Also, my current understanding is that risk compensation is definitely not the large majority of investment returns. The last chapter of Cochrane’s Asset Pricing text has a great discussion of the topic. The main conclusion is that explaining returns via risk exposure requires unrealistically high levels of risk aversion—like, one or two orders of magnitude above the risk aversion levels implied by other activities.
What’s the competing explanation?
Haven’t looked at the historical numbers, but in recent times it seems like (i) with log utility and a naive model of “future=past,” optimal leverage is around 2x, (ii) most investors are much more risk averse than log utility (even for casino risk). So it seems like things basically add up here for most of the market. Was the situation an order of magnitude different in the past?
Cochrane mainly talks about this in the context of the equity premium. His main answer is “we don’t know why there’s an equity premium, we’ve tried the obvious risk-aversion models and they don’t make sense.”
The key issue is not just “most investors are much more risk averse than log utility”, but how much more risk averse exactly. Cochrane tries to back out the curvature of the utility function (measured as γ=−cd2udc2/dudc, where c is consumption) based on observed market parameters, and he shows that γ needs to be around 50. For sense of scale, log utility would imply γ=1, and γ in the range of 1 to 5 is typical in theoretical models—that’s the sort of risk aversion you’d expect to see e.g. in a casino.γ=50 would imply some bizarre things—for example, assuming real consumption growth of around 1% annually with 1% std dev, the risk free rate should be around 40%. (Cochrane has a bunch more discussion of weird things implied by very high risk aversion, and looks at some variations of the basic model as well. I don’t know it well enough to expound on the details.)
Personally, I suspect that the “true” answer to the problem is some combination of:
Despite using the words “log utility”, most of these are actually second-order expansion models which don’t account for the tail behavior or details of “bankruptcy” (i.e. margin calls).
Most of these models ignore the Volker fence and functionally-similar reserve requirements on banks—factors which we would expect to dramatically lower the rates on bonds and other low-reserve-requirement assets relative to stocks.
… but I haven’t gotten around to building and solving models for these yet; my interest is more on the market microstructure end of things.
What’s the competing explanation?
Haven’t looked at the historical numbers, but in recent times it seems like (i) with log utility and a naive model of “future=past,” optimal leverage is around 2x, (ii) most investors are much more risk averse than log utility (even for casino risk). So it seems like things basically add up here for most of the market. Was the situation an order of magnitude different in the past?
Cochrane mainly talks about this in the context of the equity premium. His main answer is “we don’t know why there’s an equity premium, we’ve tried the obvious risk-aversion models and they don’t make sense.”
The key issue is not just “most investors are much more risk averse than log utility”, but how much more risk averse exactly. Cochrane tries to back out the curvature of the utility function (measured as γ=−cd2udc2/dudc, where c is consumption) based on observed market parameters, and he shows that γ needs to be around 50. For sense of scale, log utility would imply γ=1, and γ in the range of 1 to 5 is typical in theoretical models—that’s the sort of risk aversion you’d expect to see e.g. in a casino.γ=50 would imply some bizarre things—for example, assuming real consumption growth of around 1% annually with 1% std dev, the risk free rate should be around 40%. (Cochrane has a bunch more discussion of weird things implied by very high risk aversion, and looks at some variations of the basic model as well. I don’t know it well enough to expound on the details.)
Personally, I suspect that the “true” answer to the problem is some combination of:
Despite using the words “log utility”, most of these are actually second-order expansion models which don’t account for the tail behavior or details of “bankruptcy” (i.e. margin calls).
Most of these models ignore the Volker fence and functionally-similar reserve requirements on banks—factors which we would expect to dramatically lower the rates on bonds and other low-reserve-requirement assets relative to stocks.
… but I haven’t gotten around to building and solving models for these yet; my interest is more on the market microstructure end of things.