Consider a dividend-paying stock currently priced at $49.02. The stock will pay a $2 dividend per share in one year. The 1-year spot rate is 4%, and a 1-year futures contract on the stock, maturing immediately after the dividend is paid, is trading at $50.96.

Suppose you borrow funds to buy 500 shares of the stock and short 500 one-year futures contracts on the stock.

What is the closest value to the payoff of your portfolio at maturity (combining all positions)?

Enter your final answer rounded to two decimal places. For example, enter 1.23 if your answer is $1.234, and enter -1.23 if your answer is -$1.234.

Question

Consider a dividend-paying stock currently priced at $49.02. The stock will pay a $2 dividend per share in one year. The 1-year spot rate is 4%, and a 1-year futures contract on the stock, maturing immediately after the dividend is paid, is trading at $50.96.

Suppose you borrow funds to buy 500 shares of the stock and short 500 one-year futures contracts on the stock.

What is the closest value to the payoff of your portfolio at maturity (combining all positions)?

Enter your final answer rounded to two decimal places. For example, enter 1.23 if your answer is $1.234, and enter -1.23 if your answer is -$1.234.

BlackTom AI · Accepted Answer

Step 1: Restate the key financial details. You buy 500 shares at 49.02 per share using borrowed funds, and you short 500 futures contracts maturing just after the dividend is paid. The stock pays a dividend of 2 per share in one year. The 1-year spot rate is 4%, and the 1-year futures price is 50.96. You need the payoff of the entire portfolio at maturity.

Step 2: Compute the amount borrowed and the repayment at maturity. The cost to purchase 500 shares is 500 × 49.02 = 24,510. This is financed with a loan that must be repaid with interest after one year: 24,510 × (1 + 0.04) = 25,490.40.

Step 3: Identify the cash flows at maturity from the stock position and the dividend. At t = 1, you own 500 shares, so you have the stock value 500 × S1. You also receive a total dividend of 500 × 2 = 1,000 at t = 1.

Step 4: Determine the payoff from the 500 short futures contracts. Short futures payoff is (F0 − S1) per future contract, so for 500 contracts the payoff is 500 × (50.96 − S1) = 25,480 − 500 × S1.

Step 5: Combine all components to get the total payoff at maturity. Sum stock value, dividend, and futures payoff, then subtract the loan repayment:
- Stock value: 500 × S1
- Futures payoff: 25,480 − 500 × S1
- Dividend: 1,000
- Loan repayment: 25,490.40
Total payoff = (500 × S1) + (25,480 − 500 × S1) + 1,000 − 25,490.40

Step 6: Observe that the S1 terms cancel out: 500 × S1 − 500 × S1 = 0. What remains is 25,480 + 1,000 − 25,490.40.

Step 7: Perform the arithmetic: 25,480 + 1,000 = 26,480. Then 26,480 − 25,490.40 = 989.60.

Step 8: Note the result is independent of the unknown future stock price S1, meaning the payoff is pinned down by the initial prices, dividend, and the futures price via the replication setup.

Answer (rounded to two decimals): 989.60

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