# BT2012.P2.09. KMV's merton model for credit risk

Discussion in 'BT Part 2' started by David Harper CFA FRM, Apr 29, 2012.

1. ### David Harper CFA FRMDavid Harper CFA FRM (test)

Question:

9. A firm's assets have a current value, V(0), of $20.0 billion. The face value of firm's short-term debt is$8.0 billion and its long-term debt is $10.0 billion. The expected return on the firm's asset value, R(V), is +10.0% with volatility, sigma[R(V)], of 20.0% per annum. The riskless rate is 4.0% per annum. Your colleague wants to estimate the distance-to-default (DD) but, instead of the traditional Merton model, wants to try to mimic the KMV approach which counts only 50% of the long-term debt toward the default point. To simplify the analysis, we assume both the short-term and long-term debt will mature in one (1) year. Assuming the future distribution of the firm's asset value is lognormally distributed, where the realized drift is according to a geometric not arithmetic average, what is the lognormal distance to default? a. 1.3, which corresponds to 1.8 normal returns-based DD b. 2.0, which corresponds to 2.6 normal returns-based DD c. 3.3, which corresponds to 3.7 normal returns-based DD d. 5.2, which corresponds to 5.5 normal returns-based DD Answer: 9. B. 2.0, which corresponds to 2.6 normal returns-based DD The expected future firm value =$20*exp[(10% - 0.5*20%^2)*1] = $21.67 billion. As the default point =$8.0 short-term + 50% of $10.0 =$13.0,
The lognormal DD = ($21.67 -$13.00)/(21.67*20%) = 2.000
...which is consistent with a normal returns-based DD of [LN(20/13) + (10% - 0.5*20%^2)*1]/[20%*SQRT(1)] = 2.554

Last edited: May 7, 2014
2. ### shanlaneActive Member

Hello,

Do we always use the "future" value * sigma in the denominator? In the problem like this from this year's practice test they use the current asset value. I have seen it both ways, but it is really inconsistent.

Thanks!

Shannon

3. ### David Harper CFA FRMDavid Harper CFA FRM (test)

Hi Shannon, Yes, technically it should (absolutely, no doubt) be future value sigma in the denominator. You can find the rational in my explanation at http://www.bionicturtle.com/forum/threads/merton-model-a-summary-of-the-issues.5646 ... Distance to default is, unlike risk-neutral idea in derivatives valuation, an estimation of the future, physical distribution of the asset (based on its expected return), it is not a present value calculation. Thanks,

4. ### shanlaneActive Member

So does that mean that this is another possible mistake in the 2012 GARP practice exam?

Thanks!

Shannon

5. ### David Harper CFA FRMDavid Harper CFA FRM (test)

I guess you refer to question 6? (I have not vetted their 2012 practice paper fully yet ...)
I didn't see this question before.
Yes, that's just a terrible mistake on their part, I am embarrassed for them. I am definitely posting to the errata.
(to be candid, as this is protected forum area: I could see the 2012 practice exams were sloppy, so I didn't annotate them...)

6. ### shanlaneActive Member

It certainly was. There were a few things we had to infer that were not necessarily obvious, like the fact that the lookback in question 1 was a floating or fixed. Didnt give a strike so it was kind of obvious, but it was still strange that it did not tell us.

Thanks again!

Shannon

• Like x 2
7. ### RK17New Member

I think there is a small typo in the last line of the solution (highlighted in red). I noticed while looking through the sample exam posted today.

"... [LN(20/13) + (10% - 0.5*20%^2)*1]/[20%*SQRT(1)] = 2.554 "

• Like x 1
8. ### David Harper CFA FRMDavid Harper CFA FRM (test)

RK17 - Yes, agreed. THANK YOU!

• Like x 1
9. ### sleepybirdActive Member

I think there's another typo:
The expected future firm value = $20*exp[(10% - 0.5*20%^2)*1] =$21.67 billion.
The lognormal DD = ($21.76 -$13.00)/(21.67*20%) = 2.000

10. ### David Harper CFA FRMDavid Harper CFA FRM (test)

Hi Sleepybird, Yes, agreed, THANK YOU, it does not impact the result of 2.000, but I transposed. Fixed above
(cc: Suzanne Evans : submitted to errata_frm for PDF correction)

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11. ### trabala38Active Member

Hi David,

A quick-one just before the exam:

While the DD computed in log-normal terms is very clear, I am not sure to follow for the DD computed in normal returns terms? Could you explain briefly? Especially the denominator part, where only the volatility and SQRT(time) appears.

Thank you.

trabala38

12. ### David Harper CFA FRMDavid Harper CFA FRM (test)

Hi trabala,

I think it is all covered in my detailed explain at http://www.bionicturtle.com/forum/threads/merton-model-a-summary-of-the-issues.5646/

i.e., In (log) normal return terms, DD is essentially similar to d2 in BSM, with the important difference that the risk-free rate is replaced by expected asset return (real world drift) so the numerator is intuitively giving a number of standard deviations form the future expected firm value to the default threshold, which the denominator then standardizes into a (unit) standard standard deviation; i.e., for use in the standard normal CDF N(.)

actually my brief youtube video might help, see below. Good luck tomorrow, please share back!

13. ### kerrNew Member

Do we assume T=1 for any DD problem?
Thanks

14. ### David Harper CFA FRMDavid Harper CFA FRM (test)

Hi @kerr No, it is absolutely a mistake in my question (that escaped notice for too long, sorry). As the Merton model includes (at least KMV-type variants) can include both long-term and short-term debt, the long-term debt maturity can be any horizon (e.g., 5 years has been used on a previous exam). This question needs to specify that the long-term debt matures in one year (i.e., implying it would have originated previously). Thank you!

Inserted "To simplify the analysis, we assume both the short-term and long-term debt will mature in one (1) year."
#revisepdf

Last edited: May 7, 2014