Mean-Variance Analysis If the investor s objective is to Maximize the Expected Rate of Return for a given level of Risk (or, Minimize Risk for a given level of Expected Rate of Return), and If the investor defines Risk as the Variance (or, standard deviation) of the expected returns, then Mean-Variance Analysis should be used.
rocedure for onducting Mean-Variance Analysis: 1. Determine the possible assets to invest in. Determine each asset s expected rate of returns and standard deviations 3. Determine the correlation between all assets 4. onstruct the efficient frontier by varying the weights of each asset in the portfolio ---- do this for all possible weights (e.g., 40% asset 1 & 60% asset ; 41% asset 1 & 59% asset, etc. etc.) 5. Determine the investor s subjective views on the trade-off between risk and return. In theory, we do this with indifference curves. 6. Find the portfolio on the efficient frontier where the additional unit of risk leads to an increase in expected return that just matches with the investor s preferences. Graphically, this is where the investor s indifference curve is just tangent to the efficient frontier.
Assume two assets have the following characteristics. Asset 1 Asset Return 30% 10% Standard Deviation 40% 0% orrelation oefficient -0.5 The following three portfolios have been constructed from the two assets. Discuss the portfolios. ortfolio X: 40% Asset 1 and 60% Asset ortfolio Y: 10% Asset 1 and 90% Asset ortfolio Z: 30% Asset 1 and 70% Asset
Assume two assets have the following characteristics. Asset 1 Asset Return 30% 10% Standard Deviation 40% 0% orrelation oefficient -0.5 The following three portfolios have been constructed from the two assets. Discuss the portfolios. ortfolio X: 40% Asset 1 and 60% Asset E ( R) = (40% 30%) + (60% 10%) = 1% + 6% = 18% σ = (.40.40) + (.60.0) + (.4)(.6)(.4)(.)(.5) =.056 +.0144.019 =.008 σ = σ =.008 =.144 14.4%
ortfolio Y: 10% Asset 1 and 90% Asset E ( R) = (10% 40%) + (90% 10%) = 3% + 9% = 1% σ = (.10.40) + (.90.0) + (.1)(.9)(.4)(.)(.5) =.068 σ = σ =.068 =.1637 16.37% ortfolio Z: 30% Asset 1 and 70% Asset E ( R) = (30% 40%) + (70% 10%) = 9% + 7% = 16% σ = (.30.40) + (.70.0) + (.3)(.7)(.4)(.)(.5) =.017 σ = σ =.017 =.1311 13.11%
Asset 1 Asset Return 30% 10% Standard Deviation 40% 0% orrelation oefficient -0.5 Summary Table Return Risk X (40/60) 18.00% 14.4% Y (10/90) 1.00% 16.37% Z (30/70) 16.00% 13.11% Now, construct the efficient frontier
Efficient Frontier 33.00% 8.00% Return 3.00% 18.00% 13.00% 8.00% 10.00% 15.00% 0.00% 5.00% 30.00% 35.00% 40.00% 45.00% Risk Now, take into account the investor s preferences for the risk-return tradeoff
Efficient Frontier 33.00% 8.00% Return 3.00% 18.00% 13.00% 8.00% 10.00% 15.00% 0.00% 5.00% 30.00% 35.00% 40.00% 45.00% Risk We have now found the most efficient portfolio for the investor. That is, the portfolio the maximize return for a given level risk and matches with the investor s preferences.
Adding the Separation Theorem to Mean-Variance Analysis The Separation Theorem allows us to separate the choice of the optimum risky portfolio from the individual s subjective preferences. The individual s preferences will now enter only in deciding what percentage of their wealth to invest in the risky portfolio. The Separation Theorem assumes that a risk-free asset exists and the investor can either lend or borrow at this rate. Keep in mind, the term risk -free asset here means that an asset with no variance in its expected rate of return. We ll need to discuss this more carefully. rocedure: 1. onstruct the efficient frontier from a mean-variance analysis. Determine the risk-free rate of return 3. onstruct the apital Allocation Line (AL) 4. Determine the optimum risky portfolio. Thus, find the apital Allocation Line that is just tangent to the efficient frontier. This line will have the highest slope possible for a given efficient frontier. 5. Based on the investor s subjective preferences determine the percentage of their wealth to invest in the optimum risky portfolio (from step 4) and what percentage to invest in the risk-free asset (from step ).
