Simple risk measure calculations for sums of positive random variables

Accepted Manuscript Simple risk measure calculations for sums of positive random variables Montserrat Guill´en, Jos´e Mar´ıa Sarabia, Faustino Prieto PII: DOI: Reference:

S0167-6687(13)00084-X http://dx.doi.org/10.1016/j.insmatheco.2013.05.007 INSUMA 1821

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Insurance: Mathematics and Economics

Received date: March 2013 Revised date: May 2013 Accepted date: 22 May 2013 Please cite this article as: Guill´en, M., Sarabia, J.M., Prieto, F., Simple risk measure calculations for sums of positive random variables. Insurance: Mathematics and Economics (2013), http://dx.doi.org/10.1016/j.insmatheco.2013.05.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights Simple risk measure calculations for sums of positive random variables     

A multivariate generalized beta distribution is presented for positive losses Marginals follow a second kind beta distribution and can be are heavy-tailed Sums of dependent losses are easily derived in this model Risk measures for the sum of marginals have simple expressions Spreadsheet calculation is illustrated using operational risk data

Manuscript Click here to view linked References

Simple risk measure calculations for sums of positive random variables Montserrat Guill´ena , Jos´e Mar´ıa Sarabiab,∗, Faustino Prietob a

Department of Econometrics, Riskcenter-IREA, University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain b Department of Economics, University of Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain

Abstract Closed-form expressions for basic risk measures, such as value-at-risk and tail value-at-risk, are given for a family of statistical distributions that are specially suitable for right-skewed positive random variables. This is useful for risk aggregation in many insurance and financial applications that model positive losses, where the Gaussian assumption is not valid. Our results provide a direct and flexible parametric approach to multivariate risk quantification, for sums of correlated positive loss distributions, that can be readily implemented in a spreadsheet. Keywords: value at risk, tail value at risk, beta distribution, heavy-tailed, multivariate loss models 1. Introduction Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are probably the most widely used risk measures in financial institutions. However, except for a ∗ Corresponding author. Tel.: +34 942 201635; Fax: +34 942 201603. E-mail address: [email protected]. JEL codes: C10, G22, G32.

Preprint submitted to Elsevier

May 16, 2013

small group of statistical distributions, they have no closed-form expressions. As a consequence, since VaR and TVaR have simple formulae for the Gaussian and the t-Student distributions, those two models have vastly dominated the assumptions that are frequently used in the literature and in practice. We concentrate on positive loss random variables and propose a set of distributions that generalize the Beta distribution. The proposed distributions are very flexible, have a Pareto tail behaviour and can accommodate to a variety of shapes. Moreover, a multivariate distribution for dependent losses is easy to obtain and risk measures for the sum of its marginal random variables has a simple closed-form expression. We will present our results and illustrate them with multivariate data on operational risk. 2. Background In risk quantification, the variance-covariance method provides an analytical solution but it is based on two simplifying assumptions. As mentioned in McNeil et al. (2005, p. 49) “First, linearization may not always offer a good approximation of the relationship between the true loss distribution and the risk-factor changes. Second, the assumption of normality is unlikely to be realistic for the distribution of the risk-factor changes.” Practitioners have widely adopted the assumption of normality because, under this framework, spreadsheet calculations are very straightforward. Typically, loss data suggest a heavier than Gaussian tail. When we initially started our work, we were motivated by the need to extend risk quantification to probability distributions other than the normal. Moreover, we were also interested in assessing the risk of a sum of random 2

