derivatives are required to check the convexity of VaR, to conduct marginal analysis of portfolios or compute optimal portfolios under VaR constraints. Such
derivatives are easy to derive for multivariate Gaussian distributions, but, in most practical applications, the joint conditional p.d.f. of asset returns is not Gaussian
Ž .
and involves complex tail dependence Embrechts et al., 1999 . The goal here is to derive analytical forms for these derivatives in a very general framework. These
expressions can be used to ease statistical inference and to perform local risk analysis.
The paper is organized as follows. In Section 2, we consider the first and second-order expansions of VaR with respect to portfolio allocation. We get
explicit expressions for the first and second-order derivatives, which are character- ized in terms of conditional moments of asset returns given the portfolio return.
This allows to discuss the convexity properties of VaR. In Section 3, we introduce the notion of VaR efficient portfolio. It extends the standard notion of mean-vari-
ance efficient portfolio by taking VaR as underlying risk measure. First-order conditions for efficiency are derived and interpreted. Section 4 is concerned with
statistical inference. We introduce kernel-based approaches for estimating the VaR, checking its convexity and determining VaR efficient portfolios. In Section
5, these approaches are implemented on real data, namely returns on two highly traded stocks on the Paris Bourse. Section 6 gathers some concluding remarks.
2. The sensitivity and convexity of VaR
2.1. Definition of the VaR We consider n financial assets whose prices at time t are denoted by p ,i
s
i, t
1, . . . , n. The value at t of a portfolio with allocations a , i s1, . . . ,n is then:
i
Ž .
n X
W a sÝ
a p sa p . If the portfolio structure is held fixed between the
f i
s1 i i, t
t
current date t and the future date t q1, the change in the market value is given
Ž . Ž .
X
Ž .
by: W a
yW a sa p q1yp .
t q1
t t
t
The purpose of VaR analysis is to provide quantitative guidelines for setting Ž
. reserve amounts or capital requirements in phase with potential adverse changes
Ž in prices see, e.g. Morgan, 1996; Wilson, 1996; Jorion, 1997; Duffie and Pan,
1997; Dowd, 1998; Stulz, 1998 for a detailed analysis of the concept of VaR and .
applications in risk management . For a loss probability level a the Value at Risk, Ž
. VaR
a, a is defined by:
t
P W a
yW a qVaR a,a -0 sa , 2.1
Ž . Ž .
Ž .
Ž .
t t
q1 t
t
where P is the conditional distribution of future asset prices given the information
t
available at time t. Such a definition assumes a continuous conditional distribution of returns. Typical values for the loss probability range from 1 to 5, depending
on the time horizon. Hence, the VaR is the reserve amount such that the global Ž
. position portfolio plus reserve only suffers a loss for a given small probability a
over a fixed period of time, here normalized to one. The VaR can be considered as an upper quantile at level 1
ya, since:
X
P ya y
VaR a, a
sa , 2.2
Ž .
Ž .
t t
q1 t
where y sp
yp .
t q1
t q1
t
At date t, the VaR is a function of past information, of the portfolio structure a and of the loss probability level a.
2.2. Gaussian case In practice, VaR is often computed under the normality assumption for price
Ž .
changes or returns , denoted as y . Let us introduce m and V , the conditional
t q1
t t
Ž .
mean and covariance matrix of this Gaussian distribution. Then from Eq. 2.2 and the properties of the Gaussian distribution, we deduce the expression of the VaR:
1 r2
X X
VaR a, a
syam q aV a z
, 2.3
Ž .
Ž .
Ž .
t t
t 1
ya
where z is the quantile of level 1
ya of the standard normal distribution. This
1 ya
expression shows the decomposition of the VaR into two components, which compensate for expected negative returns and risk, respectively.
Let us compute the first and second-order derivatives of the VaR with respect to the portfolio allocation. We get:
EVaR a, a
V a
Ž .
t t
sym q z
t 1
ya 1
r2 X
Ea aV a
Ž .
t
V a
t X
sym q VaR
a, a qam
Ž .
Ž .
X t
t t
aV a
t X
syE y Na y
syVaR a,a ,
2.4
Ž .
Ž .
t t
q1 t
q1 t
X 2
E VaR a, a
z V aaV
Ž .
t 1
ya t
t
s V y
X X
t 1
r2 X
EaEa aV a
aV a
Ž .
t t
z
1 ya
X
s V y
Na y syVaR a,a
. 2.5
Ž .
Ž .
t t
q1 t
q1 t
1 r2
X
aV a
Ž .
t
In particular, we note that these first and second-order derivatives are affine functions of the VaR with coefficients depending on the portfolio allocation, but
independent of a. In Section 2.3, we extend these interpretations of the first and second-order derivatives of the VaR in terms of first and second-order conditional
moments given the portfolio value.
2.3. General case The general expressions for the first and second-order derivatives of the VaR
are given in the property below. They are valid as soon as y has a continuous
t q1
conditional distribution with positive density and admits second-order moments.
Property 1.
Ž . i The first-order derivative of the VaR with respect to the portfolio allocation
is: EVaR
a, a
Ž .
t X
syE y Na y
syVaR a,a .
Ž .
t t
q1 t
q1 t
Ea Ž .
ii The second-order derivative of the VaR with respect to the portfolio
allocation is: E
2
VaR a, a
Ž .
t X
EaEa Elog g
a , t X
s yVaR a,a V y
Na y syVaR a,a
Ž .
Ž .
Ž .
t t
t q1
t q1
t
E z E
X
w x
y V y
Na y syz
,
t t
q1 t
q1
½ 5
E z
Ž .
z sVaR a, a
t
where g denotes the conditional p.d.f of a
X
y .
a, t t
q1
Ž .
