Estimating allocations for Value-at-Risk
portfolio optimzation
1
Arthur Charpentier
1 ENSAE/CREST,
2 UCO/IMA,
2
& Abder Oulidi
3 avenue Pierre Larousse, F-92240 Malako, [email protected].
place André Leroy, F-49000 Angers, [email protected].
1
Abstract
Value-at-Risk, despite being adopted as the standard risk measure in nance,
suers severe objections from a practical point of vue, due to a lack of convexity,
and since it does not reward diversication (which is an essential feature in portfolio
optimization). Furthermore, it is also known as having poor behavior in risk estimation (which has been justied to impose the use of parametric models, but which
induces then model errors). The aim of this paper is to chose in favour or against
the use of VaR but to add some more information to this discussion, especially from
the estimation point of view. Here we propose a simple method not only to estimate
the optimal allocation based on a Value-at-Risk minimization constraint, but also
to derive - empirical - condence intervals based on the fact that the underlying
distribution is unkown, and can be estimated based on past observations.
Keywords:
nonparametric estimation; optimal allocations; Value-at-Risk
Introduction and motivations
The problem of allocating capital among a set of risky assets can be understood as nding
a portfolio which maximizes return, and minimizes risk.
If diversication eects were
intuited early, Markowitz intitiated a formal mathematical model.
In this approach,
return is measured by the expected value of the portfolio return, and risk is quantied by
the variance. Thus is the so-called mean-variance risk management framework.
In [36] was considered the problem of selecting some
perspective: at time
0,
optimal portfolio
with a static
money is invested among several risky assets, so that at time
T,
optimal allocation is the one which minimizes
the return should be high with low risk. The
the variance of the return between now and time t, given that its expected return exceeds
a given return target
η.
This idea has been extend in a multiperiod model, with multiple
possible reallocations (note that it can also be done in a continuous time framwork, see
e.g.
[37]).
Consider for simplicty a two period model.
among several risky assets, so that at time
T = 2,
At time
0,
money is invested
the return should be high with low
risk. But here, the investor, we can sell some assets in order to buy others at time
This yield the notion of
optimal strategy, which is a set of optimal portfolios, the rst one
being the one obtained at time
t = 1,
t = 1.
t = 0,
and the second one being the reallocation at time
based on additional information observed during period of time
[0, 1].
If those models ... from a mathematical point of view, it appears quickly that there
are serious drawback from a practical point of view. Optimal allocation is related to the
joint distribution of returns, or, at least, the rst two moments: the expected returns,
and the variance-covariance matrix. But those parameters are unknown, and should be
estimated from the past. Based on 4 years of past observations, only a1, 000-sample can
be used and therefore, all the parameters can be estimated with a signicant condence
interval. And thus, an investor can only consider
2
estimated optimal portfolio.
In the two
t = 0, 30% of the money should be invested in the rst
asset, and that, after calculations at time t = 1, it appears that 32% of the money should
be invested in the rst asset: is this 2 point dierence signicant or is it simply due to
period model, assume that at time
estimation issue ? How condence should we be in the
optimality of those portfolios ?
d risky assets, and denote
by P t = (Pt,1 , ..., Pt,n ) the random vector of prices at time t (the time interval is a
week). Let X t = (Xt,1 , ..., Xt,n ) denote weekly log-returns, Xt,i = log Xt,i − log Xt−1,i .
We will assume in this paper that random vectors X t 's are independent and identically
In this paper, we will use the following notations. Consider
distributed.
This assumption is the underlying assumption in the Black and Scholes
framework, but might appears as too strong with regards to recent litterature on GARCH
processes (see e.g.). Anyway, since the goal here is simply to allocate capital from time
t
to
t+1
(static portfolios), we might assume that - as a rst order approximation - the
dependence among log-returns is more important then possible time dependence.
X =
(X1 , ..., Xn ) will then denote the standard weekly log return, and we assume that portfolio
allocation at time t = n can be performed based on n past observations, {X 1 , . . . , X n }.
In section 2, we will redine properly this notion of
optimal portfolio,
statistical issues in the simple mean-variance model in section 3.
and highligth
Section 4 will then
focus on statistical issues in a more general setting. In section 5, VaR estimation will be
briey studied, from parametric and semiparametric models (based either on a normal
assumption or on extreme value results), to nonparametric (kernel based estimators). And
nally, a simulation study will be performed in section 6, followed by a real-data analysis
in section 7.
