Table 7.3 shows the legal liability losses (measured in pounds Sterling) of a financial institution. Table 7.4 presents the statistical characteristics, and
TABLE 7.3 Legal Liability Losses for a Financial Institution £45,457.72 £319,574.15 £13,476.34 £239,542.60 £49,852.51 £12,367.19 £112,964.44 £121,936.07 £331,168.35 £186,043.30 £188,713.75 £545,837.15 £51,668.72 £102,553.18 £1,255,736.19 £25,615.32 £35,143.21 £129,854.17 £6,155.91 £22,456.23 £30,178.48 £64,264.76 £7,117.74 £56,025.11
£214,368.33 £291,854.21 £11,238.92 £84,264.07 £8,251.00 £2,925.20 £87,430.52 £378,068.07 £514,502.77 £220,825.91 £110,627.92 £52,224.61 £308,479.08 £51,360.39 £399,251.55 £180,907.21 £169,753.33 £181,396.37 £20,510.31 £25,417.34 £150,950.47 £268,416.52 £451,384.30 £25,766.68
£55,008.91 £677,797.15 £375,798.31 £7,695.82 £191,892.23 £64,517.17 £68,867.22 £22,209.74 £7,164.50 £213,582.12 £114,511.58 £27,940.46 £707,734.82 £42,013.46 £66,681.67 £241,691.87 £276,554.41 £39,293.09 £144,190.61 £13,590.82 £93,453.08 £103,060.97 £59,260.91 £108,325.56
£186,097.42 £376,122.45 £21,512.68 £125,772.44 £103,984.82 £21,475.51 £606,086.68 £189,120.36 £79,425.85 £63,243.13 £106,674.73 £14,182.63 £45,162.70 £159,030.78 £150,726.18 £161,950.80 £127,024.01 £166,502.54 £18,224.26 £98,277.79 £281,012.41 £111,249.87 £80,543.89 £140,874.19
£260,375.11 £11,020.21 £22,005.92 £46,965.45 £73,633.52 £460,861.10 £51,541.79 £40,925.99 £183,104.87 £70,100.90 £13,584.55 £4,707.90 £199,784.16 £109,489.79 £279,630.93 £171,992.38 £83,209.51 £68,929.68 £149,043.99 £82,322.75 £76,766.50 £261,539.04 £326,492.18 £36,622.85
£192,734.99 £215,612.22 £116,020.84 £44,695.30 £166,319.02 £19,460.71 £126,015.11 £255,876.89 £394,002.33 £5,136.23 £151,326.54 £2,754.23 £48,891.19 £411,160.62 £22,322.99 £21,278.36 £346,524.34 £208,969.85 £58,910.57 £58,510.25
87
TABLE 7.4 Statistical Characteristics of Legal Liability Losses
Mean £151,944.04
Median £103,522.90
Standard deviation £170,767.06
Skew 2.84
Kurtosis 12.81
Percentage
Figure 7.12 illustrates the corresponding histogram. Several interesting points are evident. First, the mean of the sample is considerably larger than the median, which is reflected in a coefficient of skew equal to 2.83.
Second, the losses are very fat tailed, with a kurtosis in excess of 12.
Since the losses are not symmetric, we would not expect them to come from a normal distribution. This is confirmed in the probability plot of Figure 7.13, for which the Anderson-Darling test statistic is 8.09. As the mean of the data lies close to the standard deviation, and given the shape of the histogram shown in Figure 7.12, we postulate that the data comes from an exponential distribution. This appears to be confirmed in the probability plot of Figure 7.14, for which the Anderson-Darling test sta- tistic is 0.392. Therefore, we conclude that an exponential distribution with α = 151,944.04 adequately describes this data. Figure 7.15 shows the fitted exponential distribution against a histogram of the actual data.
Is this the only distribution that can fit this data reasonably well? The answer is probably not. To see this, consider Figure 7.16, which shows a
60%
50%
40%
30%
20%
10%
0%
£114 , 201
£228 ,354
£342 ,508
£456 ,661
£570 ,815
£684 ,968
£799 ,122
£913 ,276
£1,02 7,429
£1,14 1,583
£1,76 9,451 Value
FIGURE 7.12 Histogram of legal event losses.
0 500000 1000000 1
5 10 20 30 40 50 60 70 80 90 95 99
Percent
Data
FIGURE 7.13 Normal Probability Plot for legal event losses.
