Estimation of Risk Contributions with MCMC
Takaaki Koike ∗and Mihoko Minami

†

February 13, 2017

arXiv:1702.03098v1 [q-fin.RM] 10 Feb 2017

Abstract
Determining risk contributions by unit exposures to portfolio-wide economic capital is an
important task in financial risk management. Despite its practical demands, computation of
risk contributions is challenging for most risk models because it often requires rare-event simulation. In this paper, we address the problem of estimating risk contributions when the total
risk is measured by Value-at-Risk (VaR). We propose a new estimator of VaR contributions,
that utilizes Markov chain Monte Carlo (MCMC) method. Unlike the existing estimators, our
MCMC-based estimator is computed by samples of conditional loss distribution given the rare
event of our interest. MCMC method enables to generate such samples without evaluating the
density of total risk. Thanks to these features, our estimator has improved sample-efficiency
compared with the crude Monte Carlo method. Moreover, our method is widely applicable to
various risk models specified by joint portfolio loss density. In this paper, we show that our
MCMC-based estimator has several attractive properties, such as consistency and asymptotic
normality. Our numerical experiment also demonstrates that, in various risk models used in
practice, our MCMC estimator has smaller bias and MSE compared with these of existing
estimators.

Keywords: Quantitative risk management; Value-at-Risk; Economic capital; Risk allocation; risk
contributions; VaR contributions; Markov chain Monte Carlo

1

Introduction

In most financial institutions, the risk of their portfolios is measured by economic capital. For the
purpose of further risk analysis, it is necessary to decompose the portfolio-wide economic capital
into the sum of risk contributions by unit exposures. This risk-redistribution process is called
capital allocation, and the allocated capitals are called risk contributions (see, e.g., Dev, 2004).
Since it is non-trivial to allocate the economic capital in an economically meaningful way, various
principles to determine the risk contributions have been proposed. Among these principles, the
Euler principle is one of the most well-established rule proposed in Tasche (1999). Its financial
justification is given from differing points of view (e.g., by Denault, 2001; Kalkbrener, 2005; Tasche,
1999). Detailed references for rationalizing the use of the Euler principle can also be found in Tasche
(2008).
Despite its good economic properties, actual computation of risk contributions poses theoretical and numerical difficulties, especially when the portfolio-wide risk is measured by Value-at-Risk
∗ Center

for Mathematics, Graduate School of Science and Technology, Keio University, htakaaki@math.keio.ac.jpi
† Department of Mathematical Engineering,
Faculty of Science and Technology, Keio University,
hmminami@math.keio.ac.jpi

1

(VaR). Although the explicit formula of VaR contributions is derived by Tasche (2001), it can rarely
be calculated analytically with a very few exceptions (such as CreditRisk+ studied in Tasche, 2004).
Therefore, Monte Carlo (MC) method is often used to obtain numerical solution. However, the MC
estimator of VaR contributions suffers from significant bias caused by sample-inefficiency and inevitable numerical modification (see, e.g., Yamai and Yoshiba, 2002 for the example). To overcome
the difficulties, several methods have been proposed in the literature. Hallerbach (2003) and Tasche
and Tibiletti (2004) proposed the approximation formulae by regarding the VaR contributions as
the best prediction of losses given the total loss, under some model assumptions. In this paper,
we call the estimator derived in such a way the generalized regression (GR) estimator. Evaluation
of the approximation error caused by the model misspecification is quite difficult. Furthermore,
it is also challenging to find a well-fitted model among losses, to accurize the approximation.
Glasserman (2005) developed importance sampling (IS) estimators in the case of credit portfolios.
Although the problem of sample-inefficiency is overcomed in the IS method, risk models where the
IS technique are available are quite limited. Consequently, the IS estimator can not be applied to
various risk models widely used in practice. Tasche (2009) proposed the Nadaraya-Watson (NW)
estimator constructed by using kernel estimation method. Although the NW estimator is available
to most risk models, incorporation of the importance sampling technique is still indispensable to
achieve efficient estimation. Moreover, the asymptotic standard error of the NW estimator can not
be computed easily.
In this paper, we propose a new estimator of VaR contributions that utilizes Markov chain
Monte Carlo (MCMC) method. Our MCMC-based estimator is available whenever the portfolio
loss model is specified by the joint density. This case contains wide varieties of risk models since
the joint density can be completely specified for all models having marginal densities and copula
density (see, e.g., Yoshiba, 2013 for various examples of this type of models). In this case, the
IS technique can hardly be available, and thus the problem of estimating VaR contributions is
generally unsettled. We study theoretical properties of our MCMC estimator, and provide a
guideline for the efficient application of MCMC to our problem. The proposed estimation method
is then carried out within some risk models often used in practice. For these risk models, we
compare the performance of the MCMC estimator with the existing estimators.
The MCMC estimator has several attractive properties. First, the MCMC estimator is consistent. Since the MCMC method has an improved sample-efficieny compared with the MC method,
we can expect the MCMC estimate to have much smaller bias than that of MC. Second, the MCMC
estimator holds asymptotic normality, and its standard error can be consistently estimated by using
methods in the theory of Markov chain. Thanks to these features, we can construct approximated
confidence interval of the true VaR contributions. Moreover, together with the sample-efficiency
of the MCMC method, we can expect that the MCMC estimator has much smaller standard
error compared with the existing methods. Finally, the MCMC estimator is free from modelmisspecification bias, and can flexibly incorporate the feature of the underlying loss distribution
into the estimation. In contrast with the GR estimator, the MCMC estimator does not use any
approximation. Therefore, the MCMC estimator is free from the approximation error caused by
the model-misspecification. Moreover, by choosing an appropriate proposal distribution of the
MCMC, we can directly capture the features of the underlying loss distributions, Thanks to these
propertoes, the MCMC estimator can have stably good performance regardless of the underlying
risk model.
The key idea behind the MCMC estimator is to generate samples directly from the joint loss
distribution given a rare event of interest. This approach is completely different from the crude MC
method in which samples are generated from the loss distribution itself. In the MC method, large
2

portion of samples have to be discarded because VaR contributions depend only on losses given the
rare event. Our method, in contrast, does not waste any samples for the estimation. Thanks to this
difference, great sample-efficiency can be achieved in the MCMC method. How can we generate
samples directly from the conditional density on the rare event? Usually it is quite difficult because
our conditional density contains the density of the total loss, which is too cumbersome to compute.
Nevertheless, MCMC method enables us to generate their samples directly by sequentially updating
the samples, without evaluating the cumbersome term on the total loss.
The paper is organized as follows. Section 2 introduces the mathematical setting of the capital
allocation problem, and explains problems for estimating VaR contributions with the existing
estimators. Section 3 is devoted to the brief introduction to Markov chain Monte Carlo method.
Several families of the proposal distributions are also introduced for efficient use of MCMC. In
Section 4, we propose the MCMC estimator, that combines the MCMC method with estimation of
VaR contributions to achieve more sample efficiency. Based on the theory of MCMC, we also study
some theoretical properties of our MCMC estimator. Numerical results and empirical studies are
presented in Section 5. We demonstrate that, for widely used risk models in practice, the MCMC
estimator has smaller bias and MSE than those of existing estimators. Through the empirical
study, we provide a guideline of incorporating the MCMC method in our framework of estimating
VaR contributions. Concluding remarks, future research and potential extensions of the MCMC
estimator are summarized in Section 6.
Throughout the paper, integration of vector-valued function is applied componentwise. Let
r×c
r×c
(E) be the set of E-valued
M (E) be the set of E-valued r × c matrices and Mr×c
+ (E) ⊂ M
r×c
(i,j)
positive definite r×c matrices. For a matrix M ∈ M (E), we write M
the (i, j)-th component
of M for i = 1, 2, . . . , r and j = 1, 2, . . . , c. For any vector x ∈ Rd , write xj the j-th component of
x.

2

Capital Allocation Problem

Throughout the paper, we consider the following portfolio loss model:
S=

d
X

Xj ,

(2.1)

j=1

where d ≥ 3 is the size of the portfolio, X1 , X2 , . . . , Xd are random variables that represent the
losses incurred by the exposures j = 1, 2, . . . , d within a fixed time period. The random variable
S defined in (2.1) stands for the portfolio-wide loss. In this paper, a positive value of loss random
variable represents a financial loss, and a negative loss is interpreted as a profit. Let Fj be the
cumulative distribution function (cdf) of Xj for j = 1, 2, . . . , d, FX be the joint cdf of the random
vector X = (X1 , X2 , . . . , Xd ) and FS be the cdf of the total loss S. Let C be a copula of FX . By
Sklar’s theorem (see, e.g., Nelsen, 2013), it holds that
FX (x) = C(F1 (x1 ), . . . , Fd (xd ))

(2.2)

for any x ∈ Rd . This formula allows us to separate the modelling of the individual marginal losses
from their dependence structure. See McNeil et al. (2015) for the benefits of this copula modelling
approach, and its application in financial risk management.
For the sake of argument, we impose two assumptions on the portfolio loss models (2.1) in this
paper. One assumption is supp(FX ) = Rd+ , or Rd , where supp(F ) is the support of a function

3

F and Rd+ := {x ∈ Rd : x ≥ 0}. The former case focuses on modelling pure losses and the
latter on modelling profits and losses (P&L). The other assumption is that, the distributions
F1 , . . . , Fd , FX , FS and C are all absolutely continuous, i.e. they all have densities f1 , . . . , fd , fX , fS
and c, respectively. In other words, we confine ourselves to continuous loss models, and discrete loss
models are beyond the scope of this paper. Although typical credit loss models such as the factor
models (see, e.g., Bluhm et al, 2016) are excluded, wide variety of loss models are still included in
the sphere of our research. Under the assumption of continuity, we have the density-form of the
Sklar’s formula:
fX (x) = c(F1 (x1 ), . . . , Fd (xd )) · f1 (x1 ) · · · fd (xd ),
(2.3)
by differentiating the both sides of (2.2).
As mentioned in Section 1, computing the risk contributions is important in the framework of
integrated risk management. In practice, a two-step procedure is conducted for determining the
risk contributions. First step is to compute the economic capital ρ(S), where ρ is a so-called risk
measure. Risk measure is a map from a loss random variable to a capital buffer that is required
to cover the loss with a certain high probability. One of the most popular risk measure is the
Value-at-Risk (VaR), defined by
VaRp (X) = inf{x ∈ R : P(X ≤ x) ≥ p},

(2.4)

where p ∈ (0, 1) is called the confidence level. Second step is to allocate the capital ρ(S) to the
individual d-exposures according to some principle. Mathematically, the capital allocation is the
problem of determining the vector (AC1 , AC2 , . . . , ACd ) that satisfies the full allocation property:
ρ(S) =

d
X

ACj .

