# Risk, required return and expected volatility - what is the relationship?

Return required from risk averse agents from risky investments are proportional to expected return variance. That is from the textbook, you take the portfolio with the highest return to standard deviation, and then you lever or dilute it to fit the return to variance requirement of your investor.

Now, shouldn't you expect future volatility indicators like the VIX to have a quadratic relationship to stock index levels? A quick look at a chart suggests more of a linear relation. Or is it so that the stock-index has a linear relation to expected volatility, but that the VIX have a quadratic relation to longer term expected volatility?

Any references to literature or explanations on this would be more than helpful!

I think you may be interested in this QJE forthcoming article by Ian Martin. The key idea of the article (page 5) is that the expected return on the market can be decomposed as $$E_t[R_{t+1}]-R_f = \frac{1}{R_f}Var^Q(R_{t+1}) + \text{extra terms}$$ As you correctly pointed out the expected return should be related with the risk neutral variance. The issue with VIX$$^2$$ is that it doesn’t measure risk neutral variance, unless we are in the standard lognormal case. As you can see from page 15 the correct way to construct a variance swap is by using a portfolio with the same weights on all puts and calls while the VIX$$^2$$ has weights proportional to the inverse of the squared strike, i.e. $$SVIX^2=\frac{1}{(T-t)R_f^2}Var^Q(R_{t+1})=\frac{2}{S_t^2}\left[\int_0^F put(K)dK+\int_F^\infty call(K)dK\right]$$ $$VIX^2=\frac{2R_f}{T-t}\left[\int_0^F \frac{1}{K^2}put(K)dK+\int_F^\infty \frac{1}{K^2}call(K)dK\right]$$
• Very interesting read @fnic, thanks. When you say: "the correct way to construct a variance swap", well, it all depends on what you would like to swap I guess. $VIX^2$ reflects the (annualised) expected realised variance of future daily returns under the risk-neutral measure (assuming these have zero mean) $$VIX^2_{0 \to T} = \frac{1}{T}\Bbb{E}^{\Bbb{Q}}_0[\langle \ln S \rangle_T]$$ while $SVIX^2$ measures the (annualised) conditional variance of the future global return $$SVIX^2_{0 \to T} = \frac{1}{T}\Bbb{V}^{\Bbb{Q}}_0[\ln(S_T/S_0)]$$ If $(S_t)_t$ is lognormal these indeed do coincide. – Quantuple Nov 18 '16 at 13:26