Optimal portfolio
Markowitz developed mean-variance analysis in the context of selecting a portfolio of
common stocks, and it has been increasingly applied to asset allocation.
But if this
mean-variance approach was historically the rst idea to formalize optimal allocations,
severall extentions in more general settings have been considered. One of the drawback
of this mean-variance approach is the symmetric attitude towards risks, which initiated
reaseach work on downside risk measures.
Denition 0.1. An optimal allocation among
n risky assets (i.e. d random log-return
X = (X1 , ..., Xd ), for a given risk measure R, is a vector of proportions ω ∗ = (ω1∗ , ..., ωd∗ ) ∈
Ω ⊂ Rn is a solution of the following optimisation program,
∗
ω ∈ argmin {R (ω 0 X) , ω ∈ Ω} ,
(1)
under constraints E (ω 0 X) ≥ η and ω 0 1 = 1
Based on the approach initiated by Von Neumann and Morgenstern (in game theory),
economic theory postulates that individuals make decisions under uncertainty by maximizing the expected value of an increasing concave utility function of consumption (see
3
[45] or [19]). Markowitz asserted that if the utility function can be approximated closely
enough by a second-order Taylor expansion over a wide range of returns, then expected
utility will be approximately equal to a function of expected value (the
ance
of returns.
mean)
and
vari-
This allows the investor's problem to be restated as a mean-variance
optimization problem where the objective function is a quadratic function of portfolio
weights.
The argument in von Neuman and Morgenstein's theory is that any preference order-
ing
dened on the set of random risks, satisfying come axioms (namely completeness,
transitivity, continuity and independence, see [45]), can equivalently be represented by
some
utility function u such that E(u(X)) ≤ E(u(Y )) holds if and only if X Y .
Hence,
the formalization of this second approach needs rst to select a certain utility function u
∗
0
and to formulate the following optimization problem, ω ∈ argmax {E(u(ω X)), ω ∈ Ω},
where it is usually required that the function
u
is concave and non-decreasing, thus rep-
resenting preferences of a risk-averse decision maker (see e.g. [45] or [19]). An alternative
formulation can be induced from [31]: dene a risk measure as R(X) = E(u(E(X) − X))
∗
0
(so that the risk measure is location free) and then ω ∈ argmin {R(ω X), ω ∈ Ω}.
And nally a third method can be derived from Yaari's dual approach (see [49]). From
early works of de Moivre ou Pascal, we know that the price of a game is the scalar product
of the probabilities and the gains, formulated later as the
fair price
(from [18]), i.e. the
expected value. In the expected value framework, risk measures were dened as
EP (u(Y )),
which is the scalar product bewteen the probabilities (P) and utility of gains (u(Y )). Note
that this function can be written through the following integral form
0
Z
Z
+∞
P(u(Y ) ≤ y)dy +
EP (u(Y )) =
−∞
P(u(Y ) > y)dy.
(2)
0
In the dual approach (see [49], [40], [47] among others) is considered the scalar product
bewteen a distorted version of probabilities (denoted
Z
0
Z
g(P(Y ≤ y)) +
Eg◦P (Y ) =
−∞
for some distortion function
g ◦ P)
and the gains (Y ),
+∞
g(P(Y > y))dy,
(3)
0
g : [0, 1] → [0, 1]
(increasing with
g(0) = 0
and
g(1) = 1).
Remark 0.2. In the case where
g is an indicator function, g(x) = 1(x > p), then
Eg◦P (Y ) = V aR(Y, p). Further, if g(x) = g(x) = min{x/p, 1}, i.e. linear from (0, 0) to
(p, 1), then Eg◦P (Y ) = T V aR(Y, p). Those two measures are dened properly in Denition
0.5.
On a formal point of view, [49] proposed a more general theoritical framework than
those distorted measures, also called
spectral risk measures in [1].