variables, which is a central problem in insurance and finance. This issue is related to the calculation of capital requirements, and it is usually set equal to a high-quantile of the sum of dependent losses. To the best of our knowledge, there are not many contributions that provide with benchmark distributions for which risk measures of the sum of positive random variables has a closed-form expression, as the one we propose. We claim that the distributions presented here can be used in a variety of practical applications and can overcome the difficulties from other approaches. Arbenz et al. (2011) recently proposed a new numerical algorithm to compute the distribution function of the sum of d dependent, non-negative random variables with a given joint distribution and from that, they can derive the usual risk measures. The algorithm is shown to converge for d ≤ 5. They also addressed calculation of the distribution function of non-decreasing functions of random variables (Arbenz et al., 2010). Modeling and combining correlated risks is a classical problem in actuarial science that goes beyond the sum of random variables. Given fixed marginal distributions and a correlation matrix, one can construct infinitely many joint distributions. From a given multivariate loss distribution, the aggregate loss distribution can sometimes be derived. Aggregation refers to any function that combines a d dimensional random vector into a univariate random variable. The sum of losses is the most frequent form of risk aggregation. Most convolutions of random variable are generally intractable and so, approximating algorithms have been proposed (see Wang, 1998). Bounds for VaR aggregation have also been studied by Embrechts et al. (2012). Embrechts et al. (2009a) have discussed sub and super-additivity of quantile based risk mea-

3

sures and show how multivariate extreme value theory yields to an ideal modeling environment. Risk aggregation is both an old problem and still a challenge in the field of applied risk analysis, where foundations need to be solid (Aven, 2012). For instance, an interesting classical application case was given by Tapiero and Jacque (1990), who referred to subsidiaries that may aggregate risks at the corporate level, acting thereby as intermediaries between the firm’s subsidiaries and commercial insurance firms. In a completely different setting, Ren (2012) proposed a multivariate aggregate loss model and provided formulas for calculating the joint moments of the present value of aggregate claims occurred at any time. In the collective risk model, the aggregate claim amount can be obtained from correlated claim amounts, as discussed by Cossette and Denuit (2000), who provided bounds of the distribution of the sum, or indicated by Embrechts and Puccetti (2006), who found bounds for functions of multivariate risks. Most often, dependence is induced via copulas, as suggested by a number of authors, like Cossette et al. (2002), who considered two individual risks, Frees and Wang (2006); Barg`es et al. (2009), who did the general case and Arbenz et al. (2012) who worked with hierarchical risks1 . Risk measures are a subject of extensive investigation (Dhaene et al., 2006; Tsanakas, 2008, 2009; Goovaerts et al., 2010, 2012). Recently, quantile un1

More generally, correlated and dependent risks have been studied by many au-

thors (Wang and Yuen, 2005; Centeno, 2005; Degen et al., 2010) and have been useful in practice, for instance when modelling portfolio credit risks (Vandendorpe et al., 2008; Frey and McNeil, 2002).

4

certainty was addressed by Alexander and Sarabia (2012). Contributions to the study of risk measures with respect to implementation are welcome by practitioners (Kaplanski and Levy, 2007; Yamai and Yoshiba, 2005; Huang et al., 2012), but many applications require intensive computational effort, mostly for historical simulation (Escanciano and Pei, 2012) and multivariate MonteCarlo simulation (Roelofs and Kennedy, 2011). Results on sums of positive dependent losses may be be useful in the context of capital requirement computations, and particularly in operational risk. Some previous results on elliptical distributions can be found in the articles by Valdez et al. (2009) and additional properties on the additivity of value at risk for heavy-tailed situations in Embrechts et al. (2009b). Bounds for the sum of dependent risks having overlapping marginals were proposed by Embrechts and Puccetti (2010). Positive random variables were also analysed using contractions by Hashorva et al. (2010). Cherubini et al. (2011) considered sums of random variables with unbounded support. Operational risk applications on severe losses were addressed by Peters et al. (2010, 2011) and Chavez-Demoulin et al. (2006). These authors do not find simple risk measure expressions. Problems related to the use of risk measures to disaggregation have also been studied. Conditional Tail Expectation for capital allocation was discussed by Asimit et al. (2011) and capital allocation in a broad sense was addressed by Gulick et al. (2011) who proposed a new rule to allocate risk capital to portfolios or divisions within a firm. Similarly, diversification and decomposition were analysed by P´erignon and Smith (2010) and Engsted et al. (2012). Non-Gaussian distributions in diversification has recently been addressed too

