## Review of Portfolio Mathematics

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Consider the problem of Humanex, a nonprofit organization deriving most of its income from the return on its endowment. Years ago, the founders of Best Candy willed a large block of Best Candy stock to Humanex with the provision that Humanex may never sell it. This block of shares now comprises 50% of Humanex's endowment. Humanex has free choice as to where to invest the remainder of its portfolio.2

The value of Best Candy stock is sensitive to the price of sugar. In years when world sugar crops are low, the price of sugar rises significantly and Best Candy suffers considerable losses. We can describe the fortunes of Best Candy stock using the following scenario analysis:

 Normal Year for Sugar Abnormal Year Bullish Bearish Stock Market Stock Market Sugar Crisis Probability .5 .3 .2 Rate of return 25% 10% -25%

To summarize these three possible outcomes using conventional statistics, we review some of the key rules governing the properties of risky assets and portfolios.

Rule 1 The mean or expected return of an asset is a probability-weighted average of its return in all scenarios. Calling Pr(s) the probability of scenario s and r(s) the return in scenario s, we may write the expected return, E(r), as

Applying this formula to the case at hand, with three possible scenarios, we find that the expected rate of return of Best Candy's stock is

Rule 2 The variance of an asset's returns is the expected value of the squared deviations from the expected return. Symbolically, ct2 = ^Pr(s)[r(s) - E(r)]2 (6.3)

Therefore, in our example aBest = .5(25 - 10.5)2 + .3(10 - 10.5)2 + .2(-25 - 10.5)2 = 357.25

The standard deviation of Best's return, which is the square root of the variance, is V357.25 = 18.9%.

Humanex has 50% of its endowment in Best's stock. To reduce the risk of the overall portfolio, it could invest the remainder in T-bills, which yield a sure rate of return of 5%. To derive the return of the overall portfolio, we apply rule 3.

Rule 3 The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with portfolio proportions as weights. This implies that the expected rate of return on a portfolio is a weighted average of the expected rate of return on each component asset.

2 The portfolio is admittedly unusual. We use this example only to illustrate the various strategies that might be used to control risk and to review some useful results from statistics.

Humanex's portfolio proportions in each asset are .5, and the portfolio's expected rate of return is

= (.5 X 10.5) + (.5 X 5) = 7.75% The standard deviation of the portfolio may be derived from rule 4.

Rule 4 When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset's standard deviation multiplied by the portfolio proportion invested in the risky asset.

The Humanex portfolio is 50% invested in Best stock and 50% invested in risk-free bills. Therefore,

By reducing its exposure to the risk of Best by half, Humanex reduces its portfolio standard deviation by half. The cost of this risk reduction, however, is a reduction in expected return. The expected rate of return on Best stock is 10.5%. The expected return on the one-half T-bill portfolio is 7.75%. Thus, while the risk premium for Best stock over the 5% rate on risk-free bills is 5.5%, it is only 2.75% for the half T-bill portfolio. By reducing the share of Best stock in the portfolio by one-half, Humanex reduces its portfolio risk premium by one-half, from 5.5% to 2.75%.

In an effort to improve the contribution of the endowment to the operating budget, Humanex's trustees hire Sally, a recent MBA, as a consultant. Researching the sugar and candy industry, Sally discovers, not surprisingly, that during years of sugar shortage, SugarKane, a big Hawaiian sugar company, reaps unusual profits and its stock price soars. A scenario analysis of SugarKane's stock looks like this:

 Normal Year for Sugar Abnormal Year Bullish Bearish Stock Market Stock Market Sugar Crisis Probability .5 .3 .2 Rate of return 1% -5% 35%

The expected rate of return on SugarKane's stock is 6%, and its standard deviation is 14.73%. Thus SugarKane is almost as volatile as Best, yet its expected return is only a notch better than the T-bill rate. This cursory analysis makes SugarKane appear to be an unattractive investment. For Humanex, however, the stock holds great promise.

SugarKane offers excellent hedging potential for holders of Best stock because its return is highest precisely when Best's return is lowest—during a sugar crisis. Consider Humanex's portfolio when it splits its investment evenly between Best and SugarKane. The rate of return for each scenario is the simple average of the rates on Best and SugarKane because the portfolio is split evenly between the two stocks (see rule 3).

 Normal Year for Sugar Abnormal Year Bullish Bearish Stock Market Stock Market Sugar Crisis Probability .5 .3 .2 Rate of return 13.0% 2.5% 5.0%

CHAPTER 6 Risk and Risk Aversion

The expected rate of return on Humanex's hedged portfolio is 8.25% with a standard deviation of 4.83%.