(2.5)

j=1

The Euler principle solves this problem by utilizing the well-known Euler rule for a function
u 7→ ρ(uT X):
d
X
∂ρ(uT X)
ρ(uT X) =
uj
,
u∈Λ
(2.6)
∂uj
j=1
where Λ ⊂ Rd \{0} is an open set such that 1 ∈ Λ and ρ is a positive homogeneous risk measure.
Define
∂ρ(uT X)
j = 1, 2, . . . , d.
(2.7)
ACρj :=
∂uj
u=1
Then the full alocation property (2.5) holds for this vector (ACρ1 , . . . ,ACρd ) by taking u = 1 in the
equation (2.6). Since VaRp is positive homogeneous, the Euler principle can be applied and its
solution is given by
VaRp

ACj

:=

∂VaRp (uT X)
∂uj

u=1

= E[Xj |X1 + · · · + Xd = VaRp (S)].

(2.8)
VaR

VaRp

See Tasche (2001) for deriving the second equality. We call this vector ACVaRp := (AC1 p , . . . , ACd
the VaR contributions. Since we focus only on this form of allocated capital in this paper, we drop
the superscript ”VaRp ” and simply denote the VaR contributions by AC=(AC1 , . . . , ACd ).
Although its good economical properties have been reported in the literature, VaR contributions can not be computed easily. Even when the joint density of the portfolio loss vector fX is
analytically specified, theoretical computation of AC is difficult since deriving the joint distribution

4

)

of (Xj , S) is challenging in general. A possible numerical method to calculate the VaR contributions is the Monte Carlo (MC) method. In the crude MC method, we consider the pseudo VaR
contributions, defined by
ACδ = E[X| S ∈ [VaRp (S) − δ, VaRp (S) + δ] ]

(2.9)

instead of the true AC defined in (2.8), for a sufficiently small δ > 0. Since the probability
P(S ∈ [VaRp (S) − δ, VaRp (S) + δ]) is positive, the right hand side of (2.9) can be written by
ACδ =

E[X · 1[S∈Aδ ] ]
P(S ∈ Aδ )

where

Aδ = [VaRp (S) − δ, VaRp (S) + δ].

(2.10)

This expression allows us to construct the estimator of the pseudo VaR contributions given by
MC

c δ,N =
AC

1
N

PN

(n)
· 1[S (n) ∈Aδ ]
n=1 X
P
N
1
n=1 1[S (n) ∈Aδ ]
N

=

N
1 X (n)
X · 1[S (n) ∈Aδ ] ,
Mδ,N n=1

(2.11)

PN
i.i.d.
i.i.d.
(n)
(n)
where Mδ,N = n=1 1[S (n) ∈Aδ ] , X (1) , . . . , X (N ) ∼ FX and {S (n) := X1 + · · · + Xd }N
n=1 ∼
FS . We call (2.11) the MC estimator. The MC estimator is used to estimate the true VaR
contributions by setting δ and N sufficiently small and large, respectively. Note that this method
is available only when δ > 0 since P(S ∈ A0 ) = P(S = VaRp (S)) = 0 by absolute continuity of FS .
Although we can compute the MC estimator whenever we can generate i.i.d. samples from the
joint loss distribution FX , it has an inevitable bias to estimate the true VaR contributions. The
bias of the MC estimator can be decomposed by
MC

c
AC
δ,N − AC = bδ (N ) + b(δ)

(2.12)

MC

c δ,N − ACδ and b(δ) = ACδ − AC. Because we generally do not know the true
where bδ (N ) = AC
AC, the second term b(δ) can not be computed easily. Therefore, the δ should be as small as
possible. However, when the δ is quite small, it is difficult to maintain a sufficient sample size
Mδ,N that keeps the first term bδ (N ) small enough. This is because E[Mδ,N ] = N · P(S ∈ Aδ ) and
P(S ∈ Aδ ) is usually much less than 1 − p. Due to this trade off relation, the two parts of the bias
are difficult to eliminate simultaneously.
To overcome this problem, several estimators have been proposed in the literature. We end up
this section by introducing some existing estimators and their problems. Firstly, the NadarayaWatson (NW) kernel estimator proposed in Tasche (2009) is defined by
NW
c φ,h,N
AC


 (n)
S
−VaRp (S)
X (n) · φ
h
 (n)

,
PN
S
−VaRp (S)
φ
n=1
h

PN
=

n=1

(2.13)

where φ is the kernel density and h > 0 is the bandwidh. Since this estimator is a slight modification of the MC estimator (2.11), it has the same problem as the MC estimator explained above.
Moreover, the bias and the standard error of the NW estimator (see, e.g. Hansen, 2009 for their
expressions) can not be computed easily because it requires the evaluation of the total loss density
fS (VaRp (S)). Secondly, Glasserman (2005) developed the importance sampling (IS) estimators
to overcome the problem of sample-inefficiency. By using the IS technique to increase samples
whose sums belong to Aδ , the bias of the IS estimators can be sufficiently small. However, risk
models that can utilize this IS technique is limited to specific credit risk models (see, e.g., Huang
5

et al, 2007 ; Mausser and Rosen, 2008). Unfortunately, the IS estimators are hardly available in
our case, where the loss model (2.1) has a joint loss density as specified in (2.3). Finallly, Hallerbach (2003) and Tasche and Tibiletti (2004) constructed estimators by supposing a generalized
regression model among the losses:
X = gβ (S) + ε,
(2.14)
where gβ (s) is a function parametrized by β ∈ Rq and ε is an error random vector such that
E[ε|S = VaRp (S)] = 0. Let β̂N be an estimator of β. Then we call the following form of the
estimator
GR
c
(2.15)
AC
gβ ,N := gβ̂N (VaRp (S))
the generalized regression (GR) estimator. Although this estimator is intuitive and easy to apply,
it also has an inevitable bias. Let g be the true function such that E[X|S = v] = g(v) and let β ∗
be the minimizer of the error term ε in some sense. If β̂N goes to β ∗ as N → +∞, the bias of the
GR estimator can be decomposed by
GR

c
AC
gβ ,N − AC = b(N ) + b(gβ ),

(2.16)

where b(N ) = gβ̂N (VaRp (S)) − gβ∗ (VaRp (S)) and b(gβ ) = gβ∗ (VaRp (S)) − g(VaRp (S)). The first
term b(N ) is caused by the estimation error and the second term b(gβ ) is by the misspecification
of the model (2.14). Since we do not know the true function g in general, the second term b(gβ ) is
difficult to evaluate. To avoid the model-misspecification, we have to choose the family of functions
{gβ : β ∈ Rq } large enough to contain g. On the other hand, estimating the parameter β is not as
easy as in the ordinary regression model. Since we usually do not have samples from FX|S=VaRp (S) ,
it is non-trivial to estimate β by minimizing the error term ε given {S = VaRp (S)}. Due to these
obstacles, it is generally difficult to evaluate the bias of the GR estimator. A notable exception is
the case when the risk model X follows an elliptical distribution. In this case, it holds that (e.g.,
Corollary 8.43 in McNeil et al., 2015)
E[X|S = VaRp (S)] − E[X] =

Cov(X, S)
· (VaRp (S) − E[S]).
Var(S)

(2.17)

Therefore, the true VaR contributions are provided when we choose gβ (x) = β0 + β1 x and
β0∗ = E[X] −

Cov(X, S)
· E[S]
Var(S)

β1∗ =

Cov(X, S)
.
Var(S)

(2.18)

Since this β ∗ is the minimizer of E[ε2 ] = E[(X − β0 − β1 S)2 ], the ordinary least squares (OLS)
OLS
estimator βN
converges to β ∗ as N goes to infinity. As a consequence, the second term b(gβ ) is
zero and the first term b(N ) can be eliminated by taking N sufficiently large.

3

General theory of MCMC

Markov chain Monte Carlo (MCMC) is a method to generate samples from a probability distribution by constructing a Markov chain whose stationary distribution is the desired one. By
allowing Markovian type dependence within the samples, MCMC enables to simulate wide variety
of distributions even if they can not be simulated directly in practice. In this section, we briefly
introduce the theory of MCMC method. We review consistency and asymptotic normality of estimators computed by Markov chain. To obtain an efficient estimator, it is important to choose an

6

appropriate proposal distribution on generating sample path of the Markov chain. In Section 3.2,
we summarize the family of proposal distributions used in this paper.

