This question is based on a claim made in both practice question 3.25 and section 3.5 of Options, Futures and Derivatives by John C. Hull, 11th edition.

The question

On July 1, an investor holds 50,000 shares of a certain stock. The market price is \$30 per share. The investor is interested in hedging against movements in the market over the next month and decides to use an index futures contract. The index futures price is currently 1,500 and one contract is for delivery of \$50 times the index. The beta of the stock is 1.3. What strategy should the investor follow? Under what circumstances will it be profitable?

The solution

A short position in $$1.3 \times \frac{50,000 \times 30}{50 \times 1,500} = 26$$ contracts is required. It will be profitable if the stock outperforms the market in the sense that its return is greater than that predicted by the capital asset pricing model.

The highlighted statement is the one which seems confusing to me. The CAPM predicts that the coefficient of regression between the percent change in the spot price of the index and the spot price of the stock is $1.3$. However, we cannot assume such a correlation between the futures price of the index. The assumption is that the futures price on the expiry date is equal to the spot price of the index on that date. Based on this, we can construct a counterexample in which the claim in the solution holds but the portfolio still does not book a profit.

Counterexample

Here is a carefully constructed counterexample where we make some assumptions on the spot price of the index as well where the stock outperforms the market, but the futures still result in an unprofitable hedge. Let us assume that the risk-free rate $R_F = 0$. Assume that the spot price of the index on July 1 is 1590 and it returns approximately 9.43%. Thus, the spot price one month later (on the date of expiry of the futures contract) is 1740. Based on the convergence assumption of spot price and futures price of the index, the futures price on the date of expiry should be 1740 as well. Let us assume now that the stock price goes up by 20% ($20\% / 9.43\% > \beta = 1.3$) and the spot or market price of the stock is \$36.

The payoff is thus the sum of the profit made by the appreciation of the stock and the loss made on the futures $$\text{Payoff} = 50,000 \times \left(\\\$36 - \\\$30\right) + 26 \times \\\$50 \times \left(1500 - 1740\right) = -\\\$12,000$$

Thus, over here, despite the portfolio outperforming the market, the investor makes a loss. If the trader had however shorted the index itself (by an ETF perhaps), the payoff would have been: $$\text{Payoff} = 50,000 \times \left(\\\$36 - \\\$30\right) + 26 \times \\\$50 \times \left(1590 - 1740\right) = $105,000$$ This leads me to believe that the claim made in the book holds only if the hedge is made using the index itself and not necessarily if index futures are used for hedging. In case index futures are used for hedging, one would need stronger assumptions on the correlation between the change in the futures price and spot price as well.

What exactly is the fault in my understanding here?

Quant

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