Sally now summarizes the reward and risk of the three alternatives:

CONCEPT CHECK ^ QUESTION 5

 Portfolio Expected Return Standard Deviation All in Best Candy 10.50% 18.90% Half in T-bills 7.75 9.45 Half in SugarKane 8.25 4.83

The numbers speak for themselves. The hedge portfolio with SugarKane clearly dominates the simple risk-reduction strategy of investing in safe T-bills. It has higher expected return and lower standard deviation than the one-half T-bill portfolio. The point is that, despite SugarKane's large standard deviation of return, it is a hedge (risk reducer) for investors holding Best stock.

The risk of individual assets in a portfolio must be measured in the context of the effect of their return on overall portfolio variability. This example demonstrates that assets with returns that are inversely associated with the initial risky position are powerful hedge assets.

Suppose the stock market offers an expected rate of return of 20%, with a standard deviation of 15%. Gold has an expected rate of return of 6%, with a standard deviation of 17%. In view of the market's higher expected return and lower uncertainty, will anyone choose to hold gold in a portfolio?

To quantify the hedging or diversification potential of an asset, we use the concepts of covariance and correlation. The covariance measures how much the returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means that they vary inversely, as in the case of Best and SugarKane.

To measure covariance, we look at return "surprises," or deviations from expected value, in each scenario. Consider the product of each stock's deviation from expected return in a particular scenario:

This product will be positive if the returns of the two stocks move together, that is, if both returns exceed their expectations or both fall short of those expectations in the scenario in question. On the other hand, if one stock's return exceeds its expected value when the other's falls short, the product will be negative. Thus a good measure of the degree to which the returns move together is the expected value of this product across all scenarios, which is defined as the covariance:

Cov(rBest, rKane) - XPr(i)[rBest(^) _ £,(rBest)][rKane(^) - E(rKane)]

In this example, with E(rBest) = 10.5% and E(rKane) = 6%, and with returns in each scenario summarized in the next table, we compute the covariance by applying equation 6.4. The co-variance between the two stocks is

Cov(rBest, rKane) - .5(25 - 10.5)(1 - 6) + .3(10 - 10.5)(-5 -- -240.5

The negative covariance confirms the hedging quality of SugarKane stock relative to Best Candy. SugarKane's returns move inversely with Best's.

 Normal Year for Sugar Abnormal Year Bullish Stock Market Bearish Stock Market Sugar Crisis Probability .5 Rate of Return (%) .2 Best Candy SugarKane 1 10 -5 CO IV) Ol Ol

An easier statistic to interpret than the covariance is the correlation coefficient, which scales the covariance to a value between -1 (perfect negative correlation) and +1 (perfect positive correlation). The correlation coefficient between two variables equals their co-variance divided by the product of the standard deviations. Denoting the correlation by the Greek letter p, we find that p(Best, SugarKane) = ^^^ rSugarKane]

CTBestCTSugarKane -m5 .86

This large negative correlation (close to -1) confirms the strong tendency of Best and SugarKane stocks to move inversely, or "out of phase" with one another.

The impact of the covariance of asset returns on portfolio risk is apparent in the following formula for portfolio variance.

Rule 5 When two risky assets with variances o2 and o2, respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance op is given by op = wfof + wfof + 2w1w2Cov(r1, r2)

In this example, with equal weights in Best and SugarKane, w1 = w2 = .5, and with oBest = 18.9%, oKane= 14.73%, and Cov(rBest, rKane) = -240.5, we find that op = (.52 X18.92) + (.52 X 14.732) + [2 X .5 X .5 X (-240.5)] = 23.3

so that oP= V23.3 = 4.83%, precisely the same answer for the standard deviation of the returns on the hedged portfolio that we derived earlier from the scenario analysis.

Rule 5 for portfolio variance highlights the effect of covariance on portfolio risk. Apositive covariance increases portfolio variance, and a negative covariance acts to reduce portfolio variance. This makes sense because returns on negatively correlated assets tend to be offsetting, which stabilizes portfolio returns.

Basically, hedging involves the purchase of a risky asset that is negatively correlated with the existing portfolio. This negative correlation makes the volatility of the hedge asset a risk-reducing feature. A hedge strategy is a powerful alternative to the simple risk-reduction strategy of including a risk-free asset in the portfolio.