5

(Desmoulins-Lebeault and Kharoubi-Rakotomalala, 2012). We argue that a simple closed-form risk measure expression is useful for loss aggregation and for risk allocation. 3. The second kind beta distribution In this section, we introduce the second kind beta distribution as a model for studying positive risks. A random variable X is said to be a second kind beta distribution, if its probability density function (PDF) can be written as: f (x; p, q, λ) =

xp−1 , x > 0, λp B(p, q)(1 + x/λ)p+q

(1)

where p and q are positive shape parameters and λ is a positive scale parameter. This distribution corresponds to the Pearson VI distribution, in the classical Pearson system of distributions. If we set p = 1 in (1), we obtain the Pareto II distribution (see Arnold (1983)), which has been used extensively in insurance (see for example Asimit et al. (2013) and Vernic (2011)). If q = 1 in (1), we obtain the inverse Lomax distribution. This distribution has been used in the context of income distribution analysis (see Slottje (1984) and Chotikapanich et al. (2007)). A random variable with PDF (1) will be represented by X ∼ B2(p, q, λ). We include some properties of the second kind beta distribution (see Kleiber and Kotz (2003)). 1. The cumulative distribution function (CDF) is given by,   x FX (x; p, q, λ) = IB , p, q , x > 0 λ+x 6

(2)

where IB(x; p, q) represents the incomplete ratio beta function defined as IB(x; p, q) =

Rx 0

tp−1 (1−t)q−1 dt B(p,q)

and B(p, q) is the Beta function.

2. The r-th moment of the second kind beta distribution is, E[X r ] =

λr Γ(p + r)Γ(q − r) , q > r. Γ(p)Γ(q)

(3)

3. The distribution of the first incomplete moment of a random variable X, which will be denoted by X(1) is given by F(1) (x) =

Rx 0

tdFX (t) . E[X]

For

the second kind beta distribution, if X ∼ B2(p, q, λ) then X(1) ∼ B2(p+ q, q − 1, λ), if q > 1. 4. The second kind beta distribution has a simple stochastic representation in terms of ratios of gamma random variables. Let us denote Up a gamma random variable with PDF f (u) =

up−1 e−u , Γ(p)

with u > 0. If Up

and Uq are independent gamma random variables, the random variable X = λ UUpq is distributed according to a second kind beta distribution with parameters p, q and λ. The stochastic representation of the second kind distribution allows one to easily simulate from this family, which is very useful when closed-forms are not available. 3.1. Risk measures for the second kind beta distribution One of the advantages of the second kind distribution is that many of the most important risk measures are available in a simple closed form. The value at risk at level α, with 0 < α < 1 of a random variable X with CDF FX (x) is defined by V aR[X; α] = inf{x ∈ R, FX (x) ≥ α}. 7

In the following lemma we include a straightforward expression for the VaR.

Lemma 1. Let X ∼ B2(p, q, λ) be a second kind beta distribution. For a probability α ∈ (0, 1), the Value at Risk is given by, V aR[X; α; λ, p, q] = λ ·

IB −1 (α; p, q) , 1 − IB −1 (α; p, q)

(4)

where IB −1 (α; p, q) denotes the inverse of the uncomplete beta distribution. In other words IB −1 (α; p, q) is the quantile function of the classical beta distribution of the first kind. Proof.



See Appendix

The VaR is a risk measure that is not coherent. The tail value at risk of X (see Acerbi and Tasche (2002)) is given by 1 T V aR[X; α] = 1−α

Z1

V aR[X; u]du,

α

which is a coherent risk measure. If X is continuos, Pr(X ≤ V aR[X; α]) = α, and then the TVaR is the conditional tail expectation T V aR[X; α] = E[X|X > V aR[X; α]]. The second expression provides a simple expression for the TVaR. Lemma 2. Let X ∼ B2(p, q, λ) be a second kind beta distribution. For a non-negative real value a we can compute the Tail Value at Risk as follows,   a/λ ; p + 1, q − 1 1 − IB 1+a/λ   , (5) T V aR[X; a; λ, p, q] = E[X] · a/λ 1 − IB 1+a/λ ; p, q

where E[X] =

λp q−1

and q > 1.

