal to zero2. Multiplying the constraint by (the Lagrangian Multiplier)3. Adding the product found in the second step with the original function f(x,y).

Therefore, setting the constraint equal to 0 will give the g(x'v) -k = 0 or k-g (x,y) = 0, now we will multiply this equation with the Lagrangian multiplier and add this result to the original function f(x,y) to get the required function F.

F (x,y. ) = f(x,y) + (k - g(x,y)).

Here F (x,y, ,) is the Lagrangian function, f(x,y) is the original objective function and g(x,y) is the constraint. Since the constraint is always set equal to zero, the product (k - g(x, y)) also equals to zero, and the addition of term does not change the value of the objective function. Critical values x, y and A, at which the function is optimized, are found by taking the partial derivatives of F with respect to all three independent variables, setting them equal to zero and solving simultaneously:

Fx (x, y, ) = 0, F (x, y, ) = 0, Fy (x, y, ) = 0

To understand the whole optimization process, we will use the following example.

Example 4.1

Optimize the function

Z = 4x2 + 3xy + 6y2

Subject to the constraint

x + y - 56

1. Setting the constraint equal to zero 56 - x - y = 0

Multiplying it by A and adding it to the objective function will form the Lagrangian function Z.

Z = 4x2 + 3xy + 6y2 + (56 - x - y) ......(i)

2. We will take the first order partial derivatives equal to zero for all three variables and solve the three resultant equations simuitaneously.

First we take the partial derivatives of the above equation with respect to x, y, . and equate it to zero.

Zx = 8x + 3y - = 0 ......(ii)

Zy = 3x + 12y- = 0 .....(iii)

Zx = 56-x-y = 0 .....(iv)

Subtracting equation (iii) from equation (ii) will eliminate and gives the following relation between

53

the two variables x and y:

5x - 9y = 0

=> x = l.8y

Substituting the value of x in the equation (iv) gives

56-1.8y-y =0

=> y = 20

By substituting the value of y in the above equation, we can calculate the value of x and which is equal to

x = 36 and = 348

Using all these three values, we can calculate the value of the original function Z.

Z = 4(36)2 + 3(36X20) + 6(20)2 + 348(56 - 36 - 20) = 4(1296) + 3(720) + 6(400) + 348(0) = 9744.

After understanding the concept of the Lagrangian multiplier, we are now ready to apply this concept for the portfolio selection problem. As we have discussed about the efficient portfolios and efficient frontier in earlier section, the next step in portfolio selection problem is to generate the set of the efficient portfolios from innumerable portfolio possibilities. Practically, it is not feasible to plot all conceivable portfolio possibilities (given a large number of securities) in the risk-return space and then delineate the efficiency frontier. So, we can use either constrained optimization with Lagrangian multiplier using calculus or the quadratic programming approach to select the optimal portfolios.

The first step in using either of these approaches is to formulate the problem. The problem of selecting the set of the efficient portfolio is referred as portfolio selection problem and it can be formulated as follows.

Minimize Z = Var (Rp) (total risk of the portfolio p)

=

Subject to the following constraints

1.

2.

3. Xi > 0 for I = 1,2 . . . .N

Where.

Xi, Xj = Proportions of fund invested in assets i and j

54

RII = Average return on the asset iRp = Required rate of return on the portfolio p

= Covariance of returns on assets i. and j.

The basic function of the above problem is to minimize the total risk of the optimal portfolio to be selected within the ambit of the given constraints. The constraints incorporated in the problem represent the following conditions:

i. The proportion of funds invested in the different assets must add up to unity.

ii. The weighted average of the returns on the different assets must provide the targeted one-period rate of return from the selected portfolio.

iii. There is no provision for short sales - negative values for one or more of the decision variable X1, X2, Xn mean that those assets have been short sold and the sale proceeds have been invested in the other assets. By introducing constraints of the Xi > 0 (non-negative constraints) we have banned short sales.

The reader must note that the constraints presented in the problem are not exhaustive. Often the constraints to be incorporated in the problem are investor specific. For example, an approved mutual fund cannot invest more than 10% of its corpus in the securities of one company; and cannot invest more than 15% of its corpus in the securities of one industry. Such regulatory constraints need to be included in the problem function. Even if there are no regulatory constraints, there can be constraints dictated by the liability structure of an institutional investor, which may require a steady stream of income to be generated every year. Therefore, a constraint which reflects a minimum current yield must be incorporated in the above problem such as

Where,

Ci represents the current income (dividend) yield on the ith asset,

D is the overall minimum current yield.

We have said that above portfolio selection problem can be tackled using either the calculus or the computational algorithms applicable to non-linear programming problems. Of the two approaches, the computational algorithms for solving non-linear Lp are more versatile because they can handle both equality and inequality types of constraints. The problem discussed above is sometimes referred as Quadratic Programming Problem (QPP) because the objective function which is the expression for the variance of the returns on the portfolio consists of second degree terms such as X i

2 and XiXj, There are standard computer packages available for solving the QPP. The UNDO (Linear Interactive Non-linear Optimizer) developed by Linus Scharge is a user friendly computer package that can be used to solve linear and quadratic programming problems. GINO (General Interactive Non-linear Optimizer) is another user friendly software package used for solving non-linear programming problems. There are also investment management related software packages available like MARKOW and SHARPE which can generate optimal portfolios using either the variance-covariance matrix or the single-index model. The example presented in the next section uses calculus.

55

Example 4.2

A fund manager has selected stocks of Ranbaxy, Infosys and BSES for a particular investor's portfolio. Accordingly, he collected the following data which pertains to the monthly average returns, standard deviations of the monthly returns and the pair wise covariances of monthly returns on the stocks of Infosys Ranbaxy, and BSES.

Equity Stock Matrix

ExpectedMonthlyReturn

StandardDeviation

(%)Variance - Covariance Matrix

Ftanbaxy Infosys BSESRaribaxy Inlosys BSES

478% 2.64%

-4.073%

14.97% 13.82% 13.99%

223.93 185.30 119.01

185.30 191.00 -26.63

119.01 -26.63 195.72

Calculate the proportions of funds to be invested in each of the three securities by fund manager so as to generate a return of 3% per month on the portfolio consisting of these securities.

To generate the optimal portfolio, first we will formulate the problem. For this purpose, we define X1, X2 and X3 as the proportions of funds to be invested in the equity stocks to Ranbaxy, Infosys and BSES. Using the data provided above the portfolio problem can be formulated as

Minimize (Total risk of Portfolio)

Z = 223.93 + 191 + 195.72

+ 2 x 185.3 x X1X2 + 2 x 119.01 X1X2 + 2 x (-26.63) X2X3 .... (i)

Subject to

4.78X1 + 2.64X2 + (-4.073)X3 = 3% .... (ii)

X1 + X2 + X3 = 1 ... (iii)

We have to incorporate the return constraint (ii) as part of the objective function (i) using the lagrangian constraint . For this, first we will express X3 in terms of X] and X2 as (1 - X] - X2), using constraint (iii) in both the equations (i) and (ii) and this replacement will transform the equation (i) into two variable equations.

Z = 223.93 + 191X22 + 195.72(1 - Xi - X2)2 + 2(185.3 XiX2) + 2 (119.01)X1(l –X1 -X2) + 2(-

26.63)X2(1-X1-X2)

= 223.93 + 191X22 + 195.72

(1 + + X22 + 2X1X2 - 2Xi - 2X2) + 370.6XiX2 + 238.02(Xi – X1

2 – X1X2)

- 53.26 (X2-X1X2- )

56

= (223.93 + 195.72 - 238.02)

+ (191 + 195.72 + 53.26)X22

+ Xi(-391.44 + 238.02) + X2(-391.44 - 53.26) + XiX2 (391.44 + 370.6 - 23H.02 + 53-26) + 195.72

= 181.63 + 439.98X22 - 153.42 X1

- 444.70 X2 + 577.28 XiX2 + 195.72

Z = 181.63 + 439.98 X22 - 153.42 X1

- 444.70 X2 + 577.28 X1X2 + 195.72

Now we will transform the constraint equation (ii) in two variables.

4.78X1 + 2.64X2 - 4.073(1 - Xi - X2) = 3 8.853X1 +6.713X2- 7.073 = 0 8.853X1 + 6.713X2 = 7.073.

As explained earlier in lagrangian multiplier approach we have to incorporate the above equation in the objective function using the lagrangian multiplier

Z = 181.63 + 439.98 - 153.42X1

- 444.70 X2 + 577.28 X1X2 + 195.72 + (8.853X1 + 6.713X2 - 7.073) .... (iv)

Thus we find that new objective function involves minimizing both the variance of portfolio returns and the deviations between the targeted return and the expected portfolio return. Now we equate the first derivative with respect to X1, X2 and equal to zero.

= 363.26 X1 + 577.28 X2 + 8.853 - 153.42 .....(v)

= 577.28 X1 +879.96 X2+6.713 -444.70 ......(vi)

= 8.853 Xi + 6.713 X2 - 7.073 ... (vii)

By simplifying the equations (v), (vi) and (vii) we get

363.26 Xi + 577,28 X2 + 8.853 = 153.42 .... (viii)

577.28 X1 + 879.96 X2 + 6.713 = 444.70 ..... (ix)

8.853 X1 + 6.713 X2 = 7.073 ..... (x)

57

Now solving the three equations, (viii), (ix) and (x) we get the required value of X1 and X2.

X1 = 0.4889

X2 = 0.4088

Therefore, X3 = (1 - 0.4889 - 0.4088) = 0.1023

Thus the optimal portfolio with an expected monthly return of 3% involves investing 48.89% of the funds in Ranbaxy's stocks, 40.88% in Infosys's stock and 10.23% in BSES's stock. The total risk associated with mis portfolio, measured in terms of standard deviation, will be 13.085%.

(Calculated by substituting the values of the decision variables X1, X2 and X3 in the original equations).

Optimal Portfolio Selection Using Sharpe’s Optimization

In this section, we will learn how to select the optimum portfolio using the Sharpe optimization model. First we present the ranking criteria that can be used to order the stocks for selection of the optimum portfolio. Next we present the technique for employing this ranking device to form an optimum portfolio.

The Formation of Optimal Portfolios

The construction of the optimal portfolio would be greatly facilitated, and the ability of the portfolio managers and security analysts to relate to the construction of the optimum portfolios greatly simplified if a single number measures the desirability of including the stock in the optimum portfolio. If any person is willing to accept the standard form of the single-index model as describing the co-movement between the securities the justification of any stock in the optimum portfolio is directly related to its excess return-to-beta ratio. Excess return is the difference between the expected return on the stock and the risk-free rate of interest such as rate of return on the government securities. The excess return-to-beta ratio measures the additional return on a stock (excess return over the risk-free rate) per unit of non-diversifiable risk. This ratio gels an easy interpretation and acceptance by security analysts and portfolio managers, because they are interested to think in terms of the relationship between potential rewards and risk. The numerator of this ratio of excess return-to-beta contains the extra return over the risk-free rate. The denominator is the measurement of the non-diversifiable risk that we are subject to by holding risky assets rather than riskless assets.

Excess return-to-beta ratio =

Where,

Ri = the expected return on stock i

RF = the return on a riskless asset

= the expected change in the rate of return on stock i associated with a 1% change in the market return

If the stocks are ranked by excess return-to-beta (from highest to lowest), the ranking represents the

58

desirability of any stocks inclusion in the portfolio. This implies that, if a particular stock with a specific ratio of Rj - Rp/ is included in the optimal portfolio, all stocks with a higher ratio will also be included. On the other hand, if a stock with a particular (Rj - Rp)/ is excluded from an optimal portfolio, all stocks with a lower ratio will be excluded. When the single-index model is assumed to represent the covariance structure of security returns, then a stock is included or excluded, depending only on the size of its excess return-to-beta ratio. The number of stocks to be selected depends on a unique cut-off rate which ensures that all stocks with higher ratios of (Ri - Rp)/ will be included and all stocks with lower ratios should be excluded. We will denote this cut-off rate by C*.

The following steps are necessary for determining which stocks are included in the optimum portfolio:

1. Calculate the excess return-to-beta ratio for each stock under consideration and the rank from the highest to lowest.

2. After ranking the securities the next step is to find out a cut-off point with the use of the following formula.

Where,

= Variance in the market index

= Variance of stock's movement that is not associated with movement of the market index. This is the stock's unsystematic risk

Ri = Expected return on stock i

RF = Risk-free rate of return

= Beta of the stock.

3. The optimum portfolio consists of investing in all stocks for which (Ri - Rp)/ is greater than a particular cut-off point C.

Ranking Securities

To illustrate the ranking process, we are taking an example. The following table gives the necessary data to apply our ranking process to determine an optimal portfolio.

Example 4.3

Security No.

Mean return Rj (%)

Beta Unsystemat

ic risk 1 20.0 1.2 202 14.0 1.0 303 12.0 2.0 40

59

4 16.0 0.9 205 24.0 1.1 156 18.0 1.1 507 19.0 0.8 168 13.0 1,3 259 11.0 1.4 30

10 9.0 1.6 10

Risk-free rate of the return is 8% and the variance of the market return is 25%.

We will start with the first step of ranking the securities by calculating the excess return-to-beta ratio which is as follows:

SecurityNo.

I

Meanreturn

Ri

Beta ExcessReturnRi -RF

ExcessReturn-to-

Beta

Rank

1 20.0 1.2 12.0 10.00 32 14.0 1.0 6.0 6.00 63 12.0 2.0 4.0 2.00 94 16.0 0.9 8.0 8.89 55 24.0 1.1 16.0 14.54 16 18.0 1.1 10.0 9.09 47 19.0 0.8 11. 0 13.75 28 13.0 1.3 5.0 3.85 79 11.0 1.4 3.0 2.14 8

10 9.0 1.6 1.0 0.63 10

After ranking the securities the next important step in measurement of the optimum portfolio is to establish a cut-off rate C*.

