Can we use kernel smoothing to estimate Value at Risk and Tail Value at Risk?

Transcription

1 Can we use kernel smoothing to estimate Value at Risk and Tail Value at Risk? Ramon Alemany, Catalina Bolancé and Montserrat Guillén Riskcenter - IREA Universitat de Barcelona ASTIN Colloquium The Hague 22nd-24th May 2013 Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

2 1 Motivation 2 Estimation alternatives: Emp, CKE and TKE 3 Double transformed kernel estimation 4 Simulation study 5 Data study 6 Conclusions Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

3 Motivation Analyze the properties of Transformed kernel estimations of Valueat-Risk (VaR) and Tail-Value-at-Risk (TVaR). For VaR estimation the Classical kernel estimator (CKE) and the Transformed kernel estimator (TKE) have been shown to be better than the empirical approach (Chen et al., 2005, and Alemany et al., 2013, respectively). Double transformed kernel density estimation (DTKE) 1 of VaR has a lower mean square error (MSE) than the Classical kernel estimation. Double transformed kernel density estimation can be applied to very extreme risk measurement (in Alemany et al., 2013, the authors reach the confidence level) when we have a loss distribution with heavy tail and is specially suitable when the sample size is large. We assess if this result holds when we estimate TVaR. 1 A particular case of transformed kernel estimation using double transformation Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

4 Motivation The aim is to estimate: and VaR α (X ) = inf {x, F X (x) α} (1) TVaR α (X ) = E (X X VaR α (X )). (2) where X is a random variable in R + representing a loss with probability distribution function (pdf) f X, and cumulative distribution function (cdf) F X. We compare empirical approach (Emp), classical kernel estimation (CKE) and double transformed kernel estimation (DTKE) using a simulation study and an application to the estimation of the risk of loss in an auto insurance portfolio. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

5 Motivation Both, VaR and TVaR, are based on the cdf, F X, associated to the loss variable. Therefore, the theoretical properties of the nonparametric estimations of VaR and TVaR are strongly related to the theoretical properties of the nonparametric estimation of cdf. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

6 Estimation alternatives: Emp, CKE and DTKE Empirical approach of cdf Let us assume that X i, i = 1,..., n denotes data observations from the loss random variable X. A natural nonparametric method to estimate cdf is the empirical distribution, F n (x) = 1 n n I (X i x), (3) i=1 where I ( ) is the indicator function, so I ( ) = 1 if condition between parentheses is true. Empirical distribution is an unbiased estimation of cdf F X but it has larger variance than alternative nonparametric approaches. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

7 Estimation alternatives: Emp, CKE and TKE Empirical approach of VaR and TVaR Then, the empirical estimate of value-at-risk is: { VaR α (X ) n = inf x, F } n (x) α, (4) and the empirical estimator of tail value-at-risk is: n i=1 TVaR X ii (X i > VaR ) α (X ) n α (X ) n = n i=1 (X I i > VaR ), (5) α (X ) n where VaR α (X ) n is defined in (4). Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

8 Estimation alternatives: Emp, CKE and TKE Empirical approach of VaR and TVaR Risk estimation from the empirical distribution is very simple, but it cannot extrapolate beyond the maximum observed data point. This is especially troublesome if the sample is not too large, and one may suspect that the probability of a loss larger than the maximum observed loss in the data sample is not zero. This can cause the empirical estimations of VaR and TVaR to underestimate the risk. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

9 Estimation alternatives: Emp, CKE and TKE Classical kernel estimation of cdf Classical kernel estimation (CKE) of a cdf, F X, is obtained integrating the classical kernel estimation of its pdf, f X. The usual expression for the kernel estimator of a cdf is: F X (x) = x f X (u)du = 1 n ( ) n i=1 K x Xi b, (6) where K( ) is the cdf of k( ) which is known as the kernel function (usually a symmetric pdf). Some examples of very common kernel functions are the Epanechnikov and the Gaussian kernel (Silverman, 1986). Parameter b is the bandwidth or the smoothing parameter. It controls the smoothness of the estimated cdf. The larger b is, the smoother the resulting cdf. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

10 Estimation alternatives: Emp, CKE and TKE Classical kernel estimation of cdf The Classical kernel cdf estimator has less variance than the empirical distribution estimator, but it has some bias which tends to zero if the sample size is large. Azzalini (1981) showed that the MSE of the CKE is less than the MSE of the empirical estimation. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