8

Proof.



See Appendix

Lemma 2 can be easily extended for other tail risk measures, such as higher moments of the worst ‘a’ events. That is, not only the average (TVaR) of the worst ‘a’ events may be found, but higher moments, variance, etc. 3.2. A multivariate version of the second kind beta distribution In this section, we introduce a multivariate distribution with dependent second kind beta marginals. Definition 1. Let Y0 , X1 , . . . , Xm be mutually independent gamma random variables with distributions Y0 ∼ Ga(q0 ) and Xi ∼ Ga(pi ), i = 1, 2, . . . , m. The multivariate second kind beta distribution is defined as,   X1 X2 Xm {Z1 , Z2, . . . , Zm } = λ1 , λ2 , . . . , λm , Y0 Y0 Y0

(6)

with q0 > 0 and λi , pi > 0, i = 1, 2, . . . , m. Note that the random variable Y0 introduces the dependence in the model. By construction, the marginal distributions are distributed according to second kind beta random variables, Zi ∼ B2(pi , q0 , λi ), i = 1, 2, . . . , m.

(7)

3.3. Risk measures for the aggregate random variable In this section, we obtain the risk aggregation random variable Am = Z1 + · · · + Zm . The distribution of this convolution again follows a second kind beta distribution.

9

Theorem 1. Let {Z1 , Z2 , . . . , Zm } be a multivariate second beta distribution defined as the stochastic representation (6). Then, if λ1 = λ2 = · · · = λm = λ, the distribution of the convolution Am = Z1 + · · · + Zm is Am ∼ B2(p1 + p2 + · · · + pm , q0 , λ)

(8)

Proof. The proof is straightforward taking into consideration the stochastic 

representation of the second kind beta distribution.

As a direct consequence of the previous theorem, if X1 , X2 , . . . , Xm is a sample of size m from a second kind beta distribution, the distribution of the P random variable Sm = m i=1 Xi is, Sm ∼ B2(m · p, q, λ).

(9)

Moreover, as the distribution of Am is again a second kind beta distribution, we can easily obtain its VaR and TVaR: V aR[Am ; α; λ, p, q0 ] = λ ·

IB −1 (α; p1 + p2 + · · · + pm , q0 ) , 1 − IB −1 (α; p1 + p2 + · · · + pm , q0 )

and T V aR[Am ; α; λ, p, q0 ] = E[Am ] ·

1 − IB



a/λ ; p1 1+a/λ

(10) 

+ · · · + pm + 1, q0 − 1  , a/λ 1 − IB 1+a/λ ; p1 + · · · + pm , q0 − 1 (11) 

respectively. 4. A simple methodology for estimating the risk of the sum of positive losses In this section, we present the estimation methodology, including a simulation study. 10

4.1. Estimation methodology We consider the estimation of the aggregated loss A2 = X1 + X2 , where X1 ∼ B2(p1 , q1 , λ) and X2 ∼ B2(p2 , q1 , λ), using marginal information on X1 and X2 . We assume that we know the sample mean, mode and sample variance of the first risk. In a first step, we solve the system of equations E[Xi ] = µi , i = 1, 2, mode[X1 ] = m1 and var[X1 ] = σ12 for p1 , p2 , q1 and λ, we obtain the following consistent estimators   µ1 2m1 + λ pˆ1 = , λ µ1 − m1 µ1 + m1 + λ qˆ1 = , µ1 − m1 2 2 ˆ = µ1 (µ1 − m1 ) − (3m1 − µ1 )σ1 , λ σ12 − µ21 + µ1 m1

(12) (13) (14)

and pˆ2 =

qˆ1 − 1 µ2 . ˆ λ

(15)