Establishing a Cut-off Rate C*

As explained earlier, all the securities whose excess-ret urn-to-risk ratios are above the cut-off rate are selected and all whose ratios are below are rejected. The value of C* is measured from the characteristics of all of the securities that belong in the optimum portfolio. To determine C* it is necessary to calculate its value as if there were different number of securities in the optimum portfolio. The value of Ci is calculated when i securities are assumed to belong to optimal portfolio because securities are ranked from the highest excess return-to-beta to lowest. We know that if a particular security belongs to the optimal portfolio, all high-ranked securities also belong the optimal to portfolio. We proceed to calculate the value of the variable Cj for each security as follows.

RankNo.

Secu-rityNo.

Beta Unsys-

tematic

Risk

Excess

Return

(Ri - RF)

ExcessReturn – to – Beta

Ci Zi

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

60

1 5 1.1 15 16.0 14.54 1.173 0.08067

1.173 0.08067

9.7207 0.2798

2 7 0.8 16 11.0 13.75 0.550 0.04000

1.723 0.12067

10.7238 0.1513

3 1 1.2 20 12.0 10.00 0.720 0.07200

2.443 0.19267

10.4998

4 6 1.1 50 10.0 9.09 0.220 0.02420

2.663 0.21687

10.3671

5 4 0.9 20 8.0 8.89 0.360 0.04050

3.023 0.25737

10.1657

6 2 1.0 30 6.0 6.00 0.200 0.03333

3.223 0.29070

9.7460

7 8 1.3 25 5.0 3.85 0.260 0.06760

3.483 0.35830

8.7446

8 9 1.4 30 3.0 2.14 0.140 0.06533

3.623 0.42363

7.8144

9 3 2.0 40 4.0 2.00 0.200 0.10000

3.823 0.52363

6.7828

10 10 1.6 10 1.0 0.63 0.160 0.25600

3.983 0.77963

4.8595

First, value of the variable Q is calculated for the first ranked securities (i = 1) and accordingly values of Q for other less-ranked securities are calculated. These Cj are candidates for the cut-off rate C. All the necessary calculations are shown in the table and the values of the C; are shown in column 11.

The value C* is that optimum value of Cj for which all securities used in the calculation of Q have excess return-to-beta above C, and all securities not used to calculate Q have excess return-to-beta below Q As evident from the table, for only security 5 and 7, the values of the excess return-to-beta ratio is greater than their values of Q. For example, excess return-to-beta ratio for securities 5 is 14.54 which is greater than the value of its C], which is equal to 9.7207. Similarly for security 7 value of excess return-to-beta ratio is greater than the value of Q. Therefore, optimum portfolio will contain only securities 5 and 7 and cut-off rate will be 10.7238. There will always be one and only one cut-off rate C.

Constructing the Optimal Portfolio

Once the cut-off rate is determined, we know which security will figure in the optimum portfolio. The next step is to calculate the proportion to be invested in each security. The proportion invested in each security is

Xi =

Where,

The second expression determines the relative investment in each security while the first expression simply gives the weights on each security so they sum to one, and thus ensures full investment. Note

that the residual variance on each security plays an important role to determine how much to invest

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in each security. Applying this formula to our example

Z1 = (1.1/15)(14.54-10.7238)

= 0.2798

Z2 = (0.8/16)(13.75 - 10.7238)

= 0.1513 2

= (0.2798 + 0.1513) - 0.4311

Percentage of fund to be invested in security 5 = 0.2798/0.4311 x 100 = 64.9%

Percentage of fund to be invested in security 7 -0.1513/0.4311 x 100- 35.1%

Dividing each Zi by the sum of Zs, we find that we should invest 64.9% of our fund in security 5 and 35.1% -of our fund in security 7.

Let us stress that this is identical to the result that would be achieved had the problem been solved using the established quadratic programming codes. However, the solution has been reached in less time with a set of relatively simple calculations. It is interesting to notice that the characteristics of a stock that make it desirable and the relative attractiveness of stocks can be determined before the calculations of the optimum portfolio are begun. The desirability of any stock is solely a function of its excess return-to-beta ratio. Thus a portfolio manager following a set of stocks can determine the relative desirability of each stock before the information from all analysts is combined and the portfolio selection begun. We have assumed throughout our discussion that all stocks have positive beta. We believe that there are sound economic reasons to expect all stocks to have positive betas and that the few negative beta stocks that are found in large samples are due to measurement errors. However, negative beta stocks and zero beta stocks can be easily incorporated in the analysis.

We know that the risk and return of a portfolio is not a simple aggregation of the risk and return of the individual securities that form the portfolio in most of the cases. Portfolio analysis deals with the calculation of risk and return of different portfolios. We shall analyze the risk and return of different portfolios that can be constructed with the help of a given set of stocks. We will also try to understand the portfolio diversification process for the risk reduction and finally, we will analyze the various portfolio management strategies.

Components of Risk and Returns

Portfolios are constructed to be held over some time period. We can calculate portfolio's expected return using the historical data or using the probability of future returns on the constituent securities. Portfolio theory is primarily concerned with the ex ante events which indicate expected future events. All portfolio decisions are for future, and hence we should consider ex ante values. Conversely, if we want to evaluate portfolio performance, we should calculate the actual return and risk for past periods i.e. ex post values. It is important to understand that ex ante values will be always projected values, while ex post values will be always actual values.

Ex ante Return of a Portfolio

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The expected return on any portfolio can be calculated as a weighted average of the individual security's expected returns. The weights used must be the proportions of total investable funds in each security. The total portfolio weight will, therefore, be 100%. For a portfolio of two securities:

Expected portfolio return (Ep) = W1 E1 + W2 E2 Where,

E1 is the expected return on security 1

W1 is the proportion of money invested in security 1

E2 is the expected return on security 2

W2 is the proportion of money invested in security 2.

Similarly, the expected return of a portfolio of n securities, Ep is given as

Ep = .... Eq. A

Where,

Ep is the portfolio return

Wi is the proportion of investment in security i

E(Ri) is the expected return on security i

n is the total number of securities in the portfolio.

The following example will clarify.

Example 4.4

Calculate the return on a portfolio, which has 40% of its fund in asset A and rest in asset B. The probable returns in different conditions of the economy are as follows:

Condition of

economy

Probability of occurrence

A's Return

B's Return

Growth 40% 16% 10%Stable 50% 9% 8%Recession 10% - 4% -2%

Solution

A's Return = (0.40)(0.16) + (0.50)(0.09) + (0.10)(-0.04) = 0.105 or 10.5%

B's Return = (0.40)(0.10) + (0.50)(0.08) + (0.10)(-0.02) = 0.078 or 7.8%

WA = 40%, WB = 60%

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Expected Return on Portfolio = 0.40 x 10.5% + 0.6 x 7.8%

= 8.88%

Regardless of the number of securities in a portfolio, or the proportions of total funds invested in each security, the expected return on the portfolio is always a weighted average of expected returns of individual securities in the portfolio.

Ex Post Return of a Portfolio

Ex post return of a portfolio is nothing but the weighted average of the historical returns of the securities held in a portfolio. Historical return of any security can be calculated as the holding period yield of that security. In general, holding period yield for the ith asset in time I can be calculated using the following formula:

Holding period yield = .... Eq. B

Where,

Pit is the current price of the security

Pit - 1 is the price of the security at the beginning of period t

Dt is the dividend received during period t.

While using the above formula, the dividend is assumed to have been received at the end of the holding period.

Example 3.5

An investor bought 100 shares of Infosys on 30th April 1999 for Rs.8000 per share. The company paid a dividend of Rs.300 on 30th April 2000. If the price of the stock, on 1st May 2000, is Rs.8500 then calculate the holding period yield to the investor for one year time horizon.

HPY =

To calculate the ex post return of any portfolio, we must calculate the historical returns of individual securities in the portfolio. After getting the values of the historical returns, we can measure the portfolio returns by multiplying the proportions of the funds invested in each security with the historical returns on each security. To understand this concept consider the following example.

Example 4.6

Find the ex post return of the portfolio using the following data.

A portfolio consists of 30% of HDFC, 40% of Reliance and 30% of ACC.

StockPrice as on 30.06.1999

(Rs.)

Price as on 01.07.2000

(Rs.)

Yearly Dividend

(Rs.)

Rate ofReturns

(Rs.)HDFC Bank 78.00 263.10 2.76 239.56%

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Reliance industries Ltd.

184.30 337.00 6.30 86.27%

Associated Cement (ACC) 188.45 -124.10 1.10 -33.56%Companies

Rate of return is computed as

HDFC =

Reliance =

ACE =

Portfolio Return

= 0.30 x 239.56% + 0.40 x 86.27%

+ 0.30(-33.56)

= 96.31%

Risk of a Portfolio

Risk is the chance that actual returns will differ from their expected values. The expected value of return can be obtained from probability estimates for ex ante data. We must know the expected distribution of returns to estimate the risk. Portfolio risk is measured by the variance (or the standard deviation) of the portfolio's return. As we explained in previous section, the expected return of the portfolio is a weighted average of the expected returns of the individual securities in the portfolio. However, the risk (as measured by the variance or standard deviation) of a portfolio is not a weighted average of the risk of the individual securities in the portfolio. Symbolically we can write

Var(Rp)

The portfolio risk depends not only on the risk of individual securities in the portfolio, but also on the correlation or covariance between the returns on the securities of the portfolio. Portfolio risk can be defined as the function of each individual security's risk and the covariances between the returns on the individual securities. If we represent the portfolio risk in terms of variance it can be stated in the following way:

Var . . . (A)

Where,

Var(Rp) = The variance of the return on the portfolio

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Var(Rj) = Variance of return on security i.

Cov(RjRj) = The covariance between the returns of securities i and j

Wi,Wj = The percentage of investable funds invested in securities i and j.

The double summation sign indicates that n (n - 1) numbers are to be added together (i.e. all possible pairs of value for i and j when i j.)

Example 4.7

From the following data, calculate the return and risk of a portfolio containing 60% of stock A and 40% of stock B.

Market condition

Probability

ECRA) E(RB)

BoomGrowth Recession

0.250.500.25

40% 20% 10%

40%30%20%

Expected return on stock A

0.25 x 40 + 0.50 x 20 + 0.25 x 10

= 10 + 10 + 2.5 - 22.5%

Expected return on stock B

= 0.25 x 40 + 0.50 x 30 + 0.25 x 20 = 30% Portfolio return = 0.60 x 22.5% + 0.40 x 30% = 25.5%

Variance of stock A's return

= 0.25 (40 - 22.5)2 + 0.50 x (20 - 22.5)2

+ 0.25 (10 - 22.5)2 = 118.75%2

Variance of stock B's return

= 0.25 (40 - 30)2 + 0.5 (30 - 30)2 + 0.25 (20 - 30)2

= 50%

CovAB = (40 - 22.5) (40 - 30) 0.25 + (20 - 22.5) (30 - 30) 0.50 + (10 - 22.5) (20 - 30) 0.25 = 75%2

Portfolio risk = 0.602 x 118.75 + 0.402 x 50 + 2 x 0.60 x 0.40 x 75

= 86.75

= 9.314%

Covariance66

The covariance on an absolute scale determines the degree of association between any two variables. In the present context the two variables are the returns for a pair of securities. Covariance can be defined as the extent to which the two variables move together. These two variables can move either in the same direction or in the opposite direction. The covariance of returns between the two securities can be:

1. Positive, indicating that the returns on the two securities will move in the same direction during a given time. If the return on one security is increasing (decreasing), then the return on the other security will also increase (decrease). In other words, both - securities should move together in the same direction. The value of the covariance will indicate the magnitude of change in a security return when there is a change in the return on the other security.

2. Negative, indicating that the return on the two securities will move in the opposite direction, i.e. the movement of their returns is inversely related. If the return on one security is increasing (decreasing), the return on the other security decreases (increases).

3. Zero, indicating that the returns on two securities do not have any relation and they are independent.

To understand, the role covariance plays in determining the portfolio risk, consider a portfolio having two stocks A and B and the proportion of the portfolio devoted to each stock is XA and XB respectively. The total risk of this portfolio, as we have discussed earlier, can be written as follows.

Notice what the covariance ( ) does. It is the expected value of the product of two deviations: the deviations of the returns on stock A from its mean return (RAi - A) and the deviations of stock B from its mean (RBi - B). In this sense, it is quite similar to variance. However, covariance is the product of deviations of two different stock returns.

The value of covariance will be large, when good or bad outcomes for the stocks A and B occur together.

In this situation, the covariance will be a large positive number for two good outcomes. When the bad outcomes for both A and B occur together, the covariance will be the product of two large negative numbers, which is positive. Therefore, occurrence of good and bad outcomes of the two stocks together will result in large value for the covariance and a higher variance for the portfolio than otherwise. However, if good outcomes for stock A are expected to be associated with bad outcomes of stock B and vice versa, the covariance will be negative. This negative covariance comes from the product of a positive deviation for one stock and negative deviation for second stock.

Clearly, covariance indicates how returns on stocks move together. If both stocks have positive and negative deviations at the same time, the covariance will be a large positive number. On the other hand, if positive and negative deviations occur at different times, the covariance will be negative. The next question that arises is when covariance will be zero? If the deviation of either of security A and B is zero, the covariance will be zero.

Since covariance is an absolute value, it is' useful to standardize the covariance between two assets by dividing it by the product of standard deviation of each asset. This standardization will produce a ratio with the same characteristic as the covariance but with a range of -1 to +1. We know that this ratio is known as the correlation coefficient. If ij indicates the correlation between securities i and j. The correlation coefficient is defined as

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......Eq. (B)

Where,

= Covariance between securities i and j

= Standard deviation of security i

= Standard deviation of security j.

The correlation coefficient can be used as a relative measure to decide movements stock returns together.

