UFDC Home  myUFDC Home  Help 



Full Text  
COMPARING ESTIMATORS OF VAR AND CVAR UNDER THE ASYMMETRIC LAPLACE DISTRIBUTION By HSIAOHSIANG HSU A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN STATISTICS UNIVERSITY OF FLORIDA 2005 Copyright 2005 by HsiaoHsiang Hsu To my parents, brother, and husband ACKNOWLEDGMENTS I would like to especially thank my advisor, Dr. Alexandre Trindade. He guided me through all the research, and gave me invaluable advice, suggestions and comments. This thesis could never have been done without his help. I also want to thank Dr. Ramon C. Littell and Dr. Ronald Randles for serving on my committee and providing valuable comments. I am also grateful to Yun Zhu for her support and patience. Finally, I would like to thank my families for their unwaving affection and encouragement. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ......... .................................................................................... iv LIST OF TABLES ...................................................... vii LIST OF FIGURES ............ ......... ......................... viii A B ST R A C T .......... ..... ...................................................................................... x CHAPTER 1 IN T R O D U C T IO N ............................................................................. .............. ... 2 DEFINITIONS, PROPERTIES AND ESTIMATIONS.......................... .................. D definitions ........................................4 Definition 2.1: VaR (Value at Risk)....................... ... ... ...............4 Definition 2.2: CVaR (Conditional Value at Risk) ...........................................4 Definition 2.3: AL distribution (The Asymmetric Laplace distribution)..............5 B asic P properties ............................. ...................................... ... ....... 7 Proposition 2.1: The Coefficient of Skewness ............................................... 7 Proposition 2.2: The Coefficient of (excess) Kurtosis ......................................7 Proposition 2.3: The Quantiles .................. ..................... ...........7 Proposition 2.4: VaR and CVaR for the AL distribution ...................................8 E stim action s ............................................................................. 8 Param etric E stim action .................. ...................................... ..................... 8 Maximum Likelihood Estimation ( MLE ) ........................................9 Method of Moments Estimation (MME) ...................................................11 Sem iparam etric Estim ation ...................................................... ..... .......... 12 N onparam etric E stim ation........................................................ ............... 15 3 SIM U L A T IO N ST U D Y .................................................................. .....................16 S im u la tio n ............................................................................................................. 1 6 C om prison .............................................................................................................16 Y A L (1, 0 .8 1) ............................................................17 YAL(0, 1, 1) ................................ ......... 20 Y A L (1, 1.2, 1) ............................................................2 1 V 4 EM PIRICAL APPLICA TION S ............................................. .......................... 26 D a ta ...............................................................................2 6 In te re st R ate s .................................................................................................. 2 6 E x ch an g e R ate s .............................................................................................. 3 1 Comparison ............................ .................... 34 5 CONCLUSION..................... ..................36 APPENDIX THE PROBABILITIES OF THE MLES OF VAR AND CVAR BEING D E G E N E R A T E ....................................................................................37 L IST O F R EFER EN CE S ..................................... .................................................................38 B IO G R A PH IC A L SK E T C H ........................................................................................ 40 LIST OF TABLES Table page 31. Summary of related parameters of YAL(0,0.8,1). [n=200, reps=500].....................17 32. The MSEs of VaR and CVaR at different confidence levels: a=0.9, 0.95 and 0.99 of YAL(0, 0.8,1). [n=200, reps=500]............ ........... ..................... ...18 33. Summary of related parameters ofYAL(0,1,1). [n=200, reps=500].........................20 34. The MSEs of VaR and CVaR at different confidence levels: a=0.9, 0.95 and 0.99 of YAL(0,1,1). [n=200, reps=500]......................... ................ ............ 21 35. Summary of related parameters of Y(0,1.2,1). [n=200, reps=500] ........................22 36. The MSEs of VaR and CVaR at different confidence levels: a=0.9, 0.95 and 0.99 of YAL(0,1.2,1). [n=200, reps=500]..................... ... .................. ......... 22 41. Summary statistics for the interest rates, after taking logarithm conversion..............26 42. Summary statistics for the exchange rates, after taking logarithm conversion. .........31 LIST OF FIGURES Figure p 21. VaR and CVaR for the possible losses of a portfolio........................................5 22. Asymmetric Laplace densities with 0=0, z=1, and K =2, 1.25, 1, 0.8,0.5...................6 23. The probabilities of the MLEs of VaR and CVaR being degenerate at different confidence levels ........ ................... ......... ................ ............. 10 24. Estimating tail index by plotting log n ,(k) where n=200 in this case. The largest value of m, 98, gives a roughly straight line, and the slope of the line is 1.865421. R 2 =0.96............................................... ............... 