Consider a dividend paying stock whose current price is $49.02. The stock will pay a dividend of $2 per share in 1 year. The 1-year spot rate is 4% (effective annual rate) and a futures contract on the stock that matures in 1 year (right after the stock pays its dividend) is trading at $50.96.

Suppose you buy 500 shares of the stock using only borrowed money. Then, you sell 500 futures contracts (each contract is on one share of the stock).

What is the closest value to the payoff of your position at maturity?

Round your answer to two decimal places. If your answer is "123.4567", enter it as 123.46.

Question

Consider a dividend paying stock whose current price is $49.02. The stock will pay a dividend of $2 per share in 1 year. The 1-year spot rate is 4% (effective annual rate) and a futures contract on the stock that matures in 1 year (right after the stock pays its dividend) is trading at $50.96.

Suppose you buy 500 shares of the stock using only borrowed money. Then, you sell 500 futures contracts (each contract is on one share of the stock).

What is the closest value to the payoff of your position at maturity?

Round your answer to two decimal places. If your answer is "123.4567", enter it as 123.46.

BlackTom AI · Accepted Answer

We need to analyze the cash flows and holdings at maturity for the described strategy.
First, note the initial setup: you buy 500 shares of the stock at 49.02 each, financed entirely by borrowing. Then you enter a short position on 500 futures contracts, each on one share, with a futures price of 50.96.
At maturity (1 year), several events occur in sequence:
- The stock pays a dividend of 2 per share just after the futures contract matures.
- The futures contracts settle. Since you are short futures, the payoff per share is (F - S_T), where F is the initial futures price (50.96) and S_T is the stock price at maturity. Importantly, you still hold the 500 shares of stock when the futures settle.
- You repay the borrowed funds (the loan used to finance the stock purchase) with interest at the 1-year spot rate of 4%. The loan principal is 49.02 × 500 = 24,510, and at maturity you owe 24,510 × (1 + 0.04) = 25,490.4.
Next, compute the cash flows at maturity:
- Dividend: you receive 2 per share on 500 shares = 1,000 in cash.
- Futures payoff: you are short 500 contracts, so total payoff from futures is 500 × (50.96 − S_T). This cash is realized regardless of the stock’s final price because the futures settle to cash.
- Stock value: you still own 500 shares, worth S_T in total at maturity.
Therefore, the combined value of your positions at maturity before loan repayment is 500×S_T (stock value) + 500×(50.96 − S_T) (futures payoff) + 1,000 (dividend) = 500×50.96 + 1,000 = 25,480 + 1,000 = 26,480, a constant amount independent of S_T.
Subtract the loan repayment of 25,490.4 to obtain your net payoff: 26,480 − 25,490.4 = 989.6.
Thus, the payoff of the position at maturity is 989.60, rounded to two decimals as required.
In summary, the key idea is that the stock position plus the short futures produce a constant combined payoff equal to 500×F = 25,480, and adding the dividend raises the total to 26,480. After repaying the loan (25,490.4), the net payoff comes out to 989.60.

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