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 $f(\cdot)$ and $g(\cdot)$, 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.