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# Finance Questions: Stock Prices, Gold Value

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A stock is expected to pay a dividend of \$2 per share in one month and in four months. The stock price is \$40 and the risk-free rate of interest is 6% per annum with continuous compounding for all maturities. An investor has just taken a short position in a five-month forward contract on the stock.
What are the forward price and the initial value of the forward contract?
Three months later, the price of the stock is \$48 and the risk-free rate of interest is still 6% per annum. What are the forward price and the value of the short position in the forward contract?
Make sure that you show or explain all calculations. Make sure you answer all questions above.

Suppose the current spot price for gold is \$800 per ounce. The risk-free interest rate available to all investors for borrowing or lending is 0.50% per month (monthly compounding). Forward contracts are available to buy or sell gold for delivery in 1 year; the forward price for gold is \$890 per ounce. You have a large inventory of gold.
Assume that storage costs for gold are zero. Is there an arbitrage opportunity? If you answer "YES," then show step by step how you would make a profit and calculate the profit per ounce of gold. If you answer "NO," then show why there is no arbitrage opportunity.

Now assume that the present value of the storage cost for gold is \$100 per ounce for one year of storage. Is there an arbitrage opportunity? If you answer "YES," then show step by step how you would make a profit and calculate the profit per ounce of gold. If you answer "NO," then show why there is no arbitrage opportunity.
Make sure that you show or explain all calculations. Make sure you answer all questions above.

#### Solution Preview

A stock is expected to pay a dividend of \$2 per share in one month and in four months. The stock price is \$40 and the risk-free rate of interest is 6% per annum with continuous compounding for all maturities. An investor has just taken a short position in a five-month forward contract on the stock.
What are the forward price and the initial value of the forward contract?
The present value of the dividends is

I = \$2e^(−0.06×1/12) + \$2e^(−0.06×4/12) = \$1.9900+\$1.9604 = \$3.9504
The forward price is therefore
F0 = (40 − 3.9504)e(0.06×5/12) = \$47.22
The value at origination of a forward contract is zero.
Three months later, the ...

\$2.19

## 19 Questions on Derivatives: beta of a hedge fund, put option, CAPM, Capital Asset Pricing Model, investors, risk averse, portfolios, mean, variance, highest return portfolio, efficient frontier, abnormal returns, equity mutual funds, riskless return, options position, time to maturity, interest rates, convexity of a puttable bond, futures contracts, gold, initial margin, margin calls, volatility, Palm, 3COM, financial markets, beta calculation, value of a put option, exercise price, maturity, arbitrage opportunity, riskless return, call, 6-month forward rate, Canadian dollars, US dollars, implied volatility, historical volatility, SPX, floating to fixed swap, LIBOR, swap counterparty

For each of the following 8 statements determine if it is true, false or uncertain. You must justify your answer with a one-sentence explanation.

1. The beta of a hedge fund is usually close to one.

2. Suppose you hold a share of stock and a put option on that share. If the stock price is below the exercise price when the option expires the value of your position is the value of the stock.

3. There is a much greater outstanding volume of futures contracts as opposed to forward contracts because futures contracts are much more flexible in terms of settlement dates, amounts, etc.

4. The CAPM (Capital Asset Pricing Model) would be an accurate way to estimate the cost of equity for a company like Federal Express.

5. Assume that all investors are risk averse and select portfolios based on mean and variance. Then the highest return portfolio is always on the efficient frontier.

6. It is easier to earn "abnormal returns," i.e., returns greater than what investments of similar risk earn, by investing in financial rather than real assets.

7. If the price of a stock has gone up for ten consecutive trading days then on the eleventh day we would expect its price to decline.

Assume that we have the following monthly return data on 2 U.S. equity mutual funds. Assume the riskless return is 0.4% per month and the SPX return was 1.4% and its standard deviation was 3.0% per month over this period. You must show your computation to get any credit for your answer.

Fund A Fund B
Realized return 0.012 0.015
Standard deviation of return 0.06 0.08
Beta 1.5 2

8. Give an example of an options position that increases in value as the time to maturity decreases.

9. If interest rates rise, the convexity of a puttable bond will increase ______ an otherwise equivalent non-puttable bond.
a. more than
b. the same as
c. less than
d. actually the convexity will decrease

10. Assume you are short 2 futures contracts for 100 ounces of gold with initial margin of \$2,000 each; maintenance margin is \$1,500 per contract. Assume the price is initially \$500 per ounce. If the price at the end of day 1 is \$495 and \$493 on day 2, how much money do you have in your margin account (assuming no withdrawals and assuming you make all margin calls) at the start of day 3?

11. Which of the asset A or B has the higher volatility? Both calls have one year to expiration and the riskless rate is 3%.

Option Exercise price Stock price Option price
Call on asset A 95 100 13.7
Call on asset B 60 50 6.10

12. Briefly explain why you think the Palm/3COM story is relevant to the study of financial markets.

13. The following diagram shows the beta calculation for a given firm using weekly data. What can you say about its performance over the time period represented by these data? Why?

The diagram below plots the value of a put option with an exercise price of \$100 and one year to maturity.
14. Draw the value of a put option on the same security as in the previous question with an exercise price of \$100 and 6 months to maturity.

For each of the following 3 questions determine whether or not an arbitrage opportunity exists. If one exists, describe how you could exploit this arbitrage. All options are written on the same underlying stock, which has a current price of \$100 with annual standard deviation of return = .3. The riskless return is 4%.

15. Call with exercise price of 100 costs \$20; Call with exercise price of 110 costs \$13; Call with exercise price of 120 costs \$5. All three options have a one-year maturity.

16. Call with a one-year maturity and an exercise price of 100 costs \$20; put with a 2-year maturity and an exercise price of 100 costs \$16.

17. The current one-year riskless rate in the U.S. is .05 and the current one-year riskless rate in Canada is .04. The spot rate shows that a Canadian dollar is 0.846 US\$. The 6-month forward rate for converting Canadian to US dollars is 0.88.

18. Data have shown that the implied volatility of SPX puts is higher than the historical volatility of the SPX. How could you take advantage of this phenomenon?

19. Suppose you enter into a floating to fixed swap (i.e., you want to have certainty in your interest payments). The current swap rate for a 3-year maturity is LIBOR for 4.9% fixed. Assume that you are borrowing \$10 million at LIBOR. Assume the swap payments are made annually. If at the end of year 1 LIBOR is 4.0%, how much do you pay to (or receive from) the swap counterparty?

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