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Practice Problem Set #1 Note: This problem set is for you to review the content. Please do not submit it, and it will not be graded. 1. Given a stock portfolio with an expected monthly return of 1% and T-bills yielding a 0.5% monthly risk-free rate (with a standard deviation of 0): i. Calculate the continuously compounded monthly rates for both the stock portfolio and the T-bills. (Hint: 𝑒 𝑟 𝑐𝑐 = 1 + 𝑟 ) ii. If the continuously compounded return for the stock portfolio has a standard deviation of 4.4928%. Determine the monthly risk premium (excess return) and the corresponding Sharpe ratio (excess return per unit of risk)? iii. Assuming the continuously compounded excess return is normally distributed, what is the probability of the stock portfolio’s performance is below that of the T-bills (i.e., has a negative excess return)? (Hint: You can use the function NORMSDIST in Excel) iv. Over a 300-month horizon, what are the expected value and standard deviation of the excess return? v. Also, for the 300-month horizon, calculate the probability that the stock portfolio underperforms the T-bills. vi. Compare the probabilities in (iii) and (v). What can you infer from the comparison? Does it suggest that stocks are safe in the long run? Please explain your reasoning. 2. Assuming an annual management fee of 2% for an actively managed fund and 0.1% for a passively managed fund, with both yielding a 7% annual return, how much would a $1,000 investment yield after 30 years in each fund? 3. Alice decides to sell a 3-month futures contract for gold, where each contract represents 100 ounces of gold. The agreed futures price at the time of initiation is $1800 per ounce. The futures exchange uses daily mark-to-market for settling gains and losses. The initial margin is 10%, and the maintenance margin is 75% of the initial. Compute the initial and maintenance margin amount required and complete the following table. Time Futures price Daily gain Account balance Margin call Initial $1800 - - Day 1 $1810 Day 2 $1795 Day 3 $1805 Day 4 $1799 Day 5 $1815 Day 6 $1850 Day 7 $1790
4. Suppose you believe stock XYZ will increase in value from its current level of $100. However, you know your analysis could be incorrect, and it could also fall in price. Suppose a 6-month maturity call option with exercise price $100 currently sells for $10, and the interest rate for the period is 3%. Consider the following three strategies for investing $10,000: Strategy A: Invest entirely in stock XYZ, i.e., buy 100 shares of XYZ, each selling for $100. Strategy B: Invest entirely in the call option. Buy 1,000 calls, each selling for $10. (This would require 10 contracts, each for 100 shares.) Strategy C: Purchase 100 call options for $1,000. Invest the remaining $9,000 in 6-month T-bills, to earn 3% interest. The bills will grow in value from $9,000 to $9,000 × 1.03. Compute the payoffs and rates of return of each strategy in 6 months and draw a diagram for the rate of return for the following possible prices of stock XYZ: $90, $95, $100, $105, $110, $115, and $120. Compare the features of the three strategies. 5. John thinks ABC Corporation’s stock price will stay within a specific range and wants to profit using a butterfly spread. A butterfly spread is an options strategy that involves using multiple options contracts to create a range of prices at which the strategy can profit. One way to create a butterfly spread is to buy one in-the-money call, one out-of-the-money call, and sell two at-the- money calls. Suppose the current stock price of ABC Corporation is $50. The available call option and the premiums are: 1. Call option with a strike price of $45, with a premium of $6.50. 2. Call option with a strike price of $50, with a premium of $3.50. 3. Call option with a strike price of $55, with a premium of $1.50. Construct a butterfly spread for John using these options, and determine the net premium, maximum profit, maximum loss, and break-even points. Draw a diagram of the profit based on the stock price at maturity.
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