If the correlation between two securities is +1, then it indicates that there is a perfect direct linear relationship between two securities. However, if the correlation is -1, then the relationship will be the inverse linear. If the correlation is zero between two securities, there is no relation between the returns of the two securities, and knowledge of the return of one security will not give any clue about the return of the other security. Combining securities whose returns have perfect positive correlation will not reduce the risk of a portfolio, instead the portfolio risk will only be the weighted average of the individual risk of the securities. As securities with perfect positive correlation are attached to a portfolio, portfolio risk remains the weighted average. There will be no risk reduction. Combining two securities with zero correlation can reduce the risk of a portfolio. If securities with zero correlation were added to a portfolio, some risk reduction can be achieved but total elimination of portfolio risk is not possible. Finally, if we make a portfolio with two securities with negative correlation, portfolio risk can be reduced. If a portfolio consists of only two securities with perfect negative correlation (-1), the risk of the portfolio can be reduced to zero. However, in the real world it is very difficult to find two securities with perfect negative correlation. Generally securities will possess some positive correlation with each other and therefore, the risk can be reduced and cannot be eliminated completely. Ideal situation for any investor to reduce his or her portfolio risk is to find securities with negative correlation or low positive correlation but investors usually encounter securities with positive correlation.

Example 4.8

Consider a portfolio of two securities with 60% investment in stock A and 40% investment in stock B and the variance of their returns are 24(%) and 36(%) respectively. Calculate the portfolio risk if coefficient of correlation between stocks A and B is

a. AB = +1b. AB = -1c. AB = 0

Where,

a. = (0.60)2 x 24 + (0.40)2 x 54 + 2 x 0.6 x 0.40

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x 1 x x

= 8.64 + 8.64 + 17.28

= 34.56(%)2

b. = (0.60)2 x 24 + (0.40)2 x 54 + 2 x

0.60 x 0.40 x (-1) x x

= 8.64 + 8.64 - 17.28

= 0(%)2

c. = (0.6)2 x 24 + (0.40)2 x 54 + 2(0.6) (0.4)

(0) x x

= 8.64 + 8.64 = 17.28(%)2

After understanding the covariance and correlation between securities as the measure of association between securities, we are now in a better position to discuss the risk of a portfolio. As we said in the previous section, the portfolio risk can be calculated using the following two factors:

1. Weighted individual security risks (the variance of each security multiplied by the percentage of investable funds placed in each security).

2. Weighted relationship between securities (the covariance between the securities returns, multiplied by the percentage of investable funds placed in each security).

As the number of the securities in a portfolio increases, the importance of each individual security's risk (variance) decreases. Let us consider a portfolio with n securities, the number of the variance terms will be n while the total number of covariance terms will be n(n - l)/2. Clearly, as n increases the number of covariance terms will increase and difference between variance and covariance terms will also rise. Let us take various values of n and calculate the number of variance and covariance terms.

n Variance term(n)

Covariance term

3 3 210 10 4550 50 1225

100 100 49501000 1000 499500

We can see that when the number of securities in a portfolio is equal to 100, the number of covariance terms are 4,950 whereas the number of variance terms are only 100. This huge number of covariance term suggests that portfolio risk will be immensely attributable to covariance factor rather than variance factor.

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We can rewrite the equation 7.3 in the following format:

Var(Rp) = Wj Wj ij SD(Rj) SD(Rj) ...Eq. (C)

Above equation represents both the variance and the covariances of the securities, because when i = j, the variances will be accounted whereas, if i j the covariances can be calculated. If we want to use the above equation, we need to calculate the variance of each security and correlation coefficients or covariances. We can calculate both covariance and the correlation coefficient using either ex, post or ex ante data. If the historical data is good estimate of the future value then, it can be used for calculating the portfolio risk. However, it must be remembered that the variance and correlation coefficients can change over time. Our discussion on the portfolio risk can be concluded by highlighting the following:

1. The measurement of portfolio risk requires information regarding the variance of individual securities and the covariance between the securities.

2. Three factors determine any portfolio risk: variances of the individual securities, the covariances between the pairs of the securities and the proportions of total fund invested in securities.

3. As the number of the securities increase in a portfolio, the impact of the covariance of the securites rather than their individual variance, affects the portfolio risk.

Systematic and Unsystematic Risk

In our earlier sections, we discussed that the variance of the portfolio is measure of its risk. According to the portfolio theory, the total risk (variance) is not the relevant risk in the portfolio context. It is necessary to understand that the risk of security when held in isolation is not equal to the amount of risk it contributes to a portfolio, when it is included in the portfolio. We are aware that the risk of a security is the sum of systematic risk and unsystematic risk. Unsystematic risk is the extent of variability in the security's return due to the specific risk attached to the firm of that particular security. Unsystematic risk is diversifiable risk, and hence this risk can be removed from the total risk of portfolio by investing in large portfolio securities. This is possible, because the firm specific risk factors are mostly random. For example, if the financial position of one company is weak, the financial health of the other company in the portfolio can be strong enough to neutralize the risk attributed by the weak financial position of the firm. However, the systematic or non-diversifiable risk cannot be diversified away completely because it depends on the factors affecting the whole market in a particular direction. For example, a steep rise in inflation in India will affect the entire market adversely and therefore, no diversification can make a portfolio free from this risk. Since the systematic risk affects the entire market, it is also known as the market risk.

We know that total risk of security is measured in terms of the variance or standard deviation of its returns. We also know that total risk consists of systematic and unsystematic risk. We will now try to segregate these two risks.

Total risk of a security i = 2i

Systematic risk of security i =

Where,

is the beta of the security i and

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is the variance of the market portfolio.

But,

Substituting the value for in the above equation, we get

Systematic risk of security i =

As we know from the relation between covariance and correlation, the above equation can be written in the following form:

Since Covim = miim

Systematic risk of security i =

=

since

Where,

is the correlation coefficient, and

is the coefficient of determination between the security i and the market portfolio. From the above

equation, it is evident that coefficient of determination is ( ) the indicator of the systematic risk. The coefficient of determination indicates the percentage of the variance explained by the variation

of return on the market index. To calculate the systematic risk of the portfolio, we should add the systematic risk of the individual securities.

Systematic Risk of the Portfolio =

Unsystematic risk of the security is the difference between the total risk and the systematic risk of the security and can be represented in the following form:

Unsystematic Risk

or, =

= 71

=

Unsystematic risk of the security is the percentage of the variance of the security's return not explained by the variance of return on the market index. This unexplained variance is also called the residual variance of the security. Unsystematic risk of a portfolio can be calculated as the total unsystematic risk of the individual security forming that portfolio.

Unsystematic risk of portfolio =

Total portfolio variance can be represented as

--(6.6)

Where,

= Variance of portfolio return

= Expected variance of index

= Variance in security not caused by its relationship to the indexXi = Proportion of the total portfolio invested in security in = Total number of stocks.

Beta of a Portfolio

We can measure the volatility of a stock by using beta. Beta measures how much the share price of a security has fluctuated in the past in relation to fluctuations in the overall market (or appropriate market index). If properly analyzed, beta indicates the fact that both market and stock returns depend on common events. The beta tells about the relationship of market and security returns. Most of the risk and return in a portfolio is directly connected to the market. Therefore, it is essential to calculate the portfolio beta. Portfolio beta is nothing but the weighted average beta of its component securities. According to William Sharpe, proper diversification and possession of sufficient number of securities can reduce the unsystematic risk of a portfolio to zero by neutralizing the unsystematic risk of the individual securities. The rest of the risk remained in the portfolio is systematic risk, caused by the market factors and cannot be diversified away by portfolio balancing. Because of this reason, the Sharpe model emphasizes a lot on the importance of the beta, which measures the systematic risk. According to the Sharpe model, the amount of the risk contributed to a portfolio by a stock can be calculated by the stock's beta coefficient. The market index will have a beta equal to 1. If beta of a stock is +1.5, it indicates that if the market return is 10%, the return on the stock will be 15%. On the other hand, if the market return is -10%, the return on the stock with beta 1.5 will be -15%. Securities with beta greater than 1 are called aggressive stocks, while stocks with beta less than 1 are viewed as defensive stocks. Negative beta stocks can help fund managers in reducing the portfolio risk beyond the unsystematic level. Efficient portfolios do not contain unsystematic risk because of the diversification, the risk of such portfolios is entirely based on their systematic risk, which is caused exclusively by the market movements. The total risk of an efficient portfolio can be calculated by the portfolio beta. We will now illustrate the calculation of the portfolio beta.

Example 4.9

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A fund manager has apprehensions that in the short-term market is going to decline, his portfolio contains Infosys (35%), ICICI Ltd. (20%), Dr. Reddy (20%), TISCO (10%) and GACL (15%). The beta of the stocks is given below. You are required to calculate the portfolio beta and suggest him the suitable alteration in the portfolio to avoid possible loss.

Stock Beta

Infosys 1.37

ICICI Ltd. 0.99

Dr. Reddy Labs Ltd. 0.91

TTSCO 1.19

GACL 0.95

Company BetaPortfolio Proportio

ns

Weighted Beta

Infosys 1.37 35% 0.4795ICICI Ltd. 0.99 20% 0.1980Dr. Reddy Labs Ltd. 0.91 20% 0.1820TISCO 1.19 10% 0.1190GACL 0.95 15% 0.1425

100% 1.1210

The portfolio beta is 1.121, which is greater than 1, therefore, if the fund manager wants to protect his fund from the forthcoming loss, he should try to reduce the beta of the portfolio, which can be done by reducing the proportion of high beta stocks like Infosys and adding the low beta stocks like Dr. Reddy in the portfolio,

Apart from readjusting beta of the stocks, stock index futures can also be used to control the beta of the portfolio. We will discuss about this feature of stock index future in chapter "Portfolio Management using Futures".

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Chapter 5Bond Portfolio Management

INTRODUCTION

In the recent past, bond management has undergone a remarkable evolution. Improvements in the technology and insights into portfolio analysis have not only enhanced the efficiency and effectiveness of management of bond portfolios but also led to the introduction of innovative strategies.

Bond portfolio management strategies can be broadly classified into passive management, semi-active management and active management. Figure 5.1 illustrates these strategies.

Figure 5.1: Bond Management Strategies

The basis of classification of bond management strategies is the nature of inputs required. Passive management is an approach which does not rely too much on forecast about future whereas active management relies too much on forecasting. The frequency of forecast and the number of variables that are forecasted are high in case of active management. Semi-active management falls in between these two approaches. In this chapter, we discuss the three types of bond management strategies and the uses of derivative instruments in bond portfolio management.

PASSIVE MANAGEMENT

Passive management, as we have seen above, is based less on expectations. That is, most of the key inputs are known at the time of investment analysis itself.

Two widely used strategies of passive management are 'Buy-and-Hold' and 'Indexing'.

Buy-and-hold Strategy

One of the simple investment strategies is to identify a security with the desired characteristics and hold it till maturity or redemption and reinvest the proceeds in similar securities. This strategy is known as buy-and-hold strategy. Buy-and-hold investors do not trade actively with the objective of increasing their returns. They buy the bond with a maturity or duration close to their investment horizon to reduce price and reinvestment risk. When a security is held till maturity, price risk is eliminated and the return on the security is controlled by the coupon payments and reinvestment rate. Therefore, cash flows over life of the security are determined by the coupon payments received and reinvested.

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Important thing in buy-and-hold approach is identifying bonds with attractive yield and maturity profiles. Investor has to choose carefully from the available bonds based on the analysis of quality, coupon level, term to maturity and important indenture provisions such as call, sinking fund features, etc. Though management of the portfolio is passive, selection of bonds is based on a careful analysis.

Buy-and-hold strategy is suitable for income maximizing investors such as pensioners, bond mutual funds, endowment funds, insurance companies, etc. Objective of these investors is to maximize yield over the investment horizon. In some cases, following active bond management strategies may be difficult because of the market impact of large cash flows of large funds.

Another feature of buy-and-hold approach is its low level risk. As we have already seen, main source of risk for bonds, interest rate risk, can be limited to reinvestment risk. Price risk is eliminated under buy-and-hold strategy because the security is held till maturity and price realized would be the same as expected. This also makes the buy-and-hold strategy attractive for risk-averse investors.

Therefore, buy-and-hold strategy will be suitable for investors with the objective of maximizing income with minimum risk.

Bond Ladder Strategy

Another form of buy-and-hold passive strategy of bond portfolio management is bond laddering. Bond laddering involves investing in bonds with several maturity dates instead of single time horizon as in the case of simple buy-and-hold strategy. This process of bond management is called laddering because of the various rungs of investment established over the maturity ladder. An illustration of bond laddering is given in table 4.1. The investments are staggered by maturity over the next five years. This staggering of maturity minimizes fluctuations in the level of current income.

Table 5.1: Buy-and-hold Bond Ladder Strategy

Issuer (Compan

y)

Credit Rating

Par Amount (Rs.)

Current Semi-annual

YTM (%)

Maturity (Year)

M A 10,00,0 5.0 2001N BBB 10,00,0 7.0 20020 AA 10,00,0 6.0 2003P AAA 10,00,0 6.0 2004Q AAA 10,00,0 7.5 2005

One of the most attractive features of the laddered buy-and-hold strategy is that there are few expectational requirements regarding future interest rate movements. By staggering the maturities of the securities, the investor is assured that money will be available for reinvestment at regular intervals. When interest rates decline the investor loses on short-term securities since the entire redemption amount has to be invested whereas he gains from the long-term investments since they remain locked at higher rates. Similarly when interest rates increase he gains from short-term investments since the redemption amount can be reinvested at higher rates whereas he loses from long-term investments since they remain locked at lower rates. Thus an evenly distributed portfolio across maturity ladder helps in offsetting the interest rate risk. However, the coupon inflows will be subjected to reinvestment risk. This lessens the pressure on the investor to make interest rate forecasts for several years in the future. Laddering also ensures better diversification. Since investments are spread over different time horizons, bond laddering ensures better diversification.

The downside is that no consideration is given to the total return potential of the portfolio.