3.1

Brief introduction to MCMC

Let E ⊂ Rd be a set and E be a σ-algebra of subsets of E. Markov chain is a sequence of E-valued
random variables (X (1) , X (2) , . . . ) satisfying the Markov property;
P(X (n+1) ∈ A|X (k) = x(k) , k ≤ n) = P(X (n+1) ∈ A|X (n) = x(n) ),

(3.1)

for all n ≥ 1, A ∈ E, and x(1) , . . . , x(n) ∈ E. Markov chain is characterized by its stochastic kernel
P : E × E → R+ , given by x × A 7→ P (x, A) := P(X (n+1) ∈ A|X (n) = x). If there exists a
R
probability distribution π such that π(A) = π(dx)P (x, A) for all x ∈ E and A ∈ E, then the π is
called the stationary distribution. See, e.g., Nummelin (2004) for general theory of Markov chain.
Markov chain Monte Carlo (MCMC) is a widely used statistical method for simulating probability distribution by generating Markov chain. MCMC is often used to estimate the quantity
Z
π(h) :=

h(x)π(dx),

(3.2)

for some distribution π and a π-measurable function h. Its MCMC estimator is given by
π̂N (h) :=

1
( h(X (1) ) + · · · + h(X (N ) ) ),
N

(3.3)

where (X (1) , . . . , X (N ) ) is an N -path of the Markov chain whose stationary distribution is π. In
the framework of MCMC, we call this π the target distribution. Since the target distribution π
is externally determined by the problem to solve, the problem of MCMC is to find the stochastic
kernel P such that it has the stationary distribution π. Additionally, we require P that its sample
path can be generated easily.
One of the most popular stochastic kernel used in practice is the Metropolis-Hastings (MH)
kernel (Metropolis et al., 1953; Hastings, 1970), defined by
P (x, dy) = p(x, y)dy + r(x)δx (y),

(3.4)

where
p(x, y)

= q(x, y)α(x, y),

α(x, y)


h
i
min π(y)q(y,x) , 1
π(x)q(x,y)
=
0

r(x)

=

1−

if π(x)q(x, y) > 0
otherwise ,

Z
p(x, y)dy,

δx is the Dirac delta function, and q : E × E → R+ is a function such that x 7→ q(x, y) is a
measurable function for all y ∈ E, and y 7→ q(x, y) is a probability density function (pdf) for all
x ∈ E. This function q is called a proposal distribution. It can be shown that the MH kernel has
a stationary distribution π (Tierney, 1994). The great popularity of the MH kernel comes from
the fact that, we can easily generate sample path of the corresponding Markov chain by using the
7

so-called MH algorithm (see e.g., Chib and Greenberg, 1995 for the simple and intuitive exposition
of the algorithm). Under the mild conditions that (i) a vector x(1) ∈ supp(π) is known, (ii) samples
from q(x, ·) can be generated for all x ∈ E , and (iii) the ratio π(y)/π(x) can be calculated for all
x, y ∈ E, we can generate N -sample path of the desired Markov chain by the following algorithm:
Algorithm 1: MH Algorithm
1. Set N > 0, q: proposal distribution, and X (1) = x(1) ∈ supp(π)
2. For n = 1, 2, . . . , N ,
(n)

3.

Generate X∗

4.

Set

"
αn :=

5.

∼ q(X (n) , · ) and U ∼ U(0, 1)
(n)
α(X (n) , X∗ )

= min

Set

(n)

(n)

X (n+1) := 1[U ≤αn ] · X∗

6. End for

(n)

π(X∗ ) q(X∗ , X (n) )
·
, 1
π(X (n) ) q(X (n) , X∗(n) )

#

+ 1[U >αn ] · X (n)

(3.5)
(3.6)

7. Return (X (1) , . . . , X (N ) ) as an N -sample path of the Markov chain with the kernel (3.4) and
the initial value X (1) = x(1)
(n)

We call αn := α(X (n) , X∗ ) in (3.5) the acceptance probability at n-th iteration. Based on the
N -sample path (X (1) , . . . , X (N ) ) generated in Algorithm 1, we can compute the MCMC estimator
π̂N (h) given by (3.3).
The MCMC estimator π̂N (h) has several attractive properties such as consistency and asymptotic normality. Firstly, the MCMC estimator is consistent (see, e.g., Theorem 1 in Nummelin
;2002), i.e.,
lim π̂N (h) = π(h)
with probability 1,
(3.7)
N →+∞

for all π-integrable functions h and all initial state X (1) = x(n) ∈ supp(π). Moreover, Central
Limit Theorem (CLT) holds under some regularity conditions (see, e.g., Chapter 17 in Meyn and
Tweedie, 2012), i.e.,
√

d

N ( π̂N (h) − π(h) ) −→ Nd (0, Σh )

where
Σh := Varπ [ h(X

(1)

)]+2

∞
X

as N → +∞,

Covπ [ h(X (1) ), h(X (1+k) ) ]

k=1

∈ (0, ∞).

(3.8)

(3.9)

Since we can rarely compute the asymptotic variance (3.9) in real situation, we have to estimate
it from the same sample path (X (1) , . . . , X (N ) ) that we used to compute π̂N (h). One simple and
popular estimator of Σh is the so-called batch means estimator (see, e.g., Geyer, 2012). For the
N -path of the Markov chain (X (1) , . . . , X (N ) ), the batch means estimator Σ̂h,N is defined by
B

Σ̂h,N =

N
LN X
( π̂N,b (h) − π̂N (h) )T ( π̂N,b (h) − π̂N (h) ),
BN − 1

(3.10)

b=1

where LN , BN are positive integers satisfying N = LN · BN , and
1
π̂N,b (h) =
LN

bLX
N −1

h(X (l) )

l=(b−1)LN

8

for b = 1, 2, . . . , BN .

(3.11)

LN is called the batch length and BN is the number of batches. Under some regularity conditions,
the batch means estimator Σ̂h,N converges to Σh with probability 1 as N → +∞ (see, e.g., Jones
et al., 2006; Vats et al., 2015). By using the asymptotic relation (3.8) and the consistency of the
batch means estimator Σ̂h,N , we can construct an approximate 95% confidence interval of the true
quantity π(h), given by
σ̂h,N
σ̂h,N
π̂N (h) − 1.96 · √ < π(h) < π̂N (h) + 1.96 · √ ,
N
N

(3.12)

(j,j)

where σ̂h,N = (Σ̂h,N , j = 1, 2, . . . , d).

3.2

Choice of the proposal distribution

When implementing the MCMC, an appropriate choice of the proposal function q is necessary.
When choosing q, it is desirable that the Markov chain characterized by (3.4) theoretically holds
the Central limit theorem (3.8), and also, its asymptotic variance (3.9) is as small as possible. Since
the asymptotic variance Σh can rarely be calculated explicitly in real situation, we usually rely
on post-implementation review, i.e. evaluating the goodness of the selected proposal distribution
after performing the MCMC. In this section, we introduce two empirical methods for checking
validity of the proposal selection. We also provide some families of the proposal distributions for
an appropriate choice of q. Theoretical verification of CLT will be carried over into Section 4.3.
In practice, there are two prevalent methods to check the validity of the proposal distribution
(see, e.g., Geyer, 2012 in details). One method is to check that the autocorrelation plots of the
marginal sample paths steadily decline . For an N -sample path (X (1) , . . . , X (N ) ) and the MCMC
estimator π̂N (h) = (π̂N,1 (h), . . . , π̂N,d (h)), we draw plots of the sample autocorrelations r̂j (h) :=
R̂j (h)/R̂j (0), where
R̂j (h) :=

N
−k
X
1
(n)
(n+k)
(hj (Xj ) − π̂N,j (h)) · (hj (Xj
) − π̂N,j (h)),
N − k n=1

(3.13)

versus the time lag k = 0, 1, 2, . . . , for j = 1, 2, . . . , d. We can expect that the asymptotic variance
Σh should be small if the autocorrelation plots steadily decline to zero as the lag increases. The
other method is to check the acceptance rate, i.e. the percentage of times a candidate is accepted.
In general, extremely low acceptance rate implies that the chain tends to be stuck to one point.
Conversely, extremely high acceptance rate occurs when the chain moves only around the mode
of the target density and does not traverse the entire support (Chib and Greenberg, 1995). The
first situation yields high asymptotic variance, and the second gives an illusionally low asymptotic
variance. To avoid such situations, it is important to check that the acceptance rate takes a
moderate value.
Typically, proposal distribution q is selected from a certain class of distributions. To find a
suitable q, several classes are in order. First, if q(x, y) = f (y − x) for some pdf f , then the
candidate X∗ is drawn according to the process
where Z ∼ f,

X∗ = X + Z,

(3.14)

and X is the current state. We call this q the random walk proposal distribution. In the case
when f is symmpetric around the origin, the acceptance probability (3.5) is given simply by
π(y)
α(x, y) = min[ π(x)
, 1]. Second, when q(x, y) = f (y) for some pdf f , then the candidate X∗ is

9

updated by
X∗ = Z,

where Z ∼ f.

