Measuring Economic Capital: Value at Risk, Expected Tail
Loss and Copula Approach
by
Jeungbo Shim, Seung-Hwan Lee, and Richard MacMinn
August 19, 2009
Please address correspondence to: Jeungbo Shim
Department of Business Administration
Illinois Wesleyan University
P.O. Box 2900
Bloomington, IL 61702
Jeungbo Shim
Assistant Professor
Illinois Wesleyan University
Bloomington, IL 61702
Tel: 309-556-3308
jshim@iwu.edu
Seung-Hwan Lee
Assistant Professor
Illinois Wesleyan University
Bloomington, IL 61702
Tel: 309-556-3421
slee2@iwu.edu
Richard MacMinn
Professor
Illinois State University
Normal, IL 61790
Tel: 309-438-7993
rmacmin@ilstu.edu
Measuring Economic Capital: Value at Risk, Expected Tail
Loss and Copula Approach
Abstract
Investigating dependence structure between variables is important when modeling
multivariate data in the study of finance and insurance risks. In this paper, we explore
these practical applications for assessing the economic capital measured by Value at
Risk and Expected Tail Loss. The procedures are based on copula functions that
provide a flexible methodology for modeling multivariate dependence. Using copulas,
we examine the impact of dependence structure between market risk and underwriting
risk of insurance portfolio on the insurer’s economic capital, particularly for the
sample of the U.S. property liability insurance industry. Special attention is given to
the issue of choosing the copula that best fits our application data. We also assess
diversification benefit by comparing the difference between the diversified VaR and
the undiversified VaR. Numerical studies show that the insurer’s capital requirements
and the level of diversification benefit may vary depending on the choice of copulas.
The result suggests that t copula is preferred to Gaussian copula in estimating
economic capital for our given data.
Keywords: Copulas; Value at Risk; Dependence; Diversification; Property-liability insurers
1. Introduction
Economic capital is taking on growing importance along with the increasing
natural and man-made catastrophes within the insurance industry. One survey
revealed that 65% of all insurance respondents calculate a form of economic capital as
part of their business practice and an additional 19% are considering calculating
economic capital.1 Despite its increasing significance, there is currently no consensus
as to how to define and calculate economic capital. We present a model for assessing
the economic capital in an insurance context.
Capital is the financial cushion that insurance companies use to absorb adverse
consequences due to catastrophic claims costs or unfavorable asset returns. 2 The
meaning of capital varies depending on the viewpoint. Although not always precisely
defined, economic capital is distinct from regulatory capital in the sense that
1
The survey implemented by Tillinghast included responses from insurance executives in over 200
global companies, including 32 respondents from North American life insurers. See the details in the
2006 Tillinghast ERM survey at www.towersperrin.com
2
Capital is used interchangeably with policyholder surplus in the insurance industry.
2
Figure 1 VaR and Economic Capital
regulatory capital is the mandatory minimum capital required by the regulators while
economic capital serves to underpin all the actual risks borne by the insurer.
Conceptually, economic capital can be expressed as protection needed to secure
survival in a worst case scenario. More specifically, economic capital can be defined
as the difference between the expected loss and a worst tolerable loss at a selected
confidence level. Expected loss is the anticipated average loss over a defined period
of time and generally imputed as a cost of doing business. Expected loss does not
contribute to the capital charge since funds equivalent to the expected loss are held in
the insurer’s reserves. Because economic capital is the level of capital that the firm
should hold to maintain its probability of default below a certain threshold ( 1 − c %),
economic capital can be viewed as the Value at Risk (VaR) at a given confidence
level. Economic capital is based on a probabilistic assessment of potential future
losses and therefore itself random. The definition of VaR and economic capital is
presented graphically in Figure 1. Because asset-side risks and insurance underwriting
risks are aggregated to determine the insurer’s total economic capital, the distribution
is based on the aggregate return. Thus, VaR is at left tail where negative returns imply
the losses of the company portfolio value.
3
To estimate total economic capital for an insurance company, standalone
economic capital calculated separately for each risk type must be aggregated. The
conventional approach for aggregation is simply to add up standalone risk capital.
This simple additive approach may be easy to implement but overlooks potentially
important diversification effects. The risks of insurance company that should be
covered under the economic capital approach can be categorized into primary
components of asset risk, liability risk, and operational risk.3 Because these risks are
less than perfectly correlated, the potential for diversification suggests that the whole
risk will be less than the simple sum of standalone risk.
In modeling multivariate risks, the commonly used measure of dependence
structure is the linear correlation. Linear correlation is a natural scalar measure of
dependence under the assumption of elliptical distributions such as the multivariate
normal. However, the joint distributions of the real world in finance and insurance are
not always the case of the multivariate normal. Embrechts et al. (2002) point out that
the presence of non-linear derivative products and skewed heavy-tailed insurance
claims invalidates the assumption of multivariate normal distribution. Therefore, the
use of linear correlation under the assumptions of normality can mislead the level of
real dependence between different risks (Wirch, 1999). Furthermore, linear
correlation is unable to fully capture the degree of dependence in the tail of the
underlying distribution where extreme events are highly dependent. Thus, linear
correlation may not be an appropriate dependence measure for heavy-tailed
distributions (Embrechts et al., 2002).
3
Asset risk is generally decomposed of credit risk (e.g. when debt obligations are not met) and market
risk (e.g. a fall in the value of an insurer's investments). Liability risk refers to insurance underwriting
risk (e.g. the risk that arises from under-estimating the liabilities from business written or inadequate
pricing). Operational risk includes a risk of system failure or malpractice.
4
An alternative and theoretically more robust method to capture a wide of range
of dependence structure amongst different risks is to use copulas. Copulas formulate a
joint multivariate distribution by combining univariate marginal distributions in such
a way that various types of dependence can be represented. Because a copula
embodies all the information about the dependence between the components of
random vector, it has several advantages as a dependence measure. For example,
copulas identify both linear and non-linear dependence between random variables.
More importantly, copulas well capture the upper tail or the lower tail behavior. Thus,
copulas can be a useful technique to analyze dependence structure in an insurance
setting where extreme events appear to occur simultaneously and asset risks and
liability risks diversify against one another to some degree.
There are several studies that model dependence structure with copulas in the
insurance literature. Among them, Frees et al. (1996) examine the effects of
dependent mortality models on annuity valuation with data from a Canadian insurer.
They estimate the probability of joint survivorship of two annuitants using a bivariate
survivorship function called Frank copula. Their estimation results provide evidence
that strong positive dependence exists between joint lives and annuity values decrease
when dependent mortality models are used. Tang and Valdez (2006) examine the
differences of the insurer’s capital requirement under different copula assumptions.
They perform simulation using historical loss ratios of five lines of business for the
aggregated Australian industry to account for possible dependencies between
insurance risks. They find that the choice of copula has a significant impact on the
multi-line insurer’s capital requirement and diversification benefit. As they point out,
one limitation of their analysis is focusing merely on insurance underwriting risks, not
considering other sources of risks such as market risk. In reality, market risk is one of
5
dominant risks for property-liability insurers, accounting for 30% of the total
economic capital of a typical insurer, while insurance underwriting risk accounts for
50% of the capital requirement (Oliver, Wyman & Company, 2001).4
We contribute to the literature in two main aspects. First, we incorporate two
major categories of risks- market risk (asset-side risk) as well as insurance
underwriting risk (liability-side risk) in modeling the insurer’s total economic capital.
According to the new solvency requirements developing across the EU, insurers are
required to hold capital against market risk, credit risk and operational risk, which are
not covered by contemporary practice.5 The current EU regime concentrates mainly
on the insurance underwriting risks. Our study will provide practitioners and
regulators with important insights in determining total capital requirements where
both asset-side risks and liability-side risks and their interactions are considered.
Second, we attempt to identify which copula is the best one for the given application
data. The selection of the copula that best fits the data is critical since the estimates of
capital requirements can be significantly different according to the dependence
structure chosen. In addition, we quantify diversification benefit for each copula
model by comparing the difference between the diversified VaR and the undiversified
VaR on the aggregated portfolio returns.
