Let’s measure short run returns from holding equities.

On a monthly basis,

Define:

equity return = change in capital value over the month + (annual) dividend yield/12

The risk-free return = 3 month Treasury yield/12 — a proxy for the 1 month bill, for which I don’t have a long enough time series.

The excess real gain is the difference between the equity return and the risk-free return, adjusted for the month-on-month change in CPI.

Then take the 10 year (trailing) average of the excess gain, compared with the 10 year trailing average of the standard deviation of the excess gain:

The first thing to note is that the excess gain averages about 7% (annualized). More interestingly, as the standard deviation falls, the excess gain *increases, *and as the standard deviation increases, the excess gain falls.

Theory would say that the excess gain is compensation for volatility, so that a higher standard deviation would lead to a higher excess gain.

But as standard deviation declines, households bid up equities causing the real gain to increase, not decrease. In an timeless model, this adjustment would happen instantly, so the gains going forward would be lower. But say we are in a sticky expectations model, in which the adjustment happens gradually. This type of friction reverses the conclusion of the model — now falling standard deviations are associated with increasing equity returns, and vice-versa.

This is another example of the (excellent) point Nick Rowe made about robustness of models. If adding a small friction reverses the conclusion of the model, then you cannot take the frictionless model seriously. If, in our world, investors only gradually come to expect falling volatility, then the equity premium becomes contravariant with volatility.

In that case, the system might have a locally self-destabilizing property, in the sense that periods of falling volatility lead to periods of increasing return on equities, which become transformed into increasing cost of equity, which ultimately results in an excessively high cost of equity, which when not met, leads to falling capital values and increasing volatility. The only friction needed is sticky expectations.

Update: fixed “covariant” to “contravariant” 🙂

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