BKM CHAPTER 7 1. Which of the following factors reflect pure market risk for a given corporation? a. Increased short-term interest rates b. Fire in the corporate warehouse c. Increased insurance costs d. Death of the CEO, e. Increased labor costs) (a) and (e) – The other three do not affect all participants in the economy. 2. When adding real estate to an asset allocation program that currently includes only stocks, bonds, and cash alternatives (risk-free-money market investments), which of the properties of real estate returns affect portfolio risk? Explain. a. Standard Deviation b. Expected Return c. Correlation with the returns of other assets (a) and (c). The portfolio risk (standard deviation) calculation now includes the variance of real estate returns and correlation between real estate and stocks the correlation between real estate and bonds. The correlation between real estate and money markets will be zero. Not (b) since the E(r) of real estate does not affect the portfolio’s risk. Note: The question refers to correlation (ρij) between real estate and the other assets. Since correlation is defined as the covariance between two assets’ returns divided by the product of the standard deviations of the assets returns (ρij = σij/σiσj), the portfolio risk formula can be stated in terms of either correlation or covariance. 3. Which of the following statements about the minimum variance portfolio of all risky securities are valid? (Assume short sales are allowed.) Explain. a. Its variance must be lower than those of all other securities or portfolios. b. Its expected return can be lower than the risk-free rate. c. It may be the optimal risky portfolio. d. It must include all individual securities. Note: In lectures, I referred to the minimum risk or minimum standard deviation portfolio. Since the standard deviation is the square root of the variance, the portfolio with the minimum standard deviation must also have the minimum variance so it is therefore the minimum risk portfolio. All three terms describe the same portfolio. (a) is valid. This is the definition of the minimum variance (aka minimum risk or minimum standard deviation) portfolio. (b) is not valid. The return of the minimum variance portfolio must exceed the return of the risk free asset since the risk, while minimized, is still greater than risk-free (zero). 1
(c) is not valid. It will not be the optimal risky portfolio because there exists another portfolio, that when combined with the risk-free assets, will produce a larger CAL slope. See your notes or Figure 7.13 on page 224. G is the Global minimum variance portfolio and P is the optimal risky portfolio. (d) is not valid. The minimum variance portfolio (or the optimal risky portfolio, or any portfolio for that matter) may be formed using zero-weights for many assets. The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows: E(r) 20% 12%
Stock Fund (S) Bond Fund (B)
σ 30% 15%
The correlation between the funds is 0.10. 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return? The parameters of the opportunity set are: E(rS) = 20%, E(rB) = 12% σS = 30%, σB = 15% ρAB = 0.10 From the standard deviations and the correlation coefficient we can generate the “covariance matrix” (note that σSB = ρSB x σS X σB): Bonds 1.00 0.10
Stocks Bonds Stocks Bonds 0.10 Bonds .0225 .0045 1.00 Stocks Stocks .0045 .0900 00 For formula for the minimum variance weight is on page 213 in the text. (I did not derive this in class.) WSMin =
B2 SB 0.0225 0.0045 0.1739 2 2 S B 2 SB 0.0900 0.0225 (2 0.0045)
WSMin = 1 0.1739 = 0.8261 Expected Return Standard Deviation of the minimum variance portfolio are: 2
E(rMin) = (0.1739)(0.20) + (0.8261)(0.12) = 13.39% σMin = [ w S2 S2 w 2B 2B 2 w S w B Cov (rS , rB )]1 / 2 = [(0.1739)2(0.30) 2 + (0.8261)2(0.15) 2 + 2(0.1739)(0.8261)(0.0045)]1/2 = 13.92% 5. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of zero to 100% in increments of 20%. WS 0% 20% 40% 60% 80% 100%
WB 100% 80% 60% 40% 20% 0%
E(r) 12.00% 13.60% 15.20% 16.80% 18.40% 20.00%
15.00% 13.94% 15.70% 19.53% 24.48% 30.00%
6. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio? (Chart of everything later.) 7. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio The optimal risky portfolio is the portfolio along the frontier that maximizes the slope of the CAL. The formula for the weight of an asset in the optimal risky portfolio is on page 217. NOTE the change from E(r) – rf = E(R): E(RS) = 0.20 – 0.08 = 0.12; E(RB) = 0.12 – 0.08 = 0.04 σS2 = 0.090; σB2 = 0.0225; σSB = 0.0045.