(3.15)

Since the candidate is drawn independently of the current position, this q is called the independent
proposal distribution. The two proposal distributions, the random walk and the independent
proposals, are widely used due to their simplicity. However, these proposal distributions often
fail to perform well when the target distribution π is heavy-tailed. To overcome this problem, we
finally introduce the mixed preconditioned Crank-Nicolson (MpCN) proposal distribution (proposed
by Kamatani, 2014). This proposal distribution updates the candidate according to the following
process:
1
1
1
X∗ = µ + ρ 2 · (X − µ) + (1 − ρ) 2 · Z − 2 · W ,
(3.16)
where Z follows the gamma distribution with shape parameter d/2 and the scale parameter
1
||Σ− 2 (X − µ)||2 /2 for µ := E[X] and Σ := Var[X], and W ∼ Nd (0, Σ). Note that, here we
introduce the standardized version of the original MpCN in Kamatani (2014). The acceptance
probability (3.5) can be written by

α(X, X∗ ) = 

π(X∗ )
·
π(X)

||Σ

− 21
1

(X − µ)||

||Σ− 2 (X∗ − µ)||

!−d


, 1 .

(3.17)

The major difference of this proposal distribution from the first two simple ones is that, not only
the mean but also the variance of the candidate changes with the current state X. Since the
MpCN proposal distribution admits larger jumps in the tail, we can expect better acceptance
rate even when π is heavy-tailed (see, e.g., Livingstone, 2015 for such position-dependent proposal
distributions).

4

Proposed method

We have shown that estimators constructed by MCMC has several attractive properties, such
as consistency and asymptotic normality. In this section, we propose a new estimator of VaR
contributions, that utilizes MCMC method to achieve efficient estimation, We also show consistency
and asymptotic normality of our MCMC-based estimator under a certain class of risk models.

4.1

The set-up

We suppose the following situation: we have an explicit form of the joint loss density fX , and
thus, we can compute the quantity fX (x) for any x ∈ Rd . We also can generate i.i.d. samples
X (1) , . . . , X (N ) from FX up to sufficiently large N . Then, the i.i.d. samples S (1) , . . . , S (N ) from
(n)
(n)
FS can be generated by taking S (n) = X1 + · · · + Xd for n = 1, 2, . . . , N . On the other
hand, suppose that we have neither an explicit form of the total loss density fS nor the way to
compute the quantity fS (s) for any s ∈ R. The situation described above typically occurs when we
model the joint loss density fX by the copula approach, i.e. specifying the marginal loss densities
f1 , f2 , . . . , fd and then specifying the copula density c. When we use this approach, the resulting
joint loss density fX is speficied by the formula (2.3).
Under the situation above, a two-step procedure (see, e.g., Glasserman, 2005) is conducted
for computing the VaR contributions (2.8). The first step is to estimate VaRp (S) and the second
step is to estimate the VaR contributions AC= E[X|S = VaRp (S)] with the VaRp (S) replaced
by the estimated one in the first step. Estimation of the VaRp (S) in the first step is conducted
10

by Monte Carlo simulation. Based on the i.i.d. samples S (1) , . . . , S (N ) from FS , the estimator of
d p (S) = S [N p] , where S [N p] is the N p-th largest sample among N samples
VaRp (S) is given by VaR
d p (S) as v in short, and regard it as a
{S (1) , . . . , S (N ) }. Hereafter we refer to the estimate of VaR
constant. In the second step, we aim at estimating AC= E[X|S = v]. According to the crude
Monte Carlo method as explained in Section 2, the VaR contributions are estimated by
MC

c δ,N =
AC
i.i.d.

N
1 X (n)
X · 1[S (n) ∈Aδ ] ,
Mδ,N n=1
(n)

(4.1)

(n)

where X (1) , . . . , X (N ) ∼ FX , S (n) := X1 + · · · + Xd for n = 1, . . . , N , Aδ = [v − δ, v + δ] and
PN
Mδ,N = n=1 1[S (n) ∈Aδ ] .
As explained in Section 2, the problem of this two-step procedure is that the estimator of VaR
contributions in the second step often has a significant bias. To address this issue, we develop a
Markov chain Monte Carlo (MCMC)-based estimator, that achieves sample efficiency and consistency, by utilizing the MCMC method. The major difference of our MCMC-based estimator from
the MC one is that, samples are generated not from the loss distribution FX itself, but from the
conditional distribution of X given the sum constraint {S = v}. Although it is almost impossible
in the MC method, the MCMC enables us to realize it.

4.2

The MCMC estimator

We start to describe the MCMC-based estimator with reformulating the problem of computing
VaR contributions. Let X 0 = (X1 , X2 , . . . , Xd0 ), where d0 = d − 1. By the full allocation property
(2.5), we have that
0
T
E[X|S = v] = ( E[X 0 |S = v], v − 1T
d0 · E[X |S = v] ) ,

(4.2)

where S = X1 + · · · + Xd . Therefore, computation of the VaR contributions AC=E[X|S = v] can
be reduced to estimate the VaR contributions of the d0 -subportfolio, given by AC0 = E[X 0 |S = v].
In our method, we estimate this quantity AC0 by generating samples directly from FX 0 |S=v . The
conditional joint density of X 0 given {S = v} can be written by
fX 0 |S=v (x0 ) =

0
fX (x0 , v − 1T
fX 0 ,S (x0 , v)
d0 x )
=
,
fS (v)
fS (v)

0

x0 ∈ R d

(4.3)

where the second equation follows from a linear transformation (X 0 , S) 7→ X. Since it is difficult
to evaluate the total loss density fS (v), we can hardly generate samples directly from fX 0 |S=v .
0
By taking E = Rd , h(x) = x and π(x) = fX 0 |S=v (x) on the discussion in Section 3, our
problem of estimating VaR contributions can be reduced to estimate π(h) = E[X 0 |S = v] in (3.2)
by MCMC. Even though we can hardly compute fX 0 |S=v itself, we can compute the acceptance
probability (3.5), given by

α(x, y) = min




fX 0 |S=v (y) · q(y, x)
fX (y, v − 1T
d0 y) · q(y, x)
, 1 = min
,
1
fX 0 |S=v (x) · q(x, y)
fX (x, v − 1T
d0 x) · q(x, y)

(4.4)

for all x, y. Note that the cumbersome term fS (v) disappears by taking the ratio of fX 0 |S=v (y) to
fX 0 |S=v (x). Therefore, under some appropriate choice of the proposal function q, we can generate
N -sample path of the Markov chain whose stationary distribution is π(x) = fX 0 |S=v (x). Based
on the sample path of the Markov chain, we can construct the MCMC estimator π̂N (h) defined by

11

(3.3). Its consistency follows directly from the general theory of Markov chain presented in Section
3. The algorithm to compute the MCMC estimator of VaR contributions is summarized as follows:
Algorithm 2: MCMC estimator of VaR contributions E[X|S = VaRp (S)]
1. Set v > 0, N > 0, q: proposal distribution, and X (1) = x(1) ∈ supp(π).
2. Perform Algorithm 1 for the given N , q, and x(1) to generate an N -sample path (X 0(1) , . . . , X 0(N ) )
of a Markov chain whose stationary distribution is fX 0 |S=v .
3. Define
0(n) T
X (n) := (X 0(n) , v − 1T
) ,
d0 X

and set
MCMC

c
AC
q,N

=

N
1 X (n)
X
N n=1

(4.5)

(4.6)

to estimate VaR contributions AC= E[X|S = v].

4.3

Some theoretical results

In this section, we provide some theoretical results to verify asymptotic normality of the MCMC
estimator (4.6). We will see that CLT holds for various risk models when we model the pure
losses. On the other hand, when we model the profits & losses, theoretical justification of CLT is
challenging except for some special cases. We provide an example of justifying CLT only when we
model P&L by some elliptical distribution.
When we model pure losses, i.e. supp(FX ) = Rd+ , then the conditional distribution FX 0 |S=v is
supported on the following bounded set, called the v-simplex:
0

Sv := {x ∈ Rd : x ≥ 0, 0 ≤ x1 + · · · + xd0 ≤ v}.

(4.7)

Thanks to the compactness of the support, CLT is verified under some mild conditions. The
following theorem is a direct consequence of well-known results in the theory of MCMC. However,
it clarifies the conditions on the marginal loss densities and the copula densities, that are easy to
be checked in the framework of joint risk modeling with copulas.
Theorem 4.1. Suppose that the joint distribution FX is supported on Rd+ , and has marginal
√
densities f1 , f2 , . . . , fd and a copula density c. Then, N -CLT holds for the MCMC estimator
(4.6) of VaR contributions if the following (C1) − (C3) hold:
(C1)  := inf x,y∈[0,v]d0 q(x, y) > 0,
(C2) fj (x) is positive and bounded above on [0, v] for all j = 1, 2, . . . , d,
(C3) c(u) is positive and bounded above on F1 ([0, v]) × · · · × Fd ([0, v]).
√
Proof. By Theorem 23 in Roberts and Rosenthal (2004), the N -CLT holds if the Markov chain
is uniformly ergodic whenever E[||X 0 ||2 |S = v] < ∞. The moment condition is satisfied since
E[Xi Xj |S = v] ≤ E[(X1 + · · · + Xd )2 |S = v] = v 2 < ∞

(4.8)

for any i, j = 1, 2, . . . , d. Thus, it suffices to show that the Markov chain is uniformly ergodic. By
Theorem 1.3 in Mengersen and Tweeide (1996), the Markov chain is uniformly ergodic if (and only
12

if) the minorization condition (see, e.g., Rosenthal, 1995) holds on the whole space Sv , i.e., there
exist a positive integer n, δ > 0 and a probability measure ν such that P n (x, A) > δ · ν(A) for all
0
0
x ∈ Sv , A ∈ Bv , where Bv := B(Rd ) ∩ Sv . By the conditions (C2), (C3), and that Sv ⊂ [0, v]d ,
we have that
l := inf π(x) > 0,
u := sup π(x) < +∞.
(4.9)
x∈Sv

x∈Sv

Using (4.9) and the condition (C1), the minorization condition can be checked as follows. For any
x ∈ Sv , define


π(y) q(y, x)
Qx := y ∈ Sv :
·
<1 .
(4.10)
π(x) q(x, y)
Then for all A ∈ Bv , we have
P (x, A) ≥

=
≥
=
Taking n = 1, δ =


u



π(y) q(y, x)
·
dy
q(x, y) · min 1,
π(x) q(x, y)
Qx


Z
π(y) q(y, x)
·
dy
+
q(x, y) · min 1,
π(x) q(x, y)
A\Qx
Z
Z
π(y)
· q(y, x)dy +
q(x, y)dy
A\Qx
Qx π(x)
Z
Z

π(y)
π(y)dy + 
dy
u Qx
u
A\Qx

π(A).
u
Z

and ν = π completes the proof.

