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# Merton Model (Structural Model)

#### Sunil Natarajan

##### Credit Analyst
Hi David,
I have some confusion regarding Merton Model in Credit Risk.
According to Stulz (Ch-18)

d1=LN(V/F*exp(-rt))/(Std Dev * t^0.5)+0.5*Std dev*t^0.5

According to Servigny (Ch-3)
d1=d2+Std Dev*t^0.5

In Market Risk
(Black Scholes Merton)

d1=(LN(V/F)+(r+(0.5*Std Dev*t^0.5))/(Std Dev*t^0.5)

Why does each of them give different formulae for the same variable d1.Does the value of d1 in each of the formulae result in same answer.

In Merton Model what is the Prob of Default.Is it N(-d2) or N(d2). In KMV what is the Prob of Default and LGD. Is Prob of Default the same as Distance to Default.Could you just explain what happens in KMV model exactly.
In KMV what is the LGD measure.

Regards,
Sunil Natrajan

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Sunil,

Short answer on the math is: they are all the same.
If you have a moment, you might look at http://www.bionicturtle.com/premium/editgrid/frm_credit_merton_model_for_pd_lgd_and_debt_equity/

Notice in the first column, I computed PD under both Stulz (upper) and de Servigny (lower), specifically to show their equivalence, that is:

Stulz PD N(d2) = N(-3) = about 0.13%
and de Servigny
PD = N(-d2) = N(-3) = same

So, on the one hand, Stulz is using d instead d1,d2 but all are mathematically going to the same place. The numerator in Stulz/de Servigny initially appears different but that is because the Black-Scholes distribution is normal (i.e., price levels lognormal = returns are normal) so it doesn't matter if you do math "above" or "below" the mean on a symmetrical distribution. So, to your first query, we can establish mathematically all are the same (Stulz, Hull, de Servigny) in to the extent they are employing Black-Scholes OPM.

For the reason above (symmetry of the normal), PD in Merton is either N(d2) or N(-d2). However, IMO de Servigny is atypical, and we are more standard to just follow Hull: PD = N(d2). Stulz (alone among our authors) does extend Merton to LGD (see rows 19-23 in the XLS) but I do not have an "intuitive" explain for it (it is a conditional shortfall). I consider its testability to be very low...

To my knowledge, KMV does not extend to LGD (I may be dated on this), our use of KMV is only for the EDF/PD. In the Merton Model, the distance to default (DD) is d2. So it is very good to see the intuition of d2:

* The numerator has two parts: LN(S/K) is the implied return already in the capital structure (e.g., "if the firm value does not grow, it would need to decline by x% to hit the threshold).
* The other part of the numerator (it is very good for an FRM candidate to see this) is the expected (geometric) return. The relevant AIM is Hull. Instead of (r), it is (r) - 1/2 the variance. "Volatility erodes returns"
* Hopefully, you see visually, how the numerator is giving an expected continuous return above the threshold by combining two pieces: current (already in cap structure) plus expected growth
* Then divide by volatility to STANDARDIZE; i.e., convert into standard normal units
* Once standardized, can use =NORMSINV()

So, PD = N(d2) = N(DD) or N(-DD) because we are wanting to find the area under the tail, which is less than X standard deviations below the expected value.

Also:
* In the Merton above, r = expected return. The above is *not* option pricing (we already did the option pricing step to find the firm value, OPM was done at that point). r = riskless as an input into the Black-Scholes, not here, here we want the expected return of the firm's assets.

* I am saying Merton here not KMV. KMV uses OPM to find the firm value (and volatility) and compute DD. But it does not use NORMSINV() or apply parameteric to get the EDF/PD. It uses historical database of DD.

* Note we have used two different types of N(d2). One with a riskless rate, another with an expected (risky) rate for the PD.

N(d2) in the Black-Scholes contains the riskless rate: per the risk neutral idea, we can use a riskless rate here and get a reality based option price. But N(d2) with a riskless rate is a risk-neutral PD and would be lower than our PD. The risk neutral idea "works" for the option price but not for PD. N(d2) with a riskless rate is INCORRECT for the Merton application of PD. As Stulz uses risky rate (expected return), if we want PD, we use the return not the risky rate.

put another way, DD is d2 in the Black-Scholes for a call option but with riskless rate replaced by expected return.

David

#### Sunil Natarajan

##### Credit Analyst
Hi David,
This means that prob of default as per Merton & KMV are both N(-d2) , is it. But in KMV you arrive at some default threshold(DP) and then at DTD. So then DTD is same as N(-d2).Under Servigny Ch-3 DTD is arrived by some equation. It is not very clear. Does Merton & KMV models apply only for listed companies.According to Stulz KMV is inspired by CAPM. How is CAPM related to KMV.
Will GARP test on all this minute details since KMV model as such is not well defined (being a Proprietory model).
* The numerator has two parts: LN(S/K) is the implied return already in the capital structure (e.g., “if the firm value does not grow, it would need to decline by x% to hit the threshold).
* The other part of the numerator (it is very good for an FRM candidate to see this) is the expected (geometric) return. The relevant AIM is Hull. Instead of (r), it is (r) - 1/2 the variance. “Volatility erodes returns”
* Hopefully, you see visually, how the numerator is giving an expected continuous return above the threshold by combining two pieces: current (already in cap structure) plus expected growth
* Then divide by volatility to STANDARDIZE; i.e., convert into standard normal units
* Once standardized, can use =NORMSINV()

Regards,
Sunil Natrajan.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Sunil,

The d2 in Hull is a good anchor. Then,

For Black-Scholes OPM, as r=riskless, N(d2) is probabilty option will be struck in a risk-neutral work (i.e., not really useful to us)

Under a Merton model, then, N(-d2) is the PD. How did N(d2) go from being un-useful in Black-Scholes to becoming useful in Merton? (r) is replaced by the expected growth rate of the asset (firm value). With that one change, d2 becomes the distance to default (DD) and N(-d2) is now a MERTON PD.

But KMV stops at the d2 (with risky rate replacing riskfree rate): KMV is not exactly Merton "all the way to the end" b/c although it gets to DD = d2, it does not then parametrically assume PD = N(-d2).
(That's why de Servigny uses a XI greek function instead of an 'N' for normal)
(But KMV is nevertheless "structural" because it infers PD from asset value)

"Does Merton & KMV models apply only for listed companies." NO, as a Merton approach, KMV needs to find/lookup firm asset value and volatility, and this is easier if they trade. But assets do not need to be traded to have fair values. Nothing in the Merton or KMV procedure requires traded prices, only "market value"

"According to Stulz KMV is inspired by CAPM. How is CAPM related to KMV." I'm frankly not a huge fan of Stulz textbook. Stuff like this. He only means that a method like CAPM is used to get the expected asset return (the same risky return that "replaces" the riskless rate above). I honestly believe this is not important; it is an example of a proprietary element of the KMV, and even more so, it is almost the least important element of the whole thing.

"Will GARP test on all this minute details since KMV model as such is not well defined (being a Proprietary model)."
Can we be specific about what is proprietary? For our purpose, the only proprietary element is that we don't really know how KMV gets from DD to EDF. That is, we can use N(d2) via Merton but KMV does not do that (as actual defaults are fatter tailed; a strick "all the way through" Merton will underestimate defaults)

So, maybe the FRM will not test all the details, but honestly, understanding the intuition of d2 "pays dividends" in the FRM. I will screencast tomorrow (Friday) on this (<10 min) so hopefully that will help.

David

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