In this paper, we focus on how different dependence structure between market
risk and underwriting risk has a substantial impact on the insurer’s total required
4
Credit risk and operational risk account for 5% and 15% of the total economic capital requirement of
a typical property liability insurer on average, a modest amount in comparison to the 55% and 25%,
respectively for a bank (Oliver, Wyman & Company, 2001).
5
The new Solvency II rules will introduce economic risk-based solvency requirements and be expected
to operate in 2012 across all EU Member States. The new regime would establish more risk-sensitive
and harmonized solvency requirements.
6
Figure 2 Process of Risk Aggregation and Economic Capital Calculation
Market Risk
Stock Marginal
•••
Rank Correlations
beteen Risks
Underwriting Risk
Mortgage Marginal
HO Marginal
Monte Carlo Simulation
(Random returns/loss)
•••
WC Marginal
Elliptical Copulas
(Gaussian, t, Cauchy)
VaR
economic capital using the sample of U.S. property-liability insurance industry.6 We
include six business lines to represent insurer’s underwriting risk and five categories
of assets are chosen as proxies for market risks. We first choose the best fitting
marginal distribution for each market risk class and business line based on historical
quarterly returns. Using estimated parameters from selected marginal distributions
and Spearman’s rank correlation coefficients as inputs in the simulation algorithm, we
then perform Monte Carlo simulation to generate random return series for each
market and underwriting risk under several different copulas. To estimate the
insurer’s economic capital, we calculate both the Value at Risk (VaR) and the
Expected Tail Loss (ETL) from the aggregated return distribution. Figure 2 illustrates
conceptually risk aggregation process that various marginal distributions of market
6
The recent report by the International Actuarial Association (IAA) and Solvency II rules generally
emphasize the importance of recognizing and modeling interdependencies of multiple risks. We do not
include credit risk and operational risk due to data limitations in this paper.
7
and underwriting risk components are combined to generate a single return
distribution- and economic capital- for the insurer.
This paper is organized as follows. Section 2 describes a brief introduction to
copula theory, and risk measures such as VaR and ETL are discussed in Section 3.
Section 4 illustrates how to measure the difference between diversified VaR and
undiversified VaR. Section 5 provides numerical examples and conclusions for the
assessment of economic capital are presented in Section 6.
2. Copula Functions as a Dependence Measure
The statistical properties and applications of copulas have been extensively
studied in the recent works (e.g.; Frees and Valdez, 1998; Breymann et al., 2003;
Demarta and McNeil, 2003; Trivedi and Zimmer, 2005). Copula functions are used to
fully capture real dependence amongst different risks. We first discuss several
approaches of dependence measures and then present basic properties of copulas, their
relationships to measures of dependence, and technical algorithms that generate
random vectors.
2.1 Dependence measures
The choice of the appropriate dependence measure is essential in determining
capital requirements. Consider an insurer with k classes of asset portfolios and
n − k lines of insurance business facing return random vector, ( X 1 ,..., X k ,..., X n ) .
T
The insurer’s total return is given by the weighted sum of X i ’s. We assume that X i
is a random return variable and has a marginal distribution, Fi ( x ) = P ( X i ≤ x ) . The
dependence between random variables, X 1 ,..., X n , is determined by their joint
distribution function
F ( x1 , ..., xn ) = P ( X 1 ≤ x1 , ..., X n ≤ xn ).
(1)
8
The traditional approach that measures the dependence is to use Pearson’s
linear correlation coefficient, assuming that random variables have jointly a
multivariate normal distribution. If ( X 1 ,..., X n )
T
has a multivariate normal
distribution and more generally an elliptical distribution, it is unproblematic to
employ the correlation matrix as a measurement of dependence structure. However,
most real world risks such as credit or operational risks are not likely to have an
elliptical distribution (Shaw, 1997). In addition, Pearson’s correlation coefficient is
influenced by extreme values, and not invariant under non-linear monotonic
increasing transformations of random variables, resulting in incomplete description of
dependence structure (Kumar and Shoukri, 2008). Thus, the use of linear correlation
in the jointly non-normal distributions may have serious deficiency in measuring real
dependency.
Two widespread concordance (global trend) measures are the Spearman’s rank
and Kendall’s rank correlations. They are non-parametric statistics used to measure
the strength of association between random variables in rank-ordered scale. It is wellknown that both Spearman’s and Kendall’s correlations are invariant under any
monotonic transformations. However, they are unlikely to fully capture extreme tail
dependence which is a major concern in quantifying capital charge. It is also very
complicated to estimate the multivariate joint probability distribution with these rank
correlations, especially when a large number of variables are involved.
An alternative to measure the dependence structure is copulas that overcome
the aforementioned limitations of the correlations. Copulas allow for any choice of a
marginal for each risk type and facilitate a more precise description of extreme
outcomes. Copulas provide a way of isolating the peculiar behavior of individual
risk’s marginal distribution from the description of dependence structure. Thus,
9
copula modeling fits well into an insurance framework, where each asset and liability
risk of the insurance company has a different distributional characteristic.
2.2 Copula Definition7
From a statistical point of view, a copula is a function that combines univariate
marginal distributions to construct a joint distribution with a specific dependence
structure. Thus it provides a way to create probability distributions to model
dependent multivariate components.
To elaborate, a copula is a function, C :[0,1]n → [0,1] , such that
1. For ui ∈ [0,1], i = 1,..., n, C (0,..., 0, ui , 0,..., 0) = 0 and C (1,...,1, ui ,1,...,1) = ui .
2. For
all
(u11 ,..., u1n ), (u21 ,..., u2 n ) ∈ [0,1]n ,
componentwise,
2
2
i1 =1
in =1
with
u1i ≤ u2i ,
i ∈ {1, ⋅⋅⋅, n},
∑ ⋅⋅⋅∑ (−1)i1+⋅⋅⋅+in C (u1i1 , ⋅⋅⋅unin ) ≥ 0.
The important aspect of copulas is summarized by Sklar’s theorem (1959), which
shows that the univariate marginal behavior and the multivariable dependence
structure can be separated from the joint multivariate distributions.
Theorem (Sklar, 1959) If F is a joint distribution function with marginal
distributions, F1 ,..., Fn , then there exists a copula C such that
F ( x1 ,..., xn ) = C ( F1 ( x1 ),..., Fn ( xn )).
Using the standard transformation method, it can be shown that Fi ( X i ),
(2)
i = 1,..., n,
has a uniform distribution defined on the interval[0,1] . Therefore, the copula can be
viewed as a multivariate function with standard uniform marginal distribution. If the
marginal distributions, F1 ,..., Fn , are continuous, then F has a unique copula. However,
if one or more marginals are not continuous, then copula for F is no longer unique.
7
The notion of copula and its ‘must-read’ literature are well discussed in Embrechts (2009).
10
Let ui be the observed value of Fi ( X i ) . Then for continuous univariate marginals, the
unique copula function is given by
C (u1 ,..., un ) = F ( F1−1 ( x1 ),..., Fn−1 ( xn )),
(3)
where F1−1 ,..., Fn−1 denote the quantile functions of the univariate marginals, F1 ,..., Fn .
From Sklar’s theorem, it is noted that any marginals can be used, since a copula links
marginals to their multivariate distribution, and it is independent of marginals as a
measure of dependence. Hence, such a copula is constructed under the assumption
that marginal distributions are known and can be consistently estimated from a given
data. A special category of copulas used in our numerical analysis are presented in the
following section.
2.3 Elliptical Copulas
The class of elliptical copulas such as Gaussian, Student-t, and Cauchy copula
is flexible since elliptical copulas allow for both positive and negative dependence.
Elliptical copulas are particularly suitable in modeling dependence structures with
multi-dimensions. Indeed, the introduction of a correlation matrix as a multidimensional parameter allows a great flexibility in terms of structure’s shape and
simulation procedures. However, a special class of copulas termed as Archimedean
copulas limits the complex nature of the dependence structure because they represent
dependence structure with a few parameters (Nelsen, 2006). Copulas of the
Archimedean family such as Gumbel and Clayton copulas cannot account for
negative dependence. 8 In this paper, we focus on elliptical copulas which are best
suited to our problem.
