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PAK Study Manual Retirement Plan Investment and

Risk Management (RPIRM) Exam Fall 2018 Edition

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PAK Study Manual for RPIRM Fall 2018

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RPIRM Exam PAK Study Manual Litterman

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Litterman-2 The Insights of Modern Portfolio Theory (MPT)

Background

Harry Markowitz introduced MPT to the finance-world in 1952. MPT is a tool to optimal portfolio construction and is more than just an academic achievement. In this reading we explore interesting insights of MPT:

Diversification of investment across less correlated assets tends to reduce overall portfolio risk. Patience in investments is rewarded and total risk should be spread relatively evenly over time. Investors shall avoid concentrated sources of risk. In constructing their portfolios, investors need to look at the expected return of each investment in

relation to the impact that it has on the risk of the overall portfolio (Its marginal contribution to portfolio risk).

A key question faced by portfolio managers is how much to invest in domestic versus international equities. In this reading, we see how MPT approaches this question.

Introduction to MPT

Risk quantifies the likelihood and the size of potential losses. Losses are painful. Investor tolerance for taking risk is limited. Each investor can only tolerate losses up to a certain size. Thus, each investor has only a limited risk appetite. The investment manager shall recognize this when advising the client. On the flip side, wealth creation depends on risks, and on allocating that risk across many assets. Therefore:

Risk shall be viewed as a scarce resource that needs to be used wisely in order to achieve wealth creation.

Risk shall be budgeted, and positioned in a way to maximize return.

Through mathematical statistics, MPT provides interesting insights and connections between risk and return.

On the return side, the mathematics is pretty simple: Monetary returns at a different point in time are additive.

On the risk side, the mathematics is not straightforward: Portfolio risk is not additive. It depends in the covariance of the individual assets in the portfolio.

To illustrate the risk side, suppose that the portfolio is made up of two assets (a domestic asset (d) and a foreign asset (f)). The weight of each asset is denoted (d or f). The correlation between the two assets is (ρ). Assuming that the volatility of the portfolio return is the adopted measure of the portfolio risk, we have:

𝑷𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐 𝑹𝒊𝒔𝒌 = 𝝈(𝑷) = �𝒅𝟐𝝈𝒅𝟐 + 𝒇𝟐𝝈𝒇𝟐 + 𝟐𝒅𝒇𝝈𝒅𝝈𝒇𝝆

Statistically speaking, we know that:

RPIRM Exam PAK Study Manual Litterman

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−1 ≤ 𝜌 ≤ +1 Not only the portfolio risk (𝜎(𝑃)) is not additive (not the sum of the risk from individual assets), but it has the potential to be reduced (in the case where the correlation between the two assets is negative). This is called the diversification benefit: When spreading the investments across less correlated assets, the risk of the portfolio tends to reduce. The rate at which return grows over time (linear function) is different from the rate at which risk grows over time (square root function): Thus, distributing risk evenly over a long time horizon is another potential free lunch for investors (patience in investments is rewarded).

Important Rule of Optimal Investing

Just like there is a trade-off between quality and cost in making everyday purchases, optimal investing depends on balancing the quality of an investment (The excess return the investment is expected to generate) against its cost (The contribution of the investment to the risk of the portfolio, as measured by the correlation of this investment to the overall portfolio). Many investors bypass these insights and view MPT as just an academic achievement. As such, they are not aware of the recommendations from MPT:

MPT suggests that investor should create a balanced portfolio with some exposure to public market securities (both domestic and international), especially in the equity markets.

Risk Tolerance/ Risk Budget

The investment manager shall have a thorough understanding of the client’s risk tolerance (The ability and willingness to absorb large swings in the value of his or her investments). Consider two situations:

1) Situation 1: Own an illiquid, concentrated position that is extremely risky. 2) Situation 2: Own a liquid, money market fund with virtually no risk.

These two situations outermost ends of the risk spectrum in the context of MPT.

Over short period of times, position 2 provides a steady return, but the problem is over time, the real returns can be risky and has historically been poor.