4
Estimation issues in the classical mean-variance framework
Several authors have pointed out that the Markowitz optimal portfolio is extremely sen-
?
sitive to parameter perturbations, and amplies the estimation errors (see [30] or [ ]
X = (X1 , . . . , Xd ). Denote
µ = E(X) and Σ = var(X). Let ω = (ω1 , . . . , ωd ) ∈ Ω ⊂ Rd denote the weights in all
0
0
risky assets. The expected return of the portfolio is E(ω X) = ω µ, and the the variance
0
0
of the portfolio is var(ω X) = ω Σω . The optimization problem is here
∗
ω ∈ argmin {var (ω 0 X) , ω ∈ Ω} ,
(4)
0
0
under constraints E (ω X) ≥ η and ω 1 = 1
among others). Consider d risky assets, with weekly returns
Recall that this mean-variance programm can be formulated dualy (as the maximization
of the expected value, with a risk constraint), since it is an optimization with an inequality
is a quadratically constrained quadratic programming problem.
Remark 0.3. The set of possible allocation is either the simplexe of Rd or the subset of
Rd such that the sum of the component is 1. In this paper, we will assume that Ω = Rd ,
i.e. short sales are allowed. This assumption will have no impact on the optimization
program, since we will estimate risk measures on a grid of possible portfolio allocations,
and the avantage is that kernel based estimators can be considered since values will not be
bounded.
In practice,
µ = [µi ]
and
Σ = [Σi,j ]
are unknown, and should be estimated. From a
the parameters governing the central tendency
and dispersion of returns are usual ly not known, however, and are often estimated or
guessed at using observed returns and other available data. In empirical applications, the
estimated parameters are used as if they were the true value. A natural idea is to use
practical point of view, as noticed in [10], empirical estimated, based on sample
{X 1 , ..., X n }.
Hence
n
b = [b
µ
µi ]
where
1X
Xi,t ,
µ
bi =
n t=1
and
n
b = [Σ
b i,j ]
Σ
Recall that
given
(5)
Σ,
b
µ
and
b
Σ
where
X
b i,j = 1
Σ
(Xi,t − µ
bi )(Xj,t − µ
bj ).
n t=1
(6)
have respectively a Gaussian and a Wishart distribution, and that
those two estimators are independent.
As mentioned in the introduction, there is a dierence between theoretical optimal
allocation (based on
µ
and
Σ)
and the sample based optimal allocation (based on
b ).
Σ
5
b
µ
and
Mean-Value-at-Risk optimization
Value-at-Risk as a risk measure
Current regulations from nance (Basle II) or insurance (Solvency II) business formulates
risk and capital requirements in terms of quantile based measures (see e.g.
[14]).
The
upper quantile of the loss distribution is called Value-at-Risk (VaR) : the 95% VaR is an
upper estimate of losses which is exceeded with a 5% probability. For a comprehensive
introduction, we refer to [28], or [20] for nancial motivations, or for a some economic
motivation of VaR, and more generally distorted risks measures. Hence, VaR provides a
common risk measure that can be used for any type of portfolio (market risk, credit risk,
insurance risk), and a a simply interpretation as possible lost money.
Denition 0.4. For a probability level α ∈ (0, 1), the α-quantile of a random variable Y
is dened as
Q(Y, α) = inf{y ∈ R, P(Y ≤ y) > α}.
Denition 0.5. For a probability level
returns X is dened as
(7)
α ∈ (0, 1), the Value-at-Risk at level α for log-
VaR(X, α) = Q(−X, α).
(8)
The Tail Value-at-Risk at level α is dened as
TVaR(X, α) = E(−X| − X > VaR(X, α)).
Tail Value-at-Risk (as called in [3]) is called expected-shortfall
expectation in [48] or conditional Value-at-Risk in [42].
(9)
in [1],
conditional tail
Historically, mostly approches to calculate VaR relied on linear approximation of the
portfolio risks, assuming also a joint normal distribution of market parameters (see e.g.
?
[ ], or the Cornish Fiher approximation in Section 0.1). The idea of Markowitz is to nd
the optimal portfolio allocation, dened as a solution of
for some risk measure
ω ∗ ∈ argmin {R (ω 0 X) , ω ∈ Ω} ,
0
0
under constraints E (ω X) ≥ η and ω 1 = 1
R
(the variance in [36], or the VaR in this paper).
(10)
Hence, the
investor want to nd a allocation which minimizes the risk for a given expected return,
with a budget contraint. Disussions on optimization issues involving VaR minimization
can be found in [35] or [29].
As mentioned in the context of coherent risk measures
by [3], VaR suers a major drawback because of nonsubadditivity : the VaR of a sum
might exceed the sum of VaR's.
Therefore, the VaR is neither convex (see e.g.