This first estimation method produces initial parameter estimates, which can be taken as starting values for the maximum likelihood method. For the second kind beta distribution, the log-likelihood function is given by log ℓ(p, q, λ) =

n X

log f (xi ; p, q, λ),

i=1

where f (x; p, q, λ) is the probability density function (PDF) defined in Eq. (1). In this second step, we obtain the maximum likelihood estimation of paˆ which maximizes the likelihood function log ℓ(p, q, λ) rameter vector (ˆ p1 ,ˆ q1 ,λ), for X1 and then, we obtain the maximum likelihood estimation of parameter ˆ for X2 . Note that pˆ2 , which maximizes the likelihood function log ℓ(p, qˆ1 , λ) we estimate three parameters by maximum likelihood for X1 and only one 11

for X2 . This method is easily implemented in practice and, as we will discuss in the illustration, provides a reasonable good fit. A straightforward generalization to the sum of more than two losses is trivial, using the same principle. Note that the standard log-likelihood maximization to obtain all parameter estimates simultaneously is feasible, although it is much more time-consuming than the procedure described in this section. 4.2. Simulation study We perform a simulation study to observe the behaviour of the proposed estimation methodology. We consider the following values of the parameters (p1 , p2 , q1 , λ): (0.5, 0.5, 2, 1); (1, 1, 2.5, 1) and (2, 2, 3, 1) - we fix the scale parameter λ at 1 with no loss of generality. We also consider different samples sizes (n, m): (600, 300), (1000, 500), (2000, 1000), and their symmetric cases, (300, 600), (500, 1000), (1000, 2000), according to the usual practice (see the following Section). For a given (p1 , p2 , q1 , λ) and (n, m), we generate a sample from B2(p1 , q1 , λ) of size n and a sample from B2(p2 , q1 , λ) of size m, and compute the corresponding maximum likelihood estimators. We replicate the process 1000 times and obtain the sample averages of them, together with their bias and root mean square error (RMSE) from the true parameters values. The simulation results are presented in Tables 1, 2 and 3. As expected, the bias and the RMSE decrease when the sample sizes increase. For small values of the sample sizes, the method is not very efficient. 5. An empirical application in operational risk In operational risk, practitioners need to measure the risk of the sum of losses observed in several categories. For that purpose, the total sum of 12

Table 1: Bias and RMSE of parameter estimates for the sum of two univariate second kind beta distribution: B2(p1 , q1 , λ), B2(p2 , q1 , λ), with p1 = p2 = 0.5, q1 = 2, λ = 1. Bias

RMSE

n

m

p1

p2

q1

λ

p1

p2

q1

λ

600

300

-0.0030

-0.0030

-0.1048

-0.0866

0.0294

0.0366

0.4967

0.4189

1000

500

-0.0023

-0.0024

-0.0552

-0.0430

0.0241

0.0287

0,3218

0.2697

2000

1000

0.0001

-0.0012

-0.0336

-0.0284

0.0164

0.0201

0.2204

0.1841

300

600

-0.0063

-0.0051

-0.3867

-0.3341

0.0458

0.0393

3.4210

3.1868

500

1000

-0.0039

-0.0027

-0.1070

-0.0882

0.0348

0.0285

0.5110

0.4245

1000

2000

-0.0009

-0.0008

-0.0553

-0.0487

0.0233

0.0205

0.3361

0.2875

Table 2: Bias and RMSE of parameter estimates for the sum of two univariate second kind beta distribution: B2(p1 , q1 , λ), B2(p2 , q1 , λ), with p1 = p2 = 1, q1 = 2.5, λ = 1. Bias

RMSE

n

m

p1

p2

q1

λ

p1

p2

q1

λ

600

300

-0.0095

-0.0091

-0.1274

-0.0757

0.0800

0.0864

0.5976

0.4063

1000

500

-0.0057

-0.0086

-0.0462

-0.0293

0.0582

0.0655

0.3402

0.2293

2000

1000

-0.0027

-0.0048

-0.0220

-0.0136

0.0426

0.0462

0.2343

0.1608

300

600

-0.0131

-0.0104

-0.2720

-0.1831

0.1099

0.1032

1.1224

0.7891

500

1000

-0.0095

-0.0083

-0.1295

-0.0856

0.0885

0.0824

0.6354

0.4412

1000

2000

-0.0022

0.0001

-0.0683

-0.0499

0.0586

0.0555

0.3841

0.2618

Table 3: Bias and RMSE of parameter estimates for the sum of two univariate second kind beta distribution: B2(p1 , q1 , λ), B2(p2 , q1 , λ), with p1 = p2 = 2, q1 = 3, λ = 1. Bias