MPT suggests that for almost all investors, neither situation is a particularly good position to be in for very

long.

MPT suggests that investors shall recognize the benefits of a balanced position between these two extremes.

The most practical insight from MPT is that investors shall avoid concentrated sources of risk.

The good news is that MPT provides a context in which one can quantify exactly how much of an overall risk budget any particular investment should consume.

RPIRM Exam PAK Study Manual Litterman

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The Central Theme of MPT

In constructing their portfolios, investors need to look at the expected return of each investment in relation to the impact that it has on the risk of the overall portfolio. Implementing this insight within a portfolio context is quite challenging:

o It is hard to quantify individual asset’s expected return and contribution to portfolio risk.

o Coming up with reasonable assumptions is problematic.

o The reading deviates from the historical approach to estimating these parameters in favor of an equilibrium approach.

o The risk contribution of any new investment depends on its correlation to the portfolio, through the

covariance matrix (The degree to which the investment moves up or down with the values of the other investment in the portfolio).

o Correlations cannot be observed directly; they are inferred from statistical data. These estimates are

notoriously unstable. The key to optimal portfolio construction is to understand the sources of risk in the portfolio and to deploy risk effectively. Understanding, measuring and monitoring the contribution of the risks of individual assets to the entire portfolio becomes a key part of the decision about how much to invest in each asset.

Central question: How much to invest in domestic versus international equities?

We will answer the central question from the perspective of two portfolios:

1) Portfolio 1: A two asset classes portfolio (domestic equity; and international equity) and 2) Portfolio 2: A multi asset classes portfolio.

Portfolio 1

The total risk of the portfolio is a function of two variables (d: the allocation to domestic equity) and (f: the allocation to international equity). We have:

𝑅(𝑑, 𝑓) = �𝑑2𝜎𝑑2 + 𝑓2𝜎𝑓2 + 2𝑑𝑓𝜎𝑑𝜎𝑓𝜌

The marginal contribution of domestic equities to portfolio risk is the limit of the function (∆𝑑(𝛿)) when (𝛿) goes to zero.

∆𝑑(𝛿) =𝜕𝑅(𝑑, 𝑓)𝜕𝑑

=𝑅(𝑑 + 𝛿, 𝑓) − 𝑅(𝑑, 𝑓)

𝛿

This limit (the marginal contribution) is simply denoted by (∆𝒅).

RPIRM Exam PAK Study Manual Litterman

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Likewise, the marginal contribution of international equity to portfolio risk is the limit of the function (∆𝑑(𝛿)) when (𝛿) goes to zero.

∆𝑓(𝛿) =𝜕𝑅(𝑑, 𝑓)𝜕𝑑

=𝑅(𝑑, 𝑓 + 𝛿) − 𝑅(𝑑, 𝑓)

𝛿

This limit (the marginal contribution) is simply denoted by (∆𝒇).

Optimal allocation in portfolio 1

Optimal allocation of the risk budget requires equities to be allocated from domestic to international markets up to the point where the ratio of expected excess returns (excess over the risk free rate) to the marginal contribution to portfolio risk is the same for both assets:

𝑒𝑑∆𝒅

=𝑒𝑓∆𝒇

Where the (e) notation is the excess return of each asset class. In fact, using first differential calculus, it is easy to prove that:

∆𝒅=𝑑𝜎𝑑2 + 𝑓𝜎𝑑𝜎𝑓𝜌

𝑅(𝑑, 𝑓)

And

∆𝒇=𝑓𝜎𝑓2 + 𝑑𝜎𝑑𝜎𝑓𝜌

𝑅(𝑑, 𝑓)

Let:

∆𝒅∆𝒇

= 𝒕𝒉𝒆 𝒓𝒂𝒕𝒊𝒐 𝒐𝒇 𝒎𝒂𝒓𝒈𝒊𝒏𝒂𝒍 𝒄𝒐𝒏𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏 𝒕𝒐 𝒑𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐 𝒓𝒊𝒔𝒌

