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# Merton model, a summary of the issues

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Shannon - Strictly speaking, the Merton gives us no direct way to do that conversion, is sort of the the point. The Merton is limited to inferring a PD from a future distribution, either 1- year or 5-year or whatever SINGLE time horizon (that is less than or equal to the maturity of the long term debt). If we want the 1-year, we need to "re-run" the Merton under a one-year drift & one-year volatility (i.e., using T = 1.0 year). Nothing stops us from running the Merton at both T = 1.0 and (eg) T = 5.0 years.

Of course, if Merton gives us a 5-year cumulative PD, we can receive the 5-year PD estimate given by Merton, assume it to be a 5-year cumulative PD (which is an approximation and technically confuses the Merton output with a cumulative PD) and further assume constant annual conditional PD, and then use: 1-year conditional PD = 1 - (1- 5yr PD)^(1/5). Or for that matter, we can infer a hazard rate = LN(1 - 5 yr cum)*-1/T. So, we have can resort to our usual mechanics, but we've left the Merton to do that. Thanks,

#### sleepybird

##### Active Member
Hi David, nevermind my question. I found out from my FRM P1 note. The difference is closely related to the difference between arithmetic mean and geometric mean. The mean return will always be slightly less than the expected return just as the geometric mean will always be slightly less than the arithmetic mean.

David,
Thanks for the great explanation above. This greatly helps me understand the topic.
However, I still don’t understand why you use the formula V(0)*exp(mu - asset variance^2/2)*T instead of V(0)*exp(mu)*T to calculate V(t). Can you expand the explanation a little further?
Thanks.

#### sleepybird

##### Active Member
Hi David,
There was a question from the 2011 GARP FRM practice exam where we were given the following:

Current market value of firm = 4,500
Expected market value of the firm one year from now = 5,000
Short term debt = 1,000
Long term debt = 1,300
Annualized volatility of firm assets = 22%
Calculate distance-to-default (DD).

Shouldn't we calculate the DD as below?
Default point = 1,000 + 0.5*1,300 = 1,650
DD=ln(5,000/1,650) / 0.22 = 5.04

But the answer was given as follows:
(5,000 - 1,650)/(5,000*0.22)=3.045

That's a big difference. Can you advise which method should we follow on the exam?
Thanks.

#### troubleshooter

##### Active Member
Distance to default is how many standard deviations (asset volatility) is the final asset value is from the default point, which is correctly shown in the answer given. This is as per Moody's KMV model. I am not sure why you are taking natural logs here. That's not KMV methodology.

#### sleepybird

##### Active Member
Hi Troubleshooter,
Thanks. I understand that "Distance to default is how many standard deviations (asset volatility) is the final asset value is from the default point".
The Moody's KMV is built on the Merton model, which assumes lognormal distribution of stock price, isn't it? KMV only differs from the Merton model by the following: 1) KMV uses their proprietary data based to map the calculated DD, instead of using the Z-table (which underestimate fat tail distributions), 2) instead of assuming one zero coupon debt, KMV calculates default threshold as STD+0.5*LTD (or the other formula if LTD/STD >1.5). Other than these 2 points, KMV and Merton models are the same. Isn't it?