In later chapters we will see that, in a rational market, hedge assets will offer relatively low expected rates of return. The perfect hedge, an insurance contract, is by design perfectly negatively correlated with a specified risk. As one would expect in a "no free lunch" world, the insurance premium reduces the portfolio's expected rate of return.

CONCEPT CHECK ^ QUESTION 6

 Bullish Bearish Stock Market Stock Market Sugar Crisis 7% -5% 20%

a. What would be its correlation with Best?

b. Is SugarKane stock a useful hedge asset now?

c. Calculate the portfolio rate of return in each scenario and the standard deviation of the portfolio from the scenario returns. Then evaluate ctp using rule 5.

d. Are the two methods of computing portfolio standard deviations consistent?

a. What would be its correlation with Best?

b. Is SugarKane stock a useful hedge asset now?

c. Calculate the portfolio rate of return in each scenario and the standard deviation of the portfolio from the scenario returns. Then evaluate ctp using rule 5.

d. Are the two methods of computing portfolio standard deviations consistent?

### SUMMARY

1. Speculation is the undertaking of a risky investment for its risk premium. The risk premium has to be large enough to compensate a risk-averse investor for the risk of the investment.

2. A fair game is a risky prospect that has a zero-risk premium. It will not be undertaken by a risk-averse investor.

3. Investors'preferences toward the expected return and volatility of a portfolio may be expressed by a utility function that is higher for higher expected returns and lower for higher portfolio variances. More risk-averse investors will apply greater penalties for risk. We can describe these preferences graphically using indifference curves.

4. The desirability of a risky portfolio to a risk-averse investor may be summarized by the certainty equivalent value of the portfolio. The certainty equivalent rate of return is a value that, if it is received with certainty, would yield the same utility as the risky portfolio.

5. Hedging is the purchase of a risky asset to reduce the risk of a portfolio. The negative correlation between the hedge asset and the initial portfolio turns the volatility of the hedge asset into a risk-reducing feature. When a hedge asset is perfectly negatively correlated with the initial portfolio, it serves as a perfect hedge and works like an insurance contract on the portfolio.

KEY TERMS

risk premium risk averse utility certainty equivalent rate risk neutral risk lover mean-variance (M-V)

criterion indifference curve hedging diversification expected return variance standard deviation covariance correlation coefficient

PROBLEMS 1. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either \$70,000 or \$200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year.

a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?

b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?

Bodie-Kane-Marcus: Investments, Fifth Edition

II. Portfolio Theory

6. Risk and Risk Aversion

PART II Portfolio Theory c. Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay?

d. Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?

2. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to bills?

3. Draw the indifference curve in the expected return-standard deviation plane corresponding to a utility level of 5% for an investor with a risk aversion coefficient of 3. (Hint: Choose several possible standard deviations, ranging from 5% to 25%, and find the expected rates of return providing a utility level of 5%. Then plot the expected return-standard deviation points so derived.)

4. Now draw the indifference curve corresponding to a utility level of 4% for an investor with risk aversion coefficient A = 4. Comparing your answers to problems 3 and 4, what do you conclude?

5. Draw an indifference curve for a risk-neutral investor providing utility level 5%.

6. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of 5% for a risk lover.

Use the following data in answering questions 7, 8, and 9.

 Utility Formula Data Expected Standard Investment Return E(r) Deviation a 1 12% 30% 2 15 50 3 21 16 4 24 21

U = E(r) - .005Act2 where A = 4 Based on the utility formula above, which investment would you select if you were risk averse with A = 4?

Based on the utility formula above, which investment would you select if you were risk neutral?

The variable (A) in the utility formula represents the:

a. investor's return requirement.

b. investor's aversion to risk.

c. certainty equivalent rate of the portfolio.

d. preference for one unit of return per four units of risk.

### CHAPTER 6 Risk and Risk Aversion

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 70 years has averaged about 8.5% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 5%. Use these values to solve problems 10 to 12.

10. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as follows:

 Wbills Windex 0 1.0 0.2 0.8 0.4 0.6 0.6 0.4 0.8 0.2 1.0 0

11. Calculate the utility levels of each portfolio of problem 10 for an investor with A = 3. What do you conclude?

12. Repeat problem 11 for an investor with A = 5. What do you conclude?

Reconsider the Best and SugarKane stock market hedging example in the text, but assume for questions 13 to 15 that the probability distribution of the rate of return on Sug-arKane stock is as follows:

 Bullish Stock Market Bearish Stock Market Sugar Crisis Probability .5 .3 .2 Rate of return 10% -5% 20%

13. If Humanex's portfolio is half Best stock and half SugarKane, what are its expected return and standard deviation? Calculate the standard deviation from the portfolio returns in each scenario.

