Here’s a thought experiment: suppose you own some claim to a company’s payouts that is solely a function of how much your company produces: if the company produces more, you get more. Suppose your brother owns some other claim on the same company: if the company produces more, he also gets more. If on average, your payout is higher when the market does well, is the same true of your brother’s payout?
Miraculously, the answer is no. It is entirely possible that even with claims defined solely on a company’s profits and increasing in those profits, one claimaint’s payouts can be positively correlated with the market while another’s can be negatively correlated. In economic terms, this amounts to saying debt betas and equity betas can have opposite signs. In mathematical terms, this amounts to saying that if you have two random variables, X and Y, and some weakly increasing functions f(.) and g(.), then the sign of Cov(g(X),Y) could be different from the sign of Cov(f(X),Y).
To me at least, this is extremely counterintuitive. It says that my brother and I could agree unconditionally that we prefer that our company go up rather than go down, but disagree on whether we benefit when the market — which only affects our payouts insofar as it affects our company — goes up.
To see what this looks like, it will be useful to simulate some data. Consider a company that produces apples and whose production bears some correlation with the production of the overall market. (We can think of production here as isomorphic to returns). Suppose for each year, we plotted the apples produced by the company against the production of the market. In the plot below, each circular point represents a year, with the x-coordinate as the market’s production that year and the y-coordinate as the firm’s production.
Clearly, market production is on average positively correlated with firm production. The dotted navy line, which slopes upward, captures this positive correlation.
Now consider the payoffs to different claimants. Let my brother be a debt-holder in the company, who owns debt with face value of 3; that is, he earns the minimum of the promised amount (3) and what the firm produces in a given year. Moreover, let me be the residual (equity) claimant, with payoffs of max{X-F,0}, where X is my firm’s production. Our payouts as a function of firm production look as follows
Observe that both are (weakly) increasing in the payouts of the company. That is, higher production by the firm leads each claimant to be at least as well off. The interesting part occurs where we combine these graphs and plot the payouts of each claimant against the market’s production:
Now the payouts to debt are negatively correlated with the market while the payouts to equity are positively correlated with the market.
What’s happening here is that when firm production is very low, i.e. when debt is risky, the market is negatively correlated with the firm’s production. But at high levels of firm production, the market is positively correlated with firm production. Conceptually you might think of this as a usually negative beta industry that becomes positive beta at high levels of market production. A tongue-in-cheek example, for illustrative purposes, is as follows: suppose you’re a law firm specializing in bankruptcy (so negative beta most of the time) but at high levels of market production you switch to corporate merger law (positive beta). The debt holders don’t see any upside past the face value of debt, so are accustomed to negative beta industry. But the equity holders will on average face payouts positively correlated with the market.
The motivation for this question came from a student who asked me if an asset beta of 0 implied an equity beta of 0. Since in a Modigliani-Miller world,
then an asset beta of 0 must imply that
My original (incorrect) thought was that since min{F,X} is weakly increasing in X and max{X-F,0} is weakly increasing in X, the numerator and denominator must have the same sign. But as it turns out, this is not always true.
In special circumstances, we can actually say something more about the conditions under which this counterintuitive occurrence can happen. Recall that Stein’s Lemma tells us that if X and Y are normally distributed (as returns often are), then
So if Cov(X,Y)>0, then as long as our increasing function g(.) is differentiable, it must be that E[g'(X)]>0 and hence Cov(g(X),Y)>0 as well. So this quirk can only occur with non-differentiable payout claims, like debt and equity (both of which are kinked).
One might wonder if the above result is a weird theoretical quirk: indeed, it took long enough to come up with a stylized empirical example just to illustrate the phenomenon. The question arises whether this occurrence ever happens in the real world.
Fortunately, this is straightforward to check using the TRACE bond database. For each bond in the database, I compute the principal-weighted monthly return across issues, merge those returns on issuer and date to CRSP monthly returns, and then — for stocks and bonds separately of the same firm — regress excess returns of each security against market excess returns. Below, I show a hexplot of the more than 17,000 betas computed over the 2002-2017 timespan. Shockingly, more than 31% of bonds feature a beta with a different sign than their stock beta over the same time period.
To see one example, consider ABB, the Swedish-Swiss multinational conglomerate, whose monthly returns we observe 2012-2017:
Here we have a slightly negative debt beta, and much larger positive equity beta (similar to the stylized example above). Unless this firm is extremely levered, this also implies the asset beta is likely positive as well.
I suspect a couple things are going on with these companies that constitute the 31%. First, at least some of the 31% can likely be attributed to statistical noise. Under a null hypothesis where both betas equal zero and where debt and equity betas are independent, we would expect them to be opposite-signed in 50% of all situations. Neither of those assumptions holds, but a more proper accounting would estimate the above regressions, bootstrap from the joint distribution of $\beta_D$ and $\beta_E$, and test the differing signs at some level of statistical significance. Second, these results suggest that betas are perhaps not the most appropriate measures for evaluating nonlinear securities. Betas are functions of second moments (covariances and variances) while debt and equity claims, particularly when debt is risky, have skewed, option-like payoffs that draw on higher-order moments. Finally, I imagine there is general market-wide pattern for investors to crowd into bonds when the market is doing poorly and to crowd into stocks when the market is doing well. This behavior would naturally lead to positive equity betas and negative debt betas.
That this phenomenon not only exists empirically but is so widespread is a fascinating feature of corporate security markets. Frequently the way that betas are taught to business school students and undergraduates is with respect to a business model (e.g. “an example of a high beta industries is a luxury good”), and where claimants of this business only matter insofar as they magnify these betas due to leverage (e.g. ”higher debt increases the magnitude of equity betas”). What we don’t teach — and what I learned today — is that changing the claim on the same business model may lead your beta to switch sign.