Merton Distance to Default

[kgolf20]

Hi David,

I tried to post this as a comment on the Merton PD screencast but I was not able to post there.

In that screencast, the expected future value of assets is 1184, the default threshold is 600, and the asset volatility is 0.25. LN(1184/600) = 68% and that is divided by .25 volatilty to get 2.72 std. devs. from default.

My question is: in the credit risk notes, I saw a Distance to Default formula where:

DD = (expected market value of assets - default threshold) / (expected value of assets)*(asset volatility)

Using this formula i get a DD = 1.97.... What am I missing?

Thanks as usual

Kyle

 

[David Harper CFA FRM]

Hi Kyle,

Right, good point, the reconciliation gets to a fundamental point that goes back to the idea of period log (lognormal) returns. Recall that LN(s1/s0) is a normally distributed return which implies that s1 or s1/s0 is lognormal.

So it is with the Merton model, this DD:
DD = (expected market value of assets - default threshold) / (expected value of assets)*(asset volatility)
is calculating a DD based on values; i.e., dollar difference/dollar volatility
as such, it is characterized by the lognormal distribution

But the de Servigny DD is using RETURNS; i.e., 68% intrinsic return plus expected growth/volatility (%)
as such, it is characterized by a normal distribution

Consequently, both DDs are true and they reconcile with:
(here I will just unpack the normal into a lognormal)
[EXP(2.72*25%)*600-600]/(1184*25%) = 1.97

Understanding this conversion is not necessary. Rather, you might just imagine that in dealing with prices/dollars, we are talking about a lognormal-turned-on-its-side...but in treating LN() returns, we can deal with the normal-on-it-side

btw, this is why the Merton at the end can use =NORMSINV() on the returns, because the LN() returns are normal. When i initially learnt this model, i was confused, how can we use NORMSINV() on a lognormal process (why not LOGINV())? Only after the 'aha' that St is lognormal but the LN(st/s0) is normal, did i feel i could finally get the merton.

Hope this helps.
David