14 31. Histogram of simulated AL (0,0.8,1) data. [n=200, reps=500]...............................18 32. The comparisons of three estimators of YAL(0, 0.8,1) at different confidence levels: a=0.9, 0.95 and 0.99. [n=200, reps=500] ...................................... 19 33. Histogram of simulated AL (0,1,1) data. [n=200, reps=500]............... .......... 20 34. The comparisons of three estimators of YAL(0, 1, 1) at different confidence levels: a=0.9, 0.95 and 0.99. [n=200, reps=500] ...................................... 21 35. Histogram of simulated AL (0,1.2,1) data. [n=200, reps=500]................................22 36. The comparisons of three estimators of Y~AL(0, 1.2, 1) at different confidence levels: a=0.9, 0.95 and 0.99. [n=200, reps=500] ...................................... 23 37. Relationships between K, the skewness parameter, and MSEs at different confidence levels, a=0.9, 0.95 and 0.99. [n=200, reps=500] .....................24 38. Relationships among MSEs of VaR and CVaR, the skewness parameter, K, and confidence level, a for the three different estimators parametricc, semiparametric, and nonparametric) .......... ..............................................25 41. Histogram and normal quantile plot of interest rates on 30year Treasury bonds, sam ple size = 202 .. ................................. ............. .... ...... .. 28 42. The QQ plot of interest rates on 30year Treasury bonds vs. fitted AL d istrib u tio n s ...................................... ............ .......... ............... 3 0 43. The difference between CVaR and VaR for 99 quantiles of interest rates on 30 year Treasury bonds (top) and and that of simulated AL distributions (with different simulation seeds)(bottom).............................................. 31 44. Histogram and normal quantile plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05, sam ple size = 1833.................................... ....................... 32 45. The QQ plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05 vs. fitted A L distrib u tion ............................................... ................ 3 3 46. The difference between CVaR and VaR for 99 quantiles of Taiwan Dollar daily exchange rates from 6/1/00 to 6/7/05 and and that of simulated AL distributions (with different simulation seeds).. ........................................34 47. The comparison of three estimators of interest rates on 30year Treasury bonds, from February 1977 to December 1993, at different confidence levels:a=0.9, 0.95 and 0.99. [n=200, reps=500] ............... .................34 48. The comparison of three estimators of Taiwan Dollar daily exchange rates,from 6/1/00 to 6/7/05, at different confidence levels: a =0.9, 0.95 and 0.99. [n=200, reps=500] ....................................... ................................ 35 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Master of Science in Statistics COMPARING ESTIMATORS OF VAR AND CVAR UNDER THE ASYMMETRIC LAPLACE DISTRIBUTION By HsiaoHsiang Hsu December 2005 Chair: Alexandre Trindade Major Department: Statistics Assessing the risk of losses in financial markets is an issue of paramount importance. In this thesis, we compare two common estimators of risk, VaR and CVaR, in terms of their mean squared errors (MSEs). Three types of estimators are considered: parametric, under the asymmetric laplace (AL) law; semiparametric by assuming Pareto tails; and ordinary nonparametric estimators, which can be expressed as Lstatistics. Parametric and nonparametric estimators have respectively the lowest and highest MSEs. By assessing two types of quantile plots on interest rate and exchange rate data, we determine that the AL distribution provides a plausible fit to these types of data. CHAPTER 1 INTRODUCTION Risk management has been an integral part of corporate finance, banking, and financial investment for a long time. Indeed, the idea has been dated to at least four decades ago, with Markowitz's pioneering work on portfolio selection [1]. However, the paper did not attract interest until twenty years after it was published. It was the financial crash of 19731974 that proved that past good performance was simply a result of bull market and that risk also had to be considered. This resulted in the increasing popularity of Markowitz's ideas on risk, portfolio performance and the benefits of diversification. In the past few years, the growth of financial market and trading activities has prompted new studies investigating reliable risk measurement techniques. The Valueat Risk (VaR) is a most popular measure of risk in either academic research or industry application. This is a dollar measure of the minimum loss that would be expected over a period of time with a given probability. For example, a VaR of one thousand dollars for one day at a probability of 0.05 means that the firm would expect to lose at least $1 thousand in one day 5 percent of the time. Or we can also express this as a probability of 0.95 that a loss will not exceed one thousand dollars. In this way, the VaR becomes a maximum loss with a given confidence level. The most influential contribution in this field has been J.P Morgan's RiskMetrics methodology, within which a multivariate normal distribution is employed to model the joint distribution of the assets in a portfolio [2]. However, the VaR approach suffers problems when the return and losses are not normally distributed which is often the case. It underestimates the losses since extreme events should happen with equally chance at each day. Obvious explanations for this finding are negative skewness and excess kurtosis in the true distribution of market returns, which cannot be accounted for by using a normal density model as in RiskMetrics. Another risk measure that avoids the problem is Conditional Value at Risk (CVaR). The concept of CVaR was first introduced by Artzuer, Delbaen, Eber, and Heath [3], and formulated as an optimization problem by Rockefellar and Uryasev [4]. CVaR is the conditional mean value of the loss exceeding VaR. It is a straightforward way to avoid serial dependency in the predicted events and thus base one's forecast on the conditional distribution of the portfolio returns