8
The class of Archimedean copulas is well discussed in Nelsen (2006). Gumbel is an appropriate
modeling choice when variables are strongly correlated in the upper tail of the joint distribution while
Clayton is suitable for strong lower tail dependence.
11
Gaussian Copula
The Gaussian copula is derived from the multivariate Gaussian distribution.
The dependence structure among the margins is described by a copula function, C ,
such that
C RG (u1 ,..., un ) = G (Φ −1 (u1 ),..., Φ −1 (un )),
(4)
where G denotes the joint distribution function of the multivariate standard normal
distribution function with linear correlation matrix R , and Φ −1 is the inverse of the
univariate standard normal distribution function. In the case of bivariate distribution, a
bivariate copula function can be written as
CRG (u1 , u2 ) = ∫
Φ −1 ( u1 )
−∞
∫
Φ −1 ( u2 )
−∞
 ( s12 − 2 R12 s1s2 + s22 ) 
1
exp
−
 ds1ds2 ,
2π (1 − R122 )1/ 2
2(1 − R122 )


(5)
where R12 is the linear correlation coefficient of the corresponding bivariate normal
distribution. Note that R = ( Rij ) with Rij =
∑
∑∑
for i, j = 1, ⋅⋅⋅, n , where
ij
ii
∑
ij
is
jj
the (i, j )th component of the covariance matrix ( ∑ ) of the distribution.
The most interesting property of the copula in our study is about tail
dependence. The Gaussian copula does not allow for extreme events to be dependent.
To illustrate the concept of tail dependence with bivariate variables, let X 1 and X 2 be
continuous random variables with marginal distribution functions F1 and F2 . The
coefficients of tail dependence are asymptotic measures of the dependence in the tails
of the bivariate distribution of X 1 and X 2 . The coefficients of lower and upper tail
dependences of X 1 and X 2 are defined as
λL = lim P ( X 2 < F2−1 (u ) | X 1 < F1−1 (u )),
u ↓0
(6)
12
λU = lim P( X 2 > F2−1 (u ) | X 1 > F1−1 (u )),
u ↑1
(7)
respectively, provided that the limit exists in the interval [0,1] . Hence the tail
dependence states the probabilities of having a high (low) extreme value of X 2 given
that a high (low) extreme value of X 1 occurs. The dependence structures are of a great
variety from one copula to another. If λU = 0 ( λL = 0 ), for example, random variables
X 1 and X 2 are said to be asymptotically independent in the upper (lower) tail, i.e., no
tail dependence, which is the case of the Gaussian copula. On the contrary, if
λU ∈ (0,1] ( λL ∈ (0,1] ), X 1 and X 2 are asymptotically dependent in the upper (lower)
tail.9
We now describe the algorithm that provides random vector generation from
the Gaussian copula C RG . Let ∑ be positive definite such that ∑ = AAT for some n × n
matrix A . If Z1 ,..., Z n denote a random sample from N (0,1) and µ + AZ is a normal
distribution with mean, µ and covariance, ∑ ,
where Z = ( Z1 , ⋅⋅⋅, Z n )T ,
Lower
triangular matrix A can be obtained using Cholesky decomposition method.
Construct the lower triangular matrix A so that the covariance matrix ∑ = AAT using
Cholesky decomposition. Then,
1. Simulate n independent standard normal random variates Z = ( Z1 ,..., Z n )T
from N (0,1).
2. Take the matrix product of A and Z , i.e., Y = AZ .
3. Set U i = Φ (Yi ) for i = 1,..., n., where Φ is the distribution function of the
standard normal.
4. Set X i = Fi −1 (U i ) for i = 1,..., n.
5. ( X 1 ,..., X n )T is the random vector with marginals, F1 ,..., Fn , and Gaussian
copula C RG .
9
See Embrechts, McNeil, and Straumann (1999) for a proof.
13
Student’s t-copula
Student’s t-copula (or simply t-copula) is implied by a multivariate Student’s t
distribution. The t-copula with v degrees of freedom can be written as
Cvt ,R (u1 ,..., un ) = tv, R (tv−1 (u1 ),..., tv−1 (un )),
(8)
where tv ,R denotes the multivariate t -distribution function, and tv is the marginal
distribution function of tv ,R (i.e. the usual univariate t-distribution function). For the
bivariate case, t-copula has the following form
C (u1 , u2 ) = ∫
t
v ,R
tv−1 ( u1 )
−∞
∫
tv−1 ( u2 )
−∞
 ( s12 − 2 R12 s1s2 + s22 ) 
1
1 +

2π (1 − R122 )1/ 2 
v(1 − R122 )

− ( v + 2) / 2
ds1ds2 ,
(9)
where R12 is the linear correlation coefficient of the corresponding bivariate tv
distribution, if v > 2 .
Unlike the Gaussian copula, t-copula generates joint extreme movements
regardless of the marginal behavior of random variables. The t-copula has both lower
and upper tail dependence, expressing dependence between extreme events. In order
to quantify such dependences with various degrees of freedom and different
correlations, we calculate coefficients of upper tail dependence ( λU ) for the t-copula
based on the property of the equation (7). It is noted in Table1 that the coefficient of
Table 1 Coefficients of Tail Dependence ( λU ) for t-copulas
a
v \ R12 -1
-0.9
-0.8
-0.5
-0.2
0
0.2
0.5
0.8
1
0 0.1024 0.1474 0.2468 0.3333 0.3968 0.4544 0.5641 0.7196
2
0 0.0171 0.0351 0.0955 0.1679 0.2254 0.2929 0.4226 0.6220
3
0 0.0032 0.0093 0.0405 0.0917 0.1393 0.2010 0.3318 0.5527
10
0
0
0
0.0002 0.0023 0.0078 0.0220 0.0845 0.2948
30
0
0
0
0
0
0
0.0001 0.0031 0.0733
0
0
0
0
0
0
0
0
0
∞
a
R12 denotes the linear correlation coefficient and v is degrees of freedom
0.9
0.8003
0.7295
0.6776
0.4643
0.2113
0
1
1
1
1
1
1
1
14
tail dependence is increasing in R12 and decreasing in v (see Embrechts et al., 2002
for a proof). Thus, t-copula is more suitable than Gaussian copula to model the case
where extreme events occur simultaneously. Furthermore, the coefficient of upper tail
dependence tends to zero as degrees of freedom ( v ) tend to infinity for R12 < 1 ,
implying that a t-copula converges to a Gaussian copula as v tends to infinity.
To simulate random vector ( X 1 ,..., X n )T from the t-copula, Cvt , R , the following
algorithm can be used (Embrechts et al., 2001):
1. Construct the lower triangular matrix A so that the covariance matrix ∑ =
AAT using Cholesky decomposition.
2. Simulate n independent standard normal random variates Z = ( Z1 ,..., Z n )T
from N (0,1).
3. Simulate a chi-squared random variates S with v degrees of freedom,
independent of Z = ( Z1 ,..., Z n )T .
4. Take the matrix product of A and Z , i.e., Y = AZ .
v
Yi for i = 1,..., n.
S
6. Set U i = tv (Ti ) for i = 1,..., n.
5. Set Ti =
7. Set X i = Fi −1 (U i ) for i = 1,..., n.
6. ( X 1 ,..., X n )T is the random vector with marginals, F1 ,..., Fn , and t-copula, Cvt ,R .
Cauchy Copula
The Cauchy copula is a special case of the t-copula where the degrees of
freedom ( v ) is equal to one. The copula generated by a multivariate Cauchy
distribution is
C1,c R (u1 ,..., un ) = t1,R (t1−1 (u1 ),..., t1−1 (un )),
(10)
where t1,R is the joint distribution function of a standard Cauchy random vector.
