Consider a non-dividend paying stock with current price $359.41. The 1-year spot rate is 1.7% and a futures contract on the stock with maturity 1 year is trading at $376.49. Suppose you borrow so that you can buy 970 shares of the stock. Moreover, you sell 970 futures contracts. The payoff of your position at maturity is

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BlackTom AI · Accepted Answer

Question restatement:
- A non-dividend paying stock is currently priced at 359.41
- 1-year spot rate r = 1.7%
- A 1-year futures contract on the stock trades at 376.49
- You borrow to buy 970 shares and simultaneously sell 970 futures contracts
- The task is to determine the payoff of this position at maturity.

Step-by-step analysis of the setup and payoff components:
1) Financing the stock purchase:
- Cost to buy 970 shares = 970 × 359.41 = 348,627.70
- You borrow this amount at the risk-free rate r = 1.7% for 1 year. At maturity, the loan repayment is:
  348,627.70 × (1 + 0.017) = 348,627.70 + 5,926.67 = 354,554.37
  (the small interest component is 348,627.70 × 0.017 = 5,926.67).

2) Position in futures:
- You are short 970 futures contracts. The futures payoff at maturity for a short position is: 970 × (F0 − S1), where F0 is the futures price set at initiation (376.49) and S1 is the stock price at maturity.
- Thus, futures payoff = 970 × (376.49 − S1).

3) Position in the stock itself at maturity:
- The 970 shares you own will be worth 970 × S1 at maturity.

4) Aggregate payoff at maturity:
- Stock payoff: 970 × S1
- Financing payoff (loan repayment): − 354,554.37 (you owe this amount)
- Futures payoff: + 970 × (376.49 − S1)
- Total payoff = (970 × S1) + (− 354,554.37) + (970 × 376.49) + (− 970 × S1)
- The 970 × S1 terms cancel out: 970 × S1 − 970 × S1 = 0
- What remains is: − 354,554.37 + 970 × 376.49

5) Compute 970 × 376.49:
- 376.49 × 970 = 376.49 × (1000 − 30) = 376,490 − 11,294.70 = 365,195.30

6) Final payoff:
- Payoff = 365,195.30 − 354,554.37 = 10,640.93

Notes on the result:
- The payoff is purely determined by the difference between the future value of the borrowed amount and the futures price, after canceling out the stock price exposure due to the offsetting positions in stock and futures.
- This computation assumes no dividends, no transaction costs, and that funding is exactly at the given 1-year spot rate.

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