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Nevertheless, although the focus on providing a steady stream of income may dampen the total return potential, this approach provides flexibility in designing a portfolio that will meet an investor's specific needs and holding period requirements. Furthermore, once the portfolio is established, very little time is needed to manage the securities. Only when a bond matures does the investor need to be actively involved. Thus, the laddered buy-and-hold approach is passive in terms of its management style.

Indexing Strategy

Another form of passive management is indexing strategy. Under this strategy, a bond portfolio is formed with the objective of replicating the performance of selected index. Performance is measured in terms of total return realized over the investment horizon. Sources of total return over the investment horizon are change in portfolio value, coupon interest received and reinvestment income.

This strategy of bond portfolio management has grown dramatically since it was first introduced in 1979 in the USA. A brief discussion of the advantages and disadvantages will be useful in understanding why indexing has assumed such significance.

ADVANTAGES AND DISADVANTAGES OF INDEXING STRATEGY

Before deciding about indexing, one should carefully analyze his investment objectives and constraints and advantages and disadvantages of indexing. Here, we briefly discuss the main advantages and disadvantages of indexing a bond portfolio.

One of the primary factors driving bond portfolio managers towards indexing is the disappointing performance of the active management strategies. Poor and inconsistent performance of the active bond portfolio managers in the past has turned the investors to index funds. In the past, returns earned by most active fund managers have lagged those of market indexes. Though some active investment managers could match or outperform the market indexes, their performance was not consistent over a period of time. Therefore, the investors naturally turned to index funds where they can obtain higher long-term returns consistently and reliable short-term performance.

Another factor driving interest in index funds is the lower advisory fee schedule. Compared with active fund management, advisory fee schedule is very attractive for indexed funds. In the USA, advisory fees for index funds range between 30 and 70 percent of advisory fees for actively managed funds. This can have substantial savings for the investors and increase the realized returns. Apart from lower advisory fee schedule, transaction costs will also be lower for the index funds. This is because of lower turnover of assets and hence fewer transactions in the portfolio.

Another advantage of indexing is the degree of control exercised by the investor. Under active management the investor has little control over the fund manager's investment decisions at any point of time. By indexing, ' the investor can specify the benchmark as well as the degree of latitude allowed for the index fund manager to deviate from the benchmark characteristics.

For example, an active fund manager can change the duration of the portfolio to any extent depending on his interest rate forecast, but index-fund manager may be constrained by say, a maximum deviation of 10 percent from the duration of the index. Thus, the investor can have a greater degree of control over investments under indexing strategy.

Finally, indexing facilitates easier and better measurement of performance of the fund manager. Performance of a fund manager is measured by comparing the total return of the portfolio with the total return of the benchmark. Most widely used benchmarks are broad market indexes. Such comparison under active investment management has two serious shortcomings. The selected index may not be the appropriate benchmark for the fund manager. Secondly, deviations of portfolio

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characteristics from the benchmark characteristics, which explain the relative performance, are not thoroughly examined. These shortcomings can be overcome by indexing. Extensive search for the appropriate index must precede establishing the index fund, which ensures identification appropriate performance benchmark. Secondly, index fund investors can focus on the deviation of return of the portfolio from that of benchmarked index and require fund managers to attribute these deviations to specific benchmark characteristics.

Although indexing has many advantages over active management of portfolio, it has some disadvantages also. One of the main disadvantages of indexing is the loss of incremental returns, which could have been generated by investing in sectors with the highest performance. By not investing in better performing sectors and securities, the opportunity cost can be substantial. Different sectors and different types of securities like treasuries, corporate bonds, mortgage- backed securities, etc. can generate incremental returns for the portfolio.

Another limitation of index funds is the rigid requirements associated with these funds. There may be attractive opportunities for investment outside the benchmark universe. If the fund manager is not allowed to invest in securities outside the universe of the benchmark index, then some attractive investment opportunities may be foregone.

SELECTING AN INDEX

Once it is decided to pursue an indexing strategy, the next step is to select a bond index to replicate. There are a number of bond indexes to choose from. Various factors such as investor's risk tolerance, investor's objectives, constraints imposed by regulators guide the decision on appropriate benchmark index.

If investor's risk tolerance is low, then the index should include more of government securities than corporate bonds. This is because corporate bonds expose the investor to credit risk whereas government securities do not have credit risk.

Objective of the investor has a major influence on selecting an appropriate index. If the objective is to minimise variability of total returns he may be biased towards choosing an index with a lower variability. On the other hand, if the investor has strong expectations about the direction of interest rate, selection of index may be biased towards an index, which is expected to yield maximum returns. If the objective of the investor is to meet certain future liability, then choosing an index with duration of the liability may be prudent.

Another important consideration in choosing an index is constraint on acceptable investments imposed by regulators, as in the case of financial institutions like banks, insurance companies, etc. These constraints may be in the form of limits on exposure to sectors, quality, etc. In such a case, choice of the index may be influenced by regulatory constraints.

INDEXING METHODOLOGIES

After selecting an appropriate index, comes the construction of portfolio that will track the index. The portfolio should be constructed in such a way as to minimize the tracking error. Tracking error is the deviation of the performance of the portfolio from that of the index. Tracking error can be caused by: {i} transaction costs in construction of the index; (ii) differences in the composition of the indexed portfolio and the index itself and (iii) discrepancies between prices used by the organization constructing the index and transaction prices paid by the indexer.

One way to construct the portfolio is to invest in all the issues in the index in the same proportion as

77

in the index. This can eliminate the tracking error resulting from differences in the composition of index and the portfolio. But this will increase tracking error resulting from transaction costs.

Another way to construct the portfolio is to invest in a sample of issues. This can substantially reduce the transaction costs and thereby tracking error resulting from transaction costs. But tracking error caused by the composition of the portfolio will increase. Therefore, it is a trade-off between the tracking errors resulting from transaction costs and composition. This needs to be kept in mind while constructing a portfolio.

Three popular methods of constructing a portfolio to replicate an index are (i) the stratified sampling or cellular approach; (ii) the optimization approach and (iii) the variance minimization approach. We will briefly discuss these approaches.

Stratified Sampling or Cellular Approach

This is the most simple and flexible approach of constructing a portfolio. Under this approach, the index is divided into sub-sectors or cells. This division can be based on various characteristics such as sector, term-to-maturity, duration, coupon, credit rating, call features, etc. Suppose that a fund manager stratifies the index based on the following characteristics:

1. Duration (2 cells) i. Up to 5 yearsii. More than 5 years

2. Credit rating (4 cells)i. Triple Aii, Double A iii. Single A iv. Triple B

3. Sector (2 cells)i Corporate ii. Treasury

Total number of cells for the index is equal to 2 x 4 x 2= 16.

After stratifying the index into cells, securities are selected so as to represent each of these cells. Securities are selected from each of these cells in such a way that the selection is representative of the particular cell. The proportion of investment in each cell depends on the percentage of the cell's market value in the index. For example, if 25 percent of the market value of the index is made up of triple A issues, then 25 percent of the indexed portfolio should be composed of triple A issues.

Optimization Approach

A more disciplined and quantitative extension of cellular approach to construction of a portfolio is optimization approach. Under this approach, the money manager seeks to construct an indexed portfolio that will match the requirements as under the cellular approach and satisfy a few other constraints and also optimize a specific objective function. Objective function can be maximization of yield, maximization of convexity or maximization of expected total return. This approach requires mathematical programming. If the objective function is linear, linear program is used and if the objective function is a quadratic function, quadratic program is used.

Variance Minimization Approach

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This is a complex approach to portfolio construction. The objective of this approach is to maximize the expected return of the indexed portfolio while minimizing the variance of tracking error in the construction of the portfolio. A quadratic program consisting of three components, an objective function, a set of constraints and a universe of securities, is solved to construct the indexed portfolio.

SEMI-ACTIVE MANAGEMENT

Apart from earning a steady flow of income from bond investments, many bond portfolio managers may require an investment to take care of a future liability. The objective of such investors is to accumulate the present value of investment over the investment horizon. This funding objective may be required by a person who needs to build wealth through investment so as to provide money for retirement, education of children, etc. Many large institutional investors, such as pension funds and insurance companies, must accumulate money in order to fund future liabilities. Two popular portfolio strategies that utilize bonds to accumulate value are (i) portfolio dedication and (ii) immunization.

Portfolio Dedication

Dedication is a strategy in which the objective is to create and maintain a bond portfolio that has a cash flow structure that exactly or closely matches the cash flow structure of a stream of current and future liabilities that must be paid. There are at least two approaches that can be used to construct a dedicated portfolio: pure cash matching and cash matching with reinvestment.

PURE CASH MATCHING

The most conservative of the dedicated portfolio strategies is that in which a bond portfolio is constructed in such a way that the cash flows (coupons, principal payments, and any principal payments through call features) exactly match the required payments for a stream of liabilities. In the strictest sense, the portfolio would be designed to preclude the need to reinvest. That is, reinvestment income would not be needed to help to fund the liability payments. Thus assuming that the future liability stream is known with some degree of certainty, the portfolio, once constructed, would need little monitoring. The easiest way to implement this approach is through dedication with zeros, i.e. purchase of zero coupon bonds whose maturities coincide with the dates on which money would be needed. However, because maturity dates for zero coupon securities may not exactly match liability payment dates, it may be difficult, if not impossible to do this. The dedication strategy will need to rely on some amount of reinvestment income to supplement the portfolio coupon and/or principal cash flows.

Example 5.1

Table 5.2 illustrates the pure cash matching strategy with zeros.

Table 5.2: Dedication with Zeros

Year Liability (Rs.)

Maturity Value (Rs.)

Current Purchase Price (Rs)

Current Annual

YTM (%)

1 5,00,000 5,00,000 4,62,963 8.002 10,00,000 10,00,000 8,49,435 8.503 15,00,000 15,00,000 11,58,275 9.004 20,00,000 20,00,000 13,91,149 9.50

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5 25,00,000 25,00,000 15,52,303 10.006 30,00,000 30,00,000 16,70,512 10.257 35,00,000 35,00,000 17,39,931 10.508 40,00,000 40,00,000 17,67,299 10.759 45,00,000 45,00,000 17,59,161 11.0010 50,00,000 50,00,000 16,83,532 11.50

Suppose liabilities of an individual for the next ten years are expected to be as given in the table 5.2. Table 5.2 illustrates how the individual can fund the liabilities using a dedicated zero-coupon bond portfolio. At the end of the year one he requires Rs.5,00,000. He can fund this liability by investing in zero-coupon bonds with a maturity value of the same amount. If the current annual YTM offered by zero-coupon bonds with one year to maturity is 8.00 percent, he needs to invest Rs.4,62,963 in the zero-coupon bonds to meet the liability. This way, the individual can finance the liability stream by investing in zeros. The current YTMs given in the last column of table 5.2, and required current investments to finance the liabilities are given in column 4. It is assumed in the illustration that the zeros are redeemed at the same time when the liability falls due.

CASH MATCHING WITH REINVESTMENT

An alternative approach to portfolio dedication is to construct a portfolio such that the cash flows plus expected reinvestment income provide the anticipated funds at the times when payments are required. This method provides greater flexibility in the choice of securities, because, now, the maturity of the bonds do not have to match with the dates at which funds are required. However, the manager faces the risk that the reinvestment returns, when combined with coupon and principal repayments, may be insufficient to meet the needs. As a result, a conservative estimate of the future reinvestment rate is usually made so as to protect against a potential shortfall.

Example 5.2

Suppose a pension fund has the following liabilities for the next 10 years.

Year Liability (Rs.) 1 10,00,000 2 10,00,000 3 15,00,000 4 20,00,000 5 25,00,000 6 30,00,000 7 35,00,000 8 40,00,000 9 45,00,000

10 50,00,000

Manager of the fund is considering the following corporate bonds to construct a dedicated portfolio to take care of the expected liabilities of the pension fund for the next 10 years.

Table 5.3: Bond Universe Under Consideration for Dedication

Company Credit Rating

Term to Maturity (Yrs)

Face Value(Rs.)

Coupon Rate %(Annual)

80

ABCDEFGHIJ

AAAAAAAAAAAAAAAAAAAAAAAA

AAA

12345678910

10010010010010001000100100010001000

8.008.509.009.5010.0010.5010.7511.0011.2511.50

To simplify the procedure, it is assumed that all the bonds are currently trading at their face values, hence YTM is equal to the coupon rate. Further more, no bond has a call feature. The fund manager assumes a conservative reinvestment rate of 5 percent.

If the fund manager would like to invest in bonds with YTM of not less than 9 percent, one possible cash matching bond portfolio could be as shown in table 5.4

Table 5.4: Cash Matching Bond Portfolio

Company Investment (Rs.)

CDEFGHIJ

500000700000

110000015000002200000250000030000003200000

Table 5.5 illustrates the cash flows associated with the Cash Matching Bond Portfolio with Reinvestment

Table 5.5: Cash Flow Analysis for Cash Matching with Reinvestment

(Rs.)Year

Liability Cash Balance at

the Beginning

Interest Earned on the Cash Balance

Coupon Payments Received

Redemption

Total Cash

Available

Surplus Cash at the End

(1) (2) (3) (4) = (3) x (5) (6) (7) = (8) = (7) - 1 10,00,000 0 0 15,96,000 0 15,96,000 5.96,0002 10,00,000 5,96,000 29,800 15,96,000 0 22,21,800 12,21,8003 15,00,000 12,21,800 61,090 15,96,000 5,00,000 33,78,890 18,78,8904 20,00,000 18,78,890 93,945 15,51,000 7,00,000 42,23,835 22,23,8355 25,00,000 22,23,835 1,11,192 14,84,500 11,00,000 49,19,526 24,19,5266 30,00,000 24,19,526 1,20,976 13,74,500 15,00,000 54,15,003 24,15,0037 35,00,000 24,15,003 1,20,750 12,17,000 22,00,000 59,52,753 24,52,7538 40,00,000 24,52,753 1,22,638 9,80,500 25,00,000 60,55,890 20,55,8909 45,00,000 20,55,890 1,02,795 7,05,500 30,00,000 58,64,185 13,64,18510 50,00,000 13,64,1 68,209 3,68,000 32,00,000 50,00,394 394

Notes:

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1. Cash balance at the beginning of a period is surplus cash at the end of the previous period. This surplus cash can be invested which earns 5% p.a.

