ESTIMATING LOSS SEVERITY DISTRIBUTION: CONVOLUTION APPROACH

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1 Journal o Mathematics and Statistics 10 (): 47-54, 014 ISSN: doi: /jmssp Published Online 10 () 014 ( ESTIMATING LOSS SEVERITY DISTRIBUTION: CONVOLUTION APPROACH Ro J. Pak Department o Applied Statistics, Dankook University, Korea Received ; Revised ; Accepted ABSTRACT Financial loss can be classiied into two types such as expected loss and unexpected loss. A current deinition seeks to separate two losses rom a total loss. In this article, however, we redeine a total loss as the sum o expected and unexpended losses; then the distribution o loss can be considered as the convolution o the distributions o both expected and unexpended losses. We propose to use a convolution o normal and exponential distribution or modelling a loss distribution. Subsequently, we compare its perormance with other commonly used loss distributions. The examples o property insurance claim data are analyzed to show the applicability o this normal-exponential convolution model. Overall, we claim that the proposed model provides urther useul inormation with regard to losses compared to existing models. We are able to provide new statistical quantities which are very critical and useul. Keywords: Convolution, Heavy-Tailed Distribution, Loss, Value at Risk (VaR) 1. INTRODUCTION Operatinal risk is one o the most important risks or commercial banks; hence, it is essential to estimate the economic capital or operational risk. Operational risk is deined as the risk arising rom inadequate or ailed internal processes, people and systems or rom external event (BCBS, 001; 004a). The Basel report (BCBS, 001; 004b) suggests three methods or calculating operational risk capital charges: (i) The basic indicator approach, (ii) the standardized approach and (iii) the Advanced Measurement Approach (AMA). In particular, the Loss Distribution Approach (LDA) is the major part o the AMA. Here, estimating a loss severity distribution is one o the key processes. The parametric models in particular, such as the lognormal, the Weibull and generalized Pareto distributions, are used to estimate a particular severity distribution. Sometimes, mixture models combining the parametric models are used (Carvalho and Marinho, 007). In this article, a new type o loss severity distribution is suggested and the suggested distribution is based on the convolution o an expected loss severity distribution 47 and an unexpected loss severity distribution. Unlike the existing models, the new model provides the expected values and the VaR s or both expected and unexpected losses. Even when a uture loss is anticipated, the new model provides the conditional expected values given a loss as well as the conditional VaR have given a loss or both expected and unexpected losses. In section and 3, we briely review the deinition o loss and introduce the normal-exponential convolution model (norm-exp). Further, in section 4, two examples are investigated or veriying the useulness o the normal-exponential convolution model in estimating loss severity.. REDEFINITION OF LOSSES The Basel report (BCBS, 001a; 004b) classiies inancial losses due to operational actors into the ollowing two types: Expected losses: These are considered as the normal losses that occur requently, as part o everyday business, with low severity. Examples include losses due to accidentally miscalculated oreign exchange transactions

2 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, 014 Unexpected losses: These are the unusual losses that occur rarely and have a high severity. Examples include losses resulting rom a major raud activity The current Basel II regulatory text does not provide a description/deinition o the Expected Loss (EL). Consequently, discussion has oten luctuated between three dierent interpretations o EL (FSA, 005): A business/management style deinition in which EL is related to a uture amount o expense/loss that is predicted on the basis o past experience, e.g., or credit card raud; past experience o losses allows a projection o uture losses, which is budgeted/priced or. The events giving rise to these uture losses have not yet occurred A mathematical style deinition in which EL is equated to the mean (50th percentile) o a loss distribution A inancial accounting style deinition in which EL, which describes losses expected rom identiied events, or which a reserve has been established. A common example o this is where a large legal cost is anticipated, but the exact amount o the legal settlement is not yet known Figure 1 shows the distinction between expected and unexpected losses rom the loss distribution point o view. The demarcation is purely arbitrary; however, the expected value o loss (E[L]) is oten used as a point separating the expected loss and unexpected loss. In order to quantiy both expected and unexpected losses, it is essential to ind a loss distribution whose tail represents the unexpected losses (Neil et al., 005). Unlike the existing deinition o losses identiied by a separation (Fig. 1), we would like to redeine the (total) loss as a sum o the expected loss and unexpected loss, loss expected loss + unexpected loss, such that the distribution o loss is a convolution o the distributions o the expected loss and unexpected loss. The existing deinition says that loss is really a numerical summation o expected and unexpected losses, while we propose that loss is a sum o random variables in probabilistic sense. The general consensus is that heavy-tailed distributions it historically observed losses better than light-tailed distributions; the typical probabilistic models that have proposed to describe the severity o losses are the Lognormal, Gamma and Weibull distribution. We propose to let the expected loss distribution and the unexpected loss distribution be the truncated normal distribution and the exponential distribution, respectively (Fig. ). Fig. 1. Loss distribution 48

