11 Risk and Return

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11 Risk and ReturnInvestors purchase financial assets such as shares of stock because they desire to increase theirwealth, i.e., earn a positive rate of return on their investments. The future, however, is uncertain;

investors do not know what rate of return their investments will realize.

In finance, we assume that individuals base their decisions on what they expect to happen andtheir assessment of how likely it is that what actually occurs will be close to what they expected

to happen. When evaluating potential investments in financial assets, these two dimensions ofthe decision making process are called expected return and risk.

The concepts presented in this paper include the development of measures of expected return andrisk on an indivdual financial asset and on a portfolio of financial assets, the principle of

diversification, and the Captial Asset Pricing Model (CAPM).

11.1 Expected Return

The future is uncertain. Investors do not know with certainty whether the economy will be

growing rapidly or be in recession. As such, they do not know what rate of return theirinvestments will yield. Therefore, they base their decisions on their expectations concerning the

future.

The expected rate of return on a stock represents the mean of a probability distribution of

possible future returns on the stock. The table below provides a probability distribution for thereturns on stocks A and B.

State Probability Return onStockA Return onStockB

1 20% 5% 50%2 30% 10% 30%3 30% 15% 10%3 20% 20% -10%

In this probability distribution, there are four possible states of the world one period into the

future. For example, state 1 may correspond to a recession. A probability is assigned to eachstate. The probability reflects how likely it is that the state will occur. The sum of the

probabilities must equal 100%, indicating that something must happen. The last two columnspresent the returns or outcomes for stocks A and B that will occur in the four states.

Given a probability distribution of returns, the expected return can be calculated using thefollowing equation:

where

E[R] = the expected return on the stock,


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N = the number of states, pi = the probability of state i, and Ri = the return on the stock in state i.

Expected Return on Stocks A and B

StockA

StockB

So we see that StockB offers a higher expected return than StockA. However, that is only partof the story; we haven't yet considered risk.

11.2 Measures of Risk - Variance and Standard Deviation

Risk reflects the chance that the actual return on an investment may be very different than theexpected return. One way to measure risk is to calculate the variance and standard deviation ofthe distribution of returns.

Consider the probability distribution for the returns on stocks A and B provided below.

State Probability Return onStockA Return onStockB

1 20% 5% 50%2 30% 10% 30%3 30% 15% 10%3 20% 20% -10%

The expected returns on stocks A and B were calculated on the Expected Return page. The

expected return on Stock A was found to be 12.5% and the expected return on Stock B wasfound to be 20%.

Given an asset's expected return, its variance can be calculated using the following equation:

where N = the number of states, pi = the probability of state i, Ri = the return on the stock in state i, and E[R] = the expected return on the stock.The standard deviation is calculated as the positive square root of the variance.


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The Expected Return on a Portfolio is computed as the weighted average of the expectedreturns on the stocks which comprise the portfolio. The weights reflect the proportion of the

portfolio invested in the stocks. This can be expressed as follows:

where

E[Rp] = the expected return on the portfolio, N = the number of stocks in the portfolio, wi = the proportion of the portfolio invested in stock i, and E[Ri] = the expected return on stock i.For a portfolio consisting of two assets, the above equation can be expressed as

Expected Return on a Portfolio ofStocks A and B

Note: E[RA] = 12.5% and E[RB] = 20%

Portfolio consisting of 50% StockA and 50% StockB


12.1.2 Portfolio Variance and Standard Deviation

The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of

the stocks that make up the portfolio but also how the returns on the stocks which comprise theportfolio vary together. Two measures of how the returns on a pair of stocks vary together are thecovariance and the correlation coefficient.

The Covariance between the returns on two stocks can be calculated using the followingequation:

where

s12 = the covariance between the returns on stocks 1 and 2, N = the number of states, pi = the probability of state i, R1i = the return on stock1 in state i, E[R1] = the expected return on stock1, R2i = the return on stock2 in state i, and E[R2] = the expected return on stock2.


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The Correlation Coefficient between the returns on two stocks can be calculated using thefollowing equation:

where r12 = the correlation coefficient between the returns on stocks 1 and 2, s12 = the covariance between the returns on stocks 1 and 2, s1 = the standard deviation on stock1, and s2 = the standard deviation on stock2.Covariance and Correlation Coefficent between the Returns on Stocks A and

B

Note: E[RA] = 12.5%, E[RB] = 20%, sA = 5.12%, and sB = 20.49%.

Using either the correlation coefficient or the covariance, the Variance on a Two-AssetPortfolio can be calculated as follows:

The standard deviation on the porfolio equals the positive square root of the the variance.

Variance and Standard Deviation on a Portfolio ofStocks A and BNote: E[RA] = 12.5%, E[RB] = 20%, sA = 5.12%, sB = 20.49%, and

rAB = -1.



Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lowervariance and standard deviation than eitherStocks A or B and the portfolio has a higher expected

return than Stock A. This is the essence of Diversification, by forming portfolios some of the riskinherent in the individual stocks can be eliminated.


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13 DiversificationA portfolio formed from risky securities can have a lower standard deviation than either of theindividual securities. The benefits of diversification, i.e., the reduction in risk, depends upon the

correlation coefficient (or covariance) between the returns on the securities comprising theportfolio.

Consider stocks C and D. Stock C has an expected return of 8% and a standard deviation of10%.

Stock D has an expected return of16% and a standard deviation of20%. The concept ofdiversification will be illustrated by forming portfolios of stocks C and D under three different

assumptions regarding the correlation coefficient between the returns on stocks C and D.

13.1.1 Case 1: Correlation Coefficient = 1

The table below provides the expected return and standard deviation for portfolios formed fromstocks C and D under the assumption that the correlation coefficient between their returns equals

1.

Weight

of

Stock

CPortfolio

Expected

ReturnPortfolio

Standard

Deviation100% 8% 10%90% 8.8% 11%80% 9

.6% 12%

70% 10.4% 13%60% 11.2% 14%50% 12% 15%40% 12.8% 16%30% 13.6% 17%20% 14.4% 18%10% 15.2% 19%0% 16% 20%

Opportunity Set and Efficient Set


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Opportunity Set- The opportunity set depicts the set of riskreturn choices that can be achieved by forming a portfolio of

stocks C and D. It is represented by the entire curves plottedon the graohs on this page.

Efficient Set- The efficent set (or efficient frontier) is thepositively sloped portion of the opportunity set. It is the set

of risk return choices which offer the highest expected returnfor a given level of risk.

When the correlation coefficient between the returns on two securities is equal to +1 the returnsare said to be perfectly positively correlated. As can be seen from the table and the plot of the

opportunity set, when the returns on two securities are perfectly positively correlated, none of therisk of the individual stocks can be eliminated by diversification. In this case, forming a portfolio

of stocks C and D simply provides additional risk/return choices for investors.

13.1.2 Case 2: Correlation Coefficient = -1

The table below provides the expected return and standard deviation for portfolios formed fromstocks C and D under the assumption that the correlation coefficient between their returns equals-1.

Weight

of

Stock

CPortfolio

Expected

ReturnPortfolio

Standard

Deviation100% 8% 10%90% 8.8% 7%80% 9.6% 4%70% 10.4% 1%

66.67% 10.67% 0%60% 11.2% 2%50% 12% 5%40% 12.8% 8%30% 13.6% 11%20% 14.4% 14%10% 15.2% 17%0% 16% 20%


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When the correlation coefficient between the returns on two securities is equal to -1 the returnsare said to be perfectly negatively correlated or perfectly inversely correlated. When this is the

case, all risk can be eliminated by investing a positive amount in the two stocks. This is shown inthe table above when the weight ofStock C is 66.67%.

13.1.3 Case 3: Correlation Coefficient = 0

The table below provides the expected return and standard deviation for portfolios formed from

stocks C and D under the assumption that the correlation coefficient between their returns equals0.

Weight

of

Stock

CPortfolio

Expected

ReturnPortfolio

Standard

Deviation100% 8% 10%90% 8.8% 9.22%80% 9.6% 8.94%70% 10.4% 9.22%60% 11.2% 10%50% 12% 11.18%40% 12.8% 12.65%30% 13.6% 14.32%20% 14.4% 16.12%10% 15.2% 18.03%0% 16% 20%

When the correlation coefficient between the returns on two securities is equal to 0 the returnsare said to be uncorrelated. In this case, some risk can be eliminated via diversification. Notice

that when the weight ofStock C is between 100% and 60% the portfolios have a higher expectedreturn than Stock C and a lower standard deviation than eitherStocks C or D. This is depicted in

the graph by the inward curve in the opportunity set.


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14 Capital Asset Pricing Model (CAPM)Because investors are risk averse, they will choose to hold a portfolio of securities to takeadvantage of the benefits of Diversification. Therefore, when they are deciding whether or not to

invest in a particular stock, they want to know how the stock will contribute to the risk and

expected return of their portfolios.

The standard deviation of an individual stock does not indicate how that stock will contribute tothe risk and return of a diversified portfolio. Thus, another measure of risk is needed; a measure

of a security's systematic risk. This measure is provided by the Capital Asset Pricing Model(CAPM).

Systematic and Unsystematic Risk

An asset's total risk consists of both systematic andunsystematic risk.

Systematic risk, which is also called market risk or

undiversifiable risk, is the portion of an asset's risk thatcannot be eliminated via diversification. The systematic riskindicates how including a particular asset in a diversified

portfolio will contribute to the riskiness of the portfolio

Unsystematic risk, which is also called firm-specific ordiversifiable risk, is the portion of an asset's total risk that

can be eliminated by including the security as part of adiversifiable portfolio.

