Negative risk premia and AI companies

22 May 2024

One of my favorite books is Cochrane’s Asset Pricing. Chapter 1 of Cochrane outlines the consumption-based capital asset pricing model (CCAPM), which is a shockingly elegant account of asset prices in terms of individual consumption preferences.

When an investor purchases an asset, they are exchanging certain consumption in the present for a variable amount of consumption in the future. These quantities are related in the utility function

\[U(c_t, c_{t + 1}) = u(c_t) + \beta \mathbb{E}_t [u(c_{t+1})],\]

where

• \(u(c)\) is a concave utility-form that represents the utility of consumption \(c\);
• \(\beta\) is a subjective discount factor representing impatience;
• \(c_t\) is the amount of consumption done in the present;
• \(c_{t+1}\) is a random variable representing consumption in the future.

From this equation it’s clear that future consumption differs from present consumption in two ways: first, it is less desirable by \(\beta\) purely because it happens in the future rather than the present; second, it is adversely harmed by volatility in \(c_{t+1}\), because Jensen’s inequality holds for \(u\).

So far, all of this sounds like the standard explanation for why risky assets trade below their expected discounted value. However, that conclusion only follows if variance in the price of the asset is the only source of risk affecting \(c_{t + 1}\).

Suppose, on the other hand, that I am worried about my job being replaced by artificial intelligence. My job pays me an income \(e_t\). If some transformational scaling breakthrough happens between periods \(t\) and \(t+1\), I will be out of a job and my income \(e_{t + 1}\) will be greatly diminished. However, Nvidia stock will probably be up a lot (after all, fab capacity can only scale up so much between \(t\) and \(t + 1\)). If I forgo \(\xi\) units of consumption to invest in Nvidia, and the return on Nvidia stock in this period is \(x_{t+1}\), then we have

\[\begin{align} c_t &= e_t - \xi \\ c_{t + 1} &= e_{t + 1} + \xi x_{t+1}. \end{align}\]

But we’ve already established that \(\mathrm{Cov}(e_{t + 1}, x_{t + 1}) < 0\), so purchasing at least some small amount of Nvidia stock reduces the variance of future consumption.¹

The exact derivation here isn’t too important (you can find it all in Cochrane), but CCAPM would argue that because the price of assets is determined by the relationship between present and future consumption, an asset that is anticorrelated with the average investor’s future consumption should be priced above its expected discounted value.

This is maybe obviously true in the case of assets anticorrelated with the market portfolio (“puts are insurance, hence overpriced”). But the stock market is not the economy. Depending on your financial situation, it is very possible for an asset’s systematic risk to be highly correlated with the market while its idiosyncratic risk is even more highly anticorrelated with your personal consumption. In other words, a gain in the S&P could be more than offset by losing your job permanently.

If an asset is perceived by the majority of investors to have a negative consumption beta, it will command a negative risk premium. Its P/E ratio will be considered absurdly high, and it will seem to discount fantastical futures that never seem to materialize. No amount of short-term financial trouble will be able to dislodge this effect. The market will appear to be failing in its function as a long-run weighing machine.

But you wouldn’t estimate odds from insurance prices, would you?