The Dynamic Price of Market Risk

Are we testing CAPM right?

In order to compute the dynamic or conditionally expected excess return on a bunch of assets, we need two ingredients. First, we need a risk-free rate. The standard practice is to take the yield prevailing at the short end of the yield curve as the risk-free rate of return. I am persuaded by Howell that, because duration risk assets like the 10-year note can serve as collateral in the wholesale funding flywheel with minimal haircuts and high rehypothecation rates, all obligations of the US taxpayers, including agency-MBS, should be considered safe assets. More precisely, I am convinced that the diagnostic criterion for a safe asset is that its price goes up in a market panic. This is true at least of both the 10-year note and the 10-year bund. So, duration risk notwithstanding, for pricing US equities, we’ll compute returns in excess of the yield on the 10-year note.

Second, we need return-forecasting factors or price-of-risk variables. These are variables that contain predictive information about future returns. They are not supposed to exist in the efficient markets paradigm. However, we have documented return predictability. Specifically, we have documented that cross-border banking flows predict dollar strength. In turn, dollar strength is known to be associated with tighter financial conditions, including wider risk spreads. The credit spread is in turn known to be a good predictor of the excess returns on corporate bonds. Finally, we have seen that the proportion of assets allocated to equities predicts next quarter’s market return. Note that half of directly and indirectly owned equities are owned by the 1 percent. And most of that money is managed by institutional investors. The marginal investor in this frame is a large institutional investor interested in getting more or less exposure to risk assets.

We have also noted that the standard capital asset pricing model (CAPM) is an empirical catastrophe. Could this cross-sectional pattern be due to some hidden but stupendous measurement error? Specifically, by assuming that the expected excess return is constant over time when it is, in fact, time-varying? In other words, that the signal in the location parameter (the risk premium) may be lost. Maybe the location of the distribution is not identified, but the variation in the series still contains valuable information? In order to estimate the dynamic risk premium, ie the expected excess return at time t+1 conditional on information at time t, we need return-forecasting factors. Here we take them to be first-differenced equity share, first-differenced credit spread obtained from Fred, and log-transformed and first-differenced global liquidity (‘cross-border claims denominated in all currencies plus local claims denominated in foreign currencies’) obtained from BIS. Our features are dictated by the predictive information contained in the diachronic variation in these series. These are our price-of-risk factors that will span the price of systematic risk.

We shall take our risk factor to be log returns on the market portfolio. So systematic risk is assumed to be the same as that in the CAPM. We show how our features allow us to extract the price of market risk. The price of risk is the dual of the risk. It is proportional to effective risk aversion — when risk appetite is high, risk premia collapse; when risk appetite collapses, risk premia get wider. Whatever our risk factor, the price of risk is going to be spanned by the price-of-risk factors. All the diachronic variation, in particular, is obtained from the return-forecasting factors. We believe that our features contain predictive information about future returns on the systematic factor. The bulk of the weight-lifting is done by investor risk appetite as measured by their equity allocation share. We use a simplified version of the Adrian et al. 3-pass estimator using Numpy matrix operations. EIV-corrected, heteroskedasticity robust standard errors will take a bit more coding. But the central estimates are worth reporting as of writing. They contain economically intuitive information.

Note that there is concern about a regime shift in the 1990s. We shall restrict our attention to three samples beginning in 1986, 1996, and 2009.

In order to compute the market risk premium, in the first pass, we project the returns of 100 stock portfolios sorted on size and value that we obtain from Kenneth French’s website on: lagged values of the price-of-risk factors and contemporaneous values of the market risk factor in the time-series. In the second pass, we project the betas (the slope of the risk factor in the first pass) onto the alphas (the intercepts of the first pass) and the slopes of the lagged price-of-risk factors in the cross-section of the hundred portfolios to obtain the loadings — the price-of-risk hyperparameters. The dynamic price of market risk is the linear combination of the price-of-risk series given by the loadings.

We find that there is a structural break in the 1990s when we examine the price of market risk over 1986-2020 — as suggested by the structural break in our main price-of-risk feature. The local minimum before the late-1990s is attained in 1987Q3.

We restrict attention to 1996-2020. We can see that the price of market risk compressed significantly in multiple quarters in the lead up to the 2000 and 2008 market crises. The price of systematic risk was high in the early 2000s; collapsed in the mid-2000s’ financial boom; expanded after the 2008 crisis; with two more shocks during the eurozone crisis in 2010-2012; before getting suppressed during 2012-2016, with the exception of the China panic in 2015; collapsed again with the so-called Trump reflation trade in 2016-2017, before reviving for a brief panic in the last quarter of 2018; and then, of course, the coronapanic in 2020Q1, before the Fed pulled out the bazooka and killed the risk premium. This graph contains an astonishing amount of information on systematic risk.

It is possible that there have been institutional changes since the GFC that have changed the data generating process. Actually, sufficient time has now elapsed since the deluge that we can restrict the sample to 2009-2020 and still have 47 observations in the time-series. We can still see the Greek drama in 2010-2012, the jitters in 2015, the Trump reflation trade in 2016-2018, the risk-off in 2018Q4, and, of course, the pandemic panic in 2020Q1 before the Fed killed it and trigged a risk-on.

So the CAPM problem can be attacked if you have superior technology in the form of return-forecasting factors that contain a lot of predictive information on the systematic factor. The main one is that identified by Howell — equity allocation. It may be possible after all to be more informed about the systematic factor than the dealer.