Similar to t-copula, the bivariate Cauchy copula is
C (u1 , u2 ) = ∫
c
1, R
t1−1 ( u1 )
−∞
∫
t1−1 ( u2 )
−∞
 ( s12 − 2 R12 s1s2 + s22 ) 
1
1 +

2π (1 − R122 )1/ 2 
(1 − R122 )

− (3/ 2)
ds1ds2 , (11)
15
t1−1 (⋅) is
where
the
inverse
of
a
standard
Cauchy
distribution
with
2
1  1 
⋅
dw . The Cauchy copula also yields tail dependence because of
−∞ π  1 + w 


t1 ( z ) = ∫
z
its relationship with the t-copulas. To simulate a random vector, ( X 1 ,..., X n )T from
the Cauchy copula, we follow the t-copula algorithm with v = 1 .
Independence Copula
The independence copula can be expressed as
C ind (u1 , ⋅⋅⋅, un ) = u1 ⋅ u2 ⋅⋅⋅ un .
(12)
From Sklar’s theorem, it is clear that the dependence structure of the continuous
random variables is given by the above (12) if and only if they are independent.
3. Risk Measures
The calculation of economic capital is a process that begins with the
quantification of the risks that any given company faces over a given time period.
Risk measures provide a magnitude of the severity of a potential loss in a portfolio
and present meaningful amounts to hold to support the risk. Thus, the capital required
for the company can be determined by the risk measure. Value-at-Risk (VaR) and
Expected Tail Loss (ETL), most prominent risk measures among regulators and
practitioners, are used to evaluate risks and to determine the capital requirements in
this paper.
3.1 Value-at-Risk
The Value-at-Risk (VaR) is a statistical-based risk measurement that
summarizes the expected maximum loss over a target horizon period, at a given level
of confidence. Consider a real-valued random variable X that represents the loss of
some risky portfolio in a specified time period, with the distribution,
F ( x) = P{ X ≤ x}, measuring the severity of the risk of holding portfolio. Then, the
16
maximum loss is simply infinity since F is unbounded in most models. Thus, the
maximum possible loss is given by inf{x : F ( x) = 1} on the real line, where ‘inf’
denotes a greatest lower bound. For each α , 0 < α < 1 , the VaR is given by the largest
number x such that the probability that the loss X does not exceed x is at least α .
Formally, the VaR with given level of α is defined as10
VaRα ( X ) = inf{x : F ( x) ≥ α }.
(13)
Assuming that the risk profile of the institution remains constant over the
horizon, VaR can be derived by reading the quantile off the actual empirical
distribution. The VaR is a commonly used risk measure in financial institutions to
assess the risk associated with any type of portfolio taking account of correlations
between risk factors.
11
VaR presents a probability statement about the potential
change in the value of a portfolio resulting from a change in market factors over a
given period of time (Crouhy et al., 2001). One of advantages of VaR is that it
aggregates all of the risks in a portfolio into a single number suitable for reporting to
regulators or disclosure in an annual report. Because VaR measures the worst amount
the firm is likely to lose, it can be very useful to determine capital requirements at the
level of the firm. Dhaene et al. (2008) find that when substituting the subadditivity
condition by the regulator’s condition, VaR is the most efficient capital requirement
in the sense that it reduces some reasonable cost function.
However, VaR has some limitations as a risk measure. For example, VaR does
not capture statistical properties of the significant loss beyond the quantile point of
interest. Thus, even if the severity of losses doubles beyond quantile point, the VaR
10
11
VaR can be expressed as VaRα ( X ) = inf{ x : S ( x ) ≤ 1 − α } with S ( x ) = 1 − F ( x ) . See
McNeil et al., (2005).
VaR information can be used to implement portfolio-wide hedging strategies that are otherwise
rarely impossible (Kuruc and Lee, 1998; Dowd, 1999). Properties of VaR have been discussed
extensively in the literature(e. g., Dowd and Blake, 2006).
17
figure may not be influenced and gives us no indication of how much (i.e., a loss in
excess of VaR) that might be.
3.2 Expected Tail Loss as a Coherent Risk Measure
Expected tail loss (ETL) is a coherent risk measure in the sense of Artzner et
al (1997, 1999).12 ETL is also known as the expected shortfall by Tasche (2001) and
Acerbi and Tasche (2002). 13 ETL calculates the conditional expected loss beyond
VaR. For a given α , ETL is defined by
ETLα ( X ) =
1
1−α
1
∫α VaR ( X )du.
u
(14)
ETLα ( X ) can be expressed as a linear combination of the corresponding quantile and
its expected shortfall (e.g., Denuit et al., 2005)
ETLα ( X ) = VaRα ( X ) +
P{ X > VaRα ( X )}
E{ X − VaRα ( X ) | X > VaRα ( X )}. (15)
1−α
ETL overcomes some drawbacks of VaR because ETL tells us what we can expect to
lose if a tail event occurs. Note that ETL is more conservative than VaR since it
averages VaR over all levels u ≥ α ,
looking further into the tail of the loss
distribution and ETLα > VaRα . Dhaene et al. (2008) showed that ETL is the optimal
capital requirement at a given confidence level when the regulator’s condition is
imposed to the class of concave distortion risk measures.
4. The Measure of Diversification Benefit
An important implication of diversification effect is to mitigate the riskiness of
the firm by pooling uncorrelated risks. The risk of the portfolio of businesses will be
less than the sum of the stand-alone risks of the businesses if the risks of business
lines are not perfectly correlated with one another (Merton and Perold, 1993). Myers
12
See e.g., Dowd and Blake (2006) and Dhaene et al., (2008) for more discussions about the properties
of coherent risk measure.
13
There is no consensus of terminology. Expected tail loss (ETL) is also called conditional VaR, tail
VaR and conditional tail expectation.
18
and Read (2001) show that if each line of business is assumed to be organized as a
stand-alone firm, total capital requirements for those lines increase because of loss of
diversification. Perold (2001) argues that diversification across business segments
diminish the firm’s deadweight cost of risk capital. If we do not consider the effect of
diversification when allocating capital, we may overestimate firm risk, leading to
overcharging capital requirements.
In general, the diversification benefits can be quantified by three key factors:
the number of risk positions (or assets), the concentration of risk factors, and the
cross-risk correlations. For the illustration purpose, we take only two assets (asset a
and asset b ). Then, the diversified portfolio variance can be expressed in terms of
covariance of weighted assets a and b as follows:
σ 2p = ( waσ a ) 2 + ( wbσ b )2 + 2wa wb ρabσ aσ b ,
(16)
where σ a and σ b represent standard deviation of asset a and asset b , respectively,
wa and wb are respective weights, such that wa + wb = 1, and ρ ab is correlation
coefficient between asset a and asset b . The equation (16) shows that portfolio
variance accounts for the risk of the individual asset ( σ a2 and σ b2 ) and correlation
( ρ ab ) between them.
If we assume the portfolio return is normally distributed, the portfolio variance
can be translated into a portfolio VaR measure in the delta-normal model. The
portfolio VaR is then
VaR p = ασ pW = α ( waσ a ) 2 + ( wbσ b ) 2 + 2wa wb ρ abσ aσ b W ,
(17)
where α is the confidence level and W represents the initial portfolio value. If the
correlation ( ρ ab ) between two assets is assumed to be zero, the portfolio VaR reduces
to
19
VaR p = (α waσ aW ) 2 + (α wbσ bW ) 2 = VaRa2 + VaRb2 .
(18)
The result shows that the portfolio risk is lower than the sum of individual stand alone
risk ( VaR p < VaRa + VaRb ).14 This reflects the fact that a portfolio of two assets will be
less risky than a portfolio consisting of either asset on its own if returns are
independent of each other. If the correlation reaches its maximum value of 1, and both
wa and wb are positive, equation (17) reduces to
VaR p = VaRa2 + VaRb2 + 2VaRaVaRb = VaRa + VaRb .
