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FIN 532 Investment Theory

Problem Set 1

Fall 2024

1 Risk, Preferences, and Asset Allocation

1. You bought 100 shares of ABC Inc.  common stock at $100 per share today at the opening of the market.   ABC  Inc.    just  announced  a  dividend  of $2.00  per  share payable in exactly one year from today.  It is widely believed that one year from now the economy will either be in a ‘recession’, a state of ‘normal growth’, or a ‘boom’ with probabilities of 30%, 40%, and 30% respectively.  After analyzing ABC Inc.  you are convinced that the price of ABC stock a year from now in these various states of the economy will be:

What are your estimated expected return and volatility over the next year to your investment in ABC stock?

2. TNC mutual fund invests 25% of their assets in IBM stock, 50% in GE stock, and 25% in T-Bills. You invested 50% of your wealth in TNC mutual fund and rest in the T-Bills. What percentage of your wealth is invested in each stock and in the T-Bills?

3.  Suppose we are in a world with two equally likely states u and d.  And we have three stocks A, B and C. Their net returns are given by the following table.

Table 1: The net returns of the stocks.

Can you find a risk-averse investor who prefers stock A (or B) to stock C? Explain.

4. Now suppose the net returns of the three stocks are given by table 2.

Table 2: The net returns of the stocks.

(a) Let ˜rA and ˜rC be the net returns of stocks A and C respectively. Can you find a random variable ˜z such that ˜rA = ˜rC + ˜z and E[˜z] = 0?

(b) Can you find a risk-averse investor who prefers stock A (or B) to stock C?

5. Consider a risky portfolio that ofers a rate of return of 15% per year with a standard deviation of 20% per year.  Suppose an investor with mean-variance preferences is indiferent between investing in the risky portfolio and investing in a risk free asset earning 8% per year.

a) What is the investor's risk aversion coefficient?

b) If allowed to invest in a combination of the risky portfolio and the risk free asset, what proportion would the investor hold in the risky portfolio?

c) What is the expected rate of return and the standard deviation of the rate of return on the optimally chosen combination?

d) What would be the investor's certainty equivalent return for the optimally chosen combination?

6. In this question, you are asked to evaluate the common portfolio advice of a 60/40 split between stocks and bonds. Suppose the expected rate of return on equities is 8%/year and the standard deviation of the return on equities is 19%/year. T-Bills earn 1%/year (assume they are riskless).

(a) What is the implied risk aversion coefficient of an investor for whom a 60/40 split is optimal?

(b) Plot the CAL along with a couple of indiference curves for the investor type identified above.

7. For this exercise, you will have to download data on equity returns from 1926 to

2022 from Kenneth French’s Data library (http://mba.tuck.dartmouth.edu/pages/ faculty/ken.french/data_library.html).  You will download data on the excess returns of stocks over T-bills; they are available near the top of the page under Fama/French 3 Factors.  You need the variable Mkt-RF. The variable is available at 4 diferent frequencies: annual, monthly, weekly and daily.

(a) Compute the mean and standard deviation of stock returns at diferent frequen- cies, including their standard errors. To make results comparable, express every- thing in an annual frequency. To a first order, this means multiplying monthly returns by 12, weekly returns by 52, and daily returns by 250 (there are approxi- mately 250 trading days in a year).

Compare your estimate of the mean and standard deviation (of annualized re- turns) across these diferent frequencies. How does the precision of your estimates (the tightness of confidence intervals change?). Discuss.

(b) For each decade, compute the mean return in the stock market, and volatility. You can use monthly data for this exercise.  Do your estimates of the mean and volatility vary across decades? Are your estimates statistically diferent?

(c) For this part, we will only use daily returns. For each year in the sample, compute the realized volatility (i.e. standard deviation) of daily market returns. Plot the resulting yearly observations. Is market volatility constant over time?

For this exercise, we will need a software package that allows you to estimate means and standard deviations, along with confidence intervals. If the software you use does not provide you with standard errors, you can consult your statistics textbook (or Wikipedia) and you can compute them manually.