Investors' beautiful equation

Chris Dillow

Chris Dillow

One question Google asks its job applicants is: “What is the most beautiful equation you have seen?” If I were applying, which I’m not, I’d reply with the consumption asset pricing equation, as discussed, for example, by John Cochrane:

ER = R x SDa x SDc x corr (a,c)

This says that the excess expected return over cash of any asset should equal the product of just four things:

- A measure of risk aversion, R. The more we hate risk - other things equal - the higher should be the expected returns needed to compensate us for it.

- The standard deviation of the asset, SDa. The riskier an asset, the higher must be its returns.

- The standard deviation of our consumption growth, SDc. If there’s a risk that our consumption will fall a lot – say because we lose our jobs or business – then we can’t afford to take on so much risk with our other assets. We thus need higher returns on them to compensate for taking their risks.

- The correlation between consumption and the asset’s returns. If an asset is likely to do badly when our consumption falls because we have hit hard times, we need higher returns on it to compensate for the risk that our wealth will fall when we most need it.

For a long time, this equation did not fit the facts of the stock market; until the mid-80s, equity returns were higher than the equation predicted. However, since then they have been more consistent with it.

The equation also fits other facts:

1. Why have house prices risen so much over the long-run? It’s because they are risky, in the sense that corr (a, c) is high; house prices fall in bad times so must offer high returns in average times.

2. Why is insurance expensive? It’s because corr(a,c) is negative: insurance policies pay out in bad times, so we tolerate an expected loss on them in normal times. This principle generalizes. If you think gold or index-linked gilts offer insurance against bad times, you should not at the same time expect good returns on them unless you think that investors are mistakenly under-pricing the assets. And you need a good reason to think everyone is wrong except you.

3. Why, for a long time, did small stocks and high yielding ones do well? It’s because they are more “cyclical” than others and are more likely to fall in recessions when consumer spending is falling – in other words, because corr(a, c) is higher than for other shares.

However, we shouldn’t just think of this equation as a description. We can also use it normatively, to tell us how we should invest.

One thing this equation says is that our demand for risky assets should not depend solely upon our appetite for risk, R. Two people might have the same risk aversion but very different weightings in risky assets if one faces lots of consumption risk (high SDc) and the other little. You should, therefore, consider your consumption risk when investing.

In particular, you should ask of any asset: how likely is it to do badly if my other sources of income do badly? I suspect that esoteric assets such as wine, stamps or classic cars should offer higher returns, because they might do badly in bad times. Such thinking also tells us that it is dangerous to hold shares in the firm you work for. Such shares have a high corr(c, a), but they don’t offer higher returns to you than to anyone else.

Now, you might think you don’t have data on your consumption volatility (or, more generally, your chances of hitting hard times) or on the correlation between an asset’s return and your consumption. True, but unimportant. Equations are not simply about hard numbers – many of which offer only pseudo-precision anyway. They are ways of organizing our thinking – of telling us what matters, and what questions to ask. And this is why this equation is so beautiful.

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