And 𝒆𝒅𝒆𝒇

= 𝑻𝒉𝒆 𝒓𝒂𝒕𝒊𝒐 𝒐𝒇 𝒆𝒙𝒄𝒆𝒔𝒔 𝒓𝒆𝒕𝒖𝒓𝒏𝒔

Let us assume that: 𝒆𝒅𝒆𝒇

<∆𝒅∆𝒇

Thus: the rate of change in the portfolio risk (𝑅(𝑑, 𝑓)) from the sale of one unit of domestic equity is (−∆𝒅). To bring the portfolio risk back up to its original level (𝑅(𝑑, 𝑓)), we need to purchase (∆𝒅

∆𝒇) units of international

equity. This purchase has an effect on the portfolio expected return. The effect can be quantified to:

−𝒆𝒅 +∆𝒅∆𝒇

× 𝒆𝒇

RPIRM Exam PAK Study Manual Litterman

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Since we initially assume that (𝒆𝒅𝒆𝒇

< ∆𝒅∆𝒇

), we see that the effect of the purchase of (∆𝒅∆𝒇

)) units of international equity

leads to an increase in expected return, while controlling for the portfolio risk ((𝑑, 𝑓)). We shall continue to increase the allocation to international equity as long as expected return increases. The only case in which the portfolio’s expected return cannot increase while holding risk constant is if the following condition holds:

−𝒆𝒅 +∆𝒅∆𝒇

× 𝒆𝒇 = 𝟎

This is equivalent to:

𝑒𝑑∆𝒅

=𝑒𝑓∆𝒇

Portfolio 2

More generally, we can consider sales and purchases of any pair of assets in a multiple asset portfolio. We have:

𝑤 = 𝑇ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝑤𝑒𝑖𝑔ℎ𝑡 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑡𝑜 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑎𝑠𝑠𝑒𝑡 𝑐𝑙𝑎𝑠𝑠𝑒𝑠

𝑅𝑖𝑠𝑘(𝑤) = 𝑇ℎ𝑒 𝑟𝑖𝑠𝑘 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑢𝑙𝑡𝑖 𝑎𝑠𝑠𝑒𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜

𝑅𝑖𝑠𝑘𝑚(𝑤; 𝛿)= 𝑇ℎ𝑒 𝑟𝑖𝑠𝑘 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑤𝑖𝑡ℎ 𝑤𝑒𝑖𝑔ℎ𝑡𝑠 𝑤 𝑎𝑛𝑑 𝑎 𝑠𝑚𝑎𝑙𝑙 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 (𝛿)𝑡𝑜 𝑡ℎ𝑒 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑚

∆𝑚= 𝑇ℎ𝑒 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑖𝑠𝑘 𝑓𝑜𝑟 𝑎𝑠𝑠𝑒𝑡 𝑚 We have:

∆𝑚=𝑅𝑖𝑠𝑘𝑚(𝑤; 𝛿) − 𝑅𝑖𝑠𝑘(𝑤)

𝛿

As earlier, in an optimal portfolio, it must be the case that for every pair of assets m and n, we have:

𝒆𝒎∆𝒎

=𝒆𝒏∆𝒏

If not, then the portfolio manager shall:

Buy the asset for which the ratio (𝒆 ∆� ) is higher, and sell the asset for which the ratio (𝒆 ∆� ) is smaller. Remark:

If for an asset (m) the expected excess return is zero (𝒆𝒎 = 𝟎), then the optimal position is one in which the marginal contribution to risk is also zero (∆𝒎= 𝟎).

Numerical Example

We illustrate the theory with a numerical example: The portfolio (of two asset classes) is as follows:

RPIRM Exam PAK Study Manual Litterman

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We calculate the portfolio statistics as follows:

𝑹(𝒅,𝒇) = 𝑻𝒉𝒆 𝒑𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐 𝒓𝒊𝒔𝒌 = �𝒅𝟐𝝈𝒅𝟐 + 𝒇𝟐𝝈𝒇𝟐 + 𝟐𝒅𝒇𝝈𝒅𝝈𝒇𝝆

