A Technical Note on Volatility Feedback

A note for those interested in the relationship between expected risk and expected return. This relationship makes itself felt for instance in the negative correlation between moves in index-option volatility and the index-value itself, and is sometimes referred to as volatility feedback. Simply put: the reward for risk should be proportional to return variance.

Motivation: consider a two-year period during which the index-return has had an annual volatility of let's say 10%, which corresponds to an annual variance of 1%. Suppose the risk-premia has been 1% annually. In total, a risk-premia of 2% should be required for a total index-return variance of 2% and a standard deviation of 14%. Now look at another two-year period with the same risk and return, only that the first year has a known return of 0%. The second year should hence have a variance of 2%, a standard deviation of 14% and a required risk-premium of 2% to make the two-year characteristics equal the first example.

In all risky periods, the ratio risk-premia to variance is constant (=1) in the example. Note that the Sharpe ratio, risk-premia to standard deviation, is higher in the period with the highest risk.

Update/Implication:

The decrease of short-term implied volatility as indicated by the "VIX"-index from 40ish to slightly about 10% from the Enron-crisis up to today may look like risk having decreased to a 4:th. However, with index risk-premia proportional to variance, they should rather be viewd as having decreased to a 16:th. In other words, the stock market has already collected most of the gains from risk-reduction since Enron fell. Don't expect any good stock-index returns before risk expectations have found reason to rise again.