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.


1 Answer 1


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.

  • $\begingroup$ Thanks. But this is all information that I have. One question though. When you say the conditional probability between $T_2$ and $T_1$, do you mean that, conditional that it survived between $T_1$ and $T_0$? If this is the case, can you please help understand why conditional probability is calculated in such way? $\endgroup$
    – augustine
    Nov 8, 2022 at 7:32
  • $\begingroup$ I think I meant "conditional on the underlying having started in state A". Then the total PD is $TP_{T,2,3}$ which can reach default along 3 routes: $TP_{T_1,2,1}\cdot TP_{T_2,1,3}$, $TP_{T_1,2,2}\cdot TP_{T_2,2,3}$, $TP_{T_1,2,3}$. Two first terms correspond to default in the second subperiod, last term - to default in the first subperiod. So the difference $TP_{T,2,3}-TP_{T_1,2,3}$ is PD in the second subperiod. $\endgroup$
    – Adam N.
    Feb 27, 2023 at 19:07

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