WS *
E ( RS ) B2 E ( RB ) SB E ( RS ) B2 E ( RB ) S2 [ E ( RS ) E (rB )] SB
[( 0.12 ) 0.0225 ] [( 0.04 ) 0.0045 ] 0.4516 (0.12 .0225 ) [0.04 0.09 ] (0.12 .0.04 ) .0045 ]
WB* = 1 0.4516 = 0.5484 The mean and standard deviation of the optimal risky portfolio (P) are: E(rP) = (0.4516)(0.20) + (0.5484)(0.12) = 15.61% σP= [ w S2 S2 w 2B 2B 2 w S w B Cov (rS , rB )]1 / 2 = [(0.4516)2(0.30) 2 + (0.5484)2(0.15) 2 + 2(0.4616)(0.5484)(0.0045)]1/2 = 16.54% 3
8. What is the reward-to-volatility ratio of the best feasible CAL? The Slope of the CAL = E(rP) – rf)/σP. This is also reward-to-volatility ratio of the CAL. The best feasible CAL is the CAL with highest reward-to-volatility ratio which is the achievable CAL with the maximum slope. The WS* equation above solves for the weight of the Stock fund in the risky portfolio P that achieves the maximum slope. The “*” indicates an optimal (in this case maximum) value. Slope of the Optimal CAL =
E ( rp ) r f
p
0.1561 0.08 0.4601 0.1654
9. You require that your combined portfolio yield an expected return of 14%, and that it be efficient (which means it is on the best feasible CAL). (a) What is the standard deviation of your portfolio? Combined portfolios on the best feasible CAL are efficient because for any given return, no other achievable allocation will have less risk (be to the left of the CAL). Another way to say this is that combined portfolios on the best feasible CAL are efficient because for any given risk, no other achievable allocation will have greater return (be above the CAL). All combined portfolios on the CAL will be combinations on P and the risk-free. Given: E(rP) = 15.61%; σP= 16.54%; rf = 8% Solve for the y that makes E(rC) = 14% then compute the σC= for that combined portfolio: E(rC) = rf + y[E(rP) – rf] y = [E(rC) – rf]/[E(rP) – rf] y = [0.14 – 0.08]/[0.1561 – 0.08] = 0.7884 σC= yσP = 0.7884(0.1654) = 13.04% Note: This is NOT y* since this is an arbitrary combined portfolio and not the optimal (utility maximizing) combined portfolio. (b) What is the proportion of the combined portfolio C invested in the T-bill fund and each of the two risky funds? We need to compute the portion of C in stocks, bonds and the risk-free: y = 78.84% (The portion of the combined portfolio in the risky stock-bond portfolio P) WS* = 45.16% (The portion of P in the stock fund) WB* = 54.84% (The portion of P in the bond fund) 4
Stocks: (y)(WS*) = (0.7884)(0.4516) = 35.60% Bonds: (y)(WB*) = (0.7884)(0.5484) = 43.24% T-Bills: 1 – y = 1 – 0.7884 = 21.16% Note the weights of the combined portfolio allocations sum to 1. 24%
Investment Opportunity Set Optimal Risky Portfolio (P)
20%
Risky Asset Efficient Frontier
16%
E(r)
Optimal CAL
12%
MinimumRisk Portfolio
8%
4%
0% 0%
4%
8%
12%
16%
20%
24%
28%
32%
σ 10. If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude? E(rP) = WSE(rS) + (1 – WS)E(rB) = 14.00% 0.14 = WS(0.20) + (1 – WS)(0.12) WS = 0.25 σP= [ w S2 S2 w 2B 2B 2 w S w B Cov (rS , rB )]1 / 2 = [(0.25)2(0.30) 2 + (0.75)2(0.15) 2 + 2(0.25)(0.75)(0.0045)]1/2 = 14.13% 5
The 14% portfolio with Stocks and Bonds and the risk-free has a risk of 13.04% is on the CAL. The 14% portfolio with just Stocks and Bonds has a risk of 14.13% and lies to the right of the CAL. It is dominated by the portfolio on the CAL. 11. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. (a) In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold any gold? If so, demonstrate graphically why one would do so. Answer Part (b) first. (b) Given the data above, re-answer (a) with the additional assumption that the correlation coefficient between gold and stocks equals 1. Draw a graph illustrating why one would or would not hold gold in one's portfolio. Could this set of assumptions for expected returns, standard deviations, and correlation represent an equilibrium for the security market? The chart below shows the best available CAL when the correlation coefficient between stocks and gold is one. Note the allocation curve between stocks and gold (the red line) is a straight line (since ρ = 1). The Optimal (steepest sloped) CAL passes through the 100% stock portfolio - so no gold is held. Contrast this with the chart below when the correlation coefficient between stocks and gold is negative 0.50. 24%
P = 100% Stocks
20% 16%
E(r)
CAL
12%
ρ=1
8%
Gold 4% 0% 0%
4%
8% 12% 16% 20% 24% 28% 32%
σ
