# 1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset

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1 Department of Economics Financial Economics University of California, Berkeley Economics 136 November 9, 2003 Fall 2006 Economics 136: Financial Economics Section Notes for Week 11 1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset This material is in BKM chapter 7. Assume for the moment that an investor must decide how to invest all of her wealth and has only two options: a risk-free asset such as Treasury Bills (T-Bills) and a risky portfolio of stocks (such as a mutual fund). Since all of her wealth must be invested, the decision that she makes can be summarized by one parameter, the fraction of her wealth that she invests in the risky portfolio, w. Since she must allocate all of her wealth to either the mutual fund or T-Bill, the fraction of her wealth invested in T-Bill must be, 1 w. If we assume a functional form for the investors objective (or utility) function, then we can determine the optimal fraction of wealth for the investor to put into the risky portfolio, w. Let R P denote the returns of the combined T-Bill and mutual fund portfolio and R MF denote the returns of the mutual fund. Assuming that people like high mean return and dislike high return variance then they would like to solve the following optimization, max w [ E[R P ] 1 2 A V ar(r P ) ] max w Then the first order condition (foc) is [ R f + w(e[r MF ] R f ) 1 2 w2 A V ar(r MF ) 0 E[R MF ] R f w A V ar(r MF ) w E[R MF ] R f A V ar(r MF ) Note: w gives the optimal fraction of wealth invested in the risky portfolio. The total portfolio which invests w in the risky portfolio and 1 w in the risk-free asset will result in the optimal portfolio which has an expected return and standard deviation such that the investor s indifference curve is tangent to the CAL at this point. This is why it is optimal. It gives the investor the highest possible utiity subject to the mean and standard deviation of investment possibilities (the CAL is the frontier of these possibilities). ] 1

2 2 Finding the Optimal Risky Prtfolio This material is in BKM chapter 8. In the last section, we took the expected return and standard deviation of the risky portfolio as given. Now we consider how to construct the best risky portfolio from a number of risky assets (2 in this case, but N in general). The notion of a best risky portfolio can have several interpretations. First, I consider finding the risky portfolio that has the smallest possible variance. Then, I look at the risky portfolio that results in the maximum invetor utility when it is subsequently combined with a risk-free asset. This will be the risky portfolio which maximizes the slope of the CAL, or equivalently, has the maximum Sharpe ratio. Assume there are only two risky assets which have returns R E and R D (think of a risky equity fund, and a risky bond (debt) fund). The expected portfolio return and standard deviation of the risky portfolio R RP are E[R RP ] E[R E ] + (1 )E[R D ] σ(r RP ) V ar(r RP ) [ w 2 EV ar(r E ) + (1 ) 2 V ar(r D ) + 2 (1 )Cov(R E, R D ) ] 1 2 To find the minimum variance portfolio we minimize the variance of the risky portfolio, [ min V ar(r RP ) min w 2 E V ar(r E ) + (1 ) 2 V ar(r D ) + 2 (1 )Cov(R E, R D ) ] the first order condtion (foc) sets dv ar(r RP ) 0. 2 V ar(r E ) + 2( 1)V ar(r D ) + 2(1 2 )Cov(R E, R D ) 0 V ar(r E ) + V ar(r D ) 2 Cov(R E, R D ) V ar(r D ) Cov(R E, R D ) w E,min var V ar(r D ) Cov(R E, R D ) V ar(r E ) + V ar(r D ) + 2Cov(R E, R D ) To find the fraction of wealth to invest in equity that will result in the risky portfolio with the maximum Sharpe ratio max S P E[R P ] R f σ(r P ) E[R E ] + (1 )E[R D ] R f [w 2 E V ar(r E) + (1 ) 2 V ar(r D ) + 2 (1 )Cov(R E, R D )] 1 2 the first order condtion (foc) sets ds P ( ) 0 ds P ( ) E[R E ] E[R D ] [we 2 V ar(r E) + (1 ) 2 V ar(r D ) + 2 (1 )Cov(R E, R D )] E[R E ] + (1 )E[R D ] R f [we 2 V ar(r E) + (1 ) 2 V ar(r D ) + 2 (1 )Cov(R E, R D )] 3 2 (2 V ar(r E ) 2(1 )V ar(r D ) + 2(1 )Cov(R E, R D )) 0 2

