# Calculating the long run average default rate when the portfolio changes during the year

The Basel rules prescribe to calculate a long run average default rate (LADR). It is stated that his rate should be calculated as the average of yearly default rates.

A first idea what be:

• look at the number of living exposures at each year end $t$ - say $L_t$
• count the number of defaults among $L_t$ from $t$ to $t+1$ - say $D_t$
• calculte $d_t = D_t/L_t$
• calculate the average $\hat{D} = \frac1n \sum_{i=1}^n d_t$

The above estimator $\hat{D}$ has nice properties as the $d_t$ do not overlap.

But ... In practice if we think of accounts or exposures that only exist for a short period of time (sometimes much less than a year) it can happen that the portfolio fluctuates significantly during a given year. In the above we miss all those living exposures and probably defaults too.

What is market practice to calculate Basel-compliant LADRs that take into account fluctuating portfolios?

I don't know the answer to this and am responding purely out of interest and idea sharing, since I like the question.

Could this work as a Parametric intensive computational statistical approach;

1) Define a universe of positions the portfolio can feasibly hold, and model each with some assumption about default over a given time period (statically at first for simplicity).

2) Initially create your portfolio with a random allocation of a subset of feasible positions (all same notional for simplicity).

3) Randomly simulate some defaults within your feasible space and record those held in your portfolio.

4) Parametrise the 'churn' of your portfolio since this characterises its fluctuation; after each period choose $N$ closed positions and $M$ new positions where $N$, $M$ are your parametrised random variables and then randomly select those numbers of closed and new positions from the relevant sets. (choose from distributions that average a constant size portfolio)

5) Repeat 3) and continue until enough periods have been measured to return a default rate over the complete period.

6) Perform enough experiments to take an expectation of the long term default rate.

Under this procedure, if the rate of default on every position was the same, I expect that mathematically the churn parameter will do nothing to the expectation of the long term default rate. It will however increase the variance of the estimator since for each trail the number of holdings can increase significantly or reduce significantly and this will impact the final result.

I recognise a strong critcism of this model is that it is parametric and will likely miscapture some facets, and of course requires calibration. The problem with your approach, although, non-parametric is that it assumes the position is held for a period of time that it may indeed not be. I'll have a think about how one might go about a non-parametric bootstrap procedure, to account for this.

• Thank you, this is in an interesting approach! As all Basel-compliant (IRBA) banks have to calculate a LADR, I assume that there is some market practice out there :) Thus I appreciate your experiment and it made me think in new ways abou this but I besides that I need market practive too. – Ric Jul 24 '18 at 6:11
• @Richard it is also entirely possible that there are numerous regulator approved methods in operation at different banks. If you have any contacts and find the answer please do post it here. – Attack68 Jul 25 '18 at 11:23