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Eroboy
03-17-2008, 11:43 PM
It seems that the recovery rate defined in Hull's book is

market value of bond at default / Face (page 483 and see example on page 485)

while in the Crouhy's book the LGD is defined as % of bond value. (see page 412 to 414 )

Thus, it seems that two books have slight definition of recovery rate.

This is not an issue if this is just one period zero coupon bond, but it seems that different definition will result slightly different answer with coupon bond depending how you interpret the recovery rate.

Is there an convention if the problem gives a recovery rate, we use the definition in Hull's book but for LGD use the definition in Crouhy's book.

BTW, has anyone verify the estimation of default intensity formula in Hull's book for multi-period zero coupon bond? (lamda=S/(1-h), where s is spread and h is recovery rate). It seems that based on formula 3 on page 413 of Crouhy, if yield curve and risk free curve is flat, all the forward intensity shall be same. Thus, Hull's approximation shall hold. However, I fail to analytically sovle the following formula to derive it directly on a continuous basis:

exp(-yT)=exp(-rT)*[1-exp(-lamda*T)]+h*[Integral(lamda*exp(-(lamda+r)*t)dt, t from 0 to T)]

where y is the bond yield,
r is risk free
lamda is average default intensity
T is the maturity

The integral is easy, but solving lamda as (y-r) requires some skills.

Also, isn't using Merton's formula under estimate the default rate for long term? Merton's formula is European option, while company can default at any time before the maturity. It seems to me that the short term estimation is okay.