3 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, 014 Fig.. Expected and unexpected losses 3. NORMAL-EXPONENTIAL CONVOLUTION I we let X and Y represent the expected loss and unexpected loss, respectively, then S represents the loss which is X+Y. Let X be normally distributed with mean µ and variance σ, but truncated at 0 and Y be exponentially distributed with mean 1/α. Then, S X+Y has a density: S(s; α, µ, σ ) αexp α σ α(s µ ) 1 µ µ s Φ + ασ Φ + ασ, σ σ where Φ( ) is a cumulative normal distribution. The s ( ) is called as the norm-exp density (Bolstad, 004). Parameter µ controls the horizontal position, parameter σ controls the width near peak and parameter α controls the length o the tail (Fig. 3). The conditional density o X given S and Y given S are: (x s) X S 1 x + a s 1 y a ϕ ϕ b b, Y S(y s) a s a a s a Φ + Φ 1 Φ + Φ 1 b b b b 49 where, a s-µ-ασ and b σ, respectively. The (conditional) expected value o the expected loss given the loss becomes: E(X S s) a s a a s a b 1 µ + ασ ϕ ϕ Φ + Φ And the (conditional) expected value o the unexpected loss given the loss then becomes: E(Y S s) s E(X S s). VaR(X S), VaR(Y S), which can be obtained by taking the percentile o the corresponding conditional loss distributions at the desired conidence. All o the mathematical expressions in this section can be obtained by closely ollowing Bolstad (004), while Bolstad (004) provided the expressions only or Y. We add some mathematical derivations about the norm-exp at the end o this article. Example 1 4. DATA ANALYSIS The Danish data on large ire insurance losses, which were collected by a Danish reinsurance company rom Jan. 3, 1980 to Dec. 31, 1990, are considered as an example.

4 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, 014 Fig. 3. Various norm-exp densities This dataset contains,167 individual losses in million Danish Krones. The dataset is accompanied with the R package itdistr and it analyzed with R packages o both itdistr and limma. To conirm the model adequacy, some tests have been conducted and statistics are listed in Table 1. The critical value o K-S test at the 5% signiicant level can be calculated by / n (Barrett and Donald, 003), which is 0.09 or this dataset. All o the models we considered does not provide perect goodness o it. Among the our models, the Lognormal its the data best; norm-exp, Gamma and Weibull ollowed in order in terms o the log likelihood and Akaike Inormation Criterion (AIC). The VaR o loss, VaR(S), based on the norm-exp is about 7.575, which is slightly larger than by the lognormal distribution but smaller than the VaR s by Gamma and Weibull (Table ). In Fig. 4, we plot a histogram o losses and estimated densities or the total, expected and unexpected losses. The density o expected losses is primarily located near 1.0 but the density o unexpected losses has a long tail. The sum o VaR(X) and VaR(Y) ( ) is also slightly larger than VaR(S) ( 7.575) that is, VaR(S) < VaR(X) + VaR(Y), as expected by the nature o VaR. VaR(S) is accounted mainly by VaR(Y) (VaR by unexpected loss). We can observe that E[X S] and VaR[X S] are being bounded when s is over 1.0 and 50 E[Y S] and VaR[Y S] are denominating E[X S] and VaR[X S] ater s is beyond 1.7, that is, i the (total) loss is larger than 1.7, the loss is taken accounted primarily by the unexpected loss. Example We deal with the Korean property insurance claim dataset as an example, which consists o the monthly total claims (losses) rom March 00 to January 009 in Korea. The claim is monthly collected by the Insurance Statistics Inormation Service in Korea. The recorded data have been adjusted in 100 billion Korean won (U.S $ Korean won). The Weibull density is best itted or the given dataset rom all criteria (Table 3). The norm-exp density is also suitable or the given data based on the K-S test. Figure 5a and b Claims seem to ollow norm-exp (0.94, , ). Weibull and gamma it the data well as a whole, but lognormal and norm-exp it the peak better than Weibull and gamma. The density o the expected claim is estimated as N(0.4888, ) and that o the unexpected claim is estimated as Exp(0.94). We have E[X] , E[Y] and E[S] Figure 5b We have VaR[X] , VaR[Y] , VaR[S] (VaR[S] < VaR[X] + VaR[Y]). The VaR based on norm-exp is the second largest among the our VaR s (Table 1).