RMSE

n

m

p1

p2

q1

λ

p1

p2

q1

λ

600

300

-0.0251

-0.0284

-0.1010

-0.0563

0.2139

0,2271

0.5330

0.3217

1000

500

-0.0122

-0.0160

-0.0664

-0.0380

0.1650

0.1741

0.3972

0.2472

2000

1000

-0.0068

-0.0095

-0.0387

-0.0224

0.1229

0.1294

0.2798

0.1769

300

600

-0.0658

-0.0565

-0.2391

-0.1438

0.3588

0.3452

1.0320

0.6521

500

1000

-0.0377

-0.0338

-0.1357

-0.0733

0.2563

0.2479

0.6267

0.3789

1000

2000

-0.0189

-0.0172

-0.0555

-0.0315

0.1731

0.1692

0.3792

0.2414

13

operational losses is usually modeled as the sum of partial sums obtained from each category, as if risk categories were independent. However, risk categories are usually dependent, because they correspond to risk classes from the same financial institution. In this example, we present a simple case with only two losses. We use two sets of data that are taken from Bolanc´e et al. (2012). These are called the public data risk no. 1 and public data risk no. 2, respectively. For the first category, we have a sample of 1, 000 observed severities and for the second category, we have 400 observations. As in Bolanc´e et al. (2012, Chapter 7), we use each of the two data sets to fit the distribution of a loss severity in each risk category and then we assume a scenario where we have n1 = 10 losses of the first category and n2 = 20 losses of the second one. We finally determine risk of the total sum. Our model allows that the sum of losses in risk no. 1 and risk no. 2 arise from the sum of beta distributions and so, they are also distributed as a beta distribution of the second kind. Our objective is to find the risk measure of the total sum, i.e. the VaR and the TVaR for the sum of 30 losses, which we will obtain easily from the closedform expression, while allowing that severities in public data risk no. 1 and public data risk no. 2 are not independent. Table 4 presents the descriptive statistics of the two data sets that were provided by Bolanc´e et al. (2012). We fit a beta distribution of the second kind with parameters (p1 , q1 and λ1 ) to the data from the public data risk no. 1. We then fit a beta distribution of the second kind with parameters (p2 , q1 and λ1 ) as described in Section 4, i.e. holding the last two parameters fixed, to the data from the public data risk no. 2. Parameter estimates

14

Table 4: Basic statistical description for operational risk data (Bolanc´e et al., 2012). Risk no.1

Risk no.2

Number of observations

1000

400

Mean

42.0594

20.8933

Median

3.471

4.293

Mode

2.55

2.9

Max

5122.14

1027.53

Min

0.002

0.003

Std. Deviation

291.963

95.9138

Variance

85242.6

9199.45

Range

5122.13

1027.52

Interquartile Range

6.2395

5.9865

Semi-interquartile range

3.11975

2.99325

are presented in Table 5. The Kolmogorov-Smirnov test method, based on bootstrap resampling (Prieto et al., 2013), rejects the null hypothesis for both risks at 1% significance level (p-value= 0.000)2 for risk no. 1 and for risk no. 2, and it also shows a better fit for risk no. 2 (KS = 0.084) than for risk no. 1 (KS = 0.105). The Q-Q plots are presented in Figure 1 and the rank-size plots (Guillen et al. (2011)) presented in Figure 2 show the univariate fit of a beta distribution of the second kind to the two sets of original data. In the case of Q-Q-plots, the upper plots correspond to public data risk no. 1, and the lower plots correspond to public data risk no. 2. We have focused on losses smaller than 400 for risk no. 1 and smaller than 300 for risk no. 2. The plots on the 2

We argue that having as many as 1, 000 observations for risk no. 1 and 400 obser-

vations for risk no. 2 implies that the null hypothesis tends to be rejected, because the power of the KS test tends to 1 as the sample size goes to infinity.

