Why an Equity Index Future Can Trade Below Its Expected Value
Summary
The document explains why an equity index futures price can be below the expected future level of the index. Under the stated setup, the expected index value reflects investors’ required expected return, while the futures price reflects the risk-free rate and dividends. With positive systematic risk and risk-averse investors, the required expected return exceeds the risk-free rate, so the expected index value is higher than the futures price.
A coin-flip example illustrates the risk premium: a risk-averse investor prefers a guaranteed payout to an equally valued uncertain payout and therefore requires extra expected compensation for risk. Equal expected returns could make a risk-neutral investor indifferent, while a genuinely risk-free index would behave like the bank account. The discussion also distinguishes expected return from recent market direction: a bear market or a run of unfavorable outcomes does not by itself establish that the expected return is below the risk-free rate. The claim depends on the assumptions of positive systematic risk and risk aversion.
Key ideas
- With positive systematic risk, risk-averse investors require an expected return above the risk-free rate.
- Under the stated pricing relations, that risk premium makes the expected future index value exceed its futures price.
- Risk-neutral investors can be indifferent between risky and risk-free assets with equal expected returns.
- A risk-free index would have a predictable future value and behave like the bank account.
- Recent bearish outcomes do not by themselves determine the asset’s expected return.
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Full text
# Future price of an index always less than expected index future value?
# Future price of an index always less than expected index future value?
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.
## Answer by Pontus Hultkrantz (score 3, accepted)
https://quant.stackexchange.com/a/70977
> 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
- Guaranteed payout of 1 USD. No risk.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.