How do you value a convertible debt?
Partial Differential Equation (PDE) Methods are probably the most robust ways of estimating the value of a convertible debt and a famous one is based on the paper by K. Tsiveriotis and C. Fernandes (the TF Model).
A quick excerpt from FinCad’s PDE model: “The main idea behind this model is that a convertible bond consists of two components, an equity component and a debt component, and these components have different default risks. The equity component has no default risk since the issuer can always issue its own stock. Thus the equity component should be discounted at the risk-free rate. Coupon and principal payments, and any put provisions depend on the issuer’s timely access to the required cash and thus introduce credit risk. Thus the debt component should be discounted at the risk-free rate plus a credit spread. To split the convertible bond (CB) up into its components, a new hypothetical security is defined, called the “cash-only part of the convertible bond” (COCB).
This hypothetical security is defined as follows: “The holder of a COCB is entitled to all cash flows, and no equity flows, that an optimally behaving holder of the corresponding convertible bond would receive.” Since the convertible bond is a derivative of the underlying equity, the COCB must also be a derivative on the underlying equity, and the value of both should follow the Black-Scholes equation.”
The next step is to develop a few partial differential equations for the CB and the COCB and to solve these PDEs simultaneously. The PDE for the CB might seem independent of that for the COCB. But the two PDEs are connected because there is a free boundary problem. The free boundary stems from the American option which allows the holder to exercise the conversion option at any time before the maturity date, and it becomes an unknown in the problem that must be solved for at each time step within the numerical method framework.
The company has given you a trick question since the answer is: it depends. Always keep in mind that the value of a convertible debt is the sum of the debt value plus the conversion option value and the key value drivers here are the moneyness, volatility, and interest rates. If the borrower is in distress then there is absolutely no value to the conversion option here and the best way is to run a recovery or asset coverage analysis to see how much the convert holders can recover from the borrower given the waterfall structure. If the company is not distress but the stock price is well below the conversion price, then there is still little value to the option and one could just use a DCF model as a proxy for the value of the convertible debt and in this case the interest rate is the key driver of the convert’s value. When the option is near the money, a more sophisticated model should be used. Bond + Call model is the most simplistic form of model that could be used to value an European convertible option or an American convertible option where the underlying stock doesn’t pay a dividend. Otherwise, for convertible debt with more complicated structures (American with underlying dividends, call option, put option, etc) there are multiple forms of binomial lattice models that can be used: the Goldman Sachs (blended discount rate) model, the Hull (trinomial) model, etc. PDE (partial differential equations) models like the TF model are also very popular among banks. All in all, these quantitative models are trying to value both the credit and equity components of a convert and capture the interactions between these components. Finally, for a convert that is deep in the money, one can use a stock + put to capture the fact that it will behave pretty like equity but also possesses a put option should the stock price drop below the conversion price.