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!