We will see typical situations where the conditions (C1) − (C3) hold. Firstly, the condition
0
(C1) holds when, for example, q(x, y) = f (y − x), f is continuous, and positive on [−v, v]d . These
conditions are satisfied for typical choices of f , such as Gaussian density or student’s t density.
Secondly, the condition (C2) also holds when, for example f1 , f2 , . . . , fd follow Pareto distributions
with the density given by
κ

fj (x) =

κj γj j
(x + γj )κj +1

κj > 0, γj > 0

for j = 1, 2, . . . , d.

(4.11)

Lastly, the condition (C3) can easily be checked for various copulas used in practice. For the use
in Section 4, we especially focus on the following two examples:
Example 4.1 (t copula). d-dimensional t copula density is of the form

ct (u) =

ν
Γ( ν+d
2 )Γ( 2 )
1
d
|R| 2 Γ( ν+1
2 )


·

1+

xT R−1 x
ν

Πdj=1 (1 +

− ν+d
2

x2j − ν+1
2
ν )

,

ν>0

(4.12)

−1
−1
where R ∈ Md×d
+ [0, 1] is a correlation matrix, x = (tν (u1 ), . . . , tν (ud )) and tν is a cdf of the
student’s t distribution with degree of freedom ν. It is clear that (4.12) satisfies the condition (C3)
if Fj (0) > 0 for j = 1, 2, 3.

Example 4.2 (π-rotated Clayton copula). d-dimensional π-rotated Clayton copula has a density

13

of the form (Yoshiba, 2013)


− θ1 −d
d
d
1
Y
X
+
d)
Γ(
cClayton
(u) = θd · θ 1
·  (1 − uj )−θ−1   (1 − uj )−θ − d + 1
,
π
Γ( θ )
j=1
j=1

0 < θ < ∞.

(4.13)
Since Clayton copula has a lower tail dependence, the π-rotated Clayton copula has an upper tail
dependence. We will see that the π-rotated Clayton copula density (4.13) satisfies the condition
(C3) under some parameter constraint. Let lj := 1 − Fj (v) ∈ (0, 1). Then, for any uj ∈ Fj ([0, v]),
it holds that 0 < lj ≤ 1 − uj ≤ 1 − Fj (0) = 1 for all j = 1, 2, . . . , d. Thus, we have
1≤
and also

d
Y

(1 − uj )−θ−1 ≤

j=1

d
Y
j=1

lj−θ−1 < ∞


− θ1 −d
− θ1 −d 
d
d
X
X

≤  (1 − uj )−θ − d + 1
≤ 1.
lj−θ − d + 1
j=1

(4.14)

(4.15)

j=1

Since v is the p-th quantile of FS , we have lj = 1 − Fj (v) ≤ 1 − p. Using this inequality,
the lower bound of (4.15) is positive if θ < log(1 − p)/ log(1 − d1 ). Putting (4.14) and (4.15)
together, the π-rotated Clayton copula density (4.13) satisfies the condition (C3) when 0 < θ <
log(1 − p)/ log(1 − d1 ).
In contrast to the pure-loss case, theoretical verification of CLT is challenging in the case when
we model P&L, i.e. supp(FX ) = Rd . Since the conditional distribution FX 0 |S=v is supported on
0
the unbounded space Rd , we need a careful investigation on the tail-behavior of the conditional
density fX 0 |S=v . When the proposal distribution is MpCN, we can justify the CLT of our MCMC
estimator for a certain class of target distributions based on the result in Kamatani (2016). As its
example, we consider the case where the underlying loss vector X follows a multivariate student’s
t distribution.
Example 4.3 (Multivariate student’s t distribution). Let X ∼ tν (µ, Σ), where ν > 2, µ ∈ Rd
and Σ ∈ Md×d
+ . The density of the multivariate student’s t distribution is given by

− ν+d
2
Γ( ν+d
(x − µ)T Σ−1 (x − µ)
2 )
p
.
fX (x) =
· 1+
1
ν
ν
|Σ| 2 Γ( 2 ) (π · d)d

(4.16)

Its conditional density fX 0 |S=v can be specified as follows. Firstly, let µ = 0 for simplicity. Write
Σ−1 =
0

0

A1
AT
2

A2
A3

!
=: A,

(4.17)

0

for A1 ∈ Md ×d (R), A2 ∈ Rd and A3 ∈ R. Then, it holds that
x
v − 1T
d0 x

!T

A1
AT
2

A2
A3

!

!

x
v − 1T
d0 x
0

= (x − w)T V (x − w) + D,
0

(4.18)
0

d ×d
T
T
where V := A1 − A2 1T
, w := V −1 (vA3 1d0 − vA2 ) ∈ Rd , and
d0 − 1d0 A2 + 1d0 1d0 ∈ M+

14

D := v 2 A3 − wT V w ∈ R. Using (4.18), we have that
fX 0 |S=v (x) ∝ fX (x, v − 1T
d0 x)

− ν+d
2
(x − w)T V (x − w) + D
∝
1+
ν

− ν+d
2
(x − w)T V (x − w)
∝
1+
.
ν+D

(4.19)

Provided ν + D > 0, X 0 |S = v follows a d0 -dimensional elliptical distribution with the location
parameter w, scale parameter V −1 and the density generator g : R+ → R+ given by

g(x) =

x
1+
ν+D

− ν+d
2
.

(4.20)

This type of distribution is called the Pearson type VII distribution (see e.g., Schmidt, 2002).
Consider the MCMC estimator (4.6) where π = fX 0 |S=v and the proposal distribution q is
√
MpCN (3.16). By Theorem 25 in Roberts and Rosenthal (2004), a N -CLT holds if the Markov
chain is geometrically ergodic and E[||X 0 ||2 |S = v] < ∞. By Proposition 3.4 in Kamatani (2016),
the Markov chain with MpCN proposal distribution is geometrically ergodic if E[||X 0 ||δ |S = v] < ∞
for some δ > 0, π(x) is strictly positive and continuous, and moreover, it is symmetrically regularly
varying, i.e.,
π(rx)
lim
= λ(x),
(4.21)
r→+∞ π(r1d )
0

0

0

d −1
d −1
for some λ : R → (0, ∞) such that λ(x) = 1 for any x ∈ SW
, where SW
:= {x ∈ Rd :
0
0
1
1
d ×d
. We will firstly see that all the moment conditions
||W − 2 x|| = ||W − 2 1d0 ||} and W ∈ M+
hold, and then, the tail condition (4.21) is also satisfied for π = fX 0 |S=v .
Write R := ||X 0 ||. It can be shown that g is regularly varying (see, e.g., Resnick, 2013) at ∞
with index α = − ν+d
2 , i.e.
ν+d
g(rx)
= x− 2
lim
x > 0.
(4.22)
r→∞ g(r)

By Proposition 3.7 in Schmidt (2002), fR|S=v is regularly varying with index −(ν + 1). Then,
by Karamata’s Theorem (see, e.g., Resnick, 2013, Theorem 0.6), FR|S=v is regularly varying with
index −ν. Therefore, E[Rδ |S = v] < ∞ holds for all δ < ν (see, e.g., Mikosch, 1999). Thus all the
moment conditions above are satisfied so long as ν > 2. In elliptical case, the tail condition (4.21)
0
is a direct consequence of (4.22). Since (x − w)T V (x − w) > 0 for all x ∈ Rd , it holds that
fX 0 |S=v (rx)
lim
=
r→+∞ fX 0 |S=v (r1d )

1

||V 2 x||

!−(ν+d)
,

1
2

||V 1d0 ||

0

x ∈ Rd .

(4.23)

Thus, taking
1

λ(x) :=

||V 2 x||

!−(ν+d)
,

1

||V 2 1d0 ||

W = V −1

(4.24)

in (4.21) yields that π = fX 0 |S=v is symmetrically regularly varying. Putting them together, we
√
can find that the MCMC estimator with MpCN proposal distribution satisfies N -CLT when the
underlying loss vector follows multivariate student’s t distribution with ν > 2 and D > −ν.

15

5

Simulation and numerical studies

We have shown that the MCMC estimator has various attractive properties for the estimation of
VaR contributions. In this section, we apply the proposed estimator to various risk models used
in practice. Our numerical experiment shows that the MCMC estimator has smaller bias and
Mean Squared Error (MSE) compared with the existing estimators that have been proposed in the
previous studies. Then, based on some numerical results, we study how to choose an appropriate
proposal distribution given risk model.