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The chart below shows the allocation curve between stocks and gold (the red line) for a correlation coefficient between stocks and gold equal to negative 0.50. It is a parabola-shaped line. Now holding some gold, an investor can achieve an optimal risky portfolio P and an accompanying CAL (New CAL in blue) that has a steeper slope than the CAL available with no gold (Old CAL in green). 24% 20%
Stocks
P
16%
E(r)
New CAL
12%
Old CAL
8%
Gold
4% 0% 0%
4%
8% 12% 16% 20% 24% 28% 32%
σ 12. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are in the table below. The correlation coefficient is -1. Suppose that it is possible to borrow at the risk-free rate, rf. What must be the value of the risk-free rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.) Stock A B
E(r) 10% 15%
σ 5% 10%
The theory is that all risk-free portfolios must earn the risk-free rate. Since A and B are perfectly negatively correlated, then the minimum risk portfolio containing these two stocks will have zero risk and the return of this risk-free portfolio will earn the risk-free rate.
Follow this procedure: 1) Calculate the weights of the minimum risk portfolio 2) Computer the return for this risk-free portfolio 3) Check to be sure it has zero risk
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1) The formula for the weight of Asset A in the minimum risk portfolio is: WAMin = (σB2 – σAB)/( σA2 + σB2 – 2σAB) Covariance = σAB = ρAB σA σA = (-1)(.05)(.10) = -0.005 WAMin = [0.102 – (-0.005)]/[0.052 + 0.102 – 2(-0.005)] = 0.6667 WBMin = 1 – 0.6667 = 0.3333 2) The formula for the return of the portfolio is: E(rP) = WAE(rA) + WBE(rB) = 0.6667(0.10) + 0.3333(0.15) = 11.67% 3) Show that this 66.67%-33.33% has zero risk by calculating the risk of the portfolio: σP = WA2σA2 + WA2σB2 + 2 WAWBσAB = (0.6667)2(0.05)2 + (0.3333)2(0.15)2 + 2(0.6667)(0.3333)(-0.005) = 0 The conclusion is – in well function markets – the risk-free rate is 11.67%. Note that there is a special case Minimum-Risk portfolio weight formula for the special case in which ρ = -1. It is equation 7.12 on page 210. The WAMin formula above works for any value of ρ. The formula above becomes equation 7.12 if you substitute ρAB σA σA for the covariance term and plug in ρAB = -1. 14. The standard deviation of the portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio. (True or false?) False. The portfolio standard deviation equals the weighted average of the componentasset standard deviations only in the special case that all assets are perfectly positively correlated. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. 16. Suppose that you have $1 million and the following two opportunities from which to construct a portfolio: Risk-free asset earning 12% per year and a risky asset with expected return of 30% per year and standard deviation of 40%. If you construct a portfolio with a standard deviation of 30%, what is its expected rate of return? y = σC/σP = 0.30/0.40 = 0.75 E(rC) = rf + y[E(rP) – rf] = 0.12 + 0.75[0.30 – 0.12] = 25.50%
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The following data are for Problems 17 through 19: The correlation coefficients between pairs of stocks are as follows: Corr(A,B) = .85 Corr(A,C) = .60 Corr(A,D) = .45 Each stock has an expected return of 8% and a standard deviation of 20% 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice): Choose D since it has the lowest correlation with A and will reduce the portfolio’s risk by the greatest amount. Note that since all stocks have the same E(r), adding any of the other stocks will not change the portfolio’s E(r). 18. Would the answer to Problem 17 change for more risk-averse or risk-tolerant investors? Explain. No, risk aversion is not a factor when choosing between risky assets. For all levels of risk aversion, lower risk is better. Extra Question: What if A = 0? What if A = 0? This would be a “risk-neutral” investor. Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8% and utility is not affected by risk. Extra Question: What if A< 0? If A < 0 then utility is increasing in risk. The person is called a “risk lover.” In this case, a person would prefer adding no assets, since adding any asset that is not perfectly positively correlated reduces risk. But if forced to add an asset, B would be preferred since it would reduce risk the least.