11 Estimation alternatives: Emp, CKE and TKE Classical kernel estimation of VaR and TVaR To estimate VaR α from CKE F X ( ), we use the Newton-Raphson method to solve the equation: F X (x) = α. (7) The kernel estimator of TVaR α is based on the kernel estimation of the survival function, ŜX (x) = 1 F X (x), so: 1 TVaR α (X ) = n(1 α) n ( VaRα (X ) X i X i (1 K b i=1 )), (8) Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

12 Estimation alternatives: Emp, CKE and TKE Classical kernel estimation of VaR and TVaR: Theoretical properties The properties of CKE of cdf are transferred to the CKE of VaR and TVaR, i.e. it has lower MSE than empirical approach, Emp. However, Chen (2008) showed that in practice these better properties cannot be transferred to the CKE of TVaR. DIFFICULTY: In expression 1 TVaR α (X ) = n(1 α) n ( VaRα (X ) X i X i (1 K b i=1 )), (9) we need to replace theoretical VaR α (X ) by its estimation, then CKE of TVaR has larger MSE than empirical approach. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

13 Estimation alternatives: Emp, CKE and TKE Transformed kernel estimation of cdf Let T ( ) be a concave transformation where Y = T (X ) and Y i = T (X i ), i = 1... n are the transformed observed losses. Then the classical kernel estimator of the transformed variable is: F T (X ) (T (x)) = 1 n n ( ) T (x) T (Xi ) K. (10) b i=1 The transformed kernel estimation (TKE) of F X (x) is: F X (x) = F T (X ) (T (x)). Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

14 Estimation alternatives: Emp, CKE and TKE Transformed kernel estimation of VaR and TVaR In order to estimate VaR α the Newton-Raphson method solves the equation F T (X ) (T (x)) = α and, once the result is obtained, the inverse of the transformation is applied. The transformed kernel estimation of TVaR α is based on the transformed kernel estimator of survival function, i.e. Ŝ X (x) = ŜT (X ) (T (x)) = 1 F T (X ) (T (x)), so, TVaR α (X ) = 1 n(1 α) n ( )) T (VaRα (X )) T (X i ) X i (1 K, b i=1 (11) Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

15 Estimation alternatives: Emp, CKE and TKE Transformed kernel estimation of VaR and TVaR: Theoretical properties Our last paper (Alemany et al., 2013) showed that the use of a suitable transformation can reduce the variance at the expense of increasing the bias of the estimation, resulting in a lower MSE. These properties are transferred to the TKE of the VaR but, as occurs with CKE, are not transferred to the TVaR estimation, given that we need to replace in TVaR α (X ) the true VaR by the estimated VaR. Can the double transformed kernel estimation (DTKE) of the authors be a good alternative? ANSWER: In some specific cases. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

16 Double transformed kernel estimation As is well known, to obtain a smoothing parameter that is asymptotically optimal, it is sufficient to minimize the A-MISE (Asymptotic mean integrated squared error): ( 1 [f 4 b4 Y (y)] 2 dy ) 2 t 2 k (t) dt 1 n b K (t) [1 K (t)] dt. Given b and the kernel k( ), the A-MISE is minimum when functional [f Y (y)]2 dy is minimum. Therefore, the proposed method is based on the transformation of the variable in order to achieve a distribution that minimizes the A-MISE, i.e. that minimizes [f Y (y)]2 dy. Terrell (1990) showed that the density of a Beta (3, 3) distribution defined on the domain [ 1, 1] minimizes [f Y (y)]2 dy, in the set of all densities with known variance. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

17 Double transformed kernel estimation Let M( ) be the cdf of the cited Beta (3, 3), the DTKE method requires an initial transformation of the data 2, T (X i ) = Z i, where we get a transformed variable distribution that is close to a Uniform (0, 1) (Buch- Larsen et al, 2005). Afterwards, the data are transformed again using the inverse of the distribution function of a Beta (3, 3), M 1 (Z i ) = Y i. The DTKE is: F M 1 (T (x)) (x) = 1 n n ( M 1 (T (x)) M 1 ) (T (X i )) K.(12) b i=1 2 With a generalized Champernowne cdf: TX (x) = ((x + c) γ c γ ) / ((x + c) γ + (M + c) γ 2c γ ) Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

