Valuing a Callable Perpetuity with Interest Rate and Credit Risk
Summary
The document considers a perpetual monthly payment obligation issued in exchange for an initial cash amount, with the issuer able to terminate it after a waiting period by repaying that original amount. It identifies the arrangement with a perpetual bond or preferred share and treats the termination right as an embedded call. It also raises the accounting question of how the liability should be recorded.
The proposed valuation approach is to specify an interest rate and, where relevant, credit model, then solve numerically with a partial differential equation, tree, or grid method. A finite horizon can approximate perpetuity value when the discounted terminal principal becomes negligible. The discussion notes that payments before the first call date can drive much of the valuation. Key limitations and risks include model and calibration error, possible suspension of preferred payments, mergers, and changes in capital structure; it does not supply a worked valuation or address detailed accounting rules.
Key ideas
- A perpetual payment stream with a redemption right can be modeled as a perpetual bond with an embedded issuer call.
- Valuation can use a stochastic interest rate or credit model with a PDE, tree, or grid method.
- A sufficiently distant finite horizon can approximate the perpetuity when discounted terminal principal is negligible.
- Payments before the first call date may be especially influential in practice.
- Credit events, payment suspension, corporate actions, and model error can materially affect value.
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# How would I value a perpetual bond with an embedded option? # How would I value a perpetual bond with an embedded option? I am trying to work out how to value the following transactions. It should be straight forward, since it breaks down into a series of well known instruments, yet I am not sure how to evaluate it: - Receive Cash payment amount of \$X - Subsequent pay out \$Y per calendar month into perpetuity - Have the option to "close out" the implied perpetuity, by paying the original received $X amount, any time after 1 year. I would like to know how to value such an instrument, which consists of effectively: - a perpetuity - an embedded option If I was to make such an instrument available to someone, how much would I sell it for? As an interesting aside, this is clearly a debt instrument, and would be recorded in the other parties 'liability column'. What value would be recorded in the books? Clearly, not the original $X ...? ## Answer by Brian B (score 9, accepted) https://quant.stackexchange.com/a/2485 There are "perpetual" bonds and preferred shares that are traded in the corporate credit markets that exactly match your conditions above. They are recorded in the 10-K at notional value $X$. The "close-out" feature is an embedded call. You should assume your favorite stochastic interest rate (and/or credit) model and run a PDE solver, tree, or other grid scheme from some time $T$ far in the future (we used to let $T$=70 years). Basically, you want $T$ far enough away that, after discounting, $X$ paid at time $T$ is negligible. Most of the time your credit model has a pretty high probability of default within 100 years. Many of these instruments have their embedded calls starting only past some date in the future, and in that case traders often tend to lend great consideration to the PV of payments only up to the first call date. The main risks to your valuation (aside from model or calibration error in the credit and interest rate models) are - Suspension of preferred coupon/dividend payments - Mergers and takeovers - Capital structure changes
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