given past information. Although CVaR has not become a standard in the finance industry, it is likely to play a major role as it currently does in the insurance industry. Therefore, in the thesis, we consider both of those two measurements for broader application. A correct statistical distribution of financial data is needed first before any proper predicative analysis can be conducted. Although the normal distribution is widely used, it has several disadvantages when applied to financial data. The first potential problem is one of statistical plausibility. The normal assumption is often justified by reference to the central limit theory, but the central limit theory applies only to the central mass of the density function, and not to its extremes. It follows that we can justify normality by reference to the central limit theory only when dealing with more central quantiles and probabilities. When dealing with the extremes, which are often the case in financial data, we should therefore not use the normal to model. Second, most financial returns have excess kurtosis. The empirical fact that the return distributions have fatter tails than normal distribution has been researched since early 1960s when Mandelbrot reported his first findings on stable (Parentian) distributions in finance [5]. Since then, several researchers have observed that practically all financial data have excess kurtosis, which is the leptokurtic phenomena. Thus, using the statistics of normal distributions to characterize the financial market is potentially very hazardous. Since Laplace distributions can account for leptokurtic and skewed data, they are natural candidates to replace normal models and processes. In this thesis, the aim is to compare parametric, semiparametric, and nonparametric estimators of VaR and CVaR random sampling from the Asymmetric Laplace distribution. To do so, we calculate their mean squared error (MSE), a popular criterion for measuring the accuracy of estimators. Broadly speaking, the best estimator should have smallest MSE. The plan of this thesis is as follows. Chapter 2 provides some background to the study by introducing some definitions and propositions related to VaR, CVaR and the Asymmetric Laplace distribution. Chapter 3 compares the parametric, semiparametric, and nonparametric three different estimators of Asymmetric Laplace distribution. Chapter 4 provides empirical analysis by using interest rates and currency exchange rates data. Chapter 5 concludes the article. Additional tables are included in the Appendix A. CHAPTER 2 DEFINITIONS, PROPERTIES AND ESTIMATIONS Let Y be a continuous realvalued random variable defined on some probability space (Q, A, P), with distribution function F(.) and density function f(.). Both the first and second moments of Y are finite. Definitions Definition 2.1: VaR (Value at Risk) The VaR refers to a particular amount of money, the maximum amount we are likely to lose over a period of time, at a specific confidence level. If positive values of Y represent losses, the VaR of Y at probability level a is defined to be the ath quantile of Y. VaR ) (Y) (Y)= F'(a) (2.1) Definition 2.2: CVaR (Conditional Value at Risk) The CVaR of Y at probability level a,is the mean of the random variable that results by turncasting Y at Ca and discarding its lower tail. CVaR,(Y) 0,(Y)= E(Y Y >) (2.2) Expanding on the definition, we obtain E(YI(Y >; )) .1 I a(Y) = ( ydF(y) = yf(y)dy (2.3) P(Y> ) a1a 1 a or we can have an equivalent definition of CVaR in terms of the quantile function of Y: S(Y) = F l(u)du (2.4) b ( ) 1  fo u SVaR Maximum loss 0 0 Probability 1 a CVaR Portfolio loss Figure 21. VaR and CVaR for the possible losses of a portfolio Definition 2.3: AL distribution (The Asymmetric Laplace distribution) Random variable Y is said to follow an Asymmetric Laplace distribution if there exist location parameter 0 e 91, scale parameter c20, and skewness parameter K>O, such that the probability density function of Y is of the form exp l y ,if y>0 ( rK2) t (2.5)1 or, the distribution function of Y is of the form :exp Iy ,if y>0 exp y if y<0 (2.6) We denote the distribution of Y by AL(O, K, c) and write Y AL(O, K, c). The mean of the distribution is given by S(2.7) Its variance is C2 = + 2 = +r2 (2.8) The value of the skewness parameter K is related to [t and r as follows, V~r 22 +2  K = (2.9) P+ +2r +//u2  and it controls the probability assigned to each side of 0. If K=1, the two probabilities are equal and the distribution is symmetric about 0. This is the standard Laplace distribution The Asymmetric Laplace Distribution :r : 10 5 0 5 10 15 20 Figure 22. Asymmetric Laplace densities with 0=0, z=1, and K =0.5, 1, and 2. Basic Properties Proposition 2.1: The Coefficient of Skewness For a distribution of an random variable Y with a finite third moment and standard deviation greater then zero, the coefficient of skewness is a measure of symmetry that is independent of scale. If Y AL(O, K, c), the coefficient of skewness, 71, is defined by 1 _3 Y = 2x K. (2.10) ( 1 2= + The coefficient of skewness is nonzero for an AL distribution. As K increases within the interval (0, oc), then the corresponding value of /1 decreases from 2 to 2. Thus, the absolute value of 7/ is bounded by two. Proposition 2.2: The Coefficient of (excess) Kurtosis For a random variable Y with a finite fourth moment, the coefficient of (excess) kurtosis can be defined as S12 2/2=6 (1/2 + 2)2 (211) It is a measure of peakness and of heaviness of the tails. If 72>0, the distribution is said to be leptokurtic (heavytailed). Otherwise, it is said to be platykurtic (lighttailed). The skewness coefficient of the AL distribution is between 3 ( the least value for asymmetric Laplace distribution when K=1) and 6 (the largest value attained for the limiting exponential distribution when K>). Proposition 2.3: The Quantiles If Y AL(O, K, c), then the qth quantile of an AL random variable is, SK 1+ K2 1+ K2 0+ log {q for q 0, +K + K 2 + T 1 log{(l+K)(lq)} for qe\K (.2 (2.12) Proposition 2.4: VaR and CVaR for the AL distribution If Y AL(O, K, T), for a > 0.5, then its