3 (E[R E ] E[R D ]) [w 2 EV ar(r E ) + (1 ) 2 V ar(r D ) + 2 (1 )Cov(R E, R D ) ] 1 2 (E[R E ] + (1 )E[R D ] R f ) (2 V ar(r E ) 2(1 )V ar(r D ) + 2(1 )Cov(R E, R D )) Fill in the steps here. Let me know if I ve made any mistakes. The next line is correct. w E (E[R D ] R f )V ar(r E ) (E[R E ] R f )Cov(R E, R D ) (E[R D ] R f )V ar(r E ) + (E[R E ] R f )V ar(r D ) (E[R D ] + E[R E ] 2R f )Cov(R E, R D ) While this first order condition is rather messy, the concept of maximizing the Sharpe ratio by finding the risky portfolio that results in the Capital Allocation Line (CAL) with the highest slope is easy to see graphically. 3 Example: BKM chapter 8, problems 1-8 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. The correlation betwenn the stock fund and the bond fund is ρ(r S, R B ) 0.1. Expected Return Standard Deviation Stock Fund (R E ) 20% 30% Bond Fund (R D ) 12% 15% T-Bill Fund (R f ) 8% 0% 1. 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? First let s find the covariance of the debt and equity fund returns Cov(R E, R D ) σ(r E )σ(r D )ρ(r E, R D ) w E,min var V ar(r D ) Cov(R E, R D ) V ar(r E ) + V ar(r D ) + 2Cov(R E, R D ) E[R RP ] ( ) % σ(r RP ) V ar(r RP ) [ ( ) ( ) ] % 2. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock funds of zero to 100% in increments of 20%. 3

4 xpected Return Standard Deviation 0 12% 15% % 13.9% % 15.7% % 19.5% % 24.5% 1 20% 30% 3. 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 risky portfolio. 4. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky porfolio. The fraction of the portfolio invested in equity is w E (E[R D ] R f )V ar(r E ) (E[R E ] R f )Cov(R E, R D ) (E[R D ] R f )V ar(r E ) + (E[R E ] R f )V ar(r D ) (E[R D ] + E[R E ] 2R f )Cov(R E, R D ) ( ) ( ) ( ) ( ) ( ) E[R RP ] wee[r E ] + (1 we)e[r D ] 16.4% σ(r RP ) [ (we) 2 V ar(r E ) + (1 we) 2 V ar(r D ) + 2wE(1 we)cov(r E, R D ) ] % 5. What is the Sharpe ratio of the best feasible CAL? S P E[R P ] R f σ(r P ) You require that your total portfolio (TP) yield an expected return of 14%, and that it be efficient on the best feasible CAL. a) What is the standard deviation of your total portfolio? 4

5 E[R T P ] 0.14 R f + w(e[r P ] R f ) w( ) w σ(r T P ) w σ(r P ) % b) What is the proportion invested in the T-Bill fund and each of the two risky funds? Proportion T-Bill Stock Fund Bond Fund ( ) If you were to use only the risky funds, and still require an expected return of 14%, what must be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio you found in 6. What do you conclude? 14% E[R E ] + (1 )E[R D ] 20% + (1 )12% 0.25 σ(r P ) [ (w E) 2 V ar(r E ) + (1 w E) 2 V ar(r D ) + 2w E(1 w E)Cov(R E, R D ) ] % The standard deviation in 6 is 8.76% which is lower than 14.1%. By combining the optimal risky portfolio with the risk-free asset it is possible to acheive the same expected return with a lower standard deviation, this can be seen graphically. 4 Example: BKM chapter 7, problem Consider the following information about a risky portfolio that you manage and a riskfree asset: E[R P ] 11%, σ(r P ) 15%, R f 5%. (a) Your client wants to invest a portion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky portfolio, P, and what proportion in the risk-free asset? 8% R f + w(e[r P ] R f ) 5% + w(11% 5%) w 0.5 5

6 (b) What will be the standard deviation of the rate of return on her portfolio? σ(r T P ) w σ(r P ) % 7.5% (c) Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse? The first client is more risk averse. 6

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