(19)
If two assets are perfectly correlated, the portfolio VaR is equal to the sum of
individual stand alone VaR. In other words, there is no benefit of risk diversification
when all correlations are unity. This is a special case because correlations are
generally not perfect. The results provide us with important insights that
diversification benefit can be measured by the difference between the diversified VaR
and the undiversified VaR. The undiversified VaR is defined as the sum of individual
VaR when all correlation are exactly unit and the diversified VaR is defined as the
portfolio VaR in all circumstances except for the special case where the correlation is
one (Jorion, 2007).15
5. Numerical Studies
We present numerical examples to explore the influence of dependence
structure on the insurer’s economic capital charge and the level of diversification
benefit under different copulas. We first discuss the process of aggregating risks
considered for this exercise. The correlation matrix and the marginal distributions
which are essential inputs for the specification of copulas are presented. We then
14
This shows that VaR is a coherent risk measure for normal and elliptical distributions in general
(Jorion, 2007)
15
We also assume that there is no short position in our portfolio.
20
discuss our simulation results. We also discuss the choice of the best copula for the
given data in this section.
5.1 Aggregating Insurance Underwriting Risk and Market Risk
We measure insurance underwriting risk using quarterly industry-wide
underwriting returns by line of insurance business. Following the literature (e.g.,
Doherty and Kang, 1988; Gron, 1994; Doherty and Garven, 1995), the underwriting
return is defined as premiums written net of the present value of incurred losses for
each quarter divided by premiums written for the quarter. The quarterly data on losses
and premiums by line is provided by the National Association of Insurance
Commissioners (NAIC). We include six business lines –Homeowners (HO), Auto
Physical Damage (APD), Auto Liability (AL), Commercial Multiple Peril (CMP),
Workers Compensation (WC), and Other Liability (OL) that account for about 85% of
total underwriting risk in terms of premium written. Each business line has a different
risk characteristic depending on the payout patterns and product coverage. Long-tail
lines of business like AL, WC, and OL are considered to be more risky and complex
since losses of long-tail lines are not paid until long after the accident has occurred
due to several factors such as loss adjustment procedures and litigated claims.
Because the products of short-tail lines such as HO, APD, and CMP are more
standardized and their losses are fully developed or ultimately paid in one or two
years, short-tail lines are considered to be less risky.
Five categories of assets –Stocks, Government Bonds (GBond), Corporate
Bonds (CBond), Real Estate, and Mortgages are selected to represent the insurer’s
market risk (or asset-side risk). As proxies for market risks of asset portfolio, the
quarterly time series of returns are obtained from the standard rate of return series:
Stocks- the total return on the Standard & Poor’s 500 stock index; Government Bond
21
-the Lehman Brothers intermediate term total return; Corporate Bond -Moody’s
corporate bond total return; Real Estate-the National Association of Real Estate
Investment Trusts (NAREIT) total return; and Mortgages-the Merrill Lynch mortgage
backed securities total return.16
To produce a distribution of overall portfolio returns, we aggregate
underwriting return series of insurance business lines and market return series of asset
classes using their respective weights. The weight for each asset class is given by the
relative amounts invested in each asset class, while the weight for each business line
is specified according to the proportion of earned premium by line of business.
Weights are constructed to sum to unity by scaling the dollar amount in each asset
class and business line by the sum of all amounts of invested asset and earned
premium considered.
Table 2 lists descriptive statistics for the return series of asset classes and
insurance business lines used in the numerical analysis and presents their weights
used in aggregating portfolio returns. Several interesting patterns are observable. The
mean return ranges from 1.8% to 2.5% for asset classes while it varies from 26.5% to
37.2% for liability lines. Stocks among asset classes show the highest volatility, 7.9%,
consistent with common perception. Homeowners’ line is more volatile than other
business lines. One important pattern to notice is that minimum returns for all asset
classes are negative. Furthermore, the last column of Table 2 shows that assets
variables account for a considerable proportion of total weights. These results imply
that the aggregated portfolio return rate will be significantly influenced by the return
rate of asset classes.
16
Employing external indices as proxies for the behavior of particular risks is common among the
banking institutions. For example, CreditMetricsTM of J.P. Morgan opted equity returns as a proxy for
asset returns since internal asset returns are not directly observable (Basel Committee on Banking
Supervision, 2003; Morone, et al., 2007).
22
Table 2 Descriptive Statistics and Weights for Asset Classes and Liability Lines
Variables
Stock
GBond
CBond
RealEstate
Mortgage
HO
APD
AL
CMP
WC
OL
Mean
SD
Min
Max
Weight
0.0233
0.0792
-0.1897
0.1930
0.2222
0.0180
0.0186
-0.0187
0.0486
0.3856
0.0193
0.0341
-0.0659
0.0713
0.1539
0.0253
0.0607
-0.1155
0.1749
0.0095
0.0192
0.0172
-0.0233
0.0536
0.0039
0.2648
0.1255
-0.2169
0.4436
0.0294
0.3724
0.0460
0.2994
0.4659
0.0500
0.3040
0.0384
0.2401
0.3825
0.0757
0.3629
0.0694
0.1179
0.4810
0.0200
0.3026
0.0812
0.1105
0.4326
0.0313
0.3545
0.0713
0.1971
0.5286
0.0185
Variables: GBond (Government Bond), CBond (Corporate Bond), HO (Homeowners), APD
(Auto Physical Damage), AL (Auto Liability), CMP (Commercial Multiple Peril), WC
(Workers Compensation), and OL (Other Liability).
5.2 Correlations
The critical parameter that needs to be estimated for the simulation algorithm
using copulas is the correlation matrix between market risks and underwriting risks.
The industry-wide correlation matrix is estimated by the historical quarterly time
series of returns from each asset class and business line over the sample period 19912002. Table 3 provides the Spearman’s rank correlation coefficients for the return
series of five asset classes and six insurance lines. Rank correlation is invariant since
the simulated numbers replicate the same rank correlations that are used to generate
them. The returns of government bonds and mortgages are strongly positively related
each other, 93 percent correlations. Of the pair-wise correlations between insurance
underwriting risks, the pair of workers compensation (WC) and auto liability (AL)
shows the highest positive correlations with 85 percent. High correlations between
them may be justified since both WC and AL have similar long-term payout patterns.
The correlation between APD and WC is negative 40 percent. It can be argued that
23
Table 3 Correlation Matrix for Asset and Liability Return Series
Variable Stock GBond CBond RealEstate Mortgage HO APD AL CMP WC OL
Stock
1.000 -0.129 0.472 0.244
-0.027 0.177 -0.065 0.400 0.108 0.303 0.035
GBond -0.129 1.000 0.121 0.100
0.933 0.002 0.018 -0.190 0.306 -0.214 0.135
CBond
0.472 0.121 1.000 0.181
0.217 0.043 0.292 -0.100 0.440 -0.102 0.181
RealEstate 0.244 0.100 0.181 1.000
0.179 -0.262 -0.230 -0.202 0.106 -0.026 -0.126
Mortgage -0.027 0.933 0.217 0.179
1.000 0.070 0.001 -0.187 0.408 -0.201 0.122
HO
0.177 0.002 0.043 -0.262
0.070 1.000 -0.092 0.430 0.435 0.258 0.374
APD
-0.065 0.018 0.292 -0.230
0.001 -0.092 1.000 -0.299 0.264 -0.404 -0.137
AL
0.400 -0.190 -0.100 -0.202
-0.187 0.430 -0.299 1.000 -0.078 0.854 0.170
CMP
0.108 0.306 0.440 0.106
0.408 0.435 0.264 -0.078 1.000 0.006 0.176
WC
0.303 -0.214 -0.102 -0.026
-0.201 0.258 -0.404 0.854 0.006 1.000 -0.008
OL
0.035 0.135 0.181 -0.126
0.122 0.374 -0.137 0.170 0.176 -0.008 1.000
Variables: GBond (Government Bond), CBond (Corporate Bond), HO (Homeowners), APD
(Auto Physical Damage), AL (Auto Liability), CMP (Commercial Multiple Peril), WC
(Workers Compensation), and OL (Other Liability).
APD line that covers damage to auto physical is less likely to be correlated with WC
line that provides coverage for bodily injuries due to job-related accidents. The
returns of government bonds, corporate bonds, real estate and mortgages are
negatively correlated with those of several business lines of insurance, suggesting that
insurer’s investment in these types of assets provides effective hedges against some
losses in insurance liability lines. This view is consistent with the findings of recent
empirical studies that the insurer’s asset returns are correlated with liability risks to
some degree (e.g., Cummins et al., 2006; Ren, et al., 2008).