𝑻𝒉𝒆 𝑻𝒐𝒕𝒂𝒍 𝒓𝒆𝒕𝒖𝒓𝒏 = 𝒅 × (𝑫𝒐𝒎𝒆𝒔𝒕𝒊𝒄 𝒆𝒒𝒖𝒊𝒕𝒚 𝒓𝒆𝒕𝒖𝒓𝒏) + 𝒇 × (𝑰𝒏𝒕𝒆𝒓𝒏𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝒆𝒒𝒖𝒊𝒕𝒚 𝒓𝒆𝒕𝒖𝒓𝒏)

𝑬𝒙𝒄𝒆𝒔𝒔 𝑹𝒆𝒕𝒖𝒓𝒏 = 𝑻𝒉𝒆 𝑻𝒐𝒕𝒂𝒍 𝒓𝒆𝒕𝒖𝒓𝒏 − 𝒕𝒉𝒆 𝒓𝒊𝒔𝒌 𝒇𝒓𝒆𝒆 𝒓𝒂𝒕𝒆

“Cash has a zero volatility and a zero correlation with all other asset classes”

Suppose that the investor wants to maximize expected return for a total portfolio volatility of 10

percent. Suppose that the portfolio starts with a 100% allocation to domestic equity (d=1 and f=0) We have:

To decrease the portfolio risk back to 10%, the investor must hold a combination of cash plus domestic equity. The allocation that will result to 10% portfolio risk is:

What happens as the investor start to sale domestic equity (that is to reduce the 66.67%) to buy international equity (increase the 0%)?

RPIRM Exam PAK Study Manual Litterman

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Let us calculate the marginal contribution to portfolio risk for domestic and international equity for the given portfolio: Since f=0, we have:

∆𝒅=𝑑𝜎𝑑2 + 𝑓𝜎𝑑𝜎𝑓𝜌

𝑅(𝑑, 𝑓) =𝑑𝜎𝑑2

𝑅(𝑑, 𝑓) = 𝟎.𝟏𝟓𝟎

And

∆𝒇=𝑓𝜎𝑓2 + 𝑑𝜎𝑑𝜎𝑓𝜌

𝑅(𝑑, 𝑓) =𝑑𝜎𝑑𝜎𝑓𝜌𝑅(𝑑,𝑓) = 𝟎.𝟏𝟎𝟒

The ratios of excess return to marginal contribution to portfolio risk are:

Since the international equity has a much higher ratio (48.08%), we buy this asset, and we sale the domestic equity. The quantities of the purchase (of international asset) and the sale of (domestic asset) that would keep the overall portfolio risk at the 10% threshold are found as:

𝛿 = −1 And

∆𝒅∆𝒇

𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑒𝑞𝑢𝑖𝑡𝑦 = 1.442

The purchase of 1.442 international equity increases the portfolio expected excess return by:

0.05 × 1.442 = 0.07215 The sale of one unit of domestic equity reduces the portfolio expected excess return by:

−1 × 0.055 = −0.055 Both result to an increase in the portfolio excess return of: 0.01715, while maintaining the portfolio risk at 10%. The portfolio manager shall continue to allocate to international equity as long as either of the following is true:

1) The ratios of excess returns to marginal contribution are not equal, 2) The total effect of the sale of domestic equity + the purchase of international equity (−𝒆𝒅 + ∆𝒅

∆𝒇× 𝒆𝒇) is

positive while the portfolio risk is level at 10%. As this goes on, the excess return on international equity (𝒆𝒇) also changes. We might consider the level of (𝒆𝒇) for which the investor would be indifferent to such a transaction. Clearly, this happens when:

RPIRM Exam PAK Study Manual Litterman

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−𝒆𝒅 +∆𝒅∆𝒇

× 𝒆𝒇 = 𝟎

3) The investor would cease to purchase international equity if the expected excess return on international equity is below the hurdle rate.

Suppose the trades have gone on to the point where the portfolio is as follows:

We plug into the formula to get:

Thus:

The portfolio risk is still 10%. Even now, the investor shall continue to purchase international equity: The ratios of (𝑒 ∆� ) are not equal. The total effect of an additional sale of domestic equity + the purchase of international equity (−𝒆𝒅 +

∆𝒅∆𝒇

× 𝒆𝒇) is now (-0.055+1.2122*0.05=0.01667).

