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?
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?
3. Real-data example
The following exercise is intended to show how to apply what we have learnt in class with real data. One way to do this is to obtain monthly price data on a market proxy (say, the S&P500) and a stock (say, Cisco) as well as monthly Treasury bill returns. Another way would be to obtain annual price data and compute annual returns.
Closing prices for S&P500 (ticker=^SPX) and Cisco (ticker=CSCO) at various frequencies (daily, weekly and monthly) can be obtained easily from the following site Yahoo! Finance. After you enter the ticker select any of the links for big charts. Just under the chart there is a link called historical quotes. Clicking here will produce the results for Cisco closing daily prices over the past three months. Towards the bottom of that page there is a link that would allow you to download the data in a spreadsheet format. Note that all the prices will be adjusted for splits and dividends which will simplify the computation of the returns.
Data on Treasury bill returns can be obtained from the Saint Louis Federal Reserve Bank which maintains a database with lots of historical financial data called FRED which can be reached by clicking here. For this example I used monthly data on 3-Month Treasury Bill rates from the secondary market. Note that those rates are annualized and in reverse order (i.e. from oldest to latest observation).
For the purposes of this example I collected 60 monthly observations on the risk-free rate (from Sep 1995 to Aug 2000) and 61 monthly closing prices for S&P500 and Cisco (from Aug 1995 to Aug 2000). Then I computed the historical realized excess returns on the S&P500 and Cisco. Both the raw realized returns, the T-bill rate and the realized excess returns can be accessed and downloaded in a spreadsheet format by clicking here.
In order to illustrate how the choice of data frequency and length influences the optimal portfolio allocations I also computed annual returns (from Aug 1991 to Aug 2000) for S&P500 and Cisco. For completeness, the raw realized returns, the T-bill rate and the realized excess returns can be found by clicking here.
Using both the monthly and annual returns supplied above do the following:
A. Compute the historical mean realized excess returns on the S&P500 and Cisco.
B. Compute the beta of Cisco.
C. Compute the variance of the S&P500 excess return.
D. Compute the idiosyncratic variance of Cisco's excess return.
E. 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 expectation equals historical mean? based on your expectation?
F. What is your expectation of the return of Cisco? Your answer can depend on the historical returns, the CAPM, analysts' reports, your knowledge of the industry or anything else, including the answer to part E above.
G. Answer part E again using your expectation in F.