Problem -

A fund manager is currently holding a portfolio composed of 4 stocks: US stock represented by S&P500 index, EU represented by Euro STOXX50 index, Australia stock represented by All Ordinary index, and China stock represented by Shanghai composite index. The total market capitalizations of the four stocks are:

The manager was thinking about including other assets to be invested with the stocks. He asks you to investigate the prospect of including energy commodities, precious metals, agricultural commodities, and treasury bonds, in the stock portfolio for the next 12 months investment.

You have been provided with a data set that includes end of month data on the four mentioned stock indexes; S&P Goldman Sachs commodity price indices: average energy price, precious metal price, agricultural product price; and US government bond index. These data are downloaded from Datastream.

The variable names in the provided dataset are as follows:

1. Literature review

Write (maximum 1 page) a review of studies that examine the set of investable products that should be added to stock portfolios. In the light of your review, explain why you think the products recommended by the manager may or may not add value to the stock portfolio. What other products do you recommend to include in the analysis?

Criteria:

- Review 5 peer-reviewed academic journal articles

- Review correctly reflects results of the reviewed studies

- The review should be succinct, consistent, coherent and informative.

2. Summary Statistics

Convert the indices into monthly log returns (i.e. continuously compounded monthly rates of return). In a table, present summary statistics that include monthly sample mean and variance of log return. In the same table, present also the mean and variance of log return, and the logarithm of the expected gross return in annualised term. In another table, present the correlation of the monthly log returns.

What implications do these statistics have for optimal portfolio management?

3. Markowitz Model

a. Efficient frontier of risky asset:

i. Describe the method to determine the efficient frontier of risky assets when short selling is allowed and when it is not allowed.

ii. In one graph, depict and label the efficient frontier derived from the 8 risky assets given in the dataset when short selling is allowed and when short selling is not allowed. Be careful to use annualized logarithm of expected gross return and annualized standard deviation of portfolio rate of return in your optimization.

iii. Suppose that the manager is currently allocating asset weights proportional to the total market capitalizations represented by the stock indexes. What are the asset weights of the current portfolio? Plot the current portfolio in the above graph.

iv. Based on the graph, comment on the risk return trade-off, the impact of the short-selling constraint, and the impact of including non-stock assets in the stock portfolio.

b. Tangency portfolio

Suppose that the manager can borrow and lend money at the risk free rate of 0.7% per year.

i. Describe the method to determine the optimal portfolio when the investor can invest in risky assets and also in the risk free asset. Apply your method to the case of 8 risky assets and the risk free rate given above (0.7% per year), when short selling is allowed.

ii. In a table, for each asset, present the asset's annualized standard deviation, annualized logarithm of expected gross return, the asset's weight in the original stock portfolio (note: original portfolio is the portfolio with only stocks and the portfolio weights are the market capitalisation weights), and the weights in the tangency portfolio. Provide comments on the weight change when non-stock assets are included in the stock portfolio.

iii. In another table, present the Sharpe ratio of the original stock portfolio and the Sharpe ratio of the optimal portfolio that you construct in part i. Also present the annualized standard deviation, annualized logarithm of expected gross return of the two portfolios. Comment on the changes in the performance of the portfolio when non-stock assets are included with stocks. Would you be able to implement the tangency portfolio in practice?

iv. Repeat part ii) and iii) for the case when short selling is not allowed.

v. Would you recommend your manager to include non-stock assets in his portfolio?

4. Black Litterman Models

Assume that the risk free rate is 0.7% per year

a. Describe the Black Litterman approach to determine the (equilibrium) expected rate of return of stocks based on the observed market portfolio.

b. Apply the Black Litterman approach to determine the equilibrium expected return of the 4 stock indexes. In a table, present the weights of the market portfolio, the logarithm of expected gross return in annualised term that is calculated based on sample mean and standard deviation of log return; and the equilibrium logarithm of expected gross return in annualised term of the four stock indexes.

c. The manager thinks that in the coming year, the expected return of US stock is higher than the corresponding equilibrium expected return by 2%. He attaches a variance of 0.0004 to the error of the view. Describe the B-L approach to combine this view with the equilibrium expected returns.

d. In a table, for each of the four stock indexes, present the logarithm of expected gross return in annualised term that is calculated based on sample mean and standard deviation of log return; the equilibrium logarithm of expected gross return in annualised term, the Black Litterman estimators that combine the equilibrium estimates and the active view.  Comment on why you think the Black Litterman estimators that you obtain are correct.

5. James-Stein Model

Assume that the risk free rate is 0.7% per year

a. Discuss the usefulness of the James-Stein (JS) theory to portfolio management. Describe how you can use James-Stein (JS) theory to provide more reliable estimates of the expected returns of non-stock indices.

b. In a table, present in annualised term the sample mean, standard deviation, grand mean, shrink factor, the JS estimators of the expected log returns, and the JS estimators of the log of the expected gross return. Comment on the change in the estimate of the expected log returns when you use JS method rather than sample mean method.

c. Use the covariance matrix calculated based on historical data, the JS estimate of the log of the expected gross return on non-stock indexes, and the equilibrium logarithm of expected gross return on stock indexes obtained in Question 4b, calculate the optimal portfolio from the 8 assets, assuming that short selling is not allowed. In a table, for each of the 8 individual assets, present the weight of the asset in the original stock portfolio, the weight of the asset in the optimal portfolio, the logarithm of expected gross return in annualised term that is calculated based on sample mean and standard deviation of log return; and the adjusted (i.e. either JS or BL) estimates of the expected returns that are used to derive the optimal weights. 

d. Based on the whole assignment, would you recommend the manager to include non-stock assets in his investment portfolio?

Need Q no 4 (a), (c) and Q 5(a), (d).

Assignment Files -

https://www.dropbox.com/s/6z9vhonc2j6biqr/Assignment%20Files.rar?dl=0