Assuming that loan values are normally distributed, the variance of loan value (in millions) around its mean is $8.9477 (squared), and its stan- dard deviation, or volatility, is the square root of the variance, equal to
$2.99 million. Thus, the 5 percent VARfor the loan is 1.65×$2.99=$4.93 million. The 1 percent VARis 2.33×$2.99=$6.97 million. However, this likely underestimates the actual or true VARof the loan because, as shown in Figure 6.3, the distribution of the loan’s value is clearly non-normal. In particular, it demonstrates a negative skew or a long-tailed downside risk.
Using the actual distribution of loan values and probabilities in Table 6.4, we can see that there is a 6.77 percent probability that the loan value will fall below $102.02, implying an “approximate” 5 percent actual VAR of $5.07 million ($107.09−$102.02=$5.07 million), and there is a 1.47 percent probability that the loan value will fall below $98.10, implying an “approximate” 1 percent actual VAR of $8.99 million ($107.09−
$98.10=$8.99). These actual VARs could be made less approximate by using linear interpolation to get at the 5 percent and 1 percent VARmea- sures. For example, because the 1.47 percentile equals $98.10 and the 0.3 percentile equals $83.64, using linear interpolation, the 1.00 percentile equals approximately $92.29. This suggests an actual 1 percent VAR of
$107.09−$92.29=$14.80 million.15
in Table 6.4, the economic capital requirement would be $14.80 million for unexpected losses plus the loan loss reserve for expected losses of $0.46 mil- lion (an amount much greater than the BIS II capital requirement).
Using the CreditMetrics approach, every loan is likely to have a differ- ent VAR and thus a different implied or economic capital requirement. If regulatory capital requirements were based on an internal model using CreditMetrics, regulators would most likely require that the VARestimate be increased using a stress-test multiplier. In particular, the 99 percent loss- of-value estimate can be expected to have a distribution. In extremely bad (catastrophic) years, a loan’s loss will exceed, by a significant margin, the 99 percent measure calculated in the previous example. Under the BIS ap- proach to market risk, this extreme loss or stress-test issue is addressed by requiring banks to multiply their VARnumber by a factor ranging between 3 and 4. Research by Boudoukh, Richardson, and Whitelaw (1995) shows (in simulation exercises) that, for some financial assets with normally dis- tributed returns, the 3-to-4 multiplication factor may well pick up extreme losses such as the mean in the tail beyond the 99th percentile.17Applying such a multiplication factor to low-quality loans would raise capital re- quirements considerably. The introduction of an internal ratings-based (IRB) approach to capital requirements makes the estimation of the appro- priate size of such a multiplication factor particularly important, given the problems of stress-testing credit risk models (see Chapter 12).
Using CreditMetrics to set capital requirements tells us nothing about the potential size of losses that exceed the VARmeasure. That is, the VAR measure is the minimumloss that will occur with a certain probability. Ex- treme Value Theory (EVT) examines the tail of the loss distribution condi- tional on the expectation that the size of the loss exceeds VAR.18Tail events are those loss events that occur rarely, but when they do, they have dramatic consequences.19Figure 6.4 depicts the size of unexpected losses when cata- strophic events occur.20Using the estimates from Table 6.4 assuming a nor- mal distribution, the 5 percent VARfor unexpected losses is $4.93 million.
We set this to be the threshold level; that is, EVT considers only the distri- bution of unexpected losses that exceed $4.93 million. However, Figure 6.4 assumes that unexpected losses beyond the 95 percent threshold level fol- low the Generalized Pareto Distribution (GPD) with “fat tails;” see Appen- dix 6.2 for derivation of the values shown in Figure 6.4. Thus, the estimated 1 percent VAR, distributed according to the GPD is larger than the nor- mally distributed 1 percent VARof $6.97 million (from Table 6.4). Under the parameter assumptions described in Appendix 6.2, the 1 percent VARfor the GPD, denoted , is $22.23 million. The Expected Short- fall, denoted , is calculated as the mean of the excess distribution of unexpected losses beyond the threshold VAR.99, which is shown as $53.53
ES.99
VAR.99
million in Figure 6.4. This would be the capital charge for the mean of the most extreme events (i.e., those in the 1 percent tail of the distribution).
As such, the amount can be viewed as the capital charge that would incorporate risks posed by extreme or catastrophic events, or alternatively, a capital charge that internally incorporates an extreme, catastrophic stress- test multiplier. Since the GPD is fat tailed, the increase in losses is quite large at high confidence levels; that is, the extreme values of (i.e., for high values of q,whereqis a risk percentile) correspond to extremely rare catastrophic events that result in enormous losses. Some have argued that the use of EVT may result in unrealistically large capital requirements [see Cruz et al. (1998)].
ESq
ES.99
FIGURE 6.4 Estimating unexpected losses using extreme value theory. Note:
ES=the expected shortfall assuming a Generalized Pareto Distribution (GPD) with fat tails.
0 Mean Distribution
of Unexpected
Losses
Probability
$4.93 $6.97 $22.23 $53.53 GPD Normal Distribribution
95%
VAR Normal
Dist.
99%
VAR Normal
Dist.
99%
VAR GPD
ES Mean of Extreme Losses Beyond the 99th Percentile
VAR under the GPD