2. Interest earned on the cash balance is equal to (cash balance available for investment x 5%)3. Coupon payment received is the sum of coupons received on all the bonds in the portfolio for the

year.For years 1 and 2:Coupon payments received =(5,00,000 x 0.09) + (7,00,000 x 0.095) +(11,00.000 x 0.10) + (15,00,000 x 0.105) +(22,00,000 x 0.1075) + (25,00,000 x 0.11) + (30,00,000 x 0.1125) + (32,00,000 x 0.1150).With redemptions, coupon payments received decrease from period 4 onwards.

4. Redemption is maturity value of the investments.5. Total cash available for a year is sum of (Cash balance at the beginning, interest earned on the

cash balance, coupon payments received and redemptions)6. Surplus cash at the end is equal to total cash available for the year less liabilities due for the year.

Advantages and Limitations of Portfolio Dedication:

The primary advantage of the dedicated portfolio strategy, when compared to other techniques used to accumulate value, is that it minimizes price (volatility) risk and reinvestment risk because it is typically structured to require a minimum amount of rebalancing and reinvestment income. However, because the technique focuses primarily on matching investment and liability flows, little attention is given to the total return potential of the portfolio. Furthermore, to the extent that securities with maturities comparable to the dates needed for cash are not available, the portfolio is exposed to the risk that the cash supplement provided by reinvestment income may be insufficient. This can occur because the portfolio manager is unable to accurately assess the future reinvestment rates.

IMMUNIZATION

A major concern to bond investors who use bonds as an investment vehicle to accumulate value is that future reinvestment rates may change, thus affecting the realized yield, and consequently, the accumulated value. The realized accumulated value may be insufficient to pay-off the required liability that the portfolio was intended to fund.

Maturity Matching

Suppose an investor has a liability of Rs. 1460.60 due after 5 years. The current semi-annual YTM is 3.5 percent. To meet the liability due after five years, the investor can invest the present value of the liability in a bond, which offers a semi-annual YTM of 3.5 percent. The value of investment should be

= Rs.1000

One option for the investor is to buy a 5-year, 3.5% (semi-annual) bond selling at par (i.e. the bond offers a semi-annual YTM of 3.5%), whose maturity matches with the maturity of the liability. The realized yield for the investor will be equal to the current YTM only, if he can reinvest all the coupon payments at 3.5% (semi-annual) over the investment horizon. If the reinvestment rate changes during the investment horizon, realized yield will not be equal to the current YTM and accumulated wealth at the end of holding period will differ from the liability due,

Example: 5.3

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Table 5.6: Maturity Matching and Interest Rate Risk 5-year, 3.5% (semi-annual)

Bond, Current Price = Rs.1000

Reinvestment rate % [semi-

annual)

Selling price (at the end of

year 5}

Coupon income (Rs.)

Reinvestment income (Rs.)

Total accumulated value (Rs.)

Realized yield (%)

0 1,000 350 0.00 1,350.00 3.051.5 1,000 350 24.60 1,374.60 3,232.5 1,000 350 42.12 1,392.12 3.363.5 1,000 350 60.60 1,410.60 3.504.5 1,000 350 80.09 1,430.09 3.645.5 1,000 350 100.64 1,450.64 3.796.5 1,000 350 122.30 1,472.30 3.947.5 1,000 350 145.16 1,495.16 4,10

Table 5.6 illustrates the accumulated value of Rs.1000 at the end of 5 years and the realized yields for various reinvestment rates. As can be seen, for the bond whose maturity period is equal to the investors holding period, increase in the market yields (reinvestment rate) will increase the realized yield and accumulated value of the investment. On the other hand, decrease in the market yields will decrease the realized yield and accumulated value of the investment will fall short of the liability. Only when the market yield remains constant through the holding period will the investor realize the promised yield and the targeted accumulated value of the investment. Therefore, maturity matching is not suitable for the investor since this strategy does not necessarily ensure the targeted yield or promised wealth because of the reinvestment problem.

Immunization

We will now see how the above probelm caused by maturity matching can be overcome by immunization.

As we know, the effect that changes in interest rates can have upon a bond's total return is interest rate risk. This interest rate risk has two components: price risk and reinvestment risk. Price risk is the uncertainty about return from selling the bond at some time in the future. Reinvestment risk is the uncertainty about return from reinvesting the coupon income received over the holding period. The changing interest rates have opposite effect on these two components of interest rate risk. When interest rate increases, return from reinvestment increases but return from selling the bond decreases. A decline in the interest rate has the opposite effect.

In maturity matching the price risk is eliminated since uncertainty about the selling price of the bond is removed by holding the bond till maturity, but reinvestment risk is not eliminated. Using the concept of duration we can immunize the portfolio from the changing interest rates and can lock promised YTM or accumulate a targeted wealth.

A portfolio is said to be immunizedi. if the realized yield or accumulated wealth at the end of holding period is at least as large as the

projected YTM or accumulated wealth.ii. its present value and duration equals the present value and duration of the stream of liabilities for

which the portfolio is designed to take care of.

LOCKING IN THE PROMISED YIELD OR ACCUMULATING A TARGETED LEVEL OF WEALTH

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It will be necessary to recall the concept of 'Macaulay Duration' here. If the Macaulay duration of a bond is equal to the investor's desired holding period or investment horizon, the investor can lock in a promised yield on the investment and accumulated wealth of the investment at the end of holding period will be equal to the targeted wealth. The same concept is applied in bond portfolio management also. For example, if your initial desired holding period is 5 years, and current semi-annual yield is 3.5 percent, you can immunize a Rs.1,000 portfolio, against future changes in interest rates by purchasing a bond whose duration equals 5 years. Further, as time passes, you should rebalance your portfolio in such a way that its duration always equals the remaining time left in your horizon. That is, after one year your portfolio duration should be 4 years. In doing this, you will lock in the initial semi-annual yield of 3.5 percent, and your accumulated value be at least as large as Rs. 1,411 [Rs. 1,411 = Rs.l,000(1.035)l0], If the purchased portfolio was acquired in order to make liability payments whose current present value is Rs. 1,000 and whose maturity is 5 years, the immunized portfolio will always have enough money to pay the required liabilities, no matter what happens to future interest rates.

Example 5.4

Table 5.7 illustrates how the investor can immunize his portfolio by matching duration with holding period. The accumulated value of Rs. 1000 at the end of 5 years and the realized yield for various reinvestment rates are shown in columns 5 and 6 respectively. Whatever may be the reinvestment rate, the accumulated wealth is at least as large as the targeted wealth and the realized yield is at least as large as the initial yield of 3.5%. The key to this achievement is the neutralization of price and reinvestment risks. As we have seen, the return from price change and reinvestment move in opposite direction and by matching duration with holding period the investor can neutralize these two effects of interest rate risk. This can be observed from columns 2 and 4 of table 5.7. Changes in selling price of the bond are approximately equal to the changes in reinvestment income, but in the opposite direction. When ' selling price increases (decreases), the reinvestment income decreases (increases). Therefore, the initial yield or targeted accumulation of wealth is assured regardless of the change in interest rate.

Table 5.7: Immunization: Locking in the Promised Yield

Term: 6 years; Coupon (semi-annual): 3.5%; Duration: 5 years; Current Price - Rs.1,000

Reinvestmen Selling Coupon Reinvestm Total Realizedrate (%) price income income

(Rs.)accumulat

edyield (%)

Semi-annual)

(at the end (Rs.) valueof year 5} (Rs.)

{Rs.)0 1,070.00 350 0.00 1,420,00 3.57

1.5 1,039.12 350 24.60 1,413.72 3.522.5 1,019.27 350 42.12 1,411.39 3.513.5 1,000.00 350 60.60 1,410.60 3,504,5 981.27 350 80.09 1,411.36 3.515.5 963.07 350 100.64 1,413.71 3.526.5 945.38 350 122.30 1,417.68 3.557.5 928.18 350 145.15 1,423:33 3.59

The easiest approach to immunization is to purchase a zero coupon bond portfolio whose maturity is the same as the desired investment horizon. As we know, the duration of a zero coupon bond always equals its maturity. Further, as time passes this relationship remains the same, that is, the duration always adjusts in conjunction with the remaining time to maturity, because there are no coupon payments, and hence no required reinvestment. The investor, therefore, can lock in the compounded

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return and the accumulated value at maturity, regardless of what happens to interest rates. Although zero coupon bonds would seem to be a simple solution to the immunization problem, the difficult part is to find zero coupon bonds whose maturity exactly matches the desired holding period. It is, therefore, important to understand how the duration-immunization concept can be used for the more general case of coupon-bearing bonds.

FUNDING OF LIABILITIES

Similar to the dedicated portfolio technique, immunization can also be used to construct a bond portfolio, from which the proceeds can be used to pay liabilities such as pension payments and the like. However, unlike portfolio dedication, the immunization technique does not require that bond cash flows be matched, at least approximately, with the required liability payments.

Example 5.5

To illustrate how immunization can be used to fund a stream of liability payments, let us examine table 5.8A. Suppose you have the schedule of liability payments over the next 3 years: Rs.1000 due at the end of first two years and Rs.1250 due at the end of third year. The present value of this liability schedule, using a 5 percent, semi-annual discount rate, is Rs.2,663. Suppose that you have Rs.2,663 to invest now and you want to purchase a portfolio of bonds that will always have a value such that, if sold, will provide enough money to enable you to pay-off your liabilities at any point in time, regardless of how interest rates change. Consider the three alternatives presented in Table 5.8A: (1) a bond portfolio that provides cash flows of Rs.2,936 at the end of year 1 (column 3), (2) a bond portfolio that provides cash flow of Rs.7,065 at the end of year 10 (column 4), and (3) a bond portfolio that provides cash flows of Rs.2,195 at the end of year 1 and Rs. 1,094 at the end of year 5 (column 5). As shown in the table, all three portfolios have the same present value as the liabilities. Thus your initial net worth (assets - liabilities) is zero for all three cases. However, the durations of the three portfolios are different. In particular, the duration of portfolio 3 is the same as the duration of the liabilities.

Table 5.8A: Immunization: Funding a Stream of Liabilities

End of year 1

Liability Payments

Cash Inflows (Rs.

Investment Strategy 1

Investment Strategy 2

Investment Strategy 3

2936 21952 1000 _ _ _

3 1250 _ _ _

4 _ _ _ _

5 _ _ _ 1094

6 _ _ _ _

7 _ _ _ _

9 _ _ _ _

10 _ _ 7065 _

85

Present value

_ _ _

at a _ _ _

semi-annual _ _ _

yield of 5% 2663 2663 2663 2663

duration (Yrs)

2 1 10 2

Now, your primary concern is that your portfolio has a value sufficient to pay the liabilities at any point in time. However, because the timing of the cash flows from each of the three portfolios is not synchronized with the timing of the cash flows required for the liabilities, changes in interest rates will alter the value of your portfolio such that your portfolio may be under funded- Which of the three portfolio strategies would you choose?

To answer this question, let us examine table 5.8B, which provides the present values of the assets, liabilities, and net worths for each of the three portfolio strategies at selected required yield (discount rate) levels.

Table 5.8B

Strategy 1 Present Values (Rs.)

Strategy 2 Present Values

(Rs.)

Strategy 3 Present Values

(Rs.)Required

YieldAssets - Liabilities

= Net WorthAssets -

Liabilities = Net Worth

Assets - Liabilities = Net Worth

4.04.55.05.56.0

2714 - 2767 = -532689 - 2714 = -25

2663-2663 = 02638 -2612 = 262613-2563 = 50

3224 - 2767 = 457

2929-2714 =2152663 - 2663 = 02421 -2612 = -

2768 - 2767 = 12714-2714 =02663-2663 =0

2613 - 2612 = 12564 - 2563 = 1

As the table illustrates, if interest rates were to rise (fall) instantaneously, the present values of all of the portfolios, as well as the liabilities, will fall (rise). However, the relative effects of changes in interest rates on the three portfolios are not the same. For the first portfolio, which has a shorter duration than the liabilities, increases (decreases) in interest rates from an initial 5 percent semi-annual yield produce portfolio values that exceed (fall short of) the value of the liabilities. Thus decreases in market yields result in this portfolio not being able to pay the liabilities. The opposite result occurs for strategy 2, in which the duration is greater than the duration of the liabilities. This portfolio becomes underfunded when interest rates rise above the 5 percent yield. On the other hand, the immunized portfolio in strategy 3 always has a portfolio present value that is at least as large as the present value of the liabilities, regardless of what happens to market yields.

Whenever the portfolio's present value is equal or greater than the liability's present value, you will be able to sell the bonds and pay-off your debts. Conversely, you will not have enough money from the sale of the bonds, if the portfolio's present value is less than the present value of the liabilities. It is only, in this case, in which you maintain a bond whose present value and duration always matches the present value and duration, respectively, of the liabilities to be paid that you can be assured of being adequately funded.

Advantages and Limitations of Immunization: The primary advantage of immunization over the

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dedicated portfolio is its flexibility. It provides the investor with a tool which neutralizes the effects of price risk and reinvestment risk. Thus a greater array of portfolios can be chosen to meet the accumulation objective.

To many investors the idea of locking in a specified yield (or total return) with a target accumulation goal is especially appealing once the concepts of price and reinvestment risks are understood. Further, if current yields are very high, the notion of locking in a high return and riding it out versus taking capital gains should future yields drop (and bond values rise) is a trade-off many investors would like. Although the approach would require constant monitoring and can be considered an active approach to bond management, it is basically a defensive technique for managing portfolios.