standardization X = (Y 0)/r AL(0,K,1). Since both VaR and CVaR are translation invariant and positively homogenous [7], VaR,(Y)= 0+ rVaR,(X) (2.13) and CVaR(Y)= 0+ zCVaR,(X) (2.14) Therefore, no generality is lost by focusing on the standard case X ~ AL (0, K, 1), provided 0 and c are known. VaR and CVaR are then easily obtained. log[( +K2)( a)] ,;(X) = (2.15) KC2 and 1 ^(X) = (X) + (2.16) Estimations We now look at some of the most popular approaches to the estimation of VaR and CVaR. Parametric Estimation The parametric approach estimates the risk by fitting probability curves to the data and then inferring the VaR from the fitted curve. Maximum Likelihood Estimation ( MLE ) Consider now the most general case of estimating all three parameters. If Y AL(O, K, c), the maximum likelihood estimators (MLEs) are available in closed form [6]. Define first the functions, 8o)= 1 n 1 (2.17) 1 n g2() 0) n 1 (2.18) and h(O)= 2log1 9 (+ 2(0)]+ 31(0)2(0) (2.19) Letting the index 1 < r < n be such that h(Y,) ) h(Y,)for i =l,.....,n, the MLE of 0 is = Y(,). Provided 1 < r < n, the MLEs of( K, c) are: = 2 (r) 1 (r) 4 (2.20) ^= 2[3()(32( )1/4 r) (y /2 +2() 1/2) (2.21) (If r = 1 or r = n, the MLEs of ( K, c ) do not exist.) Defining c,, log[1+ K2 )] (2.22) the MLEs of VaR and CVaR are then obtained by equivariance, 0/Y) c/ (2.23) () )(2.24) AV2 However, after doing some experiments, we found the MLEs of VaR and CVaR will be degenerate most of the time when all three parameters are unknown (Appendix). Figure 23. shows that the probabilities of the MLEs being degenerate rise with both the sample size, n, and the skewness parameter, K . a = 0.5 a = 0.625 prob. of "; being degenerate S kappaU kappa a = 0.75 prob.of S being degenerate 5 kappa a = 0.875 prob.of degenerate kappa kappa Figure 23. The probabilities of the MLEs of VaR and CVaR being degenerate at different confidence levels There is another way to estimate the parameters when all of them are unknown. According to Ayebo and Kozubowski [8], in the case when all parameters are unknown, one can estimate the mode (0) using one of the nonparametric methods (Bickel [9] and prob.of being degenerate kappa kappa Vieu [10]) for several estimation models). After getting 0, we can apply the following formulas for K and r, assuming 0 is known, ([6], Chapter 3), to get the maximum likelihood estimates. 2= 4 () (2.25) ', = (0)V/20) X ( (1 8)+ 2 (0)). (2.26) Remark: In our analysis, we assume that the location parameter, 0, is zero. This is a reasonable assumption when the data consists of logarithmic growth rates such as interest rates, stock returns, and exchange rates [8]. Method of Moments Estimation ( MME ) The method of moments approach is also considered in the thesis. Assuming that the 0 is known, which is set to be zero in the study, the method of moments estimators of /u and r are given by ([6], Chapter 3) 1 " n S: 1 : Y (2.27) ^ = 1 2Y2 (2.28) Then, we can compute k using relation (2.9). Remark: After checking all the cases in the thesis, we found that the MLE and the MME of each parameter are almost the same. Therefore, we only calculate the MLEs for parametric estimation in the study. Semiparametric Estimation When observing financial data, e.g. stock returns, interest rates, or exchange rates, a much less restrictive assumption is to model the return distributions as having a Pareto left tail, or equivalently that the loss distribution has a Pareto right tail. This allows for the skewness and kurtosis of returns, while making no assumptions about the underlying distribution away from the tails. We follow the development of Rupport ([11], Chapter 11. )Fory> O, P(Y > y)= L(y)ya, (2.29) where L(y) is slowly varying at infinity and a is the tail index. Therefore, if y, > 0 and yo > 0 then P(Y > y) L(y) Y (2.30) (2.30) P(Y > y,) L(yo)[y now suppose that y, = VaRy (Y) and yo = VaRo (Y), where 0 < ao < a,. Then, (2.29 ) becomes 1 PY > VaRg (Y) L{VaR (Y)} VaR (Y) a ao P{Y >VaRo (Y)} L{VaRo(Y) VaR Y) (2.31) Because L is slowly varying at infinity and VaR1 (Y) and VaRo (Y) are assumed to be reasonably large, we make the approximation that L{VaR (X) 1 (2.32) L(VaR, (X)) so (2.32) simplifies to (Y) 1 a (2.33) VaRyo (Y) 1 aJ Now dropping the subscript "1" of a', we have VaR.(Y)= VaR.o (Y) 1 a (2.34) 1 a that is, 1 a (Y) ( aOY (2.35) We now extend this idea to CVaR similarly, giving CVaR,(Y)= CVaR0(Y) ~, (2.36) or we can write, (,(Y)= o(Y) (2.37) Equations (2.35) and (2.37) become semiparametric estimators of VaRY(Y) and CVaR,(Y) when VaR, (Y) and CVaRo (Y) are replaced by nonparametric estimates (2.41), (2.42) and the tail index a is estimated by the regression estimator. To see this, note that by (2.29), we have log[P(Y> y)]= log L(y)alogy. (2.38) If n is the sample size and 1 P(Y > y)) k (2.39) n log(n ) log alog(y(,). (2.40) One can then use the linearity of the plot of log Y(k)) for different k=1 m to guide the choice of m. The value of m is selecting by plotting {logl Y(k)i)) for various values k=l of m and choosing the largest value of m giving a roughly linear plot. If we fit a straight line to these points by least squares then minus the slope estimates the tail index a. For example, if a random sample Y, Y Y, ...Y,00 is drawn from the AL (0, 0.8,1) distribution, and Y Y Y ...Y denote the corresponding order statistics of the sample. (1)' (2)' (3)' (200) For getting the value of m, we first need to plot log l Y(k)) where n=200. k=1 The plotted points and the least squares line can be seen in Figure 24. 