[3] or
[20] for a survey on convex risk measures), and thus, several optimization theorems and
results can be be used. As mentioned in [42], an alternative is to consider a convex risk
measure
R
in (10), such as the Tail VaR (also called expected shortfall or conditional
6
VaR). This measure is dened as the conditional expected loss given that it exceeds
VaR. For almost normal distributions, those two measures are dierent, but the optimal
allocations almost coincide.
As mentioned in Ingersoll, they even coincide with mean-
variance optimal allocation for elliptical distributions.For very skewed distributions, it
might not be the case anymore (see e.g.
0.1 Mean-Value-at-Risk optimality and the safety rst principle
The fact that the VaR is not convex (while the variance was) yields a theoretical (and
therefore a practical) issue. In the mean variance problem, the two following programs
were equivalent,
ω ∗ ∈ argmin{ω 0 Σω}
0
0
u.c. ω µ ≥ η and ω 1 = 1
convex
⇔
ω ∗ ∈ argmax{ω 0 µ}
0
0
0
u.c. ω Σω ≤ η and ω 1 = 1
But this duality is no longer valid here,
ω ∗ ∈ argmin{VaR(ω 0 X, α)}
0
0
u.c. E(ω X) ≥ η and ω 1 = 1
nonconvex
<
ω ∗ ∈ argmax{E(ω 0 X)}
0
0
0
u.c. VaR(ω X, α) ≤ η ,ω 1 = 1
As pointed out in [9], we believe strongly that for most investors the proper order of
investigation is to consider the quality of a bond or preferred stock before considering the
yield. For most investors, if quality is inadequate, yield is irrelevant. Therefore, even
if most of actual litterature focusses on the second problem (maximizing the return, see
e.g. [4], ), we also believe that minimizing risk is the most appropriate way to express
investors behavior (see also [32] or [5]). This approach, also motivated more recently in
[2], will also be considered in this paper.
And so far, in this paper, we have introduced portfolio optimization based on Markowitz's
approach. But historically, the mean-variance has not the only principle used by investors.
the optimal bundle of assets (investment) for investors who employ
the safety rst principle is the portfolio that minimizes the probability of disaster. [43]
relates the disaster to the idea of ruin in insurance, which can also be related, with a
modern perspective of nancial problems, to Value-at-Risk. In this safety rst principle, investors are looking for portfolio which minimize disaster probabilities, with some
As mentioned in [43],
expected return constraint.
Note nally that only a few paper focus explicitely in mean-VaR optimal portfolio, as
in [4]. Most of them prefer to focus on TVaR (which present the advantage of beeing a
convex risk measure), as in [34] or [42].
7
Inference issues
In the mean variance framework, the optimal allocation could be expressed as a function
of the expected return, and the variance matrix of the logreturns. A natural estimation of
the optimal allocation is then obtained substituting the average and the empirical variance
to the theoritical values.
A practical issue is about estimation of risk measures, and more specically VaR.
All or parts of the distribution have to be estimated.
usually considered.
In practice, two techniques are
Parametric methods estimate quantiles under the assumption that
a loss distribution has a particular parametric form. The st task is then to determine
what it can be.
VaR(X, 95%)
For instance, in the case of normale distribution, the 95% quantile is
= E(X)
+ 1.64
p
var(X), and can be estimated using an estimation of
the expected value and the variance. Note that the choice of the distribution should be
guided by some diagnostics, such as graphical tests (quantile-quantile plot for instance)
and goodness of t tests. Some additional properties might also be requiered for modelling issues, such as nice aggregation properties (stable or ellipitical distributions), or
extreme value distribution (pareto distibution). Using maximum likelihood techniques,
least squares, methods of momments, etc, we estimate the parameters of the underlying
distribution and then plug the parameter estimated into the quantile equation to obtain
quantile estimates. Note that in the case of multiple risks (which is the case in this paper),
a multivariate joint distribution is necessary. If the variance of a sum of risks can just be
expressed through variances and correlations, the full joint distribution is necessary when
calculating the VaR of a sum. In that case, a very popular idea is nd parametric distribution for individual risks, and to model the dependence structure using a parametric
copula (see Embrechts, Denuit or for comprehensive introduction). Parametric technique
are simple to use, and usually do not requiere not a long computation time. But in order
to obtain consistent estimate, only recent data should be used, and therefore, because of
the small sample size (e.g.
250 trading days as we shall see on a Monte Carlo study).
