Problem Set 1
Investment Theory
Instructions: This problem set is due on 9/5 at 11:59 pm CST and is an individual assignment. Problems 1 to 5 must be handwritten. Scan your work and submit a PDF file. Problem 6 can be done in Microsoft Excel or Python. For Problem 6, submit either an Excel spreadsheet or a Jupyter notebook saved as HTML.
Problem 1 Consider a sample space with four possible outcomes \(\Omega = \{ \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4} \}.\) The table below describes the possible values of two random variables denoted by \(X\) and \(Y.\)
| Outcome | \(X\) | \(Y\) |
|---|---|---|
| \(\omega_{1}\) | 2 | 0 |
| \(\omega_{2}\) | -3 | 0 |
| \(\omega_{3}\) | 4 | 5 |
| \(\omega_{4}\) | 1 | 5 |
Describe the sets of events generated by \(X\) and \(Y.\)
Problem 2 An investor purchased a bond one year ago for $980. He received $17 in interest and sold the bond for $987. What is the holding-period return on his investment?
Problem 3 Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of 50%. The alternative risk-free investment in T-bills pays 6% per year.
- If you require a risk premium of 8% so that the discount rate is 6 + 8 = 14%, how much will you be willing to pay for the portfolio?
- Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?
- Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay?
Problem 4 The stock of company XYZ currently trades at $100 and just paid a dividend of $1.8. Suppose that your expectations regarding the stock price and dividends next year are as follows:
| State of the Market | Probability | Dividend | Price |
|---|---|---|---|
| Boom | 0.3 | $3 | $120 |
| Normal growth | 0.5 | $2 | $100 |
| Recession | 0.2 | $1 | $80 |
Compute the mean and standard deviation of the returns for company XYZ.
Problem 5 Suppose the economy can only be in one of the following two states: (i) Boom or “good” state and (ii) Recession or “bad” state. Each of the states can occur with an equal probability. At the beginning of a month, you can purchase the following two securities in the market:
- Security 1: It is currently trading at $4. At the end of the month, the stock price is expected to increase by $10 in the good state, and expected to remain unchanged in the bad state.
- Security 2: It is also currently trading at $5. This asset has payoffs that are similar to an insurance contract. It yields a positive return when the economic conditions are poor. At the end of the month, the price of security 2 is expected to remain unchanged in the good state and expected to increase by $10 in the bad state.
- Compute the expected returns of securities 1 and 2.
- Compute the standard deviations of returns for securities 1 and 2.
- Compute the covariance and the correlation between the returns of two securities.
Problem 6 Pick any stock you want that has at least 10 years of price history. Generate a plot of monthly returns and label your plot professionally. Describe how the volatility seems to vary over time. Generate a table that presents the monthly and annualized average return and volatility of the monthly returns. If you use Excel you can upload your spreadsheet to Canvas. If you use Python and Jupyter, please save your notebook as an HTML file before upload it to Canvas.