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In the following practice problem:

Is the futures price of a stock index greater than or less than the expected future value of the index? Explain your answer.

The answer given is as follows:

The futures price of a stock index is always less than the expected future value of the index. This follows from Section 5.14 and the fact that the index has positive systematic risk. For an alternative argument, let $\mu$ be the expected return required by investors on the index so that $E(S_T) = S_0e^{(\mu-q)T}$. Because $\mu > r$ and $F_0 = S_0e^{(r-q)T}$, it follows that $E(S_T) > F_0$.

Is the reason that $\mu > r$ because by definition $r$ is the rate of return least risky investment, and therefore to entice investors to invest in something more risky (a stock index), $u$ must be strictly greater? Why can't it be the case that $\mu = r$ (a hypothetical "risk-free index"), or even in the case of a bear market, $\mu < r$? I feel like I'm thinking about this wrong.

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  • $\begingroup$ You gave yourself the answer, r is risk-free rate of return. $\mu < r$ denotes an investment with negative risk. $\endgroup$
    – hroptatyr
    May 13, 2022 at 6:38

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Is the reason that μ>r because by definition r is the rate of return least risky investment, and therefore to entice investors to invest in something more risky (a stock index), u must be strictly greater?

Strictly greater because humans are risk averse by nature, they don't like uncertainty. They would prefer the bank account over buying a security with the same expected return but with uncertainty.

Consider two choices

  1. Guaranteed payout of 1 USD. No risk.
  2. Flip a coin with expected payout 1 USD (2 USD if heads else zero). Risky.

The investor will always choose 1. over 2, unless they get an extra incentive to choose 2 by increasing the expected payout.

Why can't it be the case that μ=r (a hypothetical "risk-free index")

Only if investors are risk neutral are they indifferent between a risk free and a risky investment sharing the same expected return. In reality humans are risk averse.

If the index is risk free then yes, but then it is the same as the bank account. Risk free means that you can predict the future value perfectly.

the case of a bear market, μ<r?

The direction of the market has nothing to do with the expected return, only with the uncertain part. If you flip the coin and get 3 tails in a row these outcomes don't affect the inherent expected value of flipping the coin.

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    $\begingroup$ Fully agree with the answer, just wanted to point out that the question and answer have been interpreted in the ‘real world’ probability measure. To contrast, the expectation of future index is equal to the futures price when we are in the risk neutral money market probability measure. $\endgroup$
    – dm63
    May 21, 2022 at 12:59
  • $\begingroup$ @dm63, right. Futures price equals forward price if constant risk free rate, but if stochastic rate and dependency to stock index price then probably no. $\endgroup$ May 21, 2022 at 13:35

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