Credit Risk KMV approach : Distance to Default Equation ?

[Anonymous]

Hi,
In the PRM Handbook and in the RiskPrep tutorial about KMV approach, I read the following equations of Distance to Default (DD)
DD = ( E(V1) – DPT ) / sigma
which “can be expressed another way”, as follows
DD = ( ln (V0/DPT) + (mu – sigma²/2).T ) / ( sigma.sqrt(t) )

Might be obvious but I just don’t understand how to obtain the 2nd equation from the 1st. I suppose you do this using the log-normal distribution of V, but what are the steps of the calculation ?

Thanks.

[kzg]

Credit Risk KMV approach : Distance to Default Equation ?

[Anonymous]

Kzg,

You are right, as I read it, it does look like a leap of faith from one equation to the next. Here is an explanation, hope this helps.

The formulas are practically identical. Think of the first equation as being expressed in absolute dollar terms, ie, the numerator E(V1) – DPT represents the dollar value that needs to be lost to hit the default point. Dividing that by the standard deviation (in dollars) gives you the number of standard deviations away you are from the default point.

For the second equation, think of measuring the same distance-to-default in terms of returns measured not in dollars but in percentages (continuously compounded).

Assume for a moment that the default point, DPT = $100, and the current value of assets, V0 = $120. Therefore during the next year we can afford to lose:
(a) $20, expressed as a percentage return (of the DPT), plus
(b) Whatever we will earn in the coming year (as a percentage).

The $20 when expressed as a continuously compounded return in terms of log returns is ln($120/$100) (about 18%), or =ln(V0/DPT). To this we need to add what we will earn what we will earn in the coming year. That number is ?, or to be precise ? – (?^2)/2, because volatility reduces continuously compounded returns. (See separate tutorial here that explains how volatility kills returns, though for short periods the adjustment of “– (?^2)/2” isn’t really needed. http://www.riskprep.com/all-tutorials/35-exam-1/129-volatility-returns-and-stock-prices).

So the numerator ends up being [ln(V0/DPT) + (? – (?^2)/2)T]

The denominator would be ?, which in its more pure form is expressed as ??T just in case ? is for a different time period than the one under consideration. (WHich is also why ? – (?^2)/2 ends up getting multiplied by T.)

Thus the distance to default can be expressed as

DD = [ln(V0/DPT) + (? – (?^2)/2)] / ??T

[Author]

Posts