5.3. Fitting Marginal distributions
Determining the appropriate marginal distributions with a given data set is
important along with the choice of copula because capital requirement measured by
VaR and ETL is largely dependent on the tail behavior of distribution of each risk.
Various techniques including graphical or numerical method can be used to model
marginal distribution. The choice of distributions for each asset class and liability line
is detailed.
24
Let F ( x) be the fitted distribution function of a random variable X and then
the empirical distribution function for X 1 ,..., X n is defined as
1 K
Fˆi ( x) = ∑ I ( X ji ≤ x),
K j =1
(20)
where i = 1, ⋅⋅⋅, n, and K is the number of observations for the i th random variable.
The fit of the marginal distributions can be assessed with a graphical comparison of
the specified theoretical distribution function against its empirical version, as
demonstrated in Figure 3. For an illustration, we present here the fit of the univariate
marginals using lognormal distribution for Auto Liability line. Figure 3 (left) shows
that the plot of the empirical distribution function, Fˆ ( x), is fairly close to the smooth
curve of the theoretical distribution, F ( x) . Hence we can conclude that the lognormal
distribution seems a suitable model for AL line.
Figure 3 Graphical Method to Model Marginal Distribution
An alternative method is to use the probability-probability plot (P-P plot) that
helps to determine how well a specified distribution fits a given data set. Figure 3
(right) illustrates P-P plot method using the sample data of AL line. The adoption of
lognormal distribution for AL appears reasonable since there is no significant
discrepancy between the dotted curve of empirical distribution function and the
diagonal reference line.
25
As numerical measures, Skewness and kurtosis can be used to derive a fitting
distribution. Skewness is a measure of the asymmetry of the probability distribution
where positive (negative) skewness indicates a long right (left) tail, while zero
skewness reveals symmetry around the mean. Kurtosis is a measure of whether the
data are peaked or flat relative to a normal distribution. That is, data sets with high
Kurtosis portray a distinct peak near the mean and tend to have heavy tails, whereas
data sets with low Kurtosis show a flat top with skinny tails.
Using the identical historical quarterly returns used in the estimation of
correlations and following all procedures discussed above, we present the best fitting
distributions for asset classes and business lines in Table 4.17
Table 4 Best Fitting Distributions and Parameter Estimates
Distribution
Skewness
Kurtosis
Parameter1c
Parameter2d
Gumbel(Min)a
-0.7187
0.9260
0.0589
0.0617
Normal
0.1107
-1.1021
0.0179
0.0186
Logistic
-0.5902
-0.0760
0.0193
0.0188
Logistic
0.0477
0.6335
0.0252
0.0335
Gumbel(Max)b
0.0090
-0.2239
0.0114
0.0134
-1.9608
5.2755
0.3212
0.0978
Gumbel(Min)a
Gamma
0.3427
-0.8930
65.499
0.0057
Lognormal
0.4637
-0.7658
-1.1983
0.1231
Normal
-1.4028
3.1125
0.3629
0.0694
Lognormal
-0.4540
-0.5081
-1.2381
0.3087
Gamma
-0.0537
-0.1524
24.761
0.0143
Variables: GBond (Government Bond), CBond (Corporate Bond), HO (Homeowners), APD
(Auto Physical Damage), AL (Auto Liability), CMP (Commercial Multiple Peril), WC
(Workers Compensation), and OL (Other Liability).
a
Gumbel(Min) denotes Gumbel minimum that highlights the left tail.
b
Gumbel(Max) denotes Gumbel maximum highlights the right tail.
cd
Parameter1 and Parameter 2 indicate the estimates of the parameters in the probability
density function
Variables
Stock
GBond
CBond
RealEstate
Mortgage
HO
APD
AL
CMP
WC
OL
The variables for Stock, Mortgage, and Homeowners (HO) display the extreme value
distributions where Gumbel minimum highlights the left tail (i.e., thin right tail and
thick left tail) while Gumbel maximum highlights the right tail (i.e., thin left tail and
17
We also employed the commonly used goodness-of-fit tests such as Anderson-Darling test,
Kolmogorov-Smirnov test and Chi-Square test.
26
thick right tail). Both Auto Liability (AL) and Workers Compensation (WC) best fit
the Log-normal distribution which exhibits a relatively heavier tail than Gamma
distributions of Auto Physical Damage (APD) and Other Liability (OL) lines. We
estimate the parameters for each of the selected distributions using the maximum
likelihood method that often leads to the unbiased estimates with minimum variance.
The results are provided in Table 4.
5.4 Simulation Results
We perform Monte Carlo simulation to generate random return series for each
asset class and business line by applying the procedure described in the previous
section. The scatter plots between return series and economic capital measured by
VaR and ETL are presented for each copula model. We also discuss the results of
diversification benefit in this section.
5.4.1 Scatter Plot of Simulated Return Series
We present scatter plots of the return series to observe graphically the
dependence level between asset classes and insurance business lines for the following
copulas: Gaussian copula, t-copula with degrees of freedom v =10 and v =2, Cauchy
copula and Independence copula. We simulate 5000 random values of each asset class
and business line for each of copula models considered using correlations and
parameter estimates of the marginal distributions in Table 3 and Table 4. The scatter
plots of five selected classes -Stock, GBond, HO, AL and WC are presented to
conserve the space. Each dot stands for a joint observation for each pair.
The scatter plots show that different copulas impose distinctive tail behaviors,
leading to different level of risk measures. Even with the identical rank correlations,
the level of dependence structure can be differentiated by copula types. In general, we
observe that the Gaussian copula (Figure 4) demonstrates a modest linear relationship
27
across returns in each asset and liability class and observations tend to cluster around
the center of the distribution. It is important to realize that dependencies may be
different according to the correlation parameters initially determined by historical data.
For example, the scatter plot between AL and WC shows a relatively strong linear
relationship, compared to that of other pairs. This positively sloping linear pattern is
consistent with high rank correlation (85%) between AL and WC as shown in Table 3.
However, neither upper nor lower tail dependence is detected across all pairs in the
Gaussian copula. This is justifiable because the degree of tail dependence for the
Gaussian copula is zero if the correlation of any pair is less than unity. Little evidence
of tail dependence implies that extreme events appear to occur independently under
Gaussian copula model regardless of how high a correlation we choose between asset
and liability classes.
In contrast, t-copula differs from Gaussian copula, even when their
components have the same correlation coefficients. The scatter plots of t-copula with
v =10 (Figure 5) shows that observations are more scattered toward upper or lower
tail distribution than those of Gaussian copula. This is reasonable given that t-copula
permits symmetric tail dependence. The joint observations in Figure 5, however,
appear somewhat similar to the case of Gaussian copula even if simulated from tcopula. As we noted in Section 2, this phenomenon results from the asymptotic
characteristic of the t-copula that the tail behavior of t-copula resembles that of
Gaussian copula as the degrees of freedom increase. Although 10 degrees of freedom
may not be sufficient enough for asymptotic behavior, we still observe some
similitude with the Gaussian case.
As expected for these forms of copulas, the t-copula with v =2 in Figure 6 and
Cauchy copula in Figure 7 display more pronounced non-linear dependence between
28
Figure 4 Scatter Plots of 5000 Simulated Return Series under Gaussian Copula
Figure 5 Scatter Plots of 5000 Simulated Return Series under t ( v =10) Copula
Figure 6 Scatter Plots of 5000 Simulated Return Series under t ( v =2) Copula
29
Figure 7 Scatter Plots of 5000 Simulated Return Series under Cauchy Copula
Figure 8 Scatter Plots of 5000 Simulated Return Series under Independence
Copula
asset and underwriting returns than the other types of copulas. We also observe strong
tail dependence particularly in the lower tail of the distribution, compared to the
Gaussian copula case. This suggests that extreme negative returns have a tendency to
occur together. See for example, the pair-wise plots for Stock and WC where large
losses from stock market collapse tend to be closely associated with large losses in
workers compensation from the insurer’s perspective. Such dependence is stronger as
the degrees of freedom are decreasing. The scatter plots of all pairs in Cauchy copula
are more extended to the top right corner and the bottom left corner than those of
other forms of copula. Thus, Cauchy copula generates more simultaneously extreme
30
value observations than other types of copulas. Notably, Cauchy copula that allows
for stronger non-linear dependence constructs a cross form and thus captures both
positive and negative dependence concurrently.
