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I am simulating various price path of bonds, and one issue that came up is the recovery rate.

When a bond defaults, the amount you get back recovery rate * principle. This creates a problem if the current bond price is less than the recovery rate. For example, a 1% coupon, 30 year bond with a yield of 7% should be priced at 0.18, assuming that face value is 1. Usually corporate bond have a recovery rate around 40%. This means that in the beginning the bond holder will actually want the bond to default, because then they can collect a large fraction of the principle much earlier than expected. This totally messes up pricing models as now default probability is actually a favored pricing factor.

One solution is to assume that you actually loss recovery rate * current market value. This sort of takes care of the time value of money issue, but leads to even bigger problems:

In a realistic model, bond rating changes according to the transition matrix, as a result, the yield on the bond also changes. Sometimes, a bond can fall into a really bad rating but not defaulting, like CCC or D. In these situations, the spread gets ridiculously high, like something around 50% - 200%. If you use this spread to calculate the price, you will see that the bond holder will be better off if the bond actually defaulted.

So a natural solution to this problem is to capped all losses by the recovery rate. So if the price of bond is less than the recovery rate, then the price of the bond is the recovery rate.

But again this only solves part of the problem. After a bond falls to a very bad rating, it will almost certainly default within the next few years. Since earlier we defined the loss given default as market value * (1 - recovery rate), the bond will suffer another loss. Essentially we have made the bond suffer through default twice. In other words, given two bonds, if one defaults directly, where as the other fought hard and stayed at CCC rating before defaulting, the second one will be actually worse off. This contradicts common sense.

In essence, I am trying to come up with a method that satisfy the following properties 1. Bond holder should not gain from defaults 2. Assuming all else equal, a bond that defaults later provides higher gain than a bond that defaults earlier.

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Sam, here is a related question that might be also of interest: quant.stackexchange.com/questions/8483/… – sets Jul 24 '14 at 8:25
Thanks Sets. The answer provided is very close what I am thinking: Readjust the pricing formula for the bonds so that the recoverable portion of the bond is discounted at risk free rate. Then when a bond defaults, the bond holder simply loses the unrecoverable principle and future coupons. – Sam Li Jul 25 '14 at 13:15

To add to emcor's answer, if a bond defaults, you do not automatically get the "recovery" amount immediately, you get some unknown amount at some unknown time in the future, possibly years later, and greatly depending on your particular bond's covenants and seniority. If you are trying to consistently price bonds, you might be better off implying the recovery rate(s) from the CDS term structure, other bonds and equity options or some such.

Certainly, one should do neither of your solutions, since they likely permit arbitrage into your model.

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I think the problem here is that the recovery rate is not a fixed parameter, it is only estimated from past defaults. You never know what the actual recovery rate will be, so markets change their view all the time.

If your estimated recovery rate is higher than the bond price, you can either assume that the bond is underpriced, or your estimated recovery rate is overpriced and should hence be adjusted downwards.

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Emcor makes a good point. I would add that if a bond defaults, that recovery rate is only guaranteed at maturity to my knowledge. So assuming you are sure that it will be respected and paid in due time (Argentina is a good example of things happening otherwisw), you should discount that terminal value at the risk free rate or any appropriate discount factor, which most likely will decrease the number of paths with issues.

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I thought about exactly that. The true question is what to use as the discount factor. Using risk free rate would still lead to the problem of recovering amount being larger than the current market value. There is one way out though, that is, discount the portion of the "recoverable" principle at the risk free rate and the rest at the normal rate. – Sam Li Jul 23 '14 at 22:44
The recovery rate is not guaranteed at all, at maturity or otherwise. It's an estimate of the eventual proceeds of the sale of assets of the company, as they accrue to each particular bondholder with their rights and seniority in bankruptcy. – experquisite Jul 23 '14 at 22:55
I know it can be random, and there are lawsuits you have to go through and it could take years before you receive a penny. The model I am designing is very simple. It takes only two inputs: the transition matrix and a table containing the average spreads for bonds with each credit rating. Simply something that make sense will suffice. – Sam Li Jul 23 '14 at 23:00
Well, if its very simple, you could set the recovery rate to 0? – experquisite Jul 23 '14 at 23:42
Not that simple:). I tested different values, and recovery rate is not negligible. I kind of wish it is though. – Sam Li Jul 23 '14 at 23:44

Your premises are wrong in the real world: 1. Bond holder should not gain from defaults 2. Assuming all else equal, a bond that defaults later provides higher gain than a bond that defaults earlier.

1) There are "vulture investors" who buy defaulted bonds at an ultra low price for the purpose of forcing a restructuring at a higher price. 2) While this is theoretically true, the same vulture investors will do their best to make sure that the higher gains occur earlier.

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