I see many formulas for DD:
DD = [Ln(V) - Ln(Default Threshold) + (ROA-0.5*variance)*T]/[volatility*SQRT(T)]; this is based on lognormal distribution assumption?
DD = (Expected asset return % - Default threshold % ) / volatility %
DD = (Asset value $- Liability value$)/Volality$David used the lognormal formula in the above question: Distance to the default threshold of 10 under lognormal =Ln(13.34/10)=28.8%. And this is 28.8%/9.6% = 3 standard deviation away from the default threshold. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi sleepybird, Yes, exactly: you have illustrated the issue. Actually, I wrote to GARP about it just last Wednesday, again for the 3rd or 4th time, because they are confused and their confusion continues to cause some confusion. Although this question (i.e., GARP 2011 P2.2.8) does not ask for a PD, if we want the PD for this scenario, and if we want to use the normal CDF (naturally!), then the PD is returned NOT with the 3.045 DD but rather the PD ~= N[ (LN(4500/1650) + ((5% - 0.5*20%^/2)*1)) / (22%*1) ] = N[5.039] ~= 0.000023%. I agree that the lognormal value-based DD is 3.045 which corresponds to a normal returns-based DD of 5.039375. (this can be verified with XLS 6.c.1). While (IMO) both DDs are valid as one refers to the returns-based normal and the other the value or price-based lognormal, it is confusing because we are assigned (and the literature) generally operates on the normal returns-based DD (sleepybird is right in the sense that you never see it presented lognormally), such that your LN(.) is sensible and smart. But, as I mentioned in the last Focus Review, GARP does not seem to be aware of this difference (or rather the technical nuance): all occurrences in my historical exam sample merely , as in the question above, look for the lognormal value-based DD. I imagine my focus will get this "fixed" for the 2013 exam, but possibly not by the upcoming November. (frankly, I think the "fix" is easy: just label the DD above as a value-based lognormal version ... I'd think that would be enough to save you from wanting to convert?) Please note that I am not aware of your 3rd variation, I do not think this should ever arise, unless i missed something: DD = (Expected asset return % - Default threshold % ) / volatility % ... this would appear to be the same as the normal DD, but only the default threshold is "parsed out" from within the LN(firm value/threshold), which is math possible but un-necessary i thnk But i do definitely agree with you: GARP's historical tendency is to ask the question, as above, based on a ratio of future values, and although your translation into a returns-based normal is totally valid, it is too smart (frankly) for the current exam. Thanks! #### sleepybird ##### Active Member Hi David, In contrast to the Merton model (which naively assumes single bullet debt structure), KMV uses linear combination of short term and long term debts. My question is, does this make the KMV model assume 2 default dates? KMV still uses the BSM formula, which is a pricing model for European options. Applying the same logic, using the BSM model to model probability default means the firm either defaults at T or no default, you can't default in between. Based on this, even though the KMV models uses combinations short term debt and long term debt to calculate the default threshold, this does not mean the model allows jumps (i.e., early default)? Am I correct? Thanks. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi sleepybird, Great question. (caveat: my direct experience with KMV is aged, I can't speak to recent updates). As you say, KMV employs the Merton model, up to a point, so in my view: 1. No, KMV model does not assume 2 default horizons. The combination of short & long-term debts is a "realism" that accounts for the fact that long-term debt, by definition, is not immediately due. So, while assets < liabilities is (technically) balance sheet insolvency, it is not necessarily funding illiquidity. Put another way, KMV includes all of the short-term debt (< 1 year) because if that can't be repaid, default occurs; but less than all of the long-term debt because there is still time to repay it. Therefore, the distance to default (DD) measure--which, to remind is similar to d2 in BSM but not exactly due to the use of a real-world drift expectation--assumes a single-horizon (e.g., a weakness of KMV, cited in de Servigny, is the lack of its ability to account for migrations, a general issue of structural/KMV models that do not simulate) 2. However, importantly, KMV does not finish with PD = N(-DD). Rather, KMV maps DD to a historical (empirical) database of cumulative PDs for the horizon. In this way, although the DD is limited to a horizon, it is "translated" (in the case of multi-year) into a cumulative PD that does attempt to consider the entire interval. Your question is also great because it highlights how KMV/Merton is not BSM: while BSM is used to derive the equity value, after that step, there is no risk-neutral derivative valuation. Each of: DD, Merton's PD(-DD) and KMV's DD -- map to --> historical PDs .... are physical (intiutive), future tail distribution estimations. From Peter Crosbie's excellent primeron KMV The distance-to-default is therefore calculated using the relevant three-year asset value, asset volatility and default point. The scaling of the default probability again uses the empirical default distribution mapping three-year distance-to-defaults with the cumulative default probability to three years. That is, the mapping answers the question, what proportion of firms with this three-year distance-to-default actually default within three years. The answer to this question is the three-year cumulative default probability. EDF values are annualdefault probabilities and the three-year EDF value is calculated as the equivalent average annual default probability. #### blahbe ##### New Member Hello, in your example, can you tell me how you arrive at$10 as your default threshold: (LN(13.34/10)/9.6% sigma = Z of ~ 3.0 where 3.0 is the (standardized) distance to default)
Is it the amount of ST debt divided by something?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi blahbe - I'll add a note to the original post: the pictured example borrows the example in FRM-assigned de Servigny Chapter 3 where the firm's capital structure includes long-term debt of $6.0 billion and short-term liabilities of$7.0 billion. A Merton Model approach (although it strictly speaking assumes one debt class) would use 7+6 = \$13 billion as the default threshold because that is the face (par) value of the debt; but de Servigy is illustrating a KMV-based variation where the default threshold is more realistically somewhere between short-term liabilities and total liabilities. In that KMV approach, unless LT/ST debt is > 1.5, then default threshold = ST + 0.5*LT debt, thanks,