14. What is the covariance between Best and SugarKane?

15. Calculate the portfolio standard deviation using rule 5 and show that the result is consistent with your answer to question 13.

1. The expected rate of return on the risky portfolio is \$22,000/\$100,000 = .22, or 22%. The T-bill rate is 5%. The risk premium therefore is 22% - 5% = 17%.

2. The investor is taking on exchange rate risk by investing in a pound-denominated asset. If the exchange rate moves in the investor's favor, the investor will benefit and will earn more from the U.K. bill than the U.S. bill. For example, if both the U.S. and U.K. interest rates are 5%, and the current exchange rate is \$1.50 per pound, a \$1.50 investment today can buy one pound, which can be invested in England at a certain rate of 5%, for a year-end value of 1.05 pounds. If the year-end exchange rate is \$1.60 per pound, the 1.05 pounds can be exchanged for 1.05 X \$1.60 = \$1.68 for a rate of return in dollars of 1 + r = \$1.68/\$1.50 = 1.12, or 12%, more than is available from U.S. bills. Therefore, if the investor expects favorable exchange rate movements, the U.K. bill is a speculative investment. Otherwise, it is a gamble.

SOLUTIONS TO CONCEPT CHECKS

PART II Portfolio Theory

SOLUTIONS TO CONCEPT CHECKS

3. For the A = 4 investor the utility of the risky portfolio is

U = 20 - (.005 X 4 X 202) = 12 while the utility of bills is

The investor will prefer the risky portfolio to bills. (Of course, a mixture of bills and the portfolio might be even better, but that is not a choice here.) For the A = 8 investor, the utility of the risky portfolio is

while the utility of bills is again 7. The more risk-averse investor therefore prefers the risk-free alternative.

4. The less risk-averse investor has a shallower indifference curve. An increase in risk requires less increase in expected return to restore utility to the original level. 5. Despite the fact that gold investments in isolation seem dominated by the stock market, gold still might play a useful role in a diversified portfolio. Because gold and stock market returns have very low correlation, stock investors can reduce their portfolio risk by placing part of their portfolios in gold.

6. a. With the given distribution for SugarKane, the scenario analysis looks as follows:

 Normal Year for Sugar Abnormal Year Bullish Stock Market Bearish Stock Market Sugar Crisis Probability .5 Rate of Return (%) .2 Best Candy 25 10 -25 SugarKane 7 -5 20 T-bills 5 5 5

CHAPTER 6 Risk and Risk Aversion

SOLUTIONS TO CONCEPT CHECKS

The expected return and standard deviation of SugarKane is now

^SugarKane = [.5(7 - 6)2 + .3(-5 - 6)2 + .2(20 - 6)2]1/2 = 8.72 The covariance between the returns of Best and SugarKane is

Cov(SugarKane, Best) = .5(7 - 6)(25 - 10.5) + .3(-5 - 6)(10 + .2(20 - 6)( -25 - 10.5) = -90.5

and the correlation coefficient is

P(SugarKane, Best)

Cov(SugarKane, Best)

^SugarKane^Best

The correlation is negative, but less than before (-.55 instead of - .86) so we expect that SugarKane will now be a less powerful hedge than before. Investing 50% in SugarKane and 50% in Best will result in a portfolio probability distribution of

 Probability 0.5 0.3 2 Portfolio return 16 2.5 -2.5

resulting in a mean and standard deviation of

E(rHedged portfolio) = (.5 X 16) + (.3 X 2.5) + .2(-2.5) = 8.25

^Hedged portfolio = [.5(16 - 8.25)2 + .3(2.5 - 8.25)2 + .2(-2.5 - 8.25)2]1/2 = 7.94

b. It is obvious that even under these circumstances the hedging strategy dominates the risk-reducing strategy that uses T-bills (which results in E(r) = 7.75%, ct = 9.45%). At the same time, the standard deviation of the hedged position (7.94%) is not as low as it was using the original data. c, d. Using rule 5 for portfolio variance, we would find that ct2 = (.52 X CT2Best) + (.52 X CT2Kane) + [2 X .5 X .5 X Cov(SugarKane, Best)]

= (.52 X 18.92) + (.52 X 8.722) + [2 X .5 X .5 X (-90.5)] = 63.06

which implies that ct = 7.94%, precisely the same result that we obtained by analyzing the scenarios directly. 