3 2 2 ***** 0  3 2 1 0 1 2 3 4 5 log((nk)/n) Figure 24. Estimating tail index by plotting log  Y(k) where n=200 in this case. The largest value of m, 98, gives a roughly straight line, and the slope of the line is  1.865421. R2 =0.96 A least squares line was fit to these 98 points and R2 =0.96, indicating a good fit to a straight line. The slope of the line is 1.865421, so a is 1.865421. After getting a, we can obtain the semiparametric estimators by functions (2.35) and (2.37). Nonparametric Estimation This is the least restrictive approach to the estimation of VaR and CVaR. The nonparametric approach seeks to estimate VaR or CVaR without making any assumptions about the distribution of returns and losses. The essence of the approach is that one can try to let the data speak for themselves as much as possible. (See for example [7].) When a random sample, ",,Y,, from AL distribution is available, consistent nonparametric estimators (NPEs) of VaR and CVaR taken the form of Lstatistics. If Y) <'  VaR is the ath empirical quantile, (Y) = Yk, (2.41) where k, = Lnao denotes the greatest integer less than or equal to na. The estimator of CVaR is the corresponding empirical tail mean, (1 YY) (2.42) nk, +1,=ka Theoretically, parametric approaches are more powerful than nonparametric approach, since they make use of additional information contained in the assumed density or distribution function. CHAPTER 3 SIMULATION STUDY In this chapter, we compare the three types of estimators parametricc, semiparametric, and nonparametric) of VaR and CVaR in terms of their bias, variance, and MSE. The data is generated via Monte Carlo from AL distribution. The MSEs are obtained empirically. Simulation There are several ways to generate random values from an AL distribution. Here is an example of using two i.i.d standard exponential random variables. We can generate the Y ~ AL(O,K, r) by the following algorithm. * Generate a standard exponential random variable W. * Generate a standard exponential random variable W2, independent of W. * Set Y0+ ( W KW2). 2 K * RETURN Y. Comparison In this section, we would like to compare three different approaches: parametric, semiparimetric, and nonparametric to estimate VaR and CVaR. In order to measure the goodness of those estimation procedures, using mean square error (MSE) to check their goodness. MSE is a common criterion for comparing estimators and it is composed of bias and variance. A better estimator should have smaller MSE. Besides checking the goodness of those estimators, we would also like to know how different K the skewness parameter, would affect the MSEs. Without loss of generality, here, we focus only on the standard case, Y AL(O,K,1) . Before doing the following analysis, we need first to know how to estimate the VaR and CVaR in the standard case. After some routine calculations from (2.23),(2.24), the MLEs of VaR and CVaR in the standard case are: 44(Y) (3.1) KJ2 a(Y)= U(Y)+ (3.2) Remark: Note that 4,(Y) ,(Y) = which is independent of a. Thos will form the basis of a goodnessoffit tool in Chapter 4. YAL(1, 0.8, 1) The skewness parameter, K controls the probability assigned to each side of 0. Therefore, while K = 0.8, the distribution would be moderately skewed to the right. The histogram of simulated values from AL(0,0.8,1) is shown in Figure 31, In Table below, we summarize the corresponding estimated parameters and coefficients of skewness and kurtosis for a random sample of size=200, reps=500, drawn from Y AL(0,0.8,1). Table 31. Summary of related parameters of YAL(0,0.8,1). [n=200, reps=500] K skewness kurtosis(adjusted) 0.79954 0.99336 0.134882 3.52629 Since 90%, 95% and 99% are the most common quantiles when analyzing financial data, we consider only those three confidence levels in this study. As mentioned already, the VaR and CVaR are contingent on the choice of confidence level, and will generally change when the confidence level changes. Thus, the MSEs of VaR and CVaR of different quantiles will also change correspondingly. This is illustrated in Table 32, which shows the corresponding MSEs of VaR and CVaR at the 95%, 99%, 99.5% levels of confidence. 150 100  50  I I I I I I I I I I 4 3 2 1 0 1 2 3 4 5 6 kappa=0.8 Figure 31. Histogram of simulated AL (0,0.8,1) data. [n=200, reps=500] Table 32. The MSEs of VaR and CVaR at different confidence levels: a= 0.99 of YAL(0, 0.8,1). [n=200, reps=500] =0.9, 0.95 and MSEs of VaR a =0.9 a =0.95 a =0.99 Parametric 0.0044458 0.0086326 0.023385 Semiparametric 0.014017 0.012534 0.20382 Nonparametric 0.82425 1.5001 3.9623 MSEs of CVaR a =0.9 a =0.95 a =0.99 Parametric 0.020068 0.029502 0.054907 Semiparametric 0.025519 0.054298 0.35212 Nonparametric 0.78834 1.3716 3.1509 Figure 32 illustrates the comparisons of three different estimation approaches. Because MSEs are composed of bias and variance, the comparisons of biases and variances of VaR and CVaR are also shown in this figure. From Figure 32, not 19 surprisingly, one can find that the parametric approach is the best way for estimating the VaR and CVaR according to its minimum MSE among those three approaches. The parametric approach is more powerful than the others, because it makes use of most information contained in the assumed density or distribution function. nnonparametric +P Parametric semiparametric 0.9 0.95 Alpha 2.5 2 1.5 > 1 0.5 S0 0.5 1 0.6 2 0.5 S0.4 8 0.3 S0.2  0.1 0 5 4 3 2 1 0 2.5 1.5 S0.5 0.5 1.5 I 0.9 0.95 0.99 Alpha 0.9 0.959 Alpha 0.9 0.95 0.99 Alpha 0.9 0.95 0.99 Alpha Figure 32. The comparisons of three estimators of YAL(0, 0.8,1) at different confidence levels: a=0.9, 0.95 and 0.99. [n=200, reps=500] 0.9 0.95 0.99 Alpha Aft 0.6 S0.5 > 0.4 0.3 . 0.2 0.1 0 5 4 S3 M2 1 0 k YAL(O, 1, 1) In this case, the skewness parameter, K is assumed to be 1, which means that the two probabilities are equal and the distribution is symmetric about 0, which is assumed to be 0 here. This is the standard Laplace distribution. In Figure 33, the symmetric distribution was shown very clearly; therefore, one can expect that the coefficient of skewness should be close to zero. As for the MLEs of parameters and other coefficients are demonstrated in Table 33. Table 33. Summary of related parameters ofYAL(0,1,1). [n=200, reps=500] Sskewness kurtosis(adjusted) 1.0018 0.99435 0 3 150 100  C) C: LL 50 0  I I I I I I I I I 5 4 3 2 1 0 1 2 3 4 5 kappa=1 Figure 33. Histogram of simulated AL (0,1,1) data. [n=200, reps=500] A summary of MSEs of VaR and CVaR under different estimation approaches is given in Table 34. Table 34. The MSEs of VaR and CVaR at different confidence levels: a= 0.99 of YAL(0,1,1). [n=200, reps=500] MSEs of VaR =0.9, 0.95 and a =0.9 a =0.95 a =0.99 Parametric 0.0031112 0.0057482 0.01605 