5 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, 014 Table 1. Goodness-o-it statistics Distribution Parameters -Log likelihood AIC K-S Gamma shape Rate Weibull shape scale Lognormal meanlog sdlog Norm-exp rate.599 mean sd A-D: Anderson-Darling statistic, CVM: Cramér-Von Mises statistic, K-S: Kolmogorov-Smirnov statistic Table. VaR (95%-quantile) Distribution Gamma Weibull Lognormal Norm-exp normal (expected loss) Exp (unexpected loss) VaR Table 3. Goodness-o-it test statistics and VaR (95%) Distribution Parameters -Loglikelihood A-D CVM K-S VaR Lognormal mean s. d (0.00) (0.00) (0.1) Gamma shape Rate (0.00) (0.00) (0.39) Weibull shape Scale (0.0058) (0.13) (0.60) Norm-exp rate mean s.d (0.00) (0.00) (0.5) A-D: Anderson Darling statistic, CVM: Cramér-Von Mises statistic, K-S: Kolmogorov-Smirnov statistic. p-values are in parentheses. Fig. 4. Loss, expected and unexpected losses; E[X S], VaR[E S], E[Y S] and VaR[Y S] 51

6 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, 014 (a) (b) (c) (d) Fig. 5. Related plots or the Korean property insurance claim data (a) itted densities (b) normal and exponential: VaR s (c) Expected and unexpected cla (d) E[X S], E[Y S], VaR[X S] and VaR[Y S] Figure 5c Claims decrease in the summer and increase in the winter. The same pattern is repeated every year. E[X S] and VaR[X S] are bounded by and 1.047, respectively. Figure 5d As claim increases, E[Y S] and VaR[Y S] are increasing while E[X S] and VaR[X S] are bounded by and 1.047, respectively. I a claim is less than , E[Y S] < E[X S]; i a claim is less than 1.05, VaR[Y S] < VaR[X S]. Beyond a claim o or 1.05, the inequalities are reversed. Furthermore, when a claim is larger than , both E[X S] and VaR[X S] become less than E[Y S] and VaR[Y S]. When a claim is lower than or 1.05, the expected loss dominates the unexpected loss. More speciically, it can be said that 5 a loss is as usual, as expected. When a claim is between 1.05 and , the unexpected loss tends to be over the expected loss; however, it is still under the expected boundary. Yet, when a claim is larger than , losses are taken accounted primarily by the unexpected losses. The norm-exp its the data well, especially in the peak area (Fig. 4a). Claims are mainly aected by unexpected claims while expected claims are somehow bounded (Fig. 4c). Approximately 1.0 (100 billion Korean dollars), where E[Y S] > E[X S] and VaR[Y S] > VaR[X S], is the turning point where we should begin to seriously worry about unexpected losses (Fig. 4b, d).