5.1

Numerical study of the MCMC estimator

In the simulation study, we consider four risk models which are modeled with marginal densities and
copula densities. As is often done in risk management, we adopt heavy-tailed marginal distributions
and copulas with tail dependence. In all risk models, we set the size of the portfolio d = 3. The
models are described as follows:
(1) The loss random variable Xj follows Pareto distribution (4.11) with κj = 4 and γj = 3 for all
j = 1, 2, 3. The loss vector (X1 , X2 , X3 ) has a π-rotated Clayton copula (4.13) with θ = 0.5.
(2) Xj , (j = 1, 2, 3) have the same marginals as in the case (1). However, their copula is a
student’s t copula (4.12) with ν = 4 and the correlation matrix given by


1

P = −0.5
0.3

−0.5
1
0.5


0.3

0.5 .
1

(5.1)

(3) Xj follows student’s t distribution with νj = 4, µj = 0 and σj = 1 for all j = 1, 2, 3.
(X1 , X2 , X3 ) has a π-rotated Clayton copula with θ = 0.5.
(4) The loss random vector (X1 , X2 , X3 ) follows a multivariate student’s t distribution (4.16)
with ν = 4, µ = 0, and Σ given by (5.1).
The former two models (1) and (2) treat pure losses, and the latter two models (3) and (4) handle
P&L. In all models, marginal distributions have variance 2.0. Moreover, they are heavy-tailed
with tail index 5.0. The models (1) and (3) possess homogeneous upper tail dependence with tail
coefficient λU
i,j = 0.025 (see, e.g., Joe, 2014). The models (2) and (4) have symmetric upper, lower,
L
U
L
and upper-lower tail dependences with tail coefficients λU
1,2 = λ1,2 = 0.012, λ1,3 = λ1,3 = 0.162,
L
UL
LU
UL
LU
UL
LU
λU
2,3 = λ2,3 = 0.253, λ1,2 = λ1,2 = 0.253, λ1,3 = λ1,3 = 0.029, and λ2,3 = λ2,3 = 0.012.
For each risk model, we compute several estimators of VaR contributions AC = E[X|S =
VaRp (S)] for p = 0.999, with the VaRp (S) replaced by its Monte Carlo estimator v = S [N p] .
In this setting, the condition (C3) in Theorem 4.1 holds in the risk models (1) and (2) since
log(1 − p)/ log(1 − 1/d) = 17.037 > 0.5 = θ. Also, as shown in Example 4.3, the risk model
(4) with MpCN proposal distribution satisfies CLT since D + ν = 137.935 > 0. The estimators
that we study are the Monte Carlo (MC) estimator (2.11), Nadaraya-Watson (NW) estimator
(2.13) (Tasche, 2009), Generalized regression (GR) estimator (2.15) (Tasche and Tibiletti, 2004;
Hallerbach, 2003), and the MCMC estimator (4.6) proposed in this paper:
MC

c δ,N =
AC

1
Mδ,N

N
X
n=1

NW

X (n) · 1[S (n) ∈Aδ ] ,

c φ,h,N =
AC

16

 (n) 
S
−v
(n)
X
·
φ
n=1
h


,
PN
S (n) −v
n=1 φ
h

PN

MCMC

GR

c
AC
gβ ,N = gβ̂N (v),
i.i.d.

c
AC
q,N

=

(n) i.i.d.

(n)

N
1 X (n)
X
,
N n=1 |S=v

(n)

where X (n) ∼ FX , S (n) := X1 + · · · + Xd
∼ FS , and {X|S=v }n=1,...,N is an N -sample path
of a Markov chain with stationary distribution FX|S=v .
For all estimators, we fix the sample size N = 106 . Other parameters of the estimators above
are determined as follows. First, on the MC estimator, we set δ > 0 such that the MC sample size
Mδ,N is around 103 . As in Glasserman (2003), asymptotic normality
p
MC
c δ,N − ACδ ) → Nd (0, Σδ ),
Mδ,N · (AC

as N → +∞

(5.2)

MC

c
holds for a fixed δ. Using this result, we report the estimate of AC
δ,N and its approximated
(j,j) p
standard error sM C / Mδ,N for j = 1, 2, 3, where sM C is the sample standard error defined by

sM C

v
u
u
=t

N
1 X (n) c MC T (n) c MC
(X − ACδ,N ) (X − ACδ,N ) · 1[S (n) ∈Aδ ] .
Mδ,N n=1

(5.3)

Second, on the NW estimator, we choose the kernel density φ the standard normal density. We
decide the bandwidth h > 0 according to the Silverman’s rule of thumb h = 1.06 · σ̂S · N −1/5
(Silverman, 1986). Although asymptotic normality holds for the NW estimator, its asymptotic
variance can hardly be computed because it requires the calculation of fS (v). Therefore, we report
NW
c φ,h,N . Third, on the GR estimator, we choose gβ (s) = β0 + β1 · s and its
only the estimate of AC
coefficients are estimated by
PN

(n)
− X̄N )(S (n) −
n=1 (X
PN
(n) − S̄ )2
N
n=1 (S

β̂N,1 =

S̄N )

β̂N,0 = X̄N − β̂N,1 · S̄N ,

(5.4)

PN
PN
where X̄N = N1 n=1 X (n) and S̄N = N1 n=1 S (n) . According to the general theory of OLS, it
holds under some regularity conditions that
√





(j)
β0
 → N2 (02 , σε2 Q−1 ),
−
N · 
j
(j)
(j)
β̂N,1
β1
(j)

β̂N,0



(j)

as N → +∞

(5.5)

(j)

for j = 1, 2, 3, where β̂N,k and βk is the j-th component of β̂N,k and βk for k = 1 and 2,
respectively, εj is the j-th component of the error term ε, σε2j is its conditional variance given
S(1) , . . . , S(N ) , and
T

Q := lim Y Y /N
N →∞

where

1
S (1)

Y =

···
···

1
S (N )

!T
.

(5.6)

GR

c
According to this result, we report the estimate of AC
gβ ,N and its approximated standard error
(j) √
sGR / N for j = 1, 2, 3, where
(j)
sGR

q
(1,1)
(1,2)
(2,2)
:= Σ̂GR,j + 2v · Σ̂GR,j + v 2 · Σ̂GR,j ,

(5.7)

Σ̂GR,j = σ̂ε2j · (Y T Y /N )−1 and σ̂εj is the sample standard error of the j-th residuals. Finally,
on the MCMC estimator, we choose different proposal distributions depending on the risk models
17

(1)-(4): (1) Random walk proposal q(x, y) = f (y − x) with f ∼ Nd (0, Σ̂v ), where Σ̂v := s2M C ,
for sM C defined in (5.3), (2) Independent proposal q(x, y) = f (y), where f is the density of the
Dirichlet distribution with parameters (0.2, 0.28, 0.6), (3) and (4) MpCN proposal with ρ = 0.8,
GR
GR
c g ,N 0 , v − 1T0 AC
c g ,N 0 )T and Σ := s2 . In the Algorithm 2, we set the initial state
µ = (AC
d

β

MC

β

x(1) = (v/3, v/3, v/3)T . We suppose that asymptotic normality (3.8) holds for all risk models.
Also, its asymptotic variance is estimated by the batch means estimator Σ̂N defined by (3.10).
1
Following the recommendation of Jones et al. (2006), we choose LN := bN 2 c = 103 and BN :=
MCMC
1
c
bN/LN c = bN 2 c = 103 . We report the estimate of AC
and its approximated standard error
q,N
(j,j) √
Σ̂N / N for j = 1, 2, 3.
As mentioned in Section 3.2, the validity of the MCMC method can empirically be checked by
autocorrelation plots and the acceptance rate. Fig. 1 show the autocorrelation plots of the Markov
chains generated by Algorithm 2. The acceptance rate of the MH algorithm in each risk model was

1.0

(b)
○ : X1 |S = v

4 : X2 |S = v
△
+ : X3 |S = v
＋

●

●

○ : X1 |S = v

4 : X2 |S = v
△

●

0.8

0.8

1.0

(a)
●

+ : X3 |S = v
＋

●

0.6

0.6

●

ACF

●

●
●

0.4

●

0.4

ACF
ACF

●

●

●
●
●
●
●
●
●
●
●●
●●
●●
●●●
●●●●
●●●●●●●●
●●●●●●●●●●●●●●●●

●

0

10

20

30

40

0.2
0.0

0.0

0.2

●
●
●
●
●
●
●●
●●
●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●

50

0

10

20

Lag

40

50

1.0

(d)
○ : X1 |S = v

4 : X2 |S = v
△

●

●

○ : X1 |S = v

4 : X2 |S = v
△

●

+ : X3 |S = v
＋

●

●

+ : X3 |S = v
＋

0.8

1.0

(c)
●

0.8

30
Lag

●

0.0
0

0.6
10

20

30

40

50

Lag
Lag

●
●

0.4

●
●
●
●
●
●
●
●
●
●●
●●
●●
●●
●●●●
●●●●●●●
●●●●●●●●●●●●●●●●●●

0.2

0.4

●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●●
●●
●●●
●●●
●●●●●●
●●●●●●●
●●●●●

0.0

●

ACF

●

0.2

ACF
ACF

0.6

●

0

10

20

30

40

50

Lag
Lag

Figure 1: Autocorrelation plots of the Markov chains generated by algorithm 2: (a) Pareto +
π-rotated Clayton, (b) Pareto + t copula, (c) Student’s t + π-rotated Clayton, and (d) Student’s
t + t copula. The dotted line represents y = 0.1.
(1) 0.556, (2) 0.186, (3) 0.643, and (4) 0.757. In Fig. 1, we can observe that the autocorrelation
plots steadily decline below 0.1 by the lag h around 30 for all risk models. Together with the
moderate acceptance rates, we can confirm that the choice of the proposal distributions above are
appropriate for all risk models.
Before showing the estimation results, we check the shapes of the conditional distributions
FX 0 |S=v by plotting the Markov chains generated by Algorithm 2. Fig. 2 shows the contour
plots of the generated Markov chains. According to these plots, the features of the conditional
18

(b)

20

20

X2 |S
X2 = v

X2
15
04

10

15

0.0

055

10

00
5

0.0

03
0.0
01

0.00

5

45

5

0.00
3

0.0025

0.0015

5

10

0.