However, the immunization approach does have some limitations. First, the duration of the immunized portfolio requires periodic rebalancing. This occurs for two reasons. First, as time passes, the initial investment horizon grows shorter and the duration of the bond portfolio must continually be reset to equal the current investment horizon. Second, the examples in this section have all assumed a one-time instantaneous change in yields. In reality, yields are continually changing and this, in turn, affects the duration. Therefore, immunized portfolios are subjected to the problem of rebalancing.

A second limitation for the effective use of duration is immunization risk. The immunization concept assumes that the initial term structure is flat (that is, the current spot and all one period forward rates are equal). Further, any change in the yield structure is also assumed to either raise or lower all yields by the same amount. Stated differently, when we use the yield to maturity to compound cash flows, we are assuming that the reinvestment rate is the same for all future periods. Practically speaking, the term structure is, in general, not flat, and changes in market rates do not produce parallel shifts in the yield curve. Consequently, matching the duration of the portfolio to the investment horizon will not necessarily assure that immunization will be achieved.

ACTIVE MANAGEMENT STRATEGIES

When investing to enhance the total return from the bond portfolio, the objective is to maximize the total value in each period, given the investor's risk tolerance. Because total return includes price appreciation, coupon income, and reinvestment returns, accomplishing this objective may force the investor to trade-off one form of return for another in the hope that the total return will be increased.

Two types of strategies that seem well suited for this objective are (i) portfolio shifts in anticipation of changes in the overall structure of interest rates and (ii) bond swaps, which attempt to exploit temporary aberrations in the price/yield structure. Because both types of strategies entail a great deal of time, investors seeking to increase total return can expect to be actively involved in the management of their bond portfolios.

Interest Rate Anticipation

Interest rate anticipation is perhaps the riskiest strategy for managing bonds. With interest rate anticipation, you, as an investor, make a forecast of the direction and quantum of change. Your risk is two-fold. First, you are making an estimate and you might be wrong and it could have disastrous consequences for your overall return position. Second, if you currently own a portfolio, rebalancing will change its duration, which, in turn, will alter your risk/expected-return position.

DECISION WITH RESPECT TO MATURITY

Because interest rate sensitivity is related to bond duration, the general rule for interest rate anticipation is to increase your investment in long duration bonds (i.e. long maturity and low-coupon bonds) when interest rates are expected to decline. This enhances the opportunity to increase total

87

return in the short run through price appreciation. Alternatively, if interest rates are expected to rise, moving into shorter-duration bonds (i.e. short maturity and high-coupon bonds) aids in preserving capital, which, in turn, can stabilize or increase the total return in a market with falling prices. These guidelines may seem straightforward, once you have decided on which direction you think interest rates will move, but there are several factors to consider before making the final choice.

To illustrate, assume that you expect that interest rates to fall. To take advantage of this expected decline, you consider increasing your holdings in long-term, low-coupon securities that are currently selling at a discount. The long duration of these bonds will make them especially sensitive to declining interest rates. However, such a move also produces a low level of income through coupons and reinvestment (at lower rates). Therefore, if you also have need for current income, you might tamper this decision and consider investing in longer-term, current-coupon bonds. Although the duration of these bonds will not produce as much price appreciation as the low coupon discount securities, the additional income, when combined with some price appreciation, may provide a better overall return, especially, if interest rates decline only slightly. Thus the decision about the type of long-duration bonds to invest in, must consider the need for current income as well as how low and how soon you think interest rates will fail. Further, regardless of your choice, you should choose marketable, highly liquid securities for ease in making the portfolio shift. This will enable you to restructure your portfolio with the greatest ease. In addition, it is recommended that you emphasize quality (e.g. Treasury securities), since the higher the quality, the more sensitive the prices are to changing interest rates.

Expectations of an increase in interest rates provide for altogether different portfolio considerations. When interest rates are expected to rise, a primary consideration for many investors is the preservation of capital, that is, the need to avoid large price declines due to increased interest rates. The natural instinct would be to move into very short-term, highly liquid investments such as money market securities whose short duration makes their values relatively insensitive to changes in market yields. As changes in interest rates usually affect the short-term yields more than long-term yields, these securities yields will quickly reflect any rate increases. Therefore, you need to consider various factors before changing duration of the portfolio based on interest rate expectations.

Mapping Returns

Suppose you buy the following bond:

Coupon : 12%

Years-to-maturity : 5

Current YTM : 10%

Coupon Payments : Semi-annual

Face value : Rs.100

Redemption at face value

Current price : Rs. 107.72

The following yield curve is observed currently:

Years- to-maturity

1.0 2.0 3.0 4.0 4.5 5.0

YTM (%) 6.0 7.0 8.0 9.0 9.5 10.0

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Assume, at the end of six months you observe the following yield curve:

Years-to-maturity

1.0 2.0 3.0 4.0 4.5 5.0

YTM (%) 5.5 6.5 7.5 8.5 9.0 9.5

The bond you bought will now have a time-to-maturity of 4.5 years and the current YTM is 9.0 percent. If you sell the bond, the price will be Rs.111.83. The total return for you over the six-month holding period will be(Change in price + Coupon earned + Interest on coupon earned)/Purchase price

= [(111.83 - 107.72) + 6 + 0]/107.72 = 10.11/107.72=0.0939= 9.39% (semi-annual)

The total return on the bond can be broken down into the following components:

Total return = C + A + R + I Where,

Coupon income, C = Coupon earned/Beginning price (BP)

Amortization of premium or discount, A - Price change on level yield curve/BP Rolling yield,

R = Price change due to slope/BP

Return on account of change in the interest rate, I - Price change due to interest shift/BP

For the above bond,

Beginning price (BP) = Rs.107.72

Ending price at 10% for 4.5 years is Rs. 107.70. This price is computed on the assumption of a flat yield curve.

Ending price at 9.5% for 4.5 years is Rs. 109.74. This price is computed on the basis of YTM of 9.5% for 4.5 years as indicated by the original yield curve.

Ending price at 9.0% for 4.5 years is Rs.111.83. This price is computed on the basis of new YTM.

From the above prices components of yield can be computed as below:

Coupon income for six months is Rs.6 Price change on level yield curve - Ending price on level yield curve - BP = 107.70 - 107.72 = -Rs.0.02

Price change due to slope = Ending price on sloped yield curve - Ending price on level yield curve= 109.74- 107.70 = Rs.2.04

Price change due to interest shift= Current market price - Ending price on sloped yield curve= 111.83 - 109.74 = Rs.2.09

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Total return on the bond =C+A+R+I C = 6/107.72 x 100 = 5.57A = -0.02/107.72 x 100 = -0.02R = 2.04/107.72 x 100 = 1.90I = 2.09/107.72 x 100 = 1.94TR = 5.57-0.02+1.90+1.94 = 9.39%

Mapping Expected Returns and Interest Rate Anticipation

Suppose you expect the rate of interest to go up and comparing two bonds: 30-year, 8 percent bond versus 3-year 8 percent bond. Our discussion on bond price volatility may suggest that you should focus on short maturity, high coupon bonds since long maturity, low coupon bonds are more volatile. But this may not be so in all the cases. As our analysis below shows, it may be in your interest to choose a 30-year bond rather than the 3-year bond.

Assume the beginning and ending yield curves as shown in the figure.

Figure 5.2 Rise in Interest Rates Forecast

Based on the expected yield curve shift, components of total returns for the two bonds are given below.

3-Year 30-YearBeginning yield (maturity 3, 30 7.0 8.5Beginning yield (maturity 2 3/4, 29 30/4) 6.8 8.4Ending yield (maturity 2 3/4, 29 3/4) 8.8 8.7Beginning prices 102.66 94,60Ending prices (7.0%, 8.5%) 102.45 94.59Ending prices (6,8%, 9.4%) 102.95 95.63Ending prices (8,8%, 8.7%) 98,06 92.578% coupon payment accrued 2.00 2.00Then,

C = 0.0195 0.0211

A = -0.0020 -0.0001

R = 0.0048 0.0110

I = -0.0476 -0.0323

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Total return = -0.0253 -0.0003

This is because, although you are forecasting a rise in interest rates, the shape of the curve is forecasted to change such that intermediate maturities will be hurt more than long-term bonds. How much you shift toward the thirty-year maturity range would depend upon your confidence about your forecast, as well as the degree of risk aversion you have for incorrect forecasts.

The analysis indicates that loss on a 30-year bond is less than the 3-year bond.

DECISION WITH RESPECT TO SECTOR

By anticipating changes in yield spreads between different sectors of bonds such as treasuries, corporates, mortgage-backed, etc. the bond manager can add incremental returns to the portfolio. These yield spreads are determined by various factors like supply-demand position of issues, earnings position of the sector, influence of macroeconomic forces on the yields, etc. If the bond manager can anticipate the factors which cause these yield spreads, he can formulate appropriate strategies to profit from any expected differentials. Assume that the yield spread between treasuries and AAA corporates is 100 basis points. If you expect this spread to increase, you can swap corporates to treasuries now and then back to corporates when your expectations materialize. With widening of the spread, relative prices of treasuries increase thereby offering scope for incremental returns. Therefore, if you can forecast the factors that determine the yield spread between different sector's bonds, you can benefit from relative price changes.

DECISION WITH RESPECT TO QUALITY

Another important decision variable in interest rate anticipation is quality. Quality spreads change because of expected changes in economic prospects. Usually, credit or quality spreads between treasury and non-treasury issues widen during economic downturn or recession and narrow during economic upturn or boom. This is because of the increasing risk premiums during economic recession. In a declining economy corporates experience declining revenues and cash flows, thereby increasing the uncertainty about their repayment capability. To induce investors buy their bonds, the yield on non-treasury issues must rise relative to treasury issues. On the other hand, during economic expansion the yield spreads decrease. In non-treasury issues itself, spreads widen during economic downturn between high quality issues and low quality issues and spreads narrow during economic upturn. A bond manager can swap between bonds based on quality and expectations about economic performance. He can focus on high quality bonds when economic performance is expected to be poor and buy lower quality bonds when economic performance is expected to be good.

DECISION WITH RESPECT TO COUPON

If the term structure is upward sloping, by shifting into very short-term securities, the investor may be sacrificing too much income in order to avoid price declines. In particular, if the increase in interest rates is not large, the investor's total return can be enhanced even more by moving into higher-coupon, intermediate-term bonds, whose values are not affected greatly by rising yields. In this scenario, cushion bonds become a viable consideration. A cushion bond is a high-coupon bond that typically has a call feature. Because of the call feature, the bond is usually priced as a shorter-term security- Since the bond sells at a premium (due to its high-coupon), moderately rising interest rates do not affect its price as much as for long-term, low-coupon bonds with longer durations, Thus by investing in a cushion bond you have a security that provides higher coupon and reinvestment income, while potentially incurring only moderate price declines relative to shorter-term securities.

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Interest rates anticipation as a method for restructuring bond portfolios involves several considerations. Not only must investors accurately predict the direction and magnitude of such movements, but they must also consider the current shape of the yield curve and how it will aftect the quality and liquidity of the securities to be chosen and of course, the income needs of their portfolio.

Bond Swaps

Another approach, the investor may take in increasing the total return to the portfolio, is the use of bond swaps. A bond swap occurs whenever an investor sells a bond and exchanges it with another. Bond swaps may be initiated for many reasons, and there are numerous types of bond swaps. For example, investors may decide to swap bonds in order to increase the current yield, quality, or liquidity of their portfolio. Alternatively, they may believe that the existing yields to maturity for two securities are out of line and may thus engage in a yield-spread swap in anticipation of realignment between the two bonds yields. Still other reasons for swapping bonds are tax considerations and expectations regarding future interest rate levels. Because the motives for swaps vary, the potential return and risks from swaps will also differ.

There are two general types of swaps widely used for the purpose of increasing the total return of a portfolio: (i) risk-neutral swaps and (ii) risk-altering swaps. A risk-neutral swap is one in which the bond exchange is expected to increase the total return, as measured by the promised yield to maturity, but one that should not be affected by a general move in interest rates or one that does not significantly affect the price risk, credit risk, or call risk of the portfolio. Examples of risk-neutral swaps are substitution and sector swaps. On the other hand, a risk-altering swap alters the market risk of portfolio and /or the credit or call risk. Examples of risk-altering bond swaps are the pure yield pick-up and the interest rate anticipation. We will briefly discuss how the substitution and pure yield pick-up swaps work.

SUBSTITUTION SWAP

A substitution swap is a swap in which securities are similar in all respects except that the bond purchased has a higher promised yield to maturity than the existing bond. In the strictest sense, the two bonds' coupon, default risk, and maturity are the same. If the two bonds are perfect substitutes, then market forces should bring the two yields back together at some point in the future. Thus the investor, by selling the lower-yield bond and purchasing the higher-yield bond, has the opportunity to increase the overall return. If the market is fairly efficient, the gain should be realized in a short period of time.

Example 5.6Table 5.9: Substitution Swap

Sell: A 20 year, 8 percent AAA Corporate yield 4 percent semi-annually bond, priced at Rs.100 Buy: A 20 year, 8 percent AAA Corporate yield 4,15 percent semi-annually. bond, priced at Rs.97.696 to Assumption: Workout period 12 months and semi-annual reinvestment rate is 4 percent.

Bond sold Bond purchased1. Investment (Rs.) 100.000 97.6962. Two coupon payments (Rs.) 8.000 8.0003. Reinvestment income (on one coupon) at 4%

(Rs.) 0.160 0.160

4. Market value after 6 months at 4% semi-annual required yield (Rs.)

100.000 100.000

5. Total rupees accrued 108.160 108.1606. Total rupee gain (5-1) 8.160 10.4647. Gain per rupee invested (6 + 1) 0.0816 0.10718. Realized semi-annual yield 4,00 5.229. Gain in basis points per year (5.22 - 4.00} x 2

x 100 244

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Table 5.9 illustrates the mechanics of a hypothetical substitution swap between two securities that are similar in all respects except that the bonds to be purchased currently carries a 4.15 percent serai-annual yield versus the 4 percent semi-annual yield on the bond to be sold. In this illustration, it is assumed that the purchased bond's price will realign with the 4 percent market yield in one year. The one-year time-frame over which the yields are expected to converge is called the workout period. As table 5.9 illustrates, to compute the incremental realized return from the swap, all three return components from both bonds must be considered. In this illustration, as is common for substitution swaps, the driving force behind the gain in yield is the price appreciation on the purchased bond. As market forces equalize the two bond's yields, the price appreciation on the purchased bond, in conjunction with the coupon and reinvestment income, generates an increase in semi-annual yield of 1.22 percent (5.22 percent - 4.00 percent = 1.22 percent). This translates into an increase of 244 basis points for the year.

