Equity Catching Down to Bonds: A Macro Level View of Relative Valuation Trading

Looks like spoos are catching down to the more general basket that defines risk appetite. My read is this will be the trade for a week or so: equities had a big jump as shorts closed out. They will now be working their way to down to consistent relative valuation.

Based on relative valuation, I would be surprised if we see serious weakness in risky credit (read high yield) for a while without a full-on equity implosion based on some crazy shock. Relative valuation is powerful attractive force in the credit universe. It is worth looking at more deeply.


The basic idea is that corporate bond prices are sandwiched between the price of default-free bonds and the price of the corporate equity. There are factors that drive this behavior. If you get a good read on those factors, you can front-run a lot of money. Thus corporate bonds have some attributes of default-free bonds by virtue of being bonds and some attributes of equity by virtue of their being subject to potential default. How closely an instrument resembles a risk-free bond profile as opposed to how much it resembles an equity profile is captured in the yield spread of corporate bonds to risk-free government bonds. There is some part of every corporate bond that is very equity-like, and another part of it that is very much a government bond with a printing press backing it up.

In some cases, a corporate bond and stock is sometimes explicitly linked through convertible bonds, when they are issued by the reference name. A convertible bond is a combination of a bond and an option derivative that allows the holder to convert the security into a specified number of common stock shares, where the option strike varies with the price of the embedded bond. Even without convertibles, there is a connection. Some bonds issued by a corporation are more senior than other bonds issued by the same entity: the former are called “senior” as opposed to “subordinated” debt. Senior bonds trade more like government bonds than subordinated bonds do. Especially when a corporation is distressed, the sub bonds trade a lot like equity. The hierarchy of these bonds and equity is called capital structure.

A corporate bond’s equity features depends on the overall riskiness of the issuing name, the term of the bond, and where the bond sits in the capital structure. How these factors interrelate can be understood in terms of market spreads and instruments. The matched maturity spread between a riskless money market account to a defaultable money market account of the reference name captures overall credit riskiness. The spread of a riskless money market account to a corporate bond yield captures term risk combined with credit risk. The closer a bond’s seniority is to equity, the more equity-like it is.

These simple dynamics make possible capital structure arbitrage (it’s not really arbitrage), or as it is called now relative valuation trading. Relative valuation trades can be implemented with a number of different strategies. Trading can be very frequent—for credit anyway—it is possible to the total bond book to turnover in a month. Relative valuation trading often used both cash bonds and synthetic instruments.


In all, it reduces to a kind of pairs trade where the timing strategy that depends on mean reversion. There are two price levels: historical realized a statistical moment derived from it. Returns can be obtained from a buy-low-sell-high strategy where entry/exit levels are set as ±1 standard deviation from the long-run mean. Other related strategies consider starting and ending a pairs trade based on the spread’s distance from its mean. Other than trading cash assets, it is also possible and economical to buy and sell derivatives written on a mean-reverting underlying.

A typical solution approach for optimal stopping problems driven by diffusion involves the analytical and numerical studies of the associated free boundary problems or variational inequalities.

Free boundary problems are an understood class of problems given an infinite time horizon where the steady state is all that matters. The snag is that the optimal stopping times associated with the capital structure arbitrage set-up need to be solved over a finite horizon, and explicit solutions are less available. It goes to show that rigor in formulation provides clarity, but also has its limits.

Since finite horizon problems aren't easily amenable to mathematical analysis, hedge fund managers depending on mean-reverting prices as opposed to steady state prices. They simultaneously take positions in two highly correlated or co-moving assets, price out the pieces of the capital structures by applying risk premia, and wait for convergence or divergence.

The optimal stopping problem remains: determining when to open and close a position. Observing the prevailing market prices, a trader can choose to enter the market immediately or wait for a future opportunity. After taking the initial position, it needs to be decided when it is best to close the position. A higher stop-loss level induces the investor to voluntarily liquidate earlier at a lower take-profit level. Moreover, the entry price is characterized by a bounded price interval that lies strictly above stop-loss level. In other words, it is optimal to wait if the current price is too high or too close to the lower stop-loss level.


While this implementation problem is primary, there are important mechanical aspects of capital structure arbitrage. Capital structure necessarily involves credit, and trading credit risk merely exchanges credit risk for asymmetric counterparty risk on the trading book. These asymmetries lurk everywhere in cap structure arb by the nature of the trades. For example, in a typical fixed-floating swap, the fixed payer has a finite and known maximum total obligation (the fixed rate times the notional times the term), while the float payer has an unknown maximum obligation. This asymmetry is very pronounced for credit default swaps, when the protection buyer’s max obligation is generally much lower than the protection seller’s.

In the case of relative valuation models, especially capital structure arb, it hasn’t been working well for a while. Stat arb in general has sucked this year. High yield in particular has sucked since fall. Boaz Weinstein isn’t alone, and it may not get better for the foreseeable future. If you look at corporate bond risk premia, there is a volatility component (the part that makes them “equity-like” measured by VIX), default risk (measured by default rates), and the remainder is liquidity risk. The latter is the problem hampering relative valuation trading.

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