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Additional Question: (Note the following question’s similarity to Question 11.)
1. Consider Risky Asset A, Risky Asset B and the risk-free asset. Given the points on the axes below representing risky assets A and B and the risk-free asset, SKETCH and LABEL the following five items if -1 < < 1: (4) The optimal combined portfolio if y* = 0.5 (1) The risky asset opportunity set if -1 < < 1 (5) The maximum obtainable utility curve (2) The optimal risky portfolio if -1 < < 1 (3) The optimal CAL
E(r) Asset A
Asset B Risk-Free
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2. Given the points on the axes below representing risky assets A and B and the risk-free asset,
SKETCH and LABEL the following five items if = 1: (4) The optimal combined portfolio if y* = 0.5 (1) The risky asset opportunity set if = 1 (5) The maximum obtainable utility curve (2) The optimal risky portfolio if = 1 (3) The optimal CAL
E(r) Asset A
Asset B Risk-Free
CFA PROBLEMS 1.
a.
Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase the risk of the portfolio, but it is possible that the increase in risk will be minimal. Suppose that, for instance, the 50 stocks in a universe have the same standard deviation ( and the correlations between each pair are identical, with correlation coefficient Then, the covariance between each pair of stocks would be 2, and the variance of an equally weighted portfolio would be: 1 n 1 2 2P 2 n n The effect of the reduction in n on the second term on the right-hand side would be relatively small (since 49/50 is close to 19/20 and 2 is smaller than 2), but the denominator of the first term would be 20 instead of 50. For example, if = 45% and = 0.2, then the standard deviation with 50 stocks would be 20.91%, and would rise to 22.05% when only 20 stocks are held. Such an increase might be acceptable if the expected return is increased sufficiently.
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b.
Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio. This entails maintaining a low correlation among the remaining stocks. For example, in part (a), with = 0.2, the increase in portfolio risk was minimal. As a practical matter, this means that Hennessy would have to spread his portfolio among many industries; concentrating on just a few industries would result in higher correlations among the included stocks.
2.
Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. Rather, the incremental benefits from additional diversification are most important when you are least diversified. Restricting Hennesey to 10 instead of 20 issues would increase the risk of his portfolio by a greater amount than would a reduction in the size of the portfolio from 30 to 20 stocks. In our example, restricting the number of stocks to 10 will increase the standard deviation to 23.81%. The 1.76% increase in standard deviation resulting from giving up 10 of 20 stocks is greater than the 1.14% increase that results from giving up 30 of 50 stocks.
3.
The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessy’s portfolio is only one of six well-diversified portfolios and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous.
4.
d.
5. 6. 7. 8. 9.
c. d. b. a. c.
Portfolio Y cannot be efficient because it is dominated by another portfolio. For example, Portfolio X has both higher expected return and lower standard deviation.
10. Since we do not have any information about expected returns, we focus exclusively on reducing variability. Stocks A and C have equal standard deviations, but the correlation of Stock B with Stock C (0.10) is less than that of Stock A with Stock B (0.90). Therefore, a portfolio comprised of Stocks B and C will have lower total risk than a portfolio comprised of Stocks A and B. 11.