Independence copula in Figure 8 provides no distinctive patterns since returns
of asset and liability classes are assumed to be independent each other. The result of
scatter plot is consistent with the assumption of Independence copula.
5.4.2 Calculation of Economic Capital and Diversification Benefit
Economic capital presented in terms of portfolio return rates is calculated
using both Value-at-Risk (VaR) and Expected Tail Loss (ETL). The risk measuresVaR and ETL represent the economic capital that the insurer should hold to keep its
probability of default below a certain threshold because they measure the worst
expected loss at a given confidence level if events develop in an adverse or
unexpected way. It is noted that the VaR measure does not state how much actual loss
will exceed the VaR figure; it simply presents the information about how likely the
worst expected loss will be exceeded. Thus, we include Expected Tail Loss (ETL) to
accommodate statistical characteristics of return distribution beyond the VaR.
We initially simulate 500,000 random returns for each asset class and business
line under different copulas. The simulated returns are then aggregated using the
weights discussed in Section 5.1 to construct a distribution of the portfolio rate of
return. Figure 9 presents the histograms of the aggregated returns. 18 There is a
substantial difference in dispersion across the types of copulas. Cauchy copula and tcopula with v =2 are more widely dispersed than other forms of copulas. In particular,
Cauchy copula and t-copula with v =2 lead to negatively skewed distribution that
extends a larger range of values in the negative return direction, compared to other
18
For illustration, 5000 simulated returns are presented. We also conducted the analysis using 500,000
simulated returns to check robustness. The same results are observed.
31
copulas. This observation is consistent with the theoretical view that under heavy tail
dependence, extreme events tend to occur simultaneously, resulting in more extreme
aggregate losses in general. Thus, we expect that the heavier tail behavior of Cauchy
and t-copula ( v =2) eventually leads to larger capital requirements.
Figure 9 Distributions of Aggregated Portfolio Return Series
Given the distribution of forward portfolio return series, VaR is readily
estimated by cutting the point on the ranked portfolio return distribution that
corresponds to the selected confidence level. ETL is simply the expected value (or
mean) of the portfolio rate of return in excess of VaR of the same ranked distribution.
For comparison purpose, we choose three confidence levels- 99%, 99.5%, and 99.9%
over a one-year time horizon.19 Especially, 99.5% confidence level is consistent with
a solvency capital requirement under the new EU solvency II regulation framework.
19
The confidence level can be viewed as the risk of insolvency during a defined time period at which
management has chosen to operate. The higher the confidence level selected, the lower the probability
of insolvency.
32
Value at Risk at 99.9% rating target is the risk measure required by the Basel
Committee on Banking Supervision.
When describing the insurer’s aggregated portfolio returns and determining
capital requirements, we are more concerned about lower tail dependence which
results in negative returns in our sample.20 Table 5 presents the undiversified VaR and
ETL which is considered a base scenario to assess the diversification benefit. There is
no significant difference in the estimates of VaR and ETL among different copulas.
This is reasonable because coefficients of tail dependence are equivalent across all
types of copulas when all correlations of asset classes and business lines are assumed
to be unity (See the Table 1).
The diversified VaR and ETL for each copula model are presented in Table 6.
As expected, we observe that there is a different impact of choice of copula on the
total capital requirements. The level of capital requirements measured by VaR and
ETL is inversely related to the number of degrees of freedom. The Cauchy copula
produces the highest risk measures in terms of the aggregate portfolio returns,
followed by the t-copulas with v =2 and v =10, and Gaussian copula in all cases. This
implies that extreme losses are likely to occur simultaneously under heavy tail
dependence structure, giving rise to the most conservative economic capital number
for the aggregated risks. The Independence copula results in the lowest VaR and ETL.
This observation is expected because Independence copula is assumed to have zero
correlations between risks. On average, the risk measures range from -5% to -7%
across all copulas.
20
Negative return rates imply the loss of portfolio value. This negative returns result from large
weights in asset classes (see Table2).
33
Table 5 Undiversified VaR and ETL
Copula
VaR(99%) ETL(99%) VaR(99.5%) ETL(99.5%) VaR(99.9%) ETL(99.9%) Average
Cauchy
-0.0713 -0.0975
-0.0896
-0.1155
-0.1323
-0.1511
-0.1096
t( v = 2 )
-0.0717 -0.0968
-0.0891
-0.1144
-0.1292
-0.1557
-0.1095
t( v = 10 )
-0.0716 -0.0963
-0.0890
-0.1133
-0.1279
-0.1522
-0.1084
Gaussian
-0.0711 -0.0956
-0.0883
-0.1124
-0.1271
-0.1500
-0.1074
-0.0891
-0.1140
-0.1275
-0.1536
-0.1087
Independence -0.0715 -0.0966
Table 6 Diversified VaR and ETL
Copula
VaR(99%) ETL(99%) VaR(99.5%) ETL(99.5%) VaR(99.9%) ETL(99.9%) Average
Cauchy
-0.0380 -0.0597
-0.0529
-0.0749
-0.0888
-0.1092
-0.0706
t( v = 2 )
-0.0362 -0.0577
-0.0512
-0.0726
-0.0855
-0.1085
-0.0686
t( v = 10 )
-0.0342 -0.0532
-0.0476
-0.0663
-0.0783
-0.0966
-0.0627
Gaussian
-0.0330 -0.0510
-0.0458
-0.0633
-0.0734
-0.0915
-0.0597
Independence -0.0258 -0.0421
-0.0372
-0.0533
-0.0634
-0.0788
-0.0501
Table 7 Diversification Benefits
Copula
VaR(99%) ETL(99%) VaR(99.5%) ETL(99.5%) VaR(99.9%) ETL(99.9%) Average
Cauchy
0.3477
0.3798
0.3712
0.3934
0.4016
0.4195
0.3918
t( v = 2 )
0.3355
0.3735
0.3649
0.3882
0.3982
0.4107
0.3853
t( v = 10 )
0.3233
0.3559
0.3485
0.3692
0.3797
0.3883
0.3665
Gaussian
0.3170
0.3479
0.3415
0.3603
0.3661
0.3789
0.3571
Independence 0.2652
0.3035
0.2945
0.3186
0.3321
0.3391
0.3155
We assess diversification benefit that insurers may pursue by constituting
various asset portfolios or by operating multiple-lines of businesses. Note that
diversification benefit is defined as the relative difference between risk measure of the
unit correlation and risk measure of the non-unit correlation. We measure the extent to
which diversification reduces overall risk by the ratio of diversified VaR to
undiversified plus diversified VaR. In Table 7, the magnitude of diversification
advantage is calculated under each copula and risk measure. We observe that there
exists a positive diversification benefit across all copulas and risk measures. The
result is consistent with the theoretical view that diversification enables firms to
reduce their overall portfolio risk and firms, then, would be required to hold less
capital to support uncertain events (Merton and Perold, 1993; Myers and Read, 2001).
34
Similar to the result of capital requirements, the use of different types of copulas has a
different effect on the level of diversification benefit. Selecting tail dependence model
with Cauchy copula offers the highest level of diversification gains (34%-42%
depending on the risk measure) while applying Independence and Gaussian gives rise
to the lowest diversification advantage (26%-33% and 31%-37%, respectively). It is
interesting to note that there exists a positive relationship between the absolute level
of risk measures and the resulting diversification benefit. We observe Copulas that
result in high risk measures are likely to be associated with high diversification gains
(see Tables 6 and 7).
Overall, we find that the choice of copula has a significant impact on the
resulting capital requirements and diversification benefits. The large variation is
primarily driven by the level of tail dependency that the copula allows for between
returns of asset classes and liability lines. The observed discrepancy of the risk
measures and diversification benefits among copulas demonstrates the importance of
choosing appropriate dependence structure to model the insurer’s aggregated risks.
