Interpreting Cumulative Default Probabilities in a Rating Migration Model

# Strange calculation for Credit risk


# Strange calculation for Credit risk












One of the measures to quantify credit risk is to calculate the Expected loss, which is typically quantifies as $EL = EAD \times PD \times LGD$

However, I have come across a somewhat strange calculation which goes as below.

Let say we have a exposure of $ \\\$1 $. Calculation time is today at $T_0$, and we are interested to calculate the Expected loss in time period $T_2$. So the direction of time period is $T_0, T_1, T_2$.

Current rating is $A$, and let say possible states of rating are $A+, A, D$, the $D$ stands for default.

We have point estimates of the `Transition matrix` at time $T_1,T_2$ are ${\left(TP\right)}_{T_1}$ and ${\left(TP\right)}_{T_2}$. Therefore, two periods `total/cumulative` estimate of `Transition matrix` is ${\left(TP\right)}_{T} = {\left(TP\right)}_{T_1} \times {\left(TP\right)}_{T_2}$. The $i,j$ -th element of ${\left(TP\right)}_{T}$ is ${\left(TP\right)}_{T,i,j},i,j=1,2,3$.

Expected loss is calculated with below formula

$\\\$1\left({\left(TP\right)}_{T,2,1} \times \left({\left(TP\right)}_{T,1,3} - {\left(TP\right)}_{T_1,1,3}\right) + {\left(TP\right)}_{T,2,2} \times \left({\left(TP\right)}_{T,2,3} - {\left(TP\right)}_{T_1,2,3}\right)\right) \times LGD$

Do you think above formula is correct to estimate the two period Expected loss? I dont see this kind of formula in any text book. So I wonder if this is requirement arise from some Regulator?

Your pointer will be highly appreciated.

Thanks for your time.

## Answer by Adam N. (score 1)

https://quant.stackexchange.com/a/73434

The $TP_{T,2,1} \times \left(TP_{T,1,3} - TP_{T_1,1,3}\right) + TP_{T,2,2} \times \left(TP_{T,2,3} - TP_{T_1,2,3}\right)$ term is clearly supposed to correspond to PD, but its actual meaning is unclear.

For example ${TP}_{T,2,1} \times \left(TP_{T,1,3} - TP_{T_1,1,3}\right)$ could be interpreted as the probability of reaching the state A+ in $T_2$ times the probability of defaulting in the second subperiod if one starts in A+ instead of A. It seems to mix and match incompatible concepts. Perhaps someone wanted to enumerate paths to default and became confused by the indices (first factor assumes we end up in A+ at $T_2$, second one assumes we start at A+ at $T_0$).

Typically, cumulative PD between $T_0$ and $T_2$ would correspond to just $TP_{T,2,3}$. Or perhaps you want the conditional PD between $T_1$ and $T_2$, which would be $TP_{T,2,3}-TP_{T_1,2,3}$, which does appear as a part of that formula, but the rest of it is confusing.

I'm not aware of any regulatory requirement to calculate PD in this fashion, although credit risk modelling is only adjacent to my area of expertise.