15

right focus on the small size losses, while the plots on the left correspond to the whole range. With respect to the rank-size plots which are displayed in Figure 2 , the left plot corresponds to public data risk no. 1 and the right plot corresponds to public data risk no. 2. These plots show that the beta distribution of the second kind fits the two data sets quite well and we do not identify that the fit for the second data set is poorer given that the two parameters are held fixed in our estimation procedure. Assuming that loss severity for public data risk no. 1 follows a beta distribution of the second kind, then the sum of for n1 losses (which we call S1n1 ) also follows a beta distribution of the second kind and VaR and TVaR can be calculated directly from expressions (10) and (11). Similarly, we apply our theoretical results to obtain the VaR and TVaR for the sum of n2 losses of public data risk no. 2, which we call (S2n2 ). Figure 3 presents VaR and TVaR for S1n1 where n1 = 10 as a function of the tolerance level α, α ∈ [0, 1].

Figure 4 presents VaR and TVaR for S2n2 where n2 = 20 as a function of the tolerance level α and finally, Figure 5 presents the same plot for the sum S1n1 + S2n1 , with a total number of losses equal to n1 + n2 = 10 + 20 = 30. Table 5: Parameter estimates from the second kind beta distribution. Risk no.1

Risk no.2

p1

4.87882

-

p2

-

5.34804

q1

1.08467

1.08467

λ1

0.76560

0.76560

As expected, Figures 3, 4 and 5 show an increasing risk curve as a function of the tolerance level and TVaR is above VaR for a given α. The risk values for fixed S1n1 + S2n1 given α, n1 and n2 can be plotted very easily simply using a 16

400

4000

300 Observed

Observed

5000

3000 2000 risk no .1

1000 0

0

1000

2000

3000

4000

200 100 0

5000

risk no .1

0

100

200

Estimated 300 250 Observed

800 Observed

400

Estimated

1000

600 400 risk no .2

200 0

300

200 150 100

risk no .2

50 0

200

400

600

800

0

1000

0

50

100

Estimated

150

200

250

300

Estimated

Figure 1: Q-Q Plot for the univariate fit of a beta distribution of the second kind to the original data sets. The plots on the right correspond to small loss sizes only.

8

8 risk no .1

risk no .2

6 log HrankL

log HrankL

6 4 2

4 2

0

0 -5

0

5

10

-5

log HsizeL

0

5

10

log HsizeL

Figure 2: Rank-size plots of the complementary of the cdf multiplied by N+1 (solid lines) for the univariate fit of a beta distribution of the second kind to the original data sets. Left: public data risk no. 1. Right: public data risk no. 2.

17

spreadsheet. As mentioned before, our methodology assumes a multivariate distribution model for loss severity where there dependence exists between loss size arising from different risk sources. Tables 6, 7 and 8 present results for several choices of n1 = (10, 15, 20, 30 and 50) and n2 = (20, 25, 30, 60 and 100). We have tolerance levels equal to 95%, 99% and 99.9%, respectively. Compared to results obtained by Bolanc´e et al. (2012), who only considered α equal to 95% and independent Weibull distributions, here loss sizes are assumed to be correlated and our distribution is much more heavy-tailed. For instance, for n1 = 10 and n2 = 20 and a tolerance level of 95%, Bolanc´e et al. (2012) obtained that VaR and TVaR for S110 + S220 are equal to 920.20 and 1088.38 respectively, while our model produces much larger estimates, namely 1765 and 23296 respectively (see the last two columns of the first row in Table 6). 6. Conclusions In practice, a standard method to compute risk measures from multivariate sources is to use a large Monte Carlo simulation. In the first place, data for individual losses within each category are generated according to suitable law. Then a number of losses are simulated from the first severity distribution and then from the second distribution, assuming independent univariate distributions. The sum of all operational losses is the variable of interest. A Monte Carlo simulation repeats this generation process until a simulated distribution of the sum of losses is obtained after a large number of repetitions. From the simulated distribution, empirical risk measures can be calculated. Dependence between marginal distributions can be achieved either by assum18

3000

2500

2000

1500

1000

TVaR

500

VaR 0 0.0

0.2

0.4

0.6

0.8

1.0

Figure 3: Estimated VaR and TVaR of the sum of ten losses from public data risk no. 1 are plotted for all possible values of α.