With respect to the illustration, few comments are in order. First, the increase in the semi-annual yield of 122 basis points or, alternatively, the 244 basis-point increase for the year occurs only during the 12-month adjustment period. To actually earn an additional annual 244 basis-point increase in compound return over the 20 year life of the bond, the investor would have to conduct a bond swap each year, for 20 years, producing an incremental 244 basis points with each swap. Put differently, over the life of the bond the 244 basis-point increase is spread over 20 years, amounting to roughly 12 basis points per year. Thus the increase in total compound return, per year, is only about 12 basis points, if this were the only swap made. Although the gain is attractive, it is short-lived.

There are risks associated with substitution swaps. First, as mentioned above, the workout period may take longer than in our illustration, perhaps even up to 20 years. When this happens, the gain is spread out over a longer period of time and it lessens the value of the swap, particularly for the investor seeking to engage in this type of switching every year. Second, interest rates may go up during the 12-month period and eliminate the price appreciation component. Thus, even with a swap of perfect substitutes, things may not workout.

PURE YIELD PICK-UP SWAP

With a pure yield pick-up swap, the investor seeks to increase the portfolio's yield to maturity by swapping out of a lower-yield bond into a higher-yield bond. Because this swap usually involves switching from a lower-coupon bond into a higher-coupon bond, the current yield is also increased. However, because higher-coupon bonds typically have call features, the swap may increase the call risk of the portfolio. In addition, for yield swaps across different rating classifications, the default risk of the portfolio may also be altered.

Example 5.7

Substitution Swap

Table 5.10: Pure Yield Pick-up Swap

Sell: A 20 year, 9 percent AAA Corporate bond, priced at Rs.95.560 to yield 4.75 percent semi-annually. Buy: A 20 year, 10 percent AA Corporate yield 5.00 bond, Priced at Rs.100 to percent semi-annually. Bond sold Bond purchased

1. Investment (Rs.) 95.560 100.0002. Coupon income (Rs.) (45 x 40) and (50 x 40) 180.000 200.000

3. Reinvestment income at 5.0% semi-annually 363.599 403.999

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4. Bond value at maturity 100,000 100,0005. Total accumulated value (2 + 3 + 4)(Rs.) 643.599 703.999

6. Realized semi-annual yield 4.88% 5.00%7. Gain in accumulated value (Rs.) 60.4008. Gain in basis points per year (5.00 -4.88) x2 x

100 24

Table 5.10 illustrates a pure yield pick-up swap. In this illustration, the investor is swapping in order to enhance both the total return (yield to maturity) and current yield by moving into a higher-coupon, higher-yield-to-maturity bond. Note that the default rating on the new bond is lower, thus increasing the credit risk of the portfolio.

The mechanics of this swap are very similar to those of the previous illustration, except that the gain in yield is measured over the entire life of the bond. In this illustration, the increase in yield of 12 basis points per 6-month period, or 24 on an annual basis, comes through the additional coupon income of Rs.20 (Rs.20 = Rs.200 - Rs. 180) and the incremental reinvestment income of Rs.40.4 (Rs.40.4 = Rs.403.999 - Rs.363.599) that the higher-yielding bond is expected to generate over the 20 years.

There are a couple of attractive features in this pure yield pick-up swap. First, the swap required no yield spread inefficiencies or forecasts of interest rates. The investor simply switched into a higher-yielding security. Second, because this swap takes a long run view of the potential gain, no assumed short-term workout period is required in order to derive the benefits from the increase in yield.

The pure yield pick-up swap, however, is not without its risks. First, in order to effect such a swap, the investor may have to accept callable bonds as well as securities with lower credit ratings. Second, because the workout period is usually for the life of the bond, achieving a target reinvestment rate will be more difficult than for a swap based on a shorter time horizon. Thus the gain in yield will be sensitive to the long run reinvestment rate.

USE OF DERIVATIVES IN BOND PORTFOLIO MANAGEMENT

The advent of derivative products like interest rate futures, options and swaps changed the scenario of bond portfolio management. Fund managers have achieved new degrees of freedom. Now it is possible for the fund manager to after the interest rate sensitivity of a bond portfolio economically and quickly. Fund managers now have the flexibility to create any risk-return trade-off profile they want, which, previously, were unavailable or too costly to create. In this section, we will briefly discuss about the uses of interest rate futures, options and swaps in bond portfolio management.

Interest Rate Futures

Interest rate futures have varied uses depending on who uses them. For bond portfolio managers, primary uses are speculating on the movement of interest rates, controlling interest rate sensitivity of the portfolio and hedging against interest rate changes.

SPECULATING ON THE MOVEMENT OF INTEREST RATES

As with bond prices, price of interest rate futures and rate of interest are inversely related. When interest rates rise, price of the futures contract decreases and when interest rates decrease, price of the futures contract rises. A portfolio manager who wants to speculate on the movements in interest rates can directly take position in the cash market. That is, buy a long-term bond, if he expects the interest rates to decline and short the bond if he expects the rate to move up. The fund manager can also take position in the futures market so as to make a profit out of his expectations. If he expects the interest

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rate to go up, he can go short on futures and square his position when his expectations are realized. On the other hand, if he expects the interest rates to decline, he can go long on futures and square his position at a later point.

There are three advantages for the fund manager to trade in futures market than cash market: (i) transaction costs for trading in futures market are less than transaction costs for trading in the cash market, (ii) high leverage offered by futures because of lower margin requirements and (iii) it is easier and faster to take position in the futures market than the cash market. The leverage advantage has also the risk of encouraging speculation. Since the margin requirements are low and position can be taken easily, speculation is easier with futures than cash market.

CONTROLLING THE INTEREST-RATE SENSITIVITY OF A PORTFOLIO

Depending on the expected interest rate movements, a bond manager may wish to change the interest rate sensitivity of the portfolio. If the interest rate is expected to increase (decrease), he would like to decrease (increase) the interest rate sensitivity of the portfolio so that the decrease (increase) in the portfolio value will be less (more). As we have seen, duration of the portfolio is a measure of interest rate sensitivity of the portfolio. Therefore, by changing the duration of the portfolio interest rate sensitivity of the portfolio can be changed. Changing duration through cash market transactions may be costly and time taking. Futures market offers an inexpensive and quick means of changing the duration.

In some cases, the fund manager may require to have duration of the portfolio, which may not be available with cash market securities. Suppose, a pension fund has liabilities with an average duration of 20 years. Now the fund manager would like to have assets also with an average duration of 20 years. If there are no cash market securities available with this duration, then the fund manager can take appropriate position in the futures market so as to achieve the targeted duration for the portfolio.

Approximate number of futures contracts (X) necessary to achieve the targeted duration for the portfolio is given by the following formula.

X =

Where,X = approximate number of futures contractsDT = target effective duration for the portfolioDI = initial effective duration for the portfolioPI - initial market value of the portfolioDF = effective duration for the futures contractPF = market value of the futures contract.

As can be seen from the formula, if the targeted duration is greater than the initial duration, X becomes positive. That is, the fund manager should go long (purchase) in futures contracts so as to increase duration for the portfolio.

HEDGING

A fund manager can use futures to hedge by taking a position in futures as a temporary substitute for transactions to be made in the cash market at a later date. If cash and futures prices move in perfect unison, any loss realized by the fund manager from one position will be offset by profit on the other

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position and previous wealth of the portfolio is preserved. This is the case of a perfect hedge where the net profit or loss from the positions is exactly as anticipated.

But in reality, prices in the cash and futures markets do not move in perfect unison. The difference between the cash market price and the futures market price is called basis and basis does not remain same from the time hedge is placed and lifted. The risk caused by unpredictable changes in the basis is called basis risk.

Basis risk may be substantially increased by cross hedging. Cross hedging is a situation where the bond to be hedged and the bond underlying the futures contract are different.Risk resulting from a cross hedge can be minimized by choosing the appropriate hedge ratio.

Hedge ratio =

where, volatility is in absolute rupee terms.Number of contracts

= Hedge ratio X

There are two types of hedges depending on whether you buy or sell futures contracts. A long (or buy) hedge is used to protect against an increase in the cash market price of the bond. If the fund manager expects substantial cash inflows shortly, he can hedge against any increase in the cash market prices or decrease in the interest rates by going long in the futures market. A short (or sell) hedge is used to protect against a decrease in the cash market price of the bond. Suppose a bond portfolio is due for liquidation in three months. Concerned about the possible increase in the interest rates or decrease in the bond prices, he goes short in the futures market. At the time of liquidation if bond prices have fallen, loss in the cash market can be offset by the gain in futures market.

Options

Interest rate options can be used to hedge an underlying position in the bonds. Two popular strategies are protective put and covered call writing. If an investor is concerned about a possible increase in interest rate, he can buy a put option. This gives the investor a right to sell the bond at the strike price. If the increase in the rate of interest forces the price of the bond below the strike price, he can sell the bond at strike price. Therefore, the investor is ensured of at least the strike price for the bond. Of course, the investor needs to pay a price for this option.

Covered call writing is not entered with the sole objective of protecting a portfolio against rising rates. If the investor does not expect the market to trade much higher or much lower than its present level, he can opt for covered call writing. Covered call writing brings in premium income that can to a certain extent protect the portfolio against rising interest rates. On the other hand, if rates fall, portfolio appreciation is limited to the strike price. Therefore, covered call writing offers some protection against the portfolio depreciation, but limits the portfolio appreciation.

Interest Rate Swaps

Fund managers can use interest rate swaps in asset/liability management. Consider a Mutual Fund whose investments are in floating rate instruments but the fund has promised a fixed rate of return for the investors. If the floating rate falls substantially, the fund may not be able to honor its fixed rate commitment to the investors. By entering into an interest rate swap, the fund manager can hedge his position by swapping floating rate for a fixed rate and will be in a better position to give the promised rate to the investors.

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In the same way, a fund manager with fixed rate assets and floating rate liabilities can enter into an interest rate swap to hedge against floating rate volatility.

Market Timing

The previous analysis focused on the capability of managers in generating superior performance by means of stock-selection techniques. Managers can also generate superior performance by timing the market correctly, that is, by assessing correctly the direction of the market and positioning the portfolio accordingly. Managers with a forecast of a declining market can decrease the beta of the equity portion of the portfolio. Conversely, a forecast of a rising market would call for an increase in the beta of the equity portion of the portfolio. One method for diagnosing the success of managers in this endeavor is to simply look directly at the way fund return behaves relative to the return of the market. This method first involves calculating a series of returns for the funds and market index over a relative performance period, and plotting these on a scatter diagram. For example, one can calculate quarterly returns for a fund and for the market index over, say, a 5-year period and plot them on a scatter diagram. Given these plots, we could then fit a characteristic Line.

If the fund did not engage in market timing, and concentrated only on stock selection, the average beta of the portfolio should be fairly constant and a plotting of fund against market return would show a linear relationship as illustrated in figure 5.2. If the manager changed the beta of the portfolio over time, but was unsuccessful in properly assessing the direction of the market, the plotting would still show a linear relationship. The unsuccessful market timing activity would merely introduce an additional scatter to the plots around the fitted relationship.

Figure 5.2: Excess return on the Erf

On the other hand, if the manager was able to successfully assess the market direction and change the portifolio beta accordingly, we would observe the sort of relationship shown in figure 5.2. When the market increases substantially, the fund has a higher than normal beta and it tends to do better than otherwise. Correspondingly, when the market declines, the fund has a lower than normal beta and it declines less than it would otherwise. This causes the plotted points to be above the linear relationship at both high and low levels of market returns and would give curvature to the scatter of points.

To more properly describe this relationship, we can fit a curve to the plots by adding a quadratic term to the simple relationship.

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rp = ap + brm + crm

Where,

rp = return of the fundrm = return on the market indexa,b,c = values to be estimated by regression analysis

The curve fitted to the plots in the following figure indicates that the value of the 'c' parameter of the quadratic term is positive. This indicates that the curve becomes steeper as one moves to the right of the diagram, that is, the funds movements are amplified on the upside and dampened on the downside relative to the market. This reveals that the fund manager was anticipating market changes accurately, and the superior performance of that fund can be attributed to skills in timing the market.