7 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, DISCUSSION Most o quantities in the above examples are not available under the current deinition o expected and unexpected losses. We are able to provide new statistical quantities which, we believe, are very critical and useul. We can provide more concrete and probabilistic relation between expected and unexpected losses on condition o (total) loss. Many representative researches (Resnick, 1997; McNeil et al., 010; Cooray and Ananada, 005) supplied lot o important results on estimating losses, but did not give much inormation about what happened on expected and unexpected losses when a loss changed or when a loss was conditioned on. By the proposed method, we can visualize how expected and unexpected losses behave as a loss moves. 6. CONCLUSION In this study, we redeine loss as a sum o expected and unexpected losses. The loss ollows a distribution, which is reerred to as the normal-exponential distribution (norm-exp). Once data is itted well by the norm-exp, we can obtain useul inormation with regard to the expected and unexpected loss unlike the other existing distributions. In this article, we only consider the exponential distribution or unexpected loss, but there are certainly the other distributions with a long tail like Lognormal and Gamma, etc. or modeling unexpected loss. Convoluting those distributions would be a good topic or uture research. 7. REFERENCES BCBS, 001. Consultative paper on operational risk. Consultative Paper, Bank o International Settlement, Basel Committee on Banking Supervision. BCBS, 004a. Working paper on the regulatory treatment o operational risk. Basel Committee on Banking Supervision. BCBS, 004b. Basel II: International convergence o capital measurement and capital standards: A revised ramework. Bank or International Settlement. Bolstad, B.M., 004. Low-level analysis o high-density oligonucleotide array data: Background, normalization and summarization. Ph.D. Thesis, University o Caliornia, Berkeley. 53 Barrett, G.F. and S.G. Donald, 003. Consistent tests or stochastic dominance. J. Economet. Society, 71: DOI: / Carvalho, A.X. and A.S. Marinho, 007. Mixture models in operational risk. Development o ObjectRisk. Cooray, K. and M. Ananda, 005. Modeling actuarial data with a composite lognormal-pareto model. Scandinavian Actuarial J., 005: DOI: / McNeil, A.J., R. Frey and P. Embrechts, 010. Quantitative Risk Management: Concepts, Techniques and Tools. 1st Edn., Princeton University Press, Princeton, ISBN-10: X, pp: 608. FSA, 005. Treatment o expected losses in capital calculations. Drat by FSA AMA Quantitative Expert Group. Financial Services Authority. Neil, N., N. Fenton and M. Tailor, 005. Using Bayesian networks to model expected and unexpected operational losses. Risk Anal., 5: PMID: Resnick, S., Discussion o the Danish data on large ire insurance losses. Astin Bulletin, 7: DOI: /AST Appendix The ollowing proo closely ollows Bolstad (004) except that densities o X and Y have been switched and so the order o integrations also is changed. Let X be normally distributed with mean µ and variance σ but truncated at 0 and let Y be exponentially distributed with mean 1/α. The joint density o X and Y is just product o densities: ( α,x; µ, σ, y) X,Y 1 x µ αexp( αy) ϕ, where 0 < x, y σ σ The total loss S is deined as SX+Y and then the joint distribution o X and S is with the absolute value o the Jacobian J 1: X,S ( α,x; µ, σ,s) 1 x µ αexp( α(s x)) ϕ,where 0 < x < s. σ σ

8 Ro J. Pak / Journal o Mathematics and Statistics 10 (): 47-54, 014 The marginal distribution o S is: s 1 x µ S( α, µ, σ,s) αexp( α(s x)) ϕ dx 0 σ σ (s µ )/ σ µ / σ (s µ )/ σ µ / σ ( ) αexp( α(s σw µ )) ϕ w dw, where w (x µ ) / σ αexp( α(s µ )) 1 w exp( ασw) exp dw, π α σ αexp( α(s µ ))exp (s µ )/ σ µ / σ (s µ )/ σ ασ µ / σ ασ 1 (w ασ) exp π α σ αexp( α(s µ ))exp where z w ασ 1 z exp dw, π α σ αexp α(s µ ) µ µ s Φ + ασ Φ + ασ σ σ dw The conditional density o X on S and the corresponding expectation are: X,S(x,s) X S(x s) (s) S 1 x µ αexp( α(s x)) ϕ σ σ 1 µ µ s αexp α σ α(s µ ) Φ + ασ Φ + ασ σ σ 1 1 exp (x ) µ ασ σ πσ µ µ s Φ + ασ Φ + ασ σ σ 1 1 exp (x ) µ ασ σ πσ s µ µ Φ ασ + Φ + ασ 1 σ σ 1 x + a s ϕ, wherea s µ ασ,b σ a s a Φ + Φ 1 b b And: 1 s x x a a s a 0 E(X S s) ϕ dx Φ 1 + Φ b b s a 1 b a (bz + a) ϕ(z)dz, a s a b Φ + Φ 1 b b a s a ϕ ϕ x a where z a + b b a s a Φ + Φ 1 b b 54