89
0 500000 1000000
10 30 50 60 70 80 90 95 97 98 99
Data
Percent
FIGURE 7.14 Exponential Probability Plot for legal event losses.
0 0.1 0.2 0.3 0.4 0.5 0.6
£114,201
£228,354
£342,508
£456,661
£570,815
£684,968
£799,122
£913,276
£1,027,429
£1,141,583
£1,141,583
Probability
Value
FIGURE 7.15 Fitted Exponential distribution(solid line) and legal event losses.
FIGURE 7.16 Weibull Probability Plot for legal event losses.
1000000 100000
10000 1000
100 99 9590 8070 6050 40 30 20 10 5 3 2 1
Data
Percent
91
Severity of Loss Probability Models
probability plot for the Weibull distribution. The plot appears to indicate that the Weibull distribution fits the data at least as well as the exponential distri- bution. Furthermore, the Anderson-Darling test statistic is only 0.267. Should we use the exponential distribution or the Weibull distribution? One argu- ment in favor of the exponential distribution is that it depends on only one parameter and is therefore more parsimonious than a Weibull distribution.
However, given that we have less than 200 observations (with more coming in the future), the added flexibility of the Weibull distribution provides an argument for choosing it.
SUMMARY
Fitting appropriate severity of loss probability models is a central task in operational risk modeling. It involves first selecting an appropriate pro- bability distribution from a wide range of possible distributions and then assessing how well the selected model explains the empirical losses. Statis- tical or graphical methods can be used to assess model fit. Although much of this work can be carried out in Excel, it will also be necessary to write addi- tional statistical functions and estimators in VBA. In the following chapter we continue with the theme of fitting probability distributions to empirical data, considering the frequency of events rather than their severity.
REVIEW QUESTIONS
1. Why is the normal probability distribution not necessarily a good choice for modeling severity of losses? Under what circumstances would you envisage using the normal distribution?
2. Given loss data which lies between the range of $0 and $75,000, what transformation do you need to apply before you could fit the standard beta distribution?
■ Calculate the mean, standard deviation, skew and kurtosis of the following loss data:
$ 4,695.11 $ 147.86 $24,757.17 $12,928.33
$ 9,073.66 $ 215.62 $13,647.10 $ 8,283.56
$19,353.32 $ 2,965.99 $ 9,510.18 $ 3,981.05
$15,669.64 $ 5,976.45 $ 1,643.43 $ 5,002.29
$ 4,354.27 $ 8,003.83 $ 664.28 $ 1,232.87
$13,817.18 $21,502.09 $ 5,128.66 $ 9,403.65
$ 384.13 $10,270.70 $ 7,993.91 $ 4,504.84
$ 8,386.20 $23,631.04 $ 1,690.75 $ 3,759.90
$17,237.93 $ 4,508.38 $ 1,915.58 $15,211.48
$ 745.39 $ 6,841.31 $ 3,385.31 $ 3,690.16
■ Given your estimates, which probability models of those given in this chapter can you rule out as being unsuitable for modeling this data?
3. Fit the beta, lognormal, and normal probability functions to the above data and determine which (if any ) of the fitted distributions is adequate for this data.
4. Explain the difference between an estimator and an estimate.
5. Suppose X follows a beta distribution. Create a worksheet to illustrate that using the following transformation Y = aX + b, we can get almost any shaped density on the interval (b, a + b).
FURTHER READING
Details of the distributions given in this chapter and many others can be found in Gumbel (1954), Aitchison and Brown (1957), Ascher (1981), Hahn and Shapiro (1967), Johnson et al. (1994, 1995), and Lewis (2003).
Further information on goodness of fit tests can be found in Press et al.
(1995).
I
CHAPTER 8
Frequency of Loss Probability Models
n the previous chapter we learned how to estimate and assess severity of loss probability models. In this chapter we focus on frequency of loss models. The entity of interest for frequency of loss modeling will be a dis- crete random variable that represents the number of OR events observed.
These events will occur with some probability p.1
Our discussion begins with three popular frequency of loss probability models: the binomial distribution, the Poisson distribution, and the negative binomial distribution. This is followed by a discussion of alternatives to these models. Since assessing the goodness of fit of a postulated frequency of loss probability model is an important issue, we introduce a formal test statistic known as the chi-squared goodness of fit test. This is followed by a detailed case study used to illustrate the process of building a frequency of loss probability model.