18 Double transformed kernel estimation Double transformed kernel estimation of VaR and TVaR Double transformed kernel estimation of VaR is obtained similarly to transformed kernel estimation and has an important advantage: it allows us to calculate the optimal bandwidth (b) that minimizes MSE, given that we know that transformed data have Beta (3, 3) distribution: b = ( m (q) 1 ) 1 1 K 2 (t) dt [ 1 2 m (q) 1 1 t2 k (t) dt ] n 1 3, (13) where m is the pdf of Beta (3, 3), m its first derivative and q is the VaR α of the Beta distribution. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

19 Double transformed kernel estimation Double transformed kernel estimation of VaR and TVaR The double transformed kernel estimation of TVaR α is based on the double transformed kernel estimator of the survival function, i.e. so, Ŝ X (x) = ŜM 1 (T (X )) ( M 1 (T (x)) ) = 1 F M 1 (T (X )) ( M 1 (T (x)) ), TVaR α (X ) = 1 n(1 α) n ( M X i (1 1 (T (VaR α (X ))) M 1 )) (T (X i )) K. b i=1 (14) Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

20 Double transformed kernel estimation Double transformed kernel estimation of VaR and TVaR ( We know that q = M (T 1 VaRα (X ))) is a quantile of Beta (3, 3) distribution. Let p = M(q) be the confidence level of q. We propose to estimate TVaR based on the theoretical quantile of Beta (3, 3), so then the DTKE of TVaR is: TVaR p (X ) = 1 n(1 p) The optimal bandwidth is: b = n ( q M X i (1 1 )) (T (X i )) K. (15) b i=1 ( qm (q) 1 ) 1 1 K 2 (t) dt [ (1 p) 1 (qm (q) + m (q)) 1 1 t2 k (t) dt ] n 1 3, (16) Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

21 Double transformed kernel estimation Double transformed kernel estimation of VaR and TVaR: Theoretical properties In Alemany et al., 2013, we showed that DTKE of VaR is better than Emp and CKE, specially when we have large sample size and the loss distribution has heavy tail. Now we show that DTKE of TVaR increases bias and reduces the variance of the empirical estimate. The difficulty is that, in some cases, reducing the variance does not compensate the increase in the bias, resulting in greater MSE of DTKE if we compare with the empirical approach. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

22 Simulation study A wide simulation study is carried out. We obtain 2000 samples of size 500 and 5000 from different distribution functions with different parameters and different tails shapes: Lognormal, Weibull and mixtures Lognormal-Pareto. To ensure that mixtures Lognormal-Pareto have finite first moment we use parameter of Pareto less than unity. Results show that our DTKE method overestimates TVaR in Lognormal and Weibull distribution. We obtain a larger positive bias which is not compensated by variance reduction. In these cases MSE of DTKE is larger than MSE of the empirical approach. We obtain very good results in mixtures Lognormal-Pareto with large sample size. In these cases MSE of DTKE is much smaller than the MSE of empirical approach when confidence level is 0.95 and Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

23 Simulation study Results of MSE of TVaR α for a mixture of Lognormal-Pareto n=500 n= % Lognormal-30% Pareto Method α = 0.95 α = α = 0.95 α = Emp DTKEx % Lognormal - 70% Pareto α = 0.95 α = α = 0.95 α = Emp DTKEx Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

24 Data study We analyze a data set containing a sample with 5,122 automobile claim costs from a Spanish insurance company for traffic accidents in This is an standard insurance data set with observations on the cost of accident claims, i.e. a large sample that looks heavy-tailed with lots of small values and a few large extremes. The original data are divided into two groups: claims from policyholders who were less than 30 years old when the accident took place, and claims from policyholders who were 30 years old or older when they had the accident that originated the claim for compensation. The first group consists of 1,061 observations in the interval of cost of claim from 1 to 126,000 and the second group contains 4,061 observations in the interval from 1 to 17,000. Costs are expressed in monetary units. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

25 Data study Summary of the younger and older policyholders claims cost data Data n Mean Median Std. Deviation Coeff. Variation All 5, , Younger 1, , Older 4, Cost of claims are expressed in monetary units. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