The Capital Asset Pricing Model (CAPM) provides an expression which relates the expectedreturn on an asset to its systematic risk. The relationship is known as the Security Market Line

(SML) equation and the measure of systematic risk in the CAPM is called Beta.

14.1.1 The Security Market Line (SML)

The SML equation is expressed as follows:

where

E[Ri] = the expected return on asset i, Rf= the risk-free rate, E[Rm] = the expected return on the market portfolio, bi = the Beta on asset i, and E[Rm]-Rf= the market risk premium.The graph below depicts the SML. Note that the slope of the SML is equal to (E[Rm]-Rf) whichis the market risk premium and that the SML intercepts the y-axis at the risk-free rate.


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In capital market equilibrium, the required return on an asset must equal its expected return.Thus, the SML equation can also be used to determine an asset's required return given its Beta.

14.1.2 The Beta (Bi)

The beta for a stock is defined as follows:

where

sim = the Covariance between the returns on asset i and the market portfolio and s2m = the Variance of the market portfolio.Note that, by definition, the beta of the market portfolio equals 1 and the beta of the risk-free

asset equals 0.

An asset's systematic risk, therefore, depends upon its covariance with the market portfolio. Themarket portfolio is the most diversified portfolio possible as it consists of every asset in the

economy held according to its market portfolio weight.

Example Problems

1. Find the expected return on a stock given that the risk-freerate is 6%, the expected return on the market portfolio is12%, and the beta of the stock is 2.

2. Find the beta on a stock given that its expected return is

16%, the risk-free rate is 4%, and the expected return on themarket portfolio is 12%.


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14.2 Critical assumptions of CAPM

The CAPM is simple and elegant. Consider the many assumptions that underlie the model. Arethey valid?

Zero transaction costs. The CAPM assumes trading is costless so investments are pricedto all fall on the capital market line. If not, some investments would hover below and above theline -- with transaction costs discouraging obvious swaps. But we know that many investments

(such as acquiring a small business) involve significant transaction costs. Perhaps the capitalmarket line is really a band whose width reflects trading costs.

Zero taxes. The CAPM assumes investment trading is tax-free and returns are unaffectedby taxes. Yet we know this to be false: (1) many investment transactions are subject to capitalgains taxes, thus adding transaction costs; (2) taxes reduce expected returns for many investors,

thus affecting their pricing of investments; (3) different returns (dividends versus capital gains,taxable versus tax-deferred) are taxed differently, thus inducing investors to choose portfolios

with tax-favored assets; (4) different investors (individuals versus pension plans) are taxeddifferently, thus leading to different pricing of the same assets.

Homogeneous investor expectations. The CAPM assumes invests have the same beliefsabout expected returns and risks of available investments. But we know that there is massive

trading of stocks and bonds by investors with different expectations. We also know that investorshave different risk preferences. Again, it may be that the capital market line is a fuzzy

amalgamation of many different investors' capital market lines.

Available risk-free assets. The CAPM assumes the existence of zero-risk securities, ofvarious maturities and sufficient quantities to allow for portfolio risk adjustments. But we knoweven Treasury bills have various risks: reinvestmentrisk-- investors may have investment

horizons beyond the T-bill maturity date; inflation risk-- fixed returns may be devalued byfuture inflation; currency risk-- the purchasing power of fixed returns may diminish compared to

that of other currencies. (Even if investors could sell assets short -- by selling an asset she doesnot own, and buying it back later, thus profiting from price declines -- this method of reducing

portfolio risk has costs and assumes unlimited short-selling ability.)

Borrowing at risk-free rates. The CAPM assumes investors can borrow money at risk-free rates to increase the proportion of risky assets in their portfolio. We know this is not true for

smaller, non-institutional investors. In fact, we would predict that the capital market line shouldbecome kinked downward for riskier portfolios ( > 1) to reflect the higher cost of risk-free

borrowing compared to risk-free lending.

Beta as full measure of risk. The CAPM assumes that risk is measured by the volatility(standard deviation) of an asset's systematic risk, relative to the volatility (standard deviation) ofthe market as a whole. But we know that investors face other risks: inflation risk-- returns may

be devalued by future inflation; and liquidity risk-- investors in need of funds or wishing tochange their portfolio's risk profile may be unable to readily sell at current market prices.

Moreover, standard deviation does not measures risk when returns are not evenly distributedaround the mean (non-bell curve). This uneven distribution describes our stock markets where

winning companies, like Dell and Walmart, have positive returns (35,000% over ten years) thatgreatly exceed losing companies' negative returns (which are capped at a 100% loss).


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15 Risk and Return Equations

Expected

Return:Variance:Standard

Deviation:Covariance:Correlation

Coefficient:Portfolio

Expected

Return:


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Two-Asset

Portfolio

Variance:Security

MarketLine

(SML):Beta:

11 Risk and Return

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