Normal DD and lognormal DD (Distance to default)

[turtle2]

David,
Can you please explain different DD used in FRM assignment readings and how are they related. In particular,
From 2010 FRM L2 Sample Question 27.3. Please explain. Thanks.
27.3 The distance-to-default in the question follows a standard normal distribution as the lognormal price distribution implies normal log returns (and so the PD becomes a simple inverse standard normal). What is the lognormal distance-to-default (bonus for showing the equivalence between the normal and lognormal DDs)? The lognormal DD uses values and is given by: lognormal DD = ($362.32 - $100) / ($362.32 * 30%) = 2.413
The normal DD (the formula in the question) is given by: normal (returns-based) DD = -1.287 / 0.671 = -1.919; i.e., NORMSINV(-1.919) = 2.75%
The normal DD can be converted into the lognormal DD: [EXP(1.919*30%*SQRT(5))*100-100]/($362.32*30%) = 2.413
Turtle 2

 

[David Harper CFA FRM]

Hi Turtle2,

It is related to the idea that the asset log return LN[S(t)/S(0)] is normally distributed which corresponds to a future asset price S(t) that is lognormally distributed. The distance-to-default gives the difference between a future expected value and the default threshold, which corresponds to a point on the distribution and, in this case, a distribution median. But this difference can be expressed two ways: in terms of a return (%) or a dollar value ($).

The formula for return(%) is dealing with the more intuitive normal returns; such that a DD = 1.92 can be converted with an inverse normal CDF (excel's NORMSDIST).

Without changing the situation, the other DD of 2.41 expresses the difference in lognormal values ; i.e., the difference between future expected firm value of $362 (actually a median on the lognormal distribution) and the $100 defaults. But this is values, so like S(t), it refers to a lognormal instead of a normal.

Put more directly, the 1.92 is standardized normal return (%) DD while the 2.41 is the correspondingn standardized lognormal future values ($) DD for the exact same setup.
… there is some follow-on in the thread for this question here http://forum.bionicturtle.com/viewthread/2554/

Hope that helps, David

 

[turtle2]

David,
Thanks for the explanation. The thread mentioned helped to derive the equivalence of two DD.
X*EXP(DD*volatile * SQRT[t]) = V0*EXP[(mu - variance/2)*T]
What should we concentrate on lognormal DD or normal DD from GARP testing perspective ?
I can not see thread which refers to 2009 FRM question on lognormal DD
http://forum.bionicturtle.com/viewthread/1392/
Error
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Any reason we can not access this ?

Thanks.

Turtle2

 

[David Harper CFA FRM]

Hi Turtle2,

Sorry about the lost thread, we lost a few in the migration to the new site. That question is still on the wiki @ http://www.bionicturtle.com/wiki/FRM2009.E2.11/

So notice in that question by GARP, they implicitly used the value-based (ie., not return-based) lognormal DD: "According to KMV, Default Point = STD + ½ LTD = 500 +1/2(300) = 650. Distance to default = (Market value of asset at time 1 - Default Point)/Annualized Asset volatility at time 1 = (1200‐650)/ (1200*0.1) = 4.58."

In regard to "What should we concentrate on lognormal DD or normal DD from GARP testing perspective?" … Safest is both as GARP has deployed both in the past two years. Strictly speaking, the returns-based question (e.g., http://forum.bionicturtle.com/viewthread/2554/) is nearer to the de Servigny assignment so that is probably more likely.

David