# Recovery rate in a structured bond

I need to model the recovery rate of a structured bond whose expected cash flows, if the issuer remains solvent, will be very low. For instance, assume that I need to estimate the recovery amount of a bond with an expected cash flow at maturity of 15% of its notional value (i.e.: the expected coupon is -85%).

In order to estimate a recovery in this type of bond:

• Should I still consider the initial bond notional and apply a recovery rate? In such case, with the usual levels of recovery of $\sim$ 0.4 the defaulting scenario would result in a higher payoff than the solvency one, which seems quite odd.

• Which other approaches could be used to estimate the recovery in these types of bonds? (e.g: calculate the recovery based on the non-default market value?)

EDIT: Information added based on comments

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I'm not sure what you're getting at with the first bullet - in the solvency case the coupon + principal is paid, in the default scenario only the recovery is paid. How are you getting a higher payoff from the default? Also, "structured bond" is quite ambiguous, could you elaborate more on the class of bond you're trying to model? –  jeff m Jul 16 '13 at 3:21
@jeffm: I am referring to a structured bond with no capital protection. In particular, due to the negative evolution of the underlying asset, this bond has an expected payoff at maturity of only 15% of its nominal value. That amount includes both principal and coupon (i.e.: the expected “coupon” is -85%). In contrast, if we use a “standard” recovery rate of 0.4, and apply that recovery to the bond notional, the cash flow in default will be 40% of the face value, which is higher than the expected payoff in the solvency scenario. –  sets Jul 16 '13 at 7:32
Is there any particular reason you want to model the recovery rate? I think you're getting offtrack by trying to apply "standard" recovery rates, not to mention it's a very poor way to estimate a cash flow imo. My advice would be to forget about estimating a recovery rate and instead go after what you really want - estimating the cash flow and then arriving at the recovery rate. Is your dataset precluding you from doing it this way? –  jeff m Jul 17 '13 at 18:40
@jeffm: I not sure I understand your reasoning. I have already estimated the future cash flow distribution and the survival probabilities. For illustration purpose, assume that future cash flows in solvency (95% probability) are 8%, 15% and 22%, all with equal probability. With this CF structure, the expected payoff at maturity is 15% of the bond notional. What I need to estimate now is the CFs in default. How would you do that? Would you use the projected cash flows as a reference, or the bond notional value? –  sets Jul 18 '13 at 7:33
These are all details you should have included in the initial post, I think I finally have an idea of what your question is - I'll do my best to answer after I get some work out of the way. –  jeff m Jul 18 '13 at 13:19
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## 1 Answer

Recovery rates are rarely "modeled" per se, in the sense that most practitioners avoid treating them as random variables. I doubt you want to buck that trend here.

As jeff m implies in the comments, it's the cash flow that you really want to know about, so you'll find it more useful to think in terms of the mechanism behind recovery.

If the issuer defaults, there's going to be a court case, and a whole bunch of lawyers are going to negotiate, under the supervision of a judge, who gets what from the remaining firm assets. Around that time, experts in distressed debt will presumably be trading this bond based on their best guess as to the outcome of those negotiations.

So, what will this bond look like as a claim in those negotiations? Well, normal bonds are often issued such that the coupon more-or-less compensates for default risk and time value of money, so as a shorthand practitioners think of the non-default value of the bond as being equal to the notional, and that's the number that influences recovery cash flow.

Savvy players will know that this bond becomes worth much less over time, so you can expect the notional won't be accepted as a baseline. Instead it will probably be viewed as future cashflows, discounted by a rate somewhere between risk-free and risky.

This is all a roundabout way of saying that I would tend to take a recovery rate $\delta$ as usual, but include a risky discount rate $z$, and then set the recovery value of the instrument, conditional on default at time $\tau$ as the time-dependent value

$$\sum_{n \ni t_n>\tau} \delta\cdot c_n\cdot \exp\left( -\int_\tau^{t_n} z(s)ds\right)$$

in my pricing formulas.

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