Proof. i The condition defining the VaR can be written as:
P X qa YVaR a,a
sa ,
Ž .
t 1
t
where X syÝ
n
a y , Y
syy . The expression of the first-order deriva-
i s2 i i,tq1
1, t q1
tive directly follows from Lemma 1 in Appendix A. Ž .
ii The second-order derivative can be deduced from the first-order expansion of the first-order derivative around a benchmark allocation a . Let us set a
sa q
o o
´ e , where ´ is a small real number and e is the canonical vector, with all
j j
components equal to zero but the jth equal to one. We deduce: EVaR
a, a
Ž .
t
w x
sE X Zq´ Ys0 qo ´ ,
Ž .
t
Ea
i
where: X
syy , Z
sya
X
y yVaR a ,a ,
Ž .
i , t q1
o t
q1 t
o
Y syy
qE y NZs0 .
j, t q1
t j, t
q1
The result follows from Lemma 3 in Appendix B.Q.E.D
2.4. ConÕexity of the VaR It may be convenient for a risk measure to be a convex function of the portfolio
allocation thus inducing incentive for portfolio diversification. From the expres- sion of the second-order derivative of the VaR, we can discuss conditions, which
ensure convexity. Let us consider the two terms of the decomposition given in Property 1. The first term is positive definite as soon as the p.d.f. of the portfolio
Ž .
price change or return is increasing in its left tail. This condition is satisfied if this distribution is unimodal, but can be violated in the case of several modes in
the tail. The second term involves the conditional heteroscedasticity of changes in asset prices given the change in the portfolio value. It is non-negative if this
conditional heteroscedasticity increases with the negative level
yz of change in the portfolio value. This expresses the idea of increasing multivariate risk in the
left tail of portfolio return. To illustrate these two components, we further discuss particular examples.
2.4.1. Gaussian distribution In the Gaussian case considered in Section 2.2, we get:
Elog g z
yzqa
X
m
Ž .
a , t t
s .
X
E z aV a
t
Therefore: Elog g
VaR a, a
qa
X
m
Ž .
a , t t
t
yVaR a,a s
Ž .
Ž .
X t
E z aV a
t
z
1 ya
s ,
1 r2
X
aV a
Ž .
t
Ž .
from Eq. 2.3 . Ž
. This positive coefficient as soon as a - 0.5 corresponds to the multiplicative
Ž .
factor observed in Eq. 2.5 . Besides, the second term of the decomposition is zero due to the conditional homoscedasticity of y given a
X
y .
t t
2.4.2. Gaussian model with unobserÕed heterogeneity The previous example can be extended by allowing for unobserved heterogene-
ity. More precisely, let us introduce an heterogeneity factor u and assume that the conditional distribution of asset price changes given the information held at time t
Ž . Ž .
has mean m u and variance V u . The various terms of the decomposition can
t t
easily be computed and admit explicit forms. For instance, we get: g
z s g
z Nu P u du,
Ž . Ž
. Ž .
H
a , t a , t
Ž .
where g z
Nu is the Gaussian distribution of the portfolio price changes given
a, t
the heterogeneity factor, and P denotes the heterogeneity distribution.
We deduce that: E
E g z
Ž .
a , t
g z
Nu P u du
Ž .
Ž .
H
a , t
Elog g z
Ž .
a , t
E z E z
s s
E z g
z
Ž .
a , t
g z
Nu P u du
Ž .
Ž .
H
a , t
E log g z
Nu
Ž .
a , t
sE ,
˜
P
E z
˜
where the expectation is taken with respect to the modified frobability P defined by:
˜
P u sg z
Nu P u g
z Nu P u du .
Ž . Ž
. Ž .
Ž .
Ž .
H
a , t a , t
Due to conditional normality, we obtain:
X
Elog g VaR
a, a qam u
Ž .
Ž .
a , t t
t
yVaR a,a sE
. 2.6
Ž .
Ž .
Ž .
˜
X t
P
E z aV u a
Ž .
t
Let us proceed with the second term of the decomposition. We get:
w
X
x w
X
x
V y Na y
syz sE V y Na y
syz,u
t t
q1 t
q1 P
t t
q1 t
q1
w
X
x
qV E y Na y
syz,u .
P t
t q1
t q1
w
X
x The conditional homoscedasticity given u, implies that V y
Na y syz,u
t t
q1 t
q1
does not depend on the level z and we deduce that: E
X
V y Na y
syz
t t
q1 t
q1
E z E
X
s V E
y Na y
syz,u
P t
t q1
t q1
E z E
V u a
Ž .
t X
s V
m u q yzyam u
. 2.7
Ž . Ž .
Ž .
Ž .
X P
t t
E z aV u a
Ž .
t
Ž .
Ž .
Ž . Let us detail formulas
2.6 and
2.7 , when m u
s0, ;u, i.e. for a
t
Ž .
conditional Gaussian random walk with stochastic volatility. From Eq. 2.6 , we deduce that:
Elog g 1
a , t
yVaR a,a sVaR a,a E
0.
Ž .
Ž .
Ž .
˜
X t
t P
E z aV a a
Ž .
t
Ž .
From Eq. 2.7 , we get: E
E V u a
Ž .
t X
y V y
Na y syz sy
V yz
X t
t q1
t q1
P
E z E z
aV u a
Ž .
t
E V u a
Ž .
t 2
sy z V
X P
E z aV u a
Ž .
t z
syz
V u a
Ž .
t
sq2 zV ,
X P
aV u a
Ž .
t
Ž .
which is non-negative for z sVaR a,a . Therefore, the VaR is convex when
t
price changes follow a Gaussian random walk with stochastic volatility.
3. VaR efficient portfolio
