Convertible bonds (CBs) can be awkward beasts. In this Learning Curve we will wrestle with a question that has been controversial, namely--what discount rate should be used in valuing CBs? There is an apparent paradox here. When out of the money, a CB looks just like an ordinary bond, and the cash flows should be discounted at LIBOR plus the issuer's credit spread. On the other hand, when in the money, the CB is very likely to be converted and looks much more like a call option on the issuer's stock. Nobody includes a credit spread in valuing stock options. So apparently the discount rate should include a credit spread when the CB is out of the money but not when it is in the money. How can one make sense out of that?
John Hull's book1 suggests a procedure. The CB is valued using a one-factor binomial or trinomial tree in which the local discount rate at each node is adjusted according to the conversion probability at that node. This achieves the desired effect of making the value consistent with the straight bond value when the conversion probability is low, and consistent with Black-Scholes when it is high, but one can't say there is anything particularly scientific about it.
A much better idea was introduced by Tsiveriotis and Fernandes (T&F)2. They reason, correctly, that credit risk arises when the issuer makes a cash payment (for example, a coupon) but not when the issuer delivers shares, which have no guaranteed value. One can therefore value the CB by maintaining two 'accounts': a cash account and a share account, discounted at LIBOR + spread and LIBOR respectively. A simple example illustrates the idea. Suppose the notional value of the CB is 1 and the exchange price is EP, so that the number of shares received on exchange is n =1/EP. If we only exchange at final maturity T and there are no intermediate coupons then the exercise value is max(1+c, nS(T)) where c is the final coupon and S(T) the share price. We can express this as
where K=(1+c)/n. The first term is cash and the second share delivery, so we take the discounted expectation at rates r+s and r respectively, where r is the LIBOR rate and s the spread, to give a time-0 value of
Denoting by p = P(S(T)>K) the conversion probability, this expression is easily rearranged to give
where BS is the Black-Scholes value of the call option. The interesting thing is the factor (p+(1-p)e-sT). This is somewhere between 1 (when p is close to 1) and e-sT (when p is close to zero), so the effective discount rate does indeed depend on the exercise probability, being close to r+s when p is small and close to r when p is large. This idea is very easy to incorporate in a tree model. We 'roll back' cash and share values separately, discounting at r+s, r respectively. The total CB value is the sum of these two values. If at some node in the tree it is optimal to convert, all the value at that node is transferred to the share account, the cash account being set to zero. This algorithm is simpler than the one suggested by John Hull and does all the right things.
Satisfactory as this may be from a practical viewpoint, how does it square with textbook finance theory, which tells us that the no-arbitrage CB value is the discounted (at rate r) value of all cash flows under the risk-neutral distribution? Since the credit spread compensates investors for default risk, we have to bring in an explicit default model to explain this. Let's take our simple example as above and suppose that the issuer defaults at a random time t that is exponentially distributed, so that the 'survival probability' is q = P(*>T) = e-hT. We also assume that the post-default value of a bond is its recovery rate R<1. Taking the underlying bond with cash flow (1+c) at time T, its value is by definition of the credit spread (1+c)e-(r+s)T, whereas its expected value under our default model is e-rT(q(1+c)+(1-q)R). These are equal when q(1+c)+(1-q)R =(1+c)e-sT, which is the calibration condition giving q in terms of s (for given R). What about the stock price? Since bondholders have first call on the assets of the issuer, the stock price jumps to 0 on default. The usual log-normal stock price model is modified to
Here N(t)=1(t>*) and S(t-) denotes the value of S just before time t. At time *, dN(*)=1 and the right-hand term in the equation ensures that S jumps down by S(*-), i.e. to zero. It stays at zero thereafter. The term -h dt is needed because we want the forward price for S to be S(0)ert. Since E[dN(t)]=h dt, the term dN(t)-h dt is a martingale, which gives us the forward price condition. The key point here is that we can rewrite (2) as
This shows that before default the growth rate of S is increased from r to r+h to compensate for the default risk.
Now we can get back to CB valuation, taking our previous example. Either (with probability q) the issuer survives beyond time T and the CB value is max(1+c, nS(T)), or (with probability 1-q) default occurs and the CB holder collects R at time T. We find that the expected discounted value is
and from the above calibration condition we see that this is equal to
This time, the effect of credit risk has appeared in different form. The bond payments are discounted at r+s, but the option value is increased because the Black-Scholes formula uses r+h, not r, and the BS value is an increasing function of interest rate. It is interesting to note that the assumed recovery rate R has almost--but not quite--disappeared from the picture (the value of h still depends on it, but not very sharply). The values given by the T&F and default models are virtually identical when R = 0, the default model giving a slightly higher value when R > 0. For general CBs it is straightforward to incorporate the default model in a tree, although this is a touch more complicated than the T&F tree implementation.
Valuing CBs is really a three-factor problem involving volatility of equity, interest rates and credit spreads. The equity component is by far the dominant source of volatility, and most people are happy with a one-factor model, for which T&F is probably the best way to deal with the credit spread. A two-factor model using, say, the Hull-White interest rate model, is not hard to put together and gives a good evaluation of interest rate-risk. On the other hand a full three-factor model is in my opinion overdoing it, if only because of the difficulty of estimating credit spread volatility.
1John C.Hull, Options, Futures and Other Derivatives, 3rd ed. Prentice Hall 1997, Section 20.5
2K.Tsiveriotis and C. Fernandez, Valuing convertible bonds with credit risk, Journal of Fixed Income 8 (1998) 95-102.
This week's Learning Curve was written by Mark Davis, a researcher with the financial and actuarial mathematics group at the Technical University of Vienna.
