Paskalis Glabadanidis and Yufeng Han
1. Optimal portfolio choice with uncorrelated returns
Suppose the expected return on the market is 14%, the riskless return is 5% and the standard deviation of the market is 30%. Suppose also that given this information we decide that the optimal mix is 25% in the market and the remainder in the riskless asset.
Now let us consider whether and how much we should buy a stock "Tuvalu" which is uncorrelated with the market (i.e. it has a beta of zero), expected return of 10% and a standard deviation of 25%. What is the optimal holding of Tuvalu, the market, and the riskless asset?
Given the initial choice between the market and the riskless asset and our knowledge about optimal portfolio weights we have that
k x (14%-5%)/(30% x 30%) = 25%
which produces a value of k = 0.25. Armed with this number we can determine the optimal holdings of Tuvalu to be
0.25 x (10%-5%)/(25% x 25%) = 20%
Hence, the optimal portfolio consists of 25% in the market, 20% in Tuvalu, and 75%-20%=55% in the riskless asset.
2. Optimal portfolio choice with correlated returns
Suppose now the expected return on the market is 15%, the riskless return is 5%, and the standard deviation of the market is 20%. Suppose also that faced with these numbers we decide to put 50% in the market and 50% in the riskless asset.
Now consider a stock "Betamin" which has a beta of 0.5, expected return of 12% and idiosyncratic standard deviation of 40%. What is the optimal holding of Betamin, the market, and the riskless asset?
Again given our initial choice we can determine the constant k which represents our risk preferences from the following equation
k x (15%-5%)/(20% x 20%) = 0.5
which we can solve for k = 0.2. Now given that Betamin is correlated with the market we have to construct a synthetic security (call it Betamin*) that is uncorrelated with it in order to use the formulas from the slides. To do that imagine a portfolio that is 100% long in Betamin, 50% short in the market and 50% long in the riskless asset. This portfolio has a beta of zero, expected return of 7% and standard deviation equal to the idiosyncratic standard deviation of Betamin of 40%. Given these numbers our optimal holding of this Betamin* is
0.2 x (7%-5%)/(40% x 40%) = 2.5%
Hence, the optimal holding of the market is 50%-0.5 x 2.5%=48.75%, of Betamin is 2.5% and 50%-2.5%+0.5 x 2.5%=48.75% in the riskless asset.
3. Real-data exampleMonthly returns dataset Annual returns dataset
Using both the monthly and annual returns supplied above do the following:
A. Compute the mean realized excess returns on the S&P500 and Cisco.
For the monthly data we obtain E(xsret(S&P500))=1.35% per month and E(xsret(Cisco))=3.25% per month. For the annual data the numbers are 12.61% per annum and 101.58% per annum, respectively.
B. Compute the beta of Cisco.
In either case we do that by regressing Cisco's excess returns on the S&P500's excess returns. For the monthly data the result is 1.436 and for the annual data it is 1.606.
C. Compute the variance of the S&P500 excess return.
For the monthly data we find that the variance of the S&P500 excess return is 0.001898 or a standard deviation of 4.357% per month. For the annual data the numbers are 0.01341 per annum and 11.58% per annum, respectively.
D. Compute the idiosyncratic variance of Cisco's excess return.
To do this we need to compute the variance of the residual from the regression in part B. above. For the monthly data the result for the variance is 0.008171 or, equivalently, a standard deviation of 9.04% per month. For the annual data we obtain a variance of 0.4529 or a standard deviation of 67.3% per annum.
Assume that when choosing only between the S&P500 and the T-bill we put 50% in the former and 50% in the latter. Now consider adding Cisco to the portfolio. What are the optimal holdings of the S&P500, Cisco and the T-bill based on the historical returns?
This answer key solves assuming the prediction equals the historical returs. Let us do this for the monthly dataset first. Given that initially we split our portfolio equally in the S&P500 and the T-bill we can solve for the risk-preference constant k to be
k = 0.5 x var(xsret(S&P500)) / E(xsret(S&P500)) = 0.0704
Now consider a synthetic security Cisco* which consists of 100% long in Cisco, 100 x 1.436 = 143.6% short in the S&P500 and 143.6% long in the T-bill. By construction, Cisco* has no market (i.e. S&P500) exposure and is, hence, uncorrelated with the S&P500. Cisco* has an excess return of 3.25% and variance equal to Cisco's idiosyncratic variance or 0.008171. Applying our formula we can compute the optimal holding of Cisco* to be
k x E(xsret(Cisco*)) / var(residual(Cisco*)) = 0.0704 x 0.0325 / 0.008171 = 27.98%
Backing out the S&P500 and T-bill holdings from this we obtain 50%-1.436x27.98%=9.81% in the S&P500 and 50%-27.98%+1.436x27.87%=62.21% in the T-bill. The optimal weight in Cisco is the same as the weight in Cisco* or 27.98%.
Similarly, for the annual data we obtain the value of k in the following way
k = 0.5 x var(xsret(S&P500)) / E(xsret(S&P500)) = 0.0532
The synthetic security Cisco* in this case consists of 100% long in Cisco, 100 x 1.606 = 160.6% short in the S&P500 and 160.6% long in the T-bill. The excess return of Cisco* is 81.33% per annum and its variance is again equal to the idiosyncratic variance of Cisco which is 0.4529. Hence, the optimal weight in Cisco* is
k x E(xsret(Cisco*)) / var(residual(Cisco*)) = 0.0532 x .8133 / .4529 = 9.56%
Finally, we back out the holding of the S&P500 to be 50% - 1.606 x 9.56% = 34.67% and the T-bill to be 50% - 9.56% + 1.606 x 9.56 = 55.78%. Again, the optimal weight in Cisco is the same as the optimal weight in Cisco* or 9.56%.
It is not surprising that we see this difference in optimal holdings given how different the estimated excess returns and standard deviations are from the two datasets. For example, the annualized excess return of Cisco from the monthly dataset is 46.74% per annum and the annualized idiosyncratic standard deviation is 31.31% per annum. Compare those numbers to the ones we obtained from the annual returns dataset.
F. What is your expectation of the return of Cisco? Your answer can depend on the historical re turns, the CAPM, analysts' reports, your knowledge of the industry or anything else, including the answer to part E above.
Each person can have a different answer to this. Most answers will probably give a smaller mean return, closer to that predicted by efficient markets and the CAPM, than are historical returns.
G. Answer part E again using your expectation in F.
This depends on your personal choice of answer to E. Most people would obtain a smaller proportion here than in part E.