26 Data study VaR results VaR α results for automobile claim cost data. α=0.95 α=0.995 Method Younger Older All Younger Older All Emp CKE DTKEx Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

27 Data study VaR results Estimated VaR α for tolerance levels (x-axis) above 99%. Comparison of three methods for All insured. Solid, dashed and dotted lines correspond to the empirical, the classical kernel and the transformed kernel estimation method, respectively. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

28 Data study VaR results Estimated VaR α for tolerance levels (x-axis) between 99% and 99.9%. Comparison of three methods for All insured. Solid, dashed and dotted lines correspond to the empirical, the classical kernel and the transformed kernel estimation method, respectively. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

29 Data study VaR results Estimated VaR α for tolerance levels (x-axis) above 99%. Value-at-Risk estimated with double transformed kernel estimation given the tolerance level. Solid line and dashed line correspond to older and younger policyholders, respectively. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

30 Data study TVaR results TVaR α results for automobile claim cost data. α=0.95 α=0.995 Method Younger Older All Younger Older All Emp CKE DTKEx Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

31 Data study TVaR results Estimated TVaR α for tolerance levels (x-axis) above 99%. Comparison of two methods for All insured. Solid and dashed lines correspond to the empirical and the transformed kernel estimation method, respectively. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

32 Data study TVaR results Estimated TVaR α for tolerance levels (x-axis) above 99%. Tail Value-at-Risk estimated with double transformed kernel estimation given the tolerance level. Solid line and dashed line correspond to older and younger policyholders, respectively. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

33 Conclusions Double transformed kernel estimation of TVaR improves empirical approach when we have heavy tailed loss distribution. DTKE allows us extrapolate quantiles of the distribution above the maximum observed with relative efficiency. We believe our approach to be a practical method for risk analysts and pricing departments. Our method can establish a distance between risk classes in terms of differences in the risk of extreme severities. Additionally, a surcharge to the a priori premium can be linked to the loss distribution of severities with the advantage of needing no distributional assumptions and of being easy to implement. Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

34 References Alemany, R., Bolancé, C. and Guillén, M., 2013, A nonparametric approach to calculating value-at-risk, Insurance: Mathematics and Economics, 52, Azzalini, A., 1981, A note on the estimation of a distribution function and quantiles by a kernel method, Biometrika, 68, Bolancé, C., 2010, Optimal inverse Beta(3,3) transformation in kernel density estimation, SORT-Statistics and Operations Research Transactions, 34, Bolancé, C., Guillén, M. and Nielsen, J.P., 2003, Kernel density estimation of actuarial loss functions, Insurance: Mathematics and Economics, 32, Bolancé, C., Guillén, M. and Nielsen, J.P., 2008, Inverse Beta transformation in kernel density estimation, Statistics& Probability Letters, 78, Bolancé, C., Guillén, M., Gustafsson, J. and Nielsen, J.P., 2012, Quantitative Operational Risk Models, Chapman & Hall/CRC Finance Series, London. Bowman, A.a and Hall, P. and Prvan, T., 1998, Bandwidth selection for smoothing of distribution function, Biometrika, 85, Buch-Larsen, T., Guillén, M., Nielsen, J.P. and Bolancé, C., 2005, Kernel density estimation for heavy-tailed distributions using the Champernowne transformation, Statistics, 39, Cai, Z. and Wang, X., 2008, Nonparametric estimation of conditional VaR and expected shortfall, Journal of Econometrics, 147, Chen, S.X., 2008, Nonparametric Estimation of Expected Shortfall, Journal of Financial Econometrics, 6, Chen, S.X. and Tang, C.Y., 2005, Nonparametric inference of Value at Risk for dependent financial returns, Journal of Financial Econometrics, 3, Silverman, B.W., 1986, Density Estimation for Statistics and Data Analysis, Chapman & Hall/CRC Finance Series, London. Terrell, G.R., 1990, The maximal smoothing principle in density estimation, Journal of the American Statistical Association, 85, Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35

35 Can we use kernel smoothing to estimate Value at Risk and Tail Value at Risk? Ramon Alemany, Catalina Bolancé and Montserrat Guillén Riskcenter - IREA Universitat de Barcelona ASTIN Colloquium The Hague 22nd-24th May 2013 Alemany, Bolancé & Guillén (Riskcenter UB) ASTIN rd May / 35