0.0
0

4

00

2

0.004

0.001

0.002

0.00
3

0

0

5e−04

0

0.006

0.0
0.

05
0.0

0.002

0.0035

0.01

25

25

30

35

(a)

15

20

25

30

35

0

5

10

15
X1

X1

25

20

(d)
0.001

15

15

20

(c)

20

0.001

10

0.003
0.003

X2

0.01
0.00

5

0.0
1

8
0.009

4
0.002

−10

0.004
0.002
0.001

−15

0

5

0.00

−5

00

0.00

0.006

0

0.01

5

5

07

1

10

X2 |S
X2 = v

6
00

0.0

0.007

0.

0.005
0.

0.006

0

5

10

15

20

X1 |SX1= v

−15

−10

−5

0

5

10

15

20

X1 |SX1= v

Figure 2: Contour plots of the Markov chains generated by algorithm 2: (a) Pareto + π-rotated
Clayton, (b) Pareto + t copula, (c) Student’s t + π-rotated Clayton, and (d) Student’s t + t copula.
When drawing the contour plots, we used subsamples, which are picked up every 100-th points of
the original Markov chains, to elliminate the dependence among samples. We plot only the first
and second variables of the 3-dimensional Markov chains since the last variable can be determined
by the sum-constraint. The gray diagonal lines in the plots represent y = −x + v, where v is
the estimate of VaRp (S). The contour plots approximately express the conditional distributions
FX 0 |S=v .
distributions FX 0 |S=v are summarized as follows:
(1) Pareto + π-rotated Clayton : the contour plot Fig. 2 (a) shows that FX 0 |S=v has a unique
mode at the center of the Simplex. Most probability of FX 0 |S=v is concentrated around the
mode, and the probability decreases as going far away from the center. Also, the contour
plot is almost symmetric at the diagonal line y = x.
(2) Pareto + t copula : Unlike the case (1), FX 0 |S=v possesses two distinct modes near the xand y-axes. Especially near the y-axis, there is a peak around which the probability changes
significantly. Also, the contour plot Fig. 2 (b) is asymmetric at the diagonal line y = x.
(3) Student’s t + π-rotated Clayton : Although X 0 |S = v can take negative values, it is supported
almost on the bounded simplex as in the case (1). The contour plot Fig. 2 (c) is unimodal
and almost symmetric around y = x. Also, the tail of FX 0 |S=v is apparently light.
(4) Student’s t + t copula : As discussed in Section 4.3, the conditional distribution FX 0 |S=v in
this case follows the Pearson type VII distribution given by (4.19). From the contour plot Fig.
2 (d), we can observe some properties of this type of distribution, such as elliptical symmetry
19

and tail-heaviness. Unlike the case (3), X 0 |S = v takes large negative values beyond the
bounded simplex.
The results of our estimation are summarized in Table 1. In the four different risk models (1)(4), we report the estimates, its approximated standard errors, biases, and Root MSE (RMSE)s of
the four different estimators MC, NW, GR, and MCMC.
In the first risk model where marginals follow Pareto distributions and they have a π-rotated
Clayton copula, true VaR contributions can be computed explicitly. The true value is obtained by
allocating the total VaR homogeneously because the marginal distributions are homogeneous and
the copula is exchangeable. In this case, we observe that the MC and the NW estimators have relatively large biases. In contrast, the GR and the MCMC estimators have smaller biases compared
with the first two. The GR estimator still has some bias caused by the model-misspecification,
although its standard error is quite small. In the viewpoint of MSE, the MCMC estimator outperforms all the other estimators.
The second risk model has the same marginal distributions as in the case (1), but their copula
is t copula having upper-lower tail dependence. Even though the true VaR contributions are not
available in this case, we can construct its 95% confidence interval based on the MCMC estimator
by using (3.12). Based on this interval of the true value, we computed the range of the bias
and RMSE for other estimators. The reported ranges of bias and RMSE in Table 1 (2) show
that the MCMC estimator significantly improves the performance of other estimators. Under the
risk model (2), all estimators suffer from bias caused by asymmetry and multi-modality of the
conditional distribution FX 0 |S=v as seen in Fig. 2 (b). Especially, in contrast with the good
performance of the GR in the case (1), the GR estimator has relatively large bias and RMSE in
this case.
The third risk model examines the case where the marginal distributions are student’s t distribution and their copula is π-rotated Clayton. The true VaR contributions can be computed
similarly as in the case (1). Thanks to symmetry and uni-modality of the conditional distribution
FX 0 |S=v , all estimators remain small bias and RMSE. Together with the results in the cases (1)
and (2), it can be found that the GR estimator performs well so long as FX 0 |S=v does not have
irregular shape, such as asymmetry and multi-modality. Also, the MCMC estimator reduces bias
and RMSE compared with the MC and the NW estimators.
The final risk model illustrates the case where the underlying portfolio loss vector follows the
multivariate student’s t distribution. In this case, the true VaR contributions is available by the
formula (2.17). In such an elliptical case, the GR estimator provides a quite accurate estimate.
Although the conditional distribution FX 0 |S=v is heavy-tailed as seen in Fig. 2 (d), the MCMC
estimator keeps high performance compared with the MC and the NW estimators. Its bias is
significantly improved compared with the MC and the NW. Additionally, its standard error and
RMSE are also lower than those of the MC estimator.
Throughout the numerical study, the MCMC estimator provided significantly small bias and
RMSE regardless of the shape of the conditional distribution FX 0 |S=v . Under the case when
FX 0 |S=v is unimodal and symmetric, the GR estimator also had good performance. On the other
hand, at least in our numerical study, the MC and the NW estimator had larger bias and RMSE
compared with other estimators.
At the end of this section, we summarize the advantages of the MCMC estimator compared
with the other estimators. First, the MCMC estimator is consistent although the other estimators
are not always. As explained in Section 2, the MC, the NW, and the GR estimators have biases
which can not be evaluated easily. In Table 1, we observed that these estimators have some

20

Table 1: Performance of the four different estimators of VaR contributions under four different
risk models† .
√
Standard error ( M SE):

Estimate of AC (Bias):
Estimator

MC

NW

GR

M CM C

MC

GR

M CM C

0.166
(0.169)
0.164
(0.497)
0.167
(0.655)

0.008
(0.033)
0.008
(0.061)
0.008
(0.029)

0.016
(0.017)
0.017
(0.017)
0.016
(0.018)

0.247
(0.250,
0.694)
0.226
(0.229,
0.687)
0.142
(0.147,
0.664)

0.010
(0.034,
0.532)
0.010
(0.010,
0.532)
0.006
(0.033,
0.532)

0.050
(0.050,
0.110)
0.031
(0.031,
0.103)
0.030
(0.030,
0.102)

(1) Pareto + π-rotated Clayton: True AC = (11.265, 11.265, 11.265)
AC1
AC2
AC3

11.298
(0.033)
10.796
(-0.469)
10.631
(-0.634)

11.550
(0.2859)
11.481
(0.216)
10.762
(-0.503)

11.297
(0.033)
11.204
(-0.061)
11.293
(0.028)

11.258
(-0.007)
11.263
(-0.001)
11.273
(0.008)

(2) Pareto + t copula: True AC is not available ‡
AC1

AC2

AC3

7.475
(0.039,
0.234)
8.156
(-0.649,
-0.526)
11.803
(-0.371,
-0.255)

6.907
(-0.529,
-0.333)
8.878
(0.072,
0.196)
12.415
(0.241,
0.357)

7.772
(0.336,
0.532)
8.715
(-0.090,
0.033)
11.710
(-0.463,
-0.347)

7.338
(-0.098,
0.098)
8.744
(-0.062,
0.062)
12.115
(-0.058,
0.058)

(3) Student’s t + π-rotated Clayton : True AC = (5.917, 5.917, 5.917)
AC1
AC2
AC3

5.919
(0.002)
5.710
(-0.208)
5.638
(-0.279)

6.046
(0.129)
6.009
(0.092)
5.702
(-0.215)

5.921
(0.003)
5.918
(0.001)
5.913
(-0.004)

5.915
(-0.002)
5.940
(0.023)
5.897
(-0.021)

0.077
(0.077)
0.077
(0.221)
0.078
(0.290)

0.005
(0.006)
0.005
(0.006)
0.005
(0.007)

0.016
(0.016)
0.016
(0.028)
0.014
(0.025)

0.007
(0.019)
0.006
(0.015)
0.002
(0.005)

0.034
(0.046)
0.033
(0.044)
0.011
(0.011)

(4) Student’s t + t copula : True AC = (2.972, 3.715, 6.686)
AC1
AC2
AC3

†

3.028
(0.056)
3.482
(-0.233)
6.526
(-0.161)