5.4.3 Which Copula is the Best?
The question is now to choose the type of copula that fits best the given data.
The selection of the best copula model is an important issue especially to practitioners.
The methods of fitting a copula to the bivariate data case are well described in Frees
and Valdez (1998). Frees and Valdez illustrate how to identify an appropriate copula
with the joint distribution of insurance company losses and claims adjustment
expenses following the procedure developed by Genest and Rivest (1993). The basic
idea behind their procedure is to compare the degree of closeness of copula
distribution and its non-parametric estimate. Specifically, for the data ( xi , y i ) ,
i = 1,..., n , let the pseudo random variable Ti ={number of ( xi , y i ) such that x j < xi
35
and
y j < yi }/( n − 1). With this random variable, define the distribution of
copula, K (t ) = P(Ti ≤ t ) for t in [0,1] , and its nonparametric estimate
∧
K (t ) =
1 n
∑ I (Ti ≤ t ),
n i =1
(21)
where I is the indicator function. Then, the copula that best fits the data would be the
∧
one that yields zero distance between K (t ) and K (t ) . In their studies (Genest and
Rivest, 1993; Frees and Valdez, 1998), the best copula is chosen by minimizing the
square of the distance such as
1
∧
∧
D( K , K ) = ∫ {K (t ) − K (t )}2 d K (t ).
0
(22)
Genest and Rivest’s procedure is somewhat limited to the family of Archimedean
copulas, because K (t ) is presented only by the functional form of the generator that
defines Archimedean copulas. In contrast to Genest and Rivest, Durreleman et al.
(2000) and Palaro and Hotta (2006) suggest a different approach that utilizes
comparison of parametric and nonparametric values of a copula, which is not limited
to the Archimedean copulas. Because nonparametric (empirical) copulas have no
fixed structure and depend upon all the data points to produce an estimate, the
nonparametric copula should be getting closer to the copula being considered as the
amount of information in the sample increases. Similar to the method of the choice of
the marginal distributions discussed earlier, the best copula can be chosen in a way
that the distance of the theoretical and empirical copulas is minimized. We extend this
method with high dimensions for the elliptical copulas as described below.
Let x1, j ,...xn , j , j = 1,..., K , be a random sample from a multivariate distribution.
Then the empirical copula function is defined by
36
Table 8 Values of the Distances
Copula
Cauchy
t( v = 2 )
t( v = 10 )
Gaussian
∧
C(
|| Cˆ − C ||L2 / K
0.0239
0.0271
0.0196
0.0275
j1
j
1 K
,..., n ) = ∑ I ( x1, j ≤ x1, ( j1 ),..., xn, j ≤ xn, ( j n ) ),
K
K
K j =1
(23)
where I is the indicator function, K is the number of observations of a variable, and
xi , ( ji ) , i = 1,...n, j = 1,..., K , denote the ji th order statistics of the i th variable. Note that
K
∑ I (⋅) in (23) implies the number of elements in the sample that satisfy the argument
j =1
of the indicator function.
In order to investigate how “close” two functions are, we need a notion of
“size” for functions. To this end, we consider the L2 (Euclidean) norm as a numerical
measure. Specifically, following Durrleman et al. (2000), the best copula is captured
by looking for the minimum distance between the (assumed) parametric copula ( C )
and the empirical copula ( Ĉ ). Using the L2 norm criteria for jk = 0,1, ⋅⋅⋅, n, the
distance is
K
K
j
j
j
j
ˆ
|| C − C ||L2 = ( ∑ ⋅⋅⋅ ∑ | C ( 1 , ⋅⋅⋅, n ) − Cˆ ( 1 , ⋅⋅⋅, n ) |2 )1/ 2 .
K
K
K
K
j1 =0
jn = 0
(24)
Note that unlike the empirical copula, the theoretical copula C is parameterized by θ
which will be replaced by its estimate in practice. We estimate the distances for each
copula using the equation (24). Table 8 shows that t copula with 10 d.f. among others
is the best fit for our given data, which minimizes the distance between the parametric
copula and the empirical copula. Given that the objective of finding the best copula is
37
to estimate the economic capital necessary to cover the simultaneous occurrence of
extreme events, we clearly see that t copula with 10 d.f. is preferable to the Gaussian
copula.
6. Concluding Remarks
Economic capital refers to the amount of risk capital that financial service
firms require to secure solvency in a worst case scenario over a certain time period at
a pre-specified confidence level. Thus, economic capital is typically calculated using
VaR. The primary value of economic capital is its application to management’s
decision making as well as to risk management. Specifically, economic capital has
become a useful tool in evaluating capital adequacy in relation to risk. To assess
capital adequacy, examiners focus on a comparison of an insurer’s available capital
with its capital needs based on the insurer’s overall risk profile. Rating agency and
shareholder considerations are also cited as principal drivers for calculating economic
capital. Moreover, it is increasingly used in management’s strategic decisions about
whether to enter or exit certain market segments by reviewing economic capital
requirements.
It is vital to reflect real dependency between risk factors in modeling
economic capital. One method of measuring dependencies that overcomes the
limitations of traditional linear correlation and that has become very popular recently
among practitioners is using copula functions. A copula is an effective statistical tool
to model dependent interrelationships between the components of distinct risks
without losing information on some important characteristics of the tail dependency.
We estimated economic capital by incorporating both market risk and
underwriting risk. The market risk is associated with a fall in the value of an insurer’s
investments while underwriting risk arises when premiums are not sufficient to cover
38
future incurred losses. Because insurers’ profits and losses primarily depend on the
performance of both investment and underwriting activity, it is important to take both
market risk and underwriting risk and their interaction into account in the economic
capital modeling.
Based on the sample of the U.S. property liability insurance industry, we
examined the impact of dependence structure between the insurer’s market and
underwriting risks on the insurer’s total required economic capital under different
copulas. The diversification benefit for each copula model is also measured by
comparing the difference between the diversified VaR and the undiversified VaR. We
simulated quarterly return rate for the selected asset classes and business lines using
copula functions and then estimated VaR and ETL from the aggregated portfolio
return distribution. As a result, we found that there is a significantly different impact
of choice of copula on the insurer’s total capital requirements measured by VaR and
ETL. Copulas that allow for greater tail dependence result in greater VaR and ETL
compared to those with smaller dependence. Specifically, the Cauchy copula produces
the highest risk measures in terms of the aggregate portfolio returns, followed by t
( v =2), t ( v =10), Gaussian and Independence copulas in all cases. The result indicates
the importance of selecting appropriate copulas and risk measures to model the capital
requirements. For example, in the case of market crash where all portfolio values fall
down simultaneously, economic capital model under Gaussian copula may not
capture the riskiness of such contingency effectively because Gaussian copula
underestimates both the thickness of the tails of marginal distribution and their
dependence structure. Thus, we attempted to identify the type of copula that fits best
the empirical data. The parametric copula that is most close to its empirical copula is
defined as the best appropriate one. We found that the distance between parametric
39
and empirical copulas becomes smallest when the dependence structure is simulated
by the t copula with 10 degrees of freedom. The result suggests that t copula is
preferred to Gaussian copula not to underestimate the economic capital for our given
data.
Not surprisingly, diversification is major rationale for insurers to engage in
investment portfolio mix and operation in multi-line businesses. Insurers can realize
capital savings by reducing portfolio risks through geographic or product line
diversification. In general, the diversification benefit increases with the number of
assets or business lines and decreases with higher correlation between risk factors. We
found a positive diversification benefit -overall benefit in terms of reduction in total
capital requirements. The potential for diversification benefit stems from the fact that
some of insurers’ asset returns are negatively correlated with insurance underwriting
returns as evidenced from a correlation matrix and thus certain risks can be offsetting.
The result is consistent with Myers and Read (2001)’ proposition that low
covariability of losses with losses on the other lines of insurance in the portfolio or
high covariability of losses with returns on the portfolio of assets results in lower
capital requirements.
40
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