Semiparametric 0.0063773 0.012763 0.17371 Nonparametric 0.42144 0.79774 2.3073 MSEs of CVaR a =0.9 a =0.95 a =0.99 Parametric 0.0031112 0.0057482 0.01605 Semiparametric 0.0063773 0.012763 0.17371 Nonparametric 0.42144 0.79774 2.3073 The comparison of different estimation approaches is shown in Figure 34. nonparametric 4Parametric semiparamtric 2.5 2 1.5 W 1 0.5 0 2.5 2 ct S1.5 2 0.5 0 0.9 0.95 0.99 Alpha 0.9 0.95 0.99 Alpha Figure 34. The comparisons of three estimators of YAL(0, 1, 1) at different confidence levels: a=0.9, 0.95 and 0.99. [n=200, reps=500] Remark: Since the results are similar to the previous one, we only demonstrate the comparison of MSEs in the following two cases. YAL(1, 1.2, 1) Now, K ,which is set to be 1.2. As the histogram of simulated AL numbers shown in Figure 35, the distribution seems to be lightly skewed left. The coefficient of skewness is therefore less than zero. Table35 illustrates some related parameters. Table 35. Summary of related Darameters of Y(0.1.2. 1. rn=200. ress=5001 150 100 50 kappa=1.2 Figure 35. Histogram of simulated AL (0,1.2,1) data. [n=200, reps=500] The comparison result of VaR and CVaR for those three approaches is illustrated in Table 36, and is the same as previous two cases: the parametric one is the best one and the nonparametric one is the worst. The comparison is also shown in Figure 36. Table 36. The MSEs of VaR and CVaR at different confidence levels: a=0.9, 0.95 and 0.99 of YAL(0,1.2,1). [n=200, reps=500] MSEs of VaR a =0.9 a =0.95 a =0.99 Parametric 0.0026005 0.0046065 0.013319 Semiparametric 0.0075317 0.017683 0.15612 Nonparametric 0.2238 0.46272 1.4338 d 1 MSEs of CVaR a =0.9 a =0.95 a =0.99 Parametric 0.013104 0.016882 0.032377 Semiparametric 0.031763 0.056587 0.23916 Nonparametric 0.21668 0.42903 1.1449 nonparametric "Parametric semiparametric 2 S1.5 S1 S0.5 0 2 1.5 S0.5 0 0.9 0.95 0.99 Alpha 0.9 0.95 0.99 Alpha Figure 36. The comparisons of three estimators of YAL(0, 1.2, 1) at different confidence levels: a=0.9, 0.95 and 0.99. [n=200, reps=500] Before moving on, it might be a good idea to pause at this point to see the relationships among the confidence level, a, the skewness parameter, K and the MSEs. From Figure 37, we could recognize that the MSEs of VaR and CvaR might fall or remain constant as K rises. Snonparametric Parametric semiparametric a =0.9 1 0.8 ; 0.6 M 0.4 2 0.2 0 0.8 1 1.2 Kappa L I 0.8 1 1.2 Kappa 1 0.8 0.6 i 0.4 2 0.2 0 a =0.95 2 2 1.5 2 1.5 0.5 0.5 0 0 0.8 1 1.2 0.8 1 1.2 Kappa Kappa a =0.99 5 5 2 2 0 0 0.8 1 1.2 0.8 1 1.2 Kappa Kappa Figure 37. Relationships between K, the skewness parameter, and MSEs at different confidence levels, a=0.9, 0.95 and 0.99. [n=200, reps=500] To form a more complete picture, we need to see how the MSEs change as we allow both those two parameters to change under different estimation approaches. The results are illustrated in Figure 38, which enables us to read off the value of the MSEs for any given combination of these two parameters. Those histograms show how the MSEs change as the underlying parameters change and convey information that the MSEs rise with a but decline with K MSEs of VaR for parametric estimator parametric mse i i 0.8 1 1.2 kappa MSEs of CVaR for parametric estimator parametric mse 0.8 1 1.2 kapp1.2 kappa MSEs of VaR for semiparametric estimator MSEs of CVaR for semiparametric estimator Semiparametric (VaR) Semiparametric (CVaR) mse I mse I "I i^ 0.8 1.2 kappa ).99 alpha U.8 p 1.2 kappa MSEs of VaR for nonparametric estimator MSEs of CVaR for nonparametric estimator nonparametric mse nonparametric mse r 0.99 0.9 0.8 1 kappa 0.8 1 kappa kappa Figure 38. Relationships among MSEs of VaR and CVaR, the skewness parameter, K, and confidence level, a for the three different estimators parametricc, semiparametric, and nonparametric) CHAPTER 4 EMPIRICAL APPLICATIONS We present in this section the interest rates and exchange rates data sets along with the quantitative analysis to determine if the AL distribution is an adequate model for the data by using goodnessoffit techniques. If the data sets do fit the AL distribution, we would like to compare the MSEs of VaR and CVaR for the three estimation approaches and to see which one is the best estimator. Data Interest Rates Table 41 reports summary statistics, including estimates of the coefficients of skewness and kurtosis. The data are the interest rates on 30year Treasury bonds on the last working days of the month. The database was downloaded from: http://finance.yahoo.com. The variable of interest is the logarithm of the interest rate ratio for two consecutive days. The data were transformed accordingly. This sample is the same as that previously consider by Kozubowski and Podgorski [12], and it goes from February 1977 through December 1993, yielding a sample size = 202. Table 41. Summary statistics for the interest rates, after taking logarithm conversion. Interest rate Mean S.D. Min Max Q1 Q3 Skewness Kurtosis 0.00046 0.01492 0.04994 0.05855 0.00933 0.00761 0.05706 1.9603 Remark: The sample goes from February 1977 through December 1993, yielding a sample size =202 Figure 41 contains a histogram and a normal quantile plot. The normal quantile plot is one of the most useful tools for assessing normality. The plot is to compare the data values with the values one would predict for a standard normal distribution. The comparison is based on the idea of quantiles. If the data came perfectly from a standard normal distribution, the theoretical and empirical quantiles would be expected to be similar. Thus, all the points would fall along a straight line. However, if the plot is markedly nonlinear, then it is doubtful those data are normally distributed. From the histogram, we can find that the data have a higher peak in the center and heavier tails than normal distribution. Since it is quite symmetric, we could expect a skewness near zero. Due to the heavier tails, we might expect the kurtosis to be larger than for a normal distribution. In fact, from the summary statistics in Table 41, the skewness is around zero and the kurtosis is near two, which indicate moderate kurtosis. Furthermore, by looking at the normal quantile plot, it is quite clear that the data do not follow normal distribution, since the dots do not quite fit to a straight line and have some outliers. Now, we have observed that the data set do not follow the normal distribution. However, there is no agreement regarding the best theoretical model for fitting the interest rates data. In a recent study, Kotz et al., [6], try to fit the skew Laplace distribution to both interest rates and currency exchange rates data because of its fattail and sharp peaks at the origin. In their experiment, they found except for a slight discrepancy in skewness, the match between empirical and theoretical values is close. Thus, in the study, we consider fitting the AL models to both interest rates and currency exchange rates. 