The internal equity hurdle rate has increased from 3.81% to reach 4.54%. Yet, it is still lower than 5%. So, the investor shall continue to allocate to internal equity.

Trades go on until the portfolio is:

RPIRM Exam PAK Study Manual Litterman

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We now have:

At this point, the hurdle rate does not justify further purchase of international equity. In fact, the portfolio is still not optimal (cause the ratios of excess returns to marginal are still not equal). Nonetheless, it is evident that the manager has over-allocated to international equity. The total effect of an additional sale of domestic equity + the purchase of international equity (−𝒆𝒅 + ∆𝒅

∆𝒇× 𝒆𝒇) is

now (-0.055+0.69111*0.05=-0.012). We refer to these hurdle rates as the implied views of the portfolio. They are implied from the assumption about excess returns of individual asset classes. Purchase of an asset class is warranted when the hurdle rate is lower than the expected excess rate of that asset class. In this example, we have not really highlighted the rule played by the correlation of the asset classes (0.65)

Portfolio 2: Introduction of a commodity asset class

We consider the following asset classes:

As earlier, the initial portfolio is 2/3 domestic equity and 1/3 commodity as follows:

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The marginal contribution of commodity to this portfolio is calculated as follows:

∆𝒄𝒐𝒎=𝐶𝑜𝑚 × 𝜎𝐶𝑜𝑚2 + 𝑑𝜎𝑑𝜎𝐶𝑜𝑚𝜌

𝑅(𝑑, 𝑓) =(0.6667) × (0.15) × (0.25) × (−0.25)

0.1= −𝟎.𝟎𝟔𝟐

The negative contribution to portfolio risk is a new phenomenon:

The correlation of commodity to other assets (domestic and international) is negative, The marginal contribution of commodity to the portfolio risk is negative (thus a risk reducer) Commodity, as an asset class is a portfolio diversifier. Thus, if we sell the domestic equity and buy more of the commodity asset class, we will reduce the

overall risk of the portfolio. Let us calculate the hurdle rate on commodity:

When (Com=0), the implied view is a negative expected excess return (-2.29%). Let us increase our holdings of commodity to 5%:

The hurdle rate is now -1.18% and the portfolio volatility is down from 10%. What happens if we further allocate to commodities (to 15%)?

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The portfolio volatility is still at 9.76%. The hurdle rate has increased to 1.26%. The marginal contribution to commodity has turned positive. The commodity asset class has changed from being a diversifier to being a source of risk (for, further

purchase of this asset class has a positive impact on portfolio risk). There is a weight in commodity for which the portfolio risk is minimized. We solve for the equation (∆𝑐= 0). This means solving for (c) such that:

𝒄𝝈𝒄𝟐 + 𝒅𝝈𝒅𝝈𝒄𝝆𝒅𝒄 = 𝟎 In fact:

Holding fixed the weights in all other assets, there is a risk-minimizing position for each asset. Weights greater than that risk-minimizing position reflect positive implied views; weights less than that risk-minimizing position reflect bearish view.

The risk-minimizing position is a function of:

1) Positions in other asset classes, 2) Volatilities and 3) Correlations.

In a multi-assets portfolio, these positions can either be positive or negative. The risk minimizing positions is at zero for asset classes that are uncorrelated to the portfolio (ρ=0).

Any asset that has a zero correlation with the portfolio and yet a positive excess return shall be added to the portfolio. Such active might include: active risk (next chapter), hedge funds, active currency overlays and global tactical asset allocation mandates.

Any position that lies between zero and the risk-minimizing position is likely to represent an opportunity for the investor.

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Practice Questions

A portfolio manager has to invest in the following asset classes:

The Starting from an initial allocation of 100% to domestic equity, the manager has been adding international asset to the point of:

Complete the table below and conclude:

Determine whether the following allocation is optimal.