Merton/KMV ND formula

[rickm123]

Hi David:

What is the difference between your PD formula you have for the KMV model verses the N(d) formula you have in the merton model when we calculate the FN(d1) – Xert(Nd2) ?

I am thinking it is the drift? Am I correct in the accessment?

thanks

 

[David Harper CFA FRM]

Hi Rick,

Again, it would be greatly helpful if you can prior search; this is a good question but has been asked and answered many times. (and you don't need to post it twice) It is among the most discussed topics and there is some rich prior discussion.

But briefly,

• The variation on BSM [your FN(d1) – Xert(Nd2)] prices the firm's equity by treating the firm's equity as a call option on the firm's assets. This is the first step in the Merton, it is derivatives pricing, and requires the risk-free rate (and does not even use the firm's expected return/drift).
• After we have the firm's equity, the second big step in Merton is to compute N(-DD), which is analogous to N(d2) in BSM, except this is physical future tail estimation of the distribution, is not derivatives pricing at all, and therefore requires the firm's expected return drift
• KMV follows Merton up to retrieving the DD, but then abandons the use of PD = N(-DD) because this assumes the future firm's asset price is lognormally distributed. Instead they map DD to historical PDs, so it's empirical rather than parametric as a final step. I hope this explains, thanks,

 

[rickm123]

Hi Rick,

Again, it would be greatly helpful if you can prior search; this is a good question but has been asked and answered many times. (and you don't need to post it twice) It is among the most discussed topics and there is some rich prior discussion.

But briefly,

• The variation on BSM [your FN(d1) – Xert(Nd2)] prices the firm's equity by treating the firm's equity as a call option on the firm's assets. This is the first step in the Merton, it is derivatives pricing, and requires the risk-free rate (and does not even use the firm's expected return/drift).
• After we have the firm's equity, the second big step in Merton is to compute N(-DD), which is analogous to N(d2) in BSM, except this is physical future tail estimation of the distribution, is not derivatives pricing at all, and therefore requires the firm's expected return drift
• KMV follows Merton up to retrieving the DD, but then abandons the use of PD = N(-DD) because this assumes the future firm's asset price is lognormally distributed. Instead they map DD to historical PDs, so it's empirical rather than parametric as a final step. I hope this explains, thanks,

Great thanks
If we calculate -d2 with the formula, is it possible to get to N(-d2) and hence PD...Dont think so unless we have excel..correct? thanks

 

[David Harper CFA FRM]

Rick, are you two people using the account? You already asked me that, it was yesterday ??!!
http://forum.bionicturtle.com/threads/merton-kmv.5631/#post-15940

again, as the exam nears, we request you try to practice some forum etiquette. It's not for me, i don't care and am happy you are here, it's only so i can answer as many relevant questions (from candidates' perspectives) as possible in the runup to the exam. The forum is less useful to other customers if the daily posts are overly redundant/circular.

Thanks,

 

[southeuro]

David,
a small question - I am a bit confused as to when/where we prefer the following formula:

future value - debt / volatility*future value

is this supposed to be some sort of a general approximation to find the DD? Thanks

 

[Alex_1]

Hi @southeuro, the formula you have listed is basically the DD (distance to default), which calculates the number of standard deviations between the mean of the asset distribution and the default threshold. Once the DD value is obtained, the probability of default is found by evaluating the DD of other firms which have also defaulted (this is what David states above with "KMV follows Merton up to retrieving the DD, but then abandons the use of PD = N(-DD) because this assumes the future firm's asset price is lognormally distributed. Instead they map DD to historical PDs, so it's empirical rather than parametric as a final step."
Hope this helps.

 

[Roshan Ramdas]

David,
a small question - I am a bit confused as to when/where we prefer the following formula:

value - debt / volatility*debt

is this supposed to be some sort of a general approximation to find the DD? Thanks

Hi,
Further to the note from Alex,......this formula is normally employed when the question provides you with the future firm / asset value as opposed to the current firm / asset value (which employs BSM to arrive at DD).
Thank you

 

[southeuro]

Thanks Alex and Roshan - it's crystal clear now