We have already done step 1. Step. Assume the rate of return on the risk-free asset is 6%. Step 3. a) What is the slope of the line connecting the risk-free asset to portfolio X? R Slope = r F 18% 6% = =. 831 σ 14.4% b) What is the equation for the line connecting the risk-free asset and portfolio X (i.e., the apital Allocation Line)? R rf AL: R = rf + σ = 6 % + (.831) σ σ Note: the risk of the omplete portfolio is σ. = W σ
ortfolio Z What is the slope of the line connecting the risk-free asset to portfolio Z? R Slope = r F 16% 6% = =. 765 σ 13.11% What is the equation for the line connecting the risk-free asset and portfolio Z (i.e., the apital Allocation Line)? AL: R = r F R r + σ F σ = 6 % + (.765) σ We see that the slope of the apital Allocation Line is greater for ortfolio X. Thus, it gives greater reward per unit of risk.
3.00% 1.00% 19.00% 17.00% 15.00% 13.00% 11.00% Return X Return Z Efficient Frontier 9.00% 7.00% 5.00% 3.00% 0.00% 5.00% 10.00% 15.00% 0.00% Suppose for illustrative purposes that the apital Allocation Line for ortfolio X is in fact just tangent to the efficient frontier. Thus, ortfolio X is the optimum risky portfolio. We would now be ready for step 5 (determining the complete portfolio based upon the investor s preferences).
3.00% 1.00% 19.00% 17.00% 15.00% 13.00% 11.00% Return X Return Z Efficient Frontier 9.00% 7.00% 5.00% 3.00% 0.00% 5.00% 10.00% 15.00% 0.00% Suppose that based upon the investor s preferences (as illustrated by the blue indifference curve now), we invest 80% of the investor s wealth in the optimum risky portfolio (portfolio X in this case) and 0% in the risk-free asset. What would be the profile (i.e., expected return and risk) of the complete portfolio?
Expected Return of the omplete ortfolio AL: R = r F R r + σ F σ = 6 % + (.831) σ Note: the risk of the omplete portfolio is σ = W. σ = (80%) (14.4%) = 11.54% Thus, R r R r + σ σ F = F = 6 % + (.831) σ = 6% + (.831)(11.54%) = 15.6% Let s check this. We know that the expected rate of return on a portfolio is the weighted average of the individual rates, so E ( R) = (80% 18%) + (0% 6%) = 14.4% + 1.% = 15.6% Let s check to see if we have the risk of the complete portfolio correct at 11.54%. σ = (.80.144) + (.0 0) + (.8)(.)(.144)(0)(0) =.013308 + 0 0 =.013308 σ = σ =.013308 =.1154 11.54%
Our complete portfolio has an expected rate of return of 15.6% and risk of 11.54%. ould we have done better with ortfolio Z? Just for fun let s see what the expected return on a complete portfolio (using Z as the risky portfolio) would be with the same level of risk that we had just found. In order to have a standard deviation (i.e., risk) on a complete portfolio of 11.54% we would need to allocate 88% of the investor s wealth to the risky portfolio Z and 1% to the risk-free asset. Thus, σ = W. σ = (88%) (13.11%) = 11.54% The expected return would be R r R r + σ σ F = F = 6 % + (.831) σ = 6% + (.765)(11.54%) = 14.8% Again, we can verify this E ( R) = (88% 16%) + (1% 6%) = 14.08% +.7% = 14.8% Thus, for the same level of risk (standard deviation of 11.54%) we get a higher expected return by using ortfolio X as our optimum risky portfolio. This is a general result in that the capital allocation line associated with the optimum risky portfolio will provide the highest expected rate of return for any given level of risk regardless of the individual s subjective preferences.
Questions? 1. Does the standard deviation of an asset and/or portfolio constitute the proper definition of risk?. Does a risk-free asset exist (i.e., zero standard deviation of expected rates of return over the relevant investment horizon)? 3. How do we obtain the investor s subjective preferences? 4. Where do the inputs come from? We need expected rates of returns and standard deviations for all possible investment assets. We also need correlation coefficients between all the possible investment assets.