6000 5000 4000 3000 2000

TVaR

1000

VaR

0 0.0

0.2

0.4

0.6

0.8

1.0

Figure 4: Estimated VaR and TVaR of the sum of twenty losses from public data risk no. 2 are plotted for all possible values of α.

10 000

8000

6000

4000

TVaR 2000

VaR 0 0.0

0.2

0.4

0.6

0.8

1.0

Figure 5: Estimated VaR and TVaR of the sum of ten losses from public data risk no. 1 and twenty losses from public data risk no. 2 are plotted for all possible values of α.

19

Table 6: V aR and T V aR at the 95% level for S1n1 , S2n2 and S1n1 + S2n2 , for a range of possible values of n1 and n2 S1n1

n1 V aR

S2n2

n2 T V aR

V aR

(n1 , n2 ) T V aR

S1n1 + S2n2 V aR

T V aR

10

553

7301

20

1212

16000

(10,20)

1765

23296

15

829

10949

25

1515

19999

(15,25)

2344

30943

20

1106

14597

30

1818

23998

(20,30)

2923

38589

30

1658

21893

60

3636

47990

(30,60)

5294

69877

50

2764

36484

100

6059

79980

(50,100)

8823

116459

Table 7: V aR and T V aR at the 99% level for S1 , S2 and S1 + S2 , for a range of possible values of n1 and n2 . S1n1

n1

S2n2

n2

V aR

T V aR

10

2501

32261

15

3751

48378

20

5001

64495

30

7500

96728

50

12499

161196

(n1 , n2 )

V aR

T V aR

20

5482

70695

25

6851

88362

30

8221

60

16441

100

27401

S1n1 + S2n2 V aR

T V aR

(10,20)

7981

102928

(15,25)

10601

136712

106029

(20,30)

13220

170496

212029

(30,60)

23940

308730

353364

(50,100)

39898

514532

Table 8: V aR and T V aR at the 99.9% level for S1 , S2 and S1 + S2 , for a range of possible values of n1 , n2 and (n1 ,n2 ). S1n1

n1

S2n2

n2

(n1 , n2 )

S1n1 + S2n2

V aR

T V aR

V aR

T V aR

10

21033

269674

20

46091

590943

(10,20)

15

31541

404395

25

57609

738620

(15,25)

89132

1142780

20

42048

539116

30

69127

886298

(20,30)

111158

1425180

30

63064

808558

60

138236

1772360

(30,60)

201282

2580690

50

105094

1347440

100

230382

2953789

(50,100)

335459

4300990

20

V aR

T V aR

67106

860384

ing a multivariate model or by introducing a copula. Choosing the model or the copula, plus computational effort are two major drawbacks. Our approach imposes a flexible multivariate structure and its major advantage is that computational burden is substantially shortened, because risk measures of sums can be obtained directly from closed-form expressions. Acknowledgements The authors thank the Spanish Ministry of Education and Innovation / FEDER grants ECO2010-15455 and ECO2012-21787-C03-01. Guill´en thanks ICREA Academia. The authors would would like to thank an anonymous reviewer for her/his comments. Appendix Proof of Lemma 1. Using (2), the VaR is the only solution of the equation   x IB ; p, q = α. λ+x If we solve previous equation for x, we obtain Equation (4).

Proof of Lemma 2. The TVaR can be expressed as, R∞ xdFX (x) , T V aR[X; α] = a 1 − FX (a) where a = V aR[X; α]. Now, using (3) for r = 1 and the first incomplete distribution of a second kind beta distribution, we obtain (5).

21

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