40 30 20 LL U 10 0 m 0.05 0.00 0.05 Interest rate 99  95  o 50 go _  ^,^  i  40  S30  0 50 _     20 Data 5 ^ Figure 41. Histogram and normal quantile plot of interest rates on 30year Treasury bonds. Remark: The sample goes from February 1977 through December 1993, yielding a sample size =202 To determine if the AL distribution functions describe the data well, we employ here the most popularly used QuantileQuantile plot or QQ plot graphical technique to examine the data set. The idea of the QuantileQuantile (QQ) plot is similar to the normal quantile plot. It is a graphical technique for determining if two data sets come from populations with a common distribution. A QQ plot is a plot of the quantiles of the first data set against the quantiles of the second data set. By a quantile, it means the fraction (or percent) of points below the given value. If the two datasets come from a population with the same distribution, the points should fall approximately along this reference line. The greater the departure from this reference line, the greater the evidence for the conclusion that the two data sets have come from populations with different distributions. The plot in Figure 42 is the QQ plot of interest rates data set and AL distributions. To obtain the QQ plot, we need to fit an AL distribution to the interest rates data. Estimate the K and by formula (2.25), and (2.26), and then plot the empirical quantiles, from the 1 st to the 99th, against the corresponding quantiles calculated from (2.12) with the MLEs for K and Z substituted in. From this plot, we can see that most of the data points fall on a straight line. It is evident even to the naked eyes that AL distributions model these data more appropriately than normal distributions. 0.5 E* 0.0* * 0.5 0.2 0.1 0.0 0.1 0.2 Quantiles of AL Figure 42 The QQ plot of interest rates on 30year Treasury bonds vs. fitted AL distributions Remarkl: We compute 99 quantiles for the QQ plot. Remark2: The sample goes from February 1977 through December 1993, yielding a sample size =202 Now, we apply another goodnessoffit test. The idea is similar to the Q_Q plot. If the distance between the nonparametric estimators of CVaR and VaR for each quantile from the interest rates data set is similar to that of AL distributions, then we can conclude that the data might fit an AL distribution. In Figure 43, we can observe that the difference between CVaR and VaR for each quantile of interest rates on 30year Treasury bond is similar to that of an AL distribution. This provides additional evidence that the interest rates data set can be plausibly explained by AL distributions. 0.05 0.04 0.03 > 0.02 0.01 0 0 IIIII 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Quantiles 0.05 0.04 S0.03 0.02 0.01 0 0.05 0.04 > 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Quantiles 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Quantiles I ~I. D D Figure 43. The difference between CVaR and VaR for 99 quantiles of interest rates on 30year Treasury bonds (top) and that of simulated AL distribution (with different simulation seeds) (bottom). Remarkl: We compute 99 quantiles for the dataset. Remark2: The sample goes from February 1977 through December 1993, yielding a sample size =202 Exchange Rates The data below consist of Taiwan dollar daily exchange rates (interbank rates) against the U.S. dollar. The historical sample covers the period June 1, 2000, to June 7, 2005 (1833 days) downloaded from: http://www.oanda.com/convert/fxhistory. The variable of interest is the logarithm of the price ratio for two consecutive days. The data were transformed accordingly, yielding n=1832 values for each currency. The summary statistics for the transformed data are showed in Table 42, including the coefficient of skewness and kurtosis. Table 42. Summary statistics for the exchange rates, after taking logarithm conversion. Currency Mean S.D. Max Q1 Q3 Skewness Kurtosis Min TWD 0.00000 0.00197 0.0222 0.02204 0.0002 0.00015 0.0012129 50.365 Remark: The sample goes from June 1, 2000 through June 7, 2005, yielding a sample size =1833 The histogram and normal quantile plot of the data are presented in Figure 44. From the histogram, it is quite obvious that the data have a high peak, which is close to 0. Due to the high peak, we might expect the kurtosis to be larger than for a normal distribution. In fact, the kurtosis is 50.63, which is extremely high. Besides, the normal quantile plot confirms our findings that the data are not normally distributed. The data do not fit a straight line, not even the 95% confidence interval. 1000 C C: o a) 0) 500 L0 0 0.02 0.01 0.00 TWD 0.02 0 02 0 01 0 00 0 01 0 02 Data Figure 44. Histogram and normal quantile plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05, sample size=1833. The remaining challenge is to confirm if AL distributions fit the data well, and we will use the same strategy as that used for the interest rates data. .i 0.2  0.1 o* 0.0  0.1 o* * 0.2 0.2 0.1 0.0 0.1 0.2 Quantiles of AL Figure 45. The QQ plot of Taiwan Dollar daily exchange rates, 6/1/00 to 6/7/05 vs. fitted AL distribution Remark: We compute 99 quantiles for the QQ plot The quantile plots of the data set with theoretical AL distributions are presented in Figure 45. We see only a slight departure from the straight line. Comparing to the normal quantile plot in Figure 44, it is quite obvious that the AL distributions fit the data much better than the normal distributions. In Figure 46, the distance between the nonparametric estimators of VaR and CVaR for each quantile (99 quantiles) of the exchange rates dataset is similar to that of AL distributions. 