** DN****SC6210 Decision and Risk Analytics The George Washington University USA**

Mean-Variance Portfolio Selection with a Risk-free Asset

The accompanying Excel file contains data on yearly rates of return for four stocks (the ticker symbols are fake). Also, added to the data set is a risk-free asset available for investment, which had a constant 2% yearly rate of return over the same period. Your task is to formulate an optimization model to construct portfolios of the 5 securities that minimize the portfolio's risk for a given level of portfolio expected return, then use your model to answer a few questions.

a) Using the variance as a measure of risk, set up an optimization model to minimize the portfolio variance for a specified portfolio rate of return. Assume no short-selling of the assets, that is, the portfolio's weights must all be non-negative.

b) Use the model you built in (a) to generate three optimal portfolios that provide expected rates of return of 3%, 4%, and 5%, respectively. Copy and paste the weights and corresponding mean return of the three portfolios so obtained in a table somewhere on the same sheet containing your model.

c) Considering only the optimal portfolios obtained in (b) that include the risk-free asset, that is, those in which the weight of the risk-free asset is non-0, show that the relative proportions of the 4 risky assets remain the same across those portfolios. For this, you can calculate the ratio of a stock's weight to the sum of the 4 stocks weights, and verify that these ratios are the same in each optimal portfolio that includes the risk-free asset. This illustrates a well-known result in finance known as the Capital Market Line.

d) An investor has $60,000 to invest and a risk tolerance of $10,000. Add cells to the model you built in (a) to calculate (i) the expected value, (ii) the variance, and (iii) the certainty equivalent

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