Fund D represents the single best addition to complement Stephenson's current portfolio, given his selection criteria. First, Fund D’s expected return (14.0 percent) has the potential to increase the portfolio’s return somewhat. Second, Fund D’s relatively low correlation with his current portfolio (+0.65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio. 12
The other three funds have shortcomings in terms of either expected return enhancement or volatility reduction through diversification benefits. Fund A offers the potential for increasing the portfolio’s return, but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits, but is expected to generate a return well below the current portfolio’s return. Fund C has the greatest potential to increase the portfolio’s return, but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification. 12. a.
Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9 0.67) + (0.1 1.25) = 0.728% ii. Cov = r OP ABC = 0.40 2.37 2.95 = 2.7966 2.80 iii. NP = [wOP2 OP2 + wABC2 ABC2 + 2 wOP wABC (CovOP , ABC)]1/2 = [(0.9 2 2.372) + (0.12 2.952) + (2 0.9 0.1 2.80)]1/2 = 2.2673% 2.27%
b.
Subscript OP refers to the original portfolio, GS to government securities, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9 0.67) + (0.1 0.042) = 0.645% ii. Cov = r OP GS = 0 2.37 0 = 0 iii. NP = [wOP2 OP2 + wGS2 GS2 + 2 wOP wGS (CovOP , GS)]1/2 = [(0.9 2 2.372) + (0.12 0) + (2 0.9 0.1 0)]1/2 = 2.133% 2.13%
c.
Adding the risk-free government securities would result in a lower beta for the new portfolio. The new portfolio beta will be a weighted average of the individual security betas in the portfolio; the presence of the risk-free securities would lower that weighted average.
d.
The comment is not correct. Although the respective standard deviations and expected returns for the two securities under consideration are equal, the covariances between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated. For instance, if the covariances are different, selecting one security over the other may result in a lower standard deviation for the portfolio as a whole. In such a case, that security would be the preferred investment, assuming all other factors are equal.
e.
i. Grace clearly expressed the sentiment that the risk of loss was more important to her than the opportunity for return. Using variance (or standard deviation) as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements. ii. Two alternative risk measures that could be used instead of variance are: Range of returns, which considers the highest and lowest expected returns in the future period, with a larger range being a sign of greater variability and therefore of greater risk. 13
Semivariance, which can be used to measure expected deviations of returns below the mean, or some other benchmark, such as zero. Either of these measures would potentially be superior to variance for Grace. Range of returns would help to highlight the full spectrum of risk she is assuming, especially the downside portion of the range about which she is so concerned. Semivariance would also be effective, because it implicitly assumes that the investor wants to minimize the likelihood of returns falling below some target rate; in Grace’s case, the target rate would be set at zero (to protect against negative returns). 13. a.
Systematic risk refers to fluctuations in asset prices caused by macroeconomic factors that are common to all risky assets; hence systematic risk is often referred to as market risk. Examples of systematic risk factors include the business cycle, inflation, monetary policy and technological changes. Firm-specific risk refers to fluctuations in asset prices caused by factors that are independent of the market, such as industry characteristics or firm characteristics. Examples of firm-specific risk factors include litigation, patents, management, and financial leverage.
b.
Trudy should explain to the client that picking only the top five best ideas would most likely result in the client holding a much more risky portfolio. The total risk of a portfolio, or portfolio variance, is the combination of systematic risk and firmspecific risk. The systematic component depends on the sensitivity of the individual assets to market movements as measured by beta. Assuming the portfolio is well diversified, the number of assets will not affect the systematic risk component of portfolio variance. The portfolio beta depends on the individual security betas and the portfolio weights of those securities. On the other hand, the components of firm-specific risk (sometimes called nonsystematic risk) are not perfectly positively correlated with each other and, as more assets are added to the portfolio, those additional assets tend to reduce portfolio risk. Hence, increasing the number of securities in a portfolio reduces firm-specific risk. For example, a patent expiration for one company would not affect the other securities in the portfolio. An increase in oil prices might hurt an airline stock but aid an energy stock. As the number of randomly selected securities increases, the total risk (variance) of the portfolio approaches its systematic variance.
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