Know your PDs from your LGDs


As Basel II implementation is nearly upon us, more and more is heard about the ‘PD” of borrowers and the ‘LGD’s of transactions. This edition of Quant’s Corner reviews what PD and LGD are all about and the characteristics of these two important parameters.


Probability of default (PD) is a measure of how likely it is that a borrower will default over a specific time period. The period normally used is one year.


Clearly borrowers that are weaker financially are more likely to default than borrowers that are in stronger financial condition. Furthermore, no matter how good the credit standing of a borrower there is always some chance that it might default over the next year, even if that probability is very small. How can these figures be estimated



    The usual approach is to look at the historic experience of defaults and project these into the future. A large portfolio of exposures is analysed, first by assigning a credit rating to each borrower in the portfolio using a consistent rating system, and then examining the number of defaults experienced amongst borrowers in each rating band in successive one year periods during the portfolio history.


    The average number of defaults in these one-year periods for, say, all the borrowers rated ‘B’s is then the average PD for B-rated borrowers.


    Note that one could also take the worst year for defaults amongst B-rated borrowers and use that as a measure of the worst case PD for the B rating band. This will be a more conservative estimate that is likely to reflect economic downturn conditions as opposed to average conditions over the economic cycle.


    Table 1 illustrates PDs for different ratings using a conventional rating scale from AAA to CCC-. What is notable about the progression is that the rating scale is not linear to the PD.


    In fact PD rises exponentially as the ratings decrease. For example, moving from AA to A increases PD by 1.27x from 0.11% to 0.14%, but moving from BB to B increases PD by 3.12x from 2.77% to 8.65%.


    Table 1
    On year default probabilities
    AAA 0.02%
    AA+ 0.02%
    AA 0.11%
    AA- 0.14%
    A+ 0.14%
    A 0.14%
    A- 0.14%
    BBB+ 0.22%
    BBB 0.22%
    BBB- 0.54%
    BB+ 1.67%
    BB 2.77%
    BB- 2.81%
    B+ 3.65%
    B 8.65%
    B- 9.56%
    CCC+ 14.83%
    CCC 19.90%
    CCC- 46.84%




    This is important because as we saw in the last edition of Quant’s Corner, PD is an important component of Raroc calculations. Keeping cost and LGD constant (at 0.2% and 45% respectively in this example), and maintaining Raroc at 18% using a Basel II AIRB measure of capital, Chart 1 shows how margin over Libor will then vary with rating using the PDs from Table 1 for one year exposure.


    The exponential rises in PD mean that Raroc calculations are extremely sensitive to the precise borrower rating at the lower rating levels.



    Naturally PD also increases at longer maturities. Even if there is only a small probability of default in any one year, over successive years the cumulative probability of default at some stage during the whole period will steadily grow.


    This is shown in Table 2. The implication of course is that longer maturities require higher margins, particularly for lower rated borrowers where the cumulative PD increases more quickly with time.