3.559
(0.587)
3.053
(-0.662)
6.736
(0.050)

2.989
(0.017)
3.701
(-0.013)
6.682
(-0.004)

3.004
(0.032)
3.687
(-0.028)
6.683
(-0.004)

0.125
(0.136)
0.112
(0.259)
0.046
(0.167)

The estimate is computed for the Monte Carlo M C, the Nadaraya Watson N W , the generalized re-

gression GR, and the Markov chain Monte Carlo M CM C estimators. The standard error is computed
except for the NW estimator. The bias and RMSE are also computed for the cases (1), (3), and (4),
where true VaR contributions are available. The sample size is N = 106 . The proposal distributions of
the MCMC estimators are (1) Random Walk, (2) Independent, (3) MpCN, and (4) MpCN.
‡
In the case (2), where true VaR contributions are not available, the range of bias and RMSE are computed
based on the 95% confidence interval of the true VaR contributions derived by the MCMC estimator (3.12).
c N , we compute the range of
Based on the confidence interval ACL < AC < ACU and the estimate AC
c N − ACU , AC
c N − ACL ). The range of RMSE is computed in the same way.
the bias (AC

21

biases even when their standard errors are sufficiently small. In contrast, the MCMC estimator
provides the accurate estimate of the VaR contributions due to its consistency. Together with the
CLT, we also can construct the confidence interval of the true VaR contributions. Second, the
MCMC estimator has great sample efficiency compared with the MC estimator. Since the MCMC
estimator generates sample not from FX but from FX 0 |S=v , no sample have to be discarded. In
addition, unlike the MC, there is no trade-off between the bias and the sample efficiency. Thanks
to these properties, the MCMC estimator can hold low standard error without increasing the
bias. Finally, the MCMC estimator can maintain high performance even when the conditional
distribution FX 0 |S=v has irregular shape, such as multi-modality and heavy-tailed. As compared
with the results in the case (2) and others in Table 1, the performance of the GR estimator highly
depends on the shape of FX 0 |S=v . For the existing estimators, it is generally difficult to reflect
the shape of FX 0 |S=v to their hyper-parameters. On the other hand, for the MCMC estimator,
we can directly capture the shape of FX 0 |S=v through the proposal distribution q. By choosing an
appropriate proposal distribution q, the MCMC estimator has stably good performance regardess
of the shape of FX 0 |S=v .

5.2

Empirical study of the proposal distribution

In Section 5.1, we have shown that the MCMC estimator improves the performance of estimating
VaR contributions. However, the performance of the MCMC estimator highly depends on the
appropriate choice of the proposal distribution q. Additionally, the choice of proposal distribution
is less apparent compared with that of hyper-parameters for the other estimators. In this section,
we firstly investigate the symptoms caused by the bad choice of q. Then, we consider how to
overcome this problem based on the numerical results.
Bad choice of q is largely classified into two cases. One is that, the proposal distribution q often
generates a candidate at which the probability measured by π is quite small. This case occurs
for example when the variance of q is much larger than that of π, or when q does not capture
the irregular shape of π as appeared in Fig. 2 (b). In such cases, the Markov chain moves quite
slowly, yielding high asymptotic standard error of the MCMC estimator. This symptom appears
as a quite low acceptance rate and high autocorrelation plots. The other case is that the proposal
distribution q generates only some parts of the whole support of the target distribution π. This
case happens for example when π has distinct local modes and the variance of q is so small that
the chain can not pass between them. In this case, the estimate has a significant bias, although
the acceptance rate and the autocorrelation plots are seemingly fine. This symptom appears as a
distorted plot of the MCMC samples whose shape is clearly different from the target distribution
π.
How can we detect and avoid such fallacious estimates? First of all, as mentioned in Section
3.2, it is indispensable to check the acceptance rate and the autocorrelation plots. Additionally,
it is also important to draw the plots of the generated Markov chain, and compare them with
the plots of the MC samples whose sums belong to Aδ = [v − δ, v + δ]. Since such MC samples
follow the distribution FX 0 |S∈Aδ , we can recognize the distortion of the generated Markov chain
by comparing with the two plots of the samples. In the case of our numerical study, Fig. 3 shows
the scatter plots of the MC samples whose sums belong to [v − δ, v + δ], overlaid on the scatter
plots of the MCMC samples. We can check in Fig. 3 that the shape of the scatter plots of the
MCMC samples bear striking resemblance to those of the MC samples for all risk models.
Finally, we found that dependence information of the underlying risk model is quite helpful
for the selection of q. When the copula C of the underlying risk model represents only upper

22

(b)

35

(a)

0

5

10

15

20

25

30

25
20
X2

5

10

15

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: MC
: MCMC

●

35

0

5

10

15

X1

20

25

X1
●

(c)

0

0

5

10

15

40
20
X2

X1 |SX1= v

●

● ●
●
●
●
●

0
20

●

60

: MC
: MCMC

−20

5

10

X1 |S
X2 = v

15

●

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−40

−20

0

20

X1 |SX1= v

40

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60

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●

Figure 3: Scatter plots of the MC (•) and the MCMC (◦) samples for different risk models: (a)
Pareto + π-rotated Clayton, (b) Pareto + t copula, (c) Student’s t + π-rotated Clayton, (d)
Student’s t + t copula. We plot the MC samples generated from FX such that their sums belong
to Aδ = [v − δ, v + δ]. In the four risk models, the value of δ are (1) 4.8, (2) 3.9, (3) 2.2, and (4) 1.7.
When drawing the scatter plots of the MCMC samples, we used subsamples, which are picked up
every 100-th points among the original Markov chains. We plot only the first and second variables
of 3-dimensional samples since the last variable is determined by the sum-constraint. The plots of
the MC, and the MCMC samples express the distribution FX 0 |S∈Aδ and FX 0 |S=v , respectively.
and/or lower tail dependence, then the target distribution π is likely to be tractable. For example
in the cases of risk models (1) and (3), where the copula C has an upper tail dependence, the
contour plots Fig. 2 (a) and (c) show that the target distribution π is unimodal and light-tailed.
These features facilitate the estimation with MCMC since simple proposal distributions such as
Random Walk proposal (3.14) perform well. Conversely, when the copula C has upper-lower tail
dependence, then the target π tends to be complicated. In the cases (2) and (4), where the copula
C has upper-lower tail dependence, the Fig. 2 (b) indicates that the π is bimodal, and also the
contour plot Fig. 2 (d) shows that the π is heavy-tailed. In such cases, careful selection of the
proposal function q is required for the good performance of the MCMC estimator. For instance,
to capture the bi-modality appeared in the contour plot Fig. 2 (b), one of the good choice of q is
an independent proposal distribution (3.15) with f the Dirichlet distribution on the v-simplex Sv .
Since Dirichlet distribution can possess two distinct modes around the edges of the simplex, we
can generate a candidate which often visits the two separated modes of the π. In the case (4), to
overcome the problem of heavy-tailedness of π seen in the contour plot Fig. 2 (d), it is necessary
to adopt some specialized MCMC for heavy-tailed target distribution, such as MpCN (3.16).
23

6

Concluding remarks

Computing VaR contributions for a continuous risk model is a difficult task in general. There is no
standard method since existing estimators have their own deficiencies. Especially, it can be seen
that the crude Monte Carlo estimator suffers from unavoidable bias to the true VaR contributions.
Our MCMC estimator proposed enables the consistent estimation of VaR contributions. Its sample
efficiency is significantly improved because the MCMC estimator is computed by samples generated
directly from the conditional density of a portfolio loss given the sum constraint. Moreover, since
the MCMC estimator can capture the features of the risk model more directly than the existing
estimators, it can maintain high performance even when the shape of the underlying loss distribution is far from elliptical. We showed some theoretical properties of the MCMC estimator, such as
consistency and asymptotic normality. The consistency follows naturally from the construction of
the estimator. We provided some conditions and example where CLT holds. By the general theory
of MCMC, its asymptotic variance can be consistently estimated. We numerically compared the
performance of various estimators of VaR contributions for typical risk models used in practice.
The simulation results showed that, in most risk models we considered, the MCMC estimator had
smaller bias and MSE compared with the existing estimators.
Although the MCMC estimator has good theoretical properties and numerically shows great
performance, some remaining issues merit future research. Firstly, although we provided guidelines
to choose good proposal distribution, they are hardly applicable when the dimension of the target
distribution is too high to visualize. For such high-dimensional cases, asymptotic analysis and
adaptive method should be developed to chooce good proposal distribution. Secondly, theoretical
investigation of the conditional joint distribution given the sum constraint is still unsatisfactory.
Our concern mainly lies with the influence of the underlying copula of a risk model to the shape
of the conditional joint distribution, such as tail behavior and multi-modality. We believe that
revealing these relations helps to find a better proposal distribution in our context of estimating
VaR contributions. Finally, our MCMC method can potentially be extended to the computation
of Expected shortfall (ES) contributions. The ES contributions can be derived as a conditional
expectation of losses given that the total loss is equal to or greater than VaR (see Tasche, 2001).
Due to its similar structure to the VaR contributions, the ES contributions can also be efficiently
estimated in an analogous method proposed in this paper. This, and other potential extensions
are currently being studied by us.

Acknowledgements
We would like to thank Paul Embrechts for his valuable comments and Kengo Kamatani for a
fruitful discussion on MCMC. Our research is supported by the Core-to-Core program (Japan
Society for the Promotion of Science: JSPS) in Keio Univeristy.

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