Figure 5.3: Excess return on the tuna Erf

Performance Attribution Analysis

The methods adopted for performance evaluation should be acceptable to both evaluator and the evaluated. The acceptability will be higher if the methods are simple, consistent and accurate. The methods described so far conform to the above and hence have become acceptable to investors and fund managers as well. Any good performance measurement should begin by examining funds in risk and return space. Then the job of analysis can further proceed. The main goal of performance attribution analysis is to find the impact of all decisions made with respect to the management of the portfolio. These include strategic policy decision, the asset allocation decision and the asset selection decisions. Strategic policy decision requires setting a policy or benchmark or normal portfolio, that illustrates the suitable asset classes for long-term portfolio investment. This is the top-most investment decision that will impact the returns of the portfolio. The impact of the decisions made at this level on performance can be estimated by comparing the returns of policy portfolio to the returns of a naive portfolio. A naive portfolio is difficult to specify but can consist assets to which investor wants to compare against. Thus a naive portfolio may contain only T-Bills, 100 percent to equity investments or to average asset allocation weighing to all real asset portfolios. We will discuss the effect of decisions made

Asset Benchmark Weight Asset Returns Policy Allocate Actual Allocati Selectio

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index representing the asset class in the policy and allocated portfolios

s Portfolio dPortfolio

Portfolio onEffect

nEffect

Policy

Actual

IndexPortfolio

Stocks

SPP 500 Stock Index

65%

55%

12.75%

11.00%

12.75% X 0.65= 8.2875%

12.75% x 0.55= 7.0125%6.5%x

11.00%0.556.05%9.5% x

(0.2358)

(0.9625%)

Bonds

JP Morgan Bond Index

25%

30%

6.50%

9.50%6.5 x 0.25 = 1.625%

6.5% x 0.30 = 1.95%

9.5% x0.30 = 2.85%

(0.1946)

0.90%

Cash Equivalents

U.S Treasury Bills

10%

15%

4.80%

5.85%4.8x 0.10 = 0.48%

4.8 x 0.15 = 0.72%

0.15 = 0.8775%

(0.2796%)

0.1575%

Total returnTotal effect

10.3925%

9.6825% 9.7775%(0.71%)

0.095%

Allocation Effect

The allocation affect estimates the impact of fund managers decision to allocate funds at proportions other than the targetted levels. This happens because of the value addition a fund manager would like to make for tactical reasons. The difference between the policy portfolio and the allocated portfolio indicates the contribution of asset allocation decision. In the above example to estimate the allocation effect, the returns of the suitable benchmarks for each asset class are compared with the policy portfolio as a whole. The impact of the allocation decisions, that is allocation effect, is calculated simply as the difference between the allocated portfolio return and the policy portfolio return. In our example, the allocation effect is equal to -0.71% (9.6825 - 10.5925). Here the negative sign indicates the manager's decision to underweight equities and overweight cash equivalents have been detected and incorporated without including the impact of individual security selection. This allocation effect reflects the difference between the return that manager would have been gained had the indexes been bought in the actual weighing (allocation portfolio return) and the return he would have received had the indexes been bought in the policy weights (policy portfolio return).

Below the investment policy level, which consists the impact of the allocation effect and selection effect in portfolio performance. The table given below contains three asset classes, weights, and benchmarks for the portfolio. For the assessment period, the actual portfolio is distributed into three assets class differently from the policy allocation to these portfolios. For each type of asset class, the returns for the benchmark and the return for the actual portfolio policy are shown. The impact of the allocation affect and selection affect can be assessed once a policy decision is taken. The policy decision specifies the benchmark indices and the proportions of funds to be allocated to each asset class. However, the fund managers actual allocation and the realized returns will be at variance from the targeted allocation and the expected returns on the benchmark indices. Let us consider the following data.

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Details of the break-up of the allocation effect:

Allocation Effect = Actual Weight -Policy Weight

X Asset return in policy portfolio - Total return on policy portfolio

Equity allocation effect

= (0.55 - 0.65) X (12.75 - 10.3925J = 0-2358

Fixed-income allocation effect

= (0.30 - 0.25) X (6.5 - 10.3925) = 0.1946

Cash equivalent allocation effect

= (0.15 - 0.10) X(4.8-10.3925) =

Total allocation effect

A fund manager enhances allocation value by allocating a larger portion in an asset class that provides better performance with respect to the total returns of the portfolio or allocating a smaller portion in an asset class showing inferior performance relative to the total return of the policy portfolio.

Selection Effect

The selection affect estimates the impact of the fund manager's decision to select stock. This is assessed by finding the difference between the return on allocated portfolio and actual portfolio. In the above example, the total selection effect is 0.095%. This total selection effect has been measured here by adding the difference between the actual portfolio return and the allocated portfolio return.

Total selection effect =

Stock selection effect + Bond selection effect + Cash selection effect= [(11.00 x 0.55) - (12.75 x 0.55)] + [(9.5 x 0.30) -(6.5 x 0.30)] + [(5.85 x 0.15) - (4.80 x 0.15)] = -0.9625% + 0.90% + 0.1575% - 0.095%

Total selection effect of 0.095% is the cumulative effect of stock selection, bond selection and cash equivalent selection by the fund manager. The selection effect reflects the ability of the portfolio manager to choose individual stock, bond and cash equivalent.

Risk-Adjusted Performance Measures: Some Issues

Let us conclude our discussion on performance measures with a discussion on the criticism leveled against the use of these measures.

Use of Market Surrogate

All measures other than Sharpens measure require the identification of a market portrfolio. Empirical studies conducted in the US market have also revealed that when commonly used NYSE based surrogates are involved such as the Dow-Jones Industrial Average, the S&P 500 or any index comparable to the NYSE composite, the performance ranking of the common (equity) stock portfolios are quite different. Hence the performance is highly dependent on the selection of market portfolio.

Limitation in Using Market Index as a Benchmark Portfolio

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It has been argued that a market index should not be used as a benchmark portfolio because it is nearly impossible for an investor to construct a portfolio whose returns replicate these on the index. This is because of the transaction costs involved in initially forming the portfolio, in restructuring the portfolio when stocks are replaced in the index; and in purchasing more shares of the stocks comprising the index when the cash dividends are received. Hence, the return on the index overstate the returns of that a passive investor can earn.

Skill or Luck

Obviously, an investor would like to know whether an apparently successful investment manager was skilled or just lucky. Unfortunately a very long time interval is needed to distinguish skill from luck on the part of the investment manager.

Validity of CAPM

The measure of portfolio performance (Jensen's measure and Treynor's measure) are based on the CAPM, which may not be the correct asset pricing model. Put differently, if assets are priced according to some other model, say the APT model, use of the beta based performance measure will be inappropriate. It must be noted that the Sharpe's measure (reward-to-variability ratio) is immune to this criticism because it uses standard deviation as a measure of risk; and does not rely on the validity or on the identification of a market portfolio.

DIVERSIFICATION

Diversification is the strategy of combining distinct asset classes in a portfolio in order to reduce overall portfolio risk. In other words, diversification is the process of selecting the asset mix so as to reduce the uncertainty in the return of a portfolio. Diversification helps to reduce risk because different investments may rise and fall independent of each other. The combinations of these assets will nullify the impact of fluctuation, thereby, reducing risk.

Most financial assets are not held in isolation, rather they are held as parts of portfolios. Banks, pension funds, insurance companies, mutual funds, and other financial institutions are required to hold diversified portfolios. Even individual investors - at least those whose security holdings constitute a significant part of their total wealth - generally hold stock portfolios, not the stock of a single firm. Why is it so? An important reason is the lowering of risk, which means risk of getting zero or negative return on some assets. If a person holds a single asset, he or she is highly dependent on the issuer firm, its success, and dividend policy, as well as on the overall current market situation. On the other side, holding a well-diversified portfolio protects a person from both market fluctuations and internal problems of issuer. A diversified portfolio helps to keep investment returns stable.

As we have learnt in the earlier sections that the portfolio risk depends not only on the variance of the individual securities in the portfolio but also on the correlation coefficient between each pair of securities.

Diversification in a portfolio can be achieved in many different ways. Individuals can diversify across one type of asset classification - such as stocks. To do this, one might purchase shares in the leading companies across many different (and unrelated) industries. Many other diversification strategies are also possible. You can diversify your portfolio across different types of assets (stocks, bonds, and real estate for example) or diversify by regional allocation (such as state, region, or country). Thousands of options exist. Luckily, in almost every effective diversification strategy, the ultimate goal is to improve returns while reducing risks.

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The following possible ways can be applied by a fund manager while considering the mode of diversification.

Diversify within an industry: Investing in a number of different stocks within the same industry does not generate a diversified portfolio since the returns of firms within an industry tend to be highly correlated. However, this is better than investing in a single stock.

Diversify across industry groups: Correlation between industries is likely to be lower than between the firms with an industry. However, some industries themselves can be highly conelated with other industries and hence diversification benefits can be maximized by selecting stocks from those industries that tend to move in opposite directions or have very little correlation with each other.

Diversify across geographical regions: Companies whose operations are in the same geographical region are subject to the same risks in terms of natural disasters and state or local tax changes. Investing in companies whose operations are not in the same geographical region can diversify these risks.

Diversify across countries: Stocks in the same country tend to be more correlated than stocks across different countries. This is because many taxation and regulatory issues apply to all stocks in a particular country. International diversification provides a means for diversifying these risks.

Diversify across asset classes: Investing across asset classes such as stocks, bonds, and real property also produces diversification benefits. The returns of two stocks tend to be more highly correlated, on average, than the returns of a stock and a bond or a stock and an investment in real estate.

Diversification across Industries

Diversification across industries refers to the diversification by any portfolio holder with the help of appropriating the fund in various industries. The industries to be chosen by any fund manager should provide the minimum required return by canceling out the risk of the individual industries. For example, assume that a fund manager has invested only in the aluminum industry. It is possible that this industry may not perform well because of lack of proper power supply. The effect of power scarcity could lead the prices of all aluminum stocks to plummet. The entire holdings of fund manager would be left at deflated level. However, if fund manager also invests in other industries such as oil, consumer durables and electronics, it is unlikely that unsystematic risks in aluminum industry will adversely affect fund value. What is more, unfortunate circumstances in the aluminum industry may result in a boom in other industries which are not affected by power crisis. If a fund manager is holding stocks of those industries, he might even benefit from the troubles of aluminum industry. Unsystematic risks can be avoided by diversifying among different industries rather than just investing in the same one.

International Diversification

If any portfolio manager tries to diversify his or her portfolio by investing across the countries, the diversification is known as international diversification. For an individual investor, it is quite difficult to adopt this kind of diversification because the regulations of different countries as well as high transaction costs attached in dealing with foreign investments. Even for all fund managers it is not possible to implement international diversification due to regulatory constraints attached with it.

Given the enormous opportunities available around the world, international diversification can be a

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beneficial strategy for big investors. To analyze this kind of diversification we have to consider the following factors:

1. Returns available in different countries2. The risk attached to each foreign market3. The correlation coefficients across international markets.

The return from a foreign investment depends on the return on the assets within its domestic market and the change in the exchange rates between the asset's own currency and the currency of the buyer's home country. Therefore, the return on the asset for a foreign buyer can differ according to the domicile of the buyer. For example, assume that the stock of Microsoft earns a return of 20% for a US investor, but the real return for investors in India or Indonesia will depend on the corresponding exchange rate between the two countries. If the rupee is depreciating against the dollar, the investment in Microsoft stock will yield greater return, however, if the rupee is appreciating against dollar, the return from such investment will produce lower returns to the Indian investor. Thus the exchange rate between security's country and the country of purchaser plays an important role in deciding the actual return available to the international purchaser.

Basically, return from a foreign investment could be segregated into the return in the security's home market and return from the changes in exchange rates.

There are two sources of risk attached to an investment in the foreign securities. Firstly, the return on an investment in foreign securities fluctuates due to change in the securities prices within the securities domestic market, and second, source of risk is the variations in exchange rates.

The risk of investing in foreign securities can be assessed using the standard deviation of securities and the correlation coefficients between two security markets. The correlation coefficients between the markets of countries, where investment has been made, play a significant role in deciding the risk of the international portfolio. If any fund manager in India invests in US and Japanese stock, the correlation between US and Japanese market should be taken into account while calculating the risk of the portfolio consisting Japanese and US stocks.

The total risk of any international portfolio can be split into domestic risk and the exchange risk. Domestic risk is indicated in the standard deviation of returns, when returns are calculated in the domestic currency. Exchange risk can be measured by assessing the variations in the exchange rates, If an Indian investor has invested his money in US stock, the risk of investing can be represented by the standard deviation of the US stock price changes in dollars and the standard deviation of changes in the rupee-dollar exchange rate. It should be noted here that variability of exchange rates should be calculated by assessing the variability of each foreign currency with respect to domestic country. Use of hedging strategy by any international investor can protect his or her portfolio against the exchange risk. If an Indian investor enters into a forward contract he can protect the value of fund.

Exchange rate fluctuations generally increase the correlation among countries returns. The risk of an international portfolio can be significantly reduced, if the portfolio risk is completely protected against the exchange risk.

Apart from the exchange risk, there are several issues attached with the investment in the foreign assets. For example, if the tax rate imposed on the foreign investment differs greatly from the domestic investment, the risk of foreign portfolio will increase. Differential tax structures are quite common for international investors. Several countries impose withholding tax on dividends received from international investments. In withholding tax arrangement, a taxable firm can get a domestic credit for the foreign tax paid, provided there is an agreement between the home country and the

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foreign country. But for a non-taxable portion of any fund's portfolio like pension assets, the withholding tax is a cost that may lower the return from the international investments. Higher transaction cost in international investments compared to the domestic investment can cause lower return from the foreign investments. Controls sometimes do not allow the full benefit to be derived by an international investor. For example, RBI did not allow FIs to take a forward contract for their portfolio investment. Similarly RBI puts a cap on FII investment in a company. These restrictions either increase the risk or reduce the return.

One form of international diversification by any domestic investor is to invest in the multinational corporations based on his or her country, but research results state that this kind of the diversification does not result in the international diversification because stock prices of MNC behave much like the stocks of domestic firms and least affected by the foreign factors.

Diversification across Asset Classes

Diversification across asset classes provides a cushion against market tremors because each asset class has different risks, rewards and tolerance to economic events. By selecting investments from different asset classes, any portfolio manager can minimize the overall portfolio risk. Securities whose price movements are opposite to each other are negatively correlated. When negatively correlated assets are combined within a portfolio, the portfolio volatility is reduced. For example, if the returns from stocks and bonds are negatively correlated, investing in both stock and bond can result in lowering the risk of the portfolio.

Diversification across asset classes works with the help of three over-arching asset classes, which are stocks, bonds (or stock and bond mutual funds), and so-called cash-equivalent securities, such as money market mutual funds so called, because they are quite safe and allow easy access to your money, much like cash. Investing in any of these securities carries some risk, but at varying levels. If any portfolio is formed using these assets with required risk/return trade off, the desirable benefits of the optimal diversification can be achieved.

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