    Table 2
    1 yr 2 yr 3 yr 4 yr 5 yr 6 yr 7 yr 8 yr 9 yr 10 yr
    AAA 0.02% 0.1% 0.1% 0.2% 0.3% 0.4% 0.5% 0.7% 0.8% 1.0%
    AA+ 0.02% 0.1% 0.1% 0.2% 0.4% 0.5% 0.7% 0.8% 1.0% 1.3%
    AA 0.1% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.7% 2.0%
    AA- 0.1% 0.3% 0.5% 0.7% 0.9% 1.1% 1.4% 1.7% 2.0% 2.3%
    A+ 0.1% 0.3% 0.5% 0.7% 1.0% 1.3% 1.6% 1.9% 2.3% 2.6%
    A 0.1% 0.3% 0.5% 0.8% 1.1% 1.5% 1.8% 2.2% 2.6% 3.1%
    A- 0.1% 0.4% 0.6% 1.0% 1.3% 1.7% 2.2% 2.6% 3.1% 3.6%
    BBB+ 0.2% 0.5% 0.9% 1.4% 1.8% 2.4% 2.9% 3.5% 4.1% 4.6%
    BBB 0.2% 0.6% 1.2% 1.8% 2.5% 3.2% 3.9% 4.7% 5.4% 6.1%
    BBB- 0.5% 1.4% 2.3% 3.4% 4.4% 5.4% 6.5% 7.3% 8.3% 9.0%
    BB+ 1.7% 3.3% 4.9% 6.4% 7.9% 9.2% 10.4% 11.4% 12.7% 13.4%
    BB 2.8% 5.3% 7.5% 9.5% 11.3% 12.8% 14.2% 15.4% 16.5% 17.4%
    BB- 2.8% 5.6% 8.4% 10.9% 13.0% 14.8% 16.5% 17.7% 18.9% 20.0%
    B+ 3.7% 7.6% 11.0% 14.1% 16.7% 18.6% 20.6% 21.7% 23.0% 23.8%
    B 8.7% 14.6% 18.7% 21.4% 23.6% 25.0% 26.2% 27.2% 28.1% 28.7%
    B- 9.6% 6.6% 21.6% 24.7% 27.2% 29.2% 30.1% 30.9% 32.0% 32.5%
    CCC+ 14.8% 23.6% 28.8% 32.2% 34.3% 35.9% 37.1% 37.7% 38.6% 38.6%
    CCC 19.9% 30.1% 36.1% 38.9% 40.8% 42.7% 42.9% 44.1% 44.9% 44.6%
    CCC 46.8% 53.6% 57.8% 59.5% 61.3% 61.9% 62.6% 62.7% 63.7% 63.1%


    Before we leave PD, we should note that as well as the probability of a borrower in the portfolio defaulting, we may also be concerned with the probability that a borrower that starts life in the portfolio at one rating subsequently has a change of financial condition and moves to another, better or worse, rating band.


    This is usually called a ‘transition’s of rating, and as well as the probability of default associated with any rating band over a one-year period, it is also common to examine the probability of transition from any one specific rating band to another specific rating band in a year period (expressed in a ‘ratings transition matrix’).


    What is left after a default



    When borrowers default it is unusual for the value of what is left to lenders to be zero (although it can happen). Residual value may be recovered in a number of different ways, including a liquidator selling the assets of the borrower to pay at least something to creditors, or a restructuring of the credit, or from selling off tied collateral or other forms of security.


    The costs of recovery must be set off against the amount recovered. These may include legal costs, costs of administrators, and the time value of recovery (for instance it may be months before a liquidator pays anything to creditors and there is no interest during that period). The amount received less the total costs is called the recovery amount, and the difference between this and the original repayment amount of the credit is called the loss given default, or LGD.


    Similarly to the estimation of PD, it is possible to estimate LGDs by examining historic experience and performing a statistical analysis of the LGDs actually encountered in the past when loans have defaulted.


    This is generally more challenging than PD estimation because whilst there is usually a reasonable population of borrowers from which to observe which ones default and which do not, when it comes to measuring LGD there are usually many fewer defaults from the same number of borrowers from which statistical observations can be made (if there were not you might wonder about the continued existence of the lender that had presided over so many defaults).


    Just like PD, LGD can be based either on an average value or on a worst case or ‘downturn’s value. The downturn value typically reflects the fact that if there are many defaults then the recovery value of credits is also likely to be lower, for instance because an economic recession has pushed larger numbers of borrowers into default and since there is a recession there are also few buyers for their residual assets.


    Basel II, which is concerned with protecting bank solvency in the medium-to-long term, requires banks to use downturn rather than average LGDs in the calculation of AIRB capital.


    LGD is typically set to reflect the characteristics of specific facility types. This is especially the case in collateralised transactions where in the event of default by the borrower the lender will take possession of the collateral and sell it to repay as much as possible of the facility.


    Under these circumstances, such as many structured commodity finance transactions, LGD has tremendous importance to the economics of credit advances to lower rated borrowers.


    As noted in previous editions of Quant Corner’, Basel II has very stringent requirements on the circumstances in which collateral can qualify for LGD reduction purposes. Chart 2 shows how a reduction in LGD due to the existence of Basel II eligible collateral markedly reduces the required margin for a ‘B’s rated borrower at constant Raroc calculated using the same basis as Chart 1.