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.