0.006  0.005  0.004  0.003 0.002 * 0.001 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Quantiles 0.006 0.06 0.005 0.005 > 0.002 0.001 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Quantiles Qantiles Figure 46. The difference between CVaR and VaR for 99 quantiles of Taiwan Dollar daily exchange rates (top), and that of simulated AL distributions (with different simulation seeds)(bottom). Comparison In the previous section, we have confirmed that both the data sets: interest rates on 30year Treasury bonds and Taiwan Dollar daily exchange rates can be plausibly modeled by AL distributions. Therefore, in the following section, a comparison of three estimation approaches: parametric, nonparametric, and semiparametric will be made to check if the results are consistent with those of Chapter 3: the parametric approach is the best estimation method, and the nonparametric is the worst. [ nonparametric Parametric * semiparametric 0.0006 0.0006 0.0005 0.0005 0.0004 0.0004 0.0003 0.0003 S0.0002 0.0002 0.0001 0.0001 0.9 0.95 0.99 0.9 0.95 0.99 Alpha Alpha Figure 47. The comparison of three estimators of interest rates on 30year Treasury bonds, from February 1977 to December 1993, at different confidence levels:a=0.9, 0.95 and 0.99. [n=200, reps=500] 0.00001 0.00001 0.000008 0.000008 0.000006 > 0.000006 W 0.000004 w 0.000004 S0.000002 0.000002 0 0 0.9 0.95 0.99 0.9 0.95 0.99 Alpha Alpha Figure 48.The comparison of three estimators of Taiwan Dollar daily exchange rates,from 6/1/00 to 6/7/05, at different confidence levels: a =0.9, 0.95 and 0.99. [n=200, reps=500] From Figure 47 and Figure 48, we can find that the results are quite consistent with those of Chapter 3. The parametric estimation approach is the best method to estimate the VaR and CVaR of the AL distribution because of the lower MSEs. Besides, we can also notice that the MSEs are getting bigger while the confidence level is rising, especially for the nonparametric approach. That is because the nonparametric approach is more sensitive for extreme values, and it may overestimate the VaR and CVaR for large a The higher confidence level means a smaller tail, a cutoff point further to the extreme and, therefore, a higher VaR and CVaR. As a result, the corresponding MSEs would become higher. CHAPTER 5 CONCLUSION In this thesis, we compare three different estimates for the risk measures: VaR and CVaR when sampling from an AL distribution: parametric, semiparametric ,and nonparametric. The standard AL case is investigated in chapter 3, and we found that in general, the parametric approach is the best estimator since it has the smallest MSEs for both VaR and CVaR. We then applied the AL distribution to interest rates and exchange rates data and find it to be a plausible fit. This is because AL distributions can account for leptokurtosis and skewness typically present in financial data sets. Finally, we compared those three approaches again based on the empirical data sets and the results are consistent with those obtained earlier. APPENDIX THE PROBABILITIES OF THE MLES OF VAR AND CVAR BEING DEGENERATE a =0.5 Sn 5 10 15 20 25 0.25 1 1 1 1 1 0.50 1 0.99 0.99 0.94 0.92 0.75 1 0.93 0.89 0.73 0.71 1 0.99 0.96 0.84 0.69 0.58 a =0.625 k n 5 10 15 20 25 0.25 1 1 1 1 1 0.50 1 0.99 0.98 0.93 0.9 0.75 1 0.94 0.87 0.72 0.59 1 0.99 0.98 0.87 0.71 0.51 a =0.75 kn 5 10 15 20 25 0.25 1 1 1 1 1 0.50 1 0.97 0.96 0.97 0.92 0.75 1 0.93 0.94 0.72 0.65 1 1 0.93 0.79 0.66 0.58 a =0.625 kn 5 10 15 20 25 0.25 1 1 1 1 1 0.50 1 0.99 0.94 0.96 0.95 0.75 1 0.94 0.89 0.8 0.66 1 1 0.95 0.85 0.73 0.63 LIST OF REFERENCES 1. Markowitz, H. 1952. Portfolio Selection, Journal of Finance, Vol. 7, page 78. 2. J.P. Morgan. 1997. RiskMetrics Technical Documents, 4th edition. New York. 3. Artzuer, P., Delbaen, F., Eber, J., and Heath, D. 1999. Coherent Measures of Risk. Mathematical Finance, Vol. 9, pages 203228. 4. Rockefeller R. T. and Uryasev S. 2000. Optimization of Conditional ValueAtRisk. The Journal of Risk, Vol. 2, No. 3, pages 2141. 5. Mandelbrot, B. 1963. The Variation of Certain Speculative Prices. Journal of Business, Vol. 36, pages 394419. 6. Kotz, S., Kozubowski, T., and Podgorski, K. 2001. The Laplace Distribution and Generalizations, A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhauser. Boston. 7. Gaivoronski, A.A. Norwegian University of Science and Technology, Norway. Pflug, G. University of Vienna, Austria. 2000. Value at Risk in Portfolio Optimization: Properties and Computational Approach, working paper. 8. Ayebo, A. and Kozubowski, T. 2003. An Asymmetric Generalization of Gaussian and Laplace Laws. Journal of Probability and Statistical Science, Vol. 1(2), pages 187210. 9. Bickel, D.R. 2002. Robust Estimators of the Mode and Skewness of Continuous data. Comput. Statist. Data Anal, Vol. 39, pages 153163. 10. Vieu, P. 1996. A Note on Density Mode Estimation, Statist. Probab. Lett, Vol. 26, pages 297307. 11. Ruppert, D. 2004. Statistics and Finance: An Introduction, pages 348352. Spring Verlag. Berlin, Germany. 12. Kozubowski, T. and Podgorski, K. 1999. A Class of Asymmetric Distributions. Actuarial Research Clearing House, Vol. 1, pages 113134. 13. Dowd, K. 2002. Measuring Market Risk. John Wiley & Sons Ltd. West Success, England. 39 14. Linden, M. 2001. A Model for Stock Return Distribution. International Journal of Finance and Economics, Vol. 6, pages 159169. BIOGRAPHICAL SKETCH HsiaoHsiang Hsu was born in Taipei, Taiwan. She received her Bachelor of Business Administration (B.B.A.) degree in international trade and finance from FuJen Catholic University, Taipei, Taiwan. In Fall 2003, she enrolled for graduate studies in the Department of Statistics at the University of Florida and will receive her Master of Science in Statistics degree in December 2005. 