But at least, this estimation error can be controled, under the assumption that the parametric model is correct : the main issue for risk managers is test wether the model is
relevant. Reliability of VaR estimates is then based on faith practitionners can have in
their internal models (see for an intersting discussion on model errors). A benchmark can
be obtained using model-free estimation of VaR : this is precisely the goal of nonparametric techniques, where no (strong) assumption about the distribution is made.
The
idea to model distribution is to use histograms, or more more sophisticated kernel based
estimators.
8
Quantile and Value-at-Risk estimation
Recall that the quantile at level
α ∈ (0, 1)
is dened as
Q(X, α) = inf{x, P(X ≤ x) > α} = FX−1 (α)
where
FX−1
denotes the generalized inverse of distribution function
(11)
FX (x) = P(X ≤ x).
The Gaussian model: a fully parametric approach
2
In the Gaussian model, if X ∼ N (µ, σ ), recall that Q(X, α) = µ + u1−α σ , where u1−α
−1
Φ (1 − α), Φ standing for the cumulative distribution of the N (0, 1) distribution (u
−1.64 if α = 90%, or u = −1.96 if α = 95%).
If X is not Gaussian, it may still be possible to use a Gaussian approximation. If
variance is nite, (X − E(X))/σ might be closer to the Gaussian distribution, and
=
=
the
the
consider the so-called Cornish-Fisher approximation (see [12] or [25]), i.e.
Q(X, α) ∼ E(X) + z1−α
p
var(X),
(12)
where
ζ2
ζ2
1
z1−α = Φ−1 (1−α)+ [Φ−1 (1−α)2 −1]+ [Φ−1 (1−α)3 −3Φ−1 (1−α)]− 1 [2Φ−1 (1−α)3 −5Φ−1 (1−α)],
6
24
36
(13)
where
ζ1
is the skewness of
ζ1 =
X,
and
ζ2
is the excess kurtosis, i.e. i.e.
E([X − E]3 )
E([X − E]2 )3/2
and
ζ1 =
E([X − E]4 )
− 3.
E([X − E]2 )2
(14)
Using extreme value results, a semi-parametric approach
Quantile estimation based on extreme value results can be seen as semi parametric. Given
a
n-sample {Y1 , . . . , Yn },
let
Y1:n ≤ Y2:n ≤ . . . ≤ Yn:n
denotes the associated order statis-
tics. One possible method is to use Pickands-Balkema-de Haan theorem (see Section 6.4.
in [17] or Section 4.6. in [6] for instance). The idea is to assume that for
Y −u
given
Y > u
u
large enough,
has a Generalized Pareto distribution with parameters
ξ
and
β,
which can be estimated using standard maximum likelihood techniques. And therefore, if
u = Yn−k:n
ξ > 0, denote by βbk and ξbk maximum likelihood estimators of the Genralized Pareto distribution of sample {Yn−k+1:n −Yn−k:n , ..., Yn:n −Yn−k:n },
for
k
large enough, and if
−ξbk
βbk n
b
Q(Y, α) = Yn−k:n +
(1 − α)
−1
k
ξbk
9
(15)
An alternative is to use Hill's estimator if
ξ > 0,
−ξbk
b α) = Yn−k:n n (1 − α)
Q(Y,
,
k
(16)
k
where
1X
log Yn+1−i:n − log Yn−k:n
ξbk =
k i=1
(if
ξ > 0).
Some nonparametric quantile estimates
The most convenient quantile estimator is based on order statistics. For instance, given
a sample of size
n = 200,
the
190th
largest value should be a relevant estimator of
More generally, the raw estimator of the αth-Value-at-Risk is then
b α) = Y[αn]:n = Xi:n = Fb−1 (i/n) such that i ≤ αn < i + 1, where Fbn is the empirical
Q(Y,
n
distribution sample. If αn is not an integer, it could be interesting to consider a weighted
d (Y, α) = λYi:n + (1 − λ)Yi+1:n . Or more generally, some weighted average of
average VaR
Z 1
n
X
order statistics can be considered,
λi Yi:n =
λu Fbn−1 (u)du, where the weigth function
the 95% quantile.
depends on
n
and
α.
0
i=1
This technique has been developed intensively (see e.g. [24], [38] or
[39]).
A completely dierent approach will be to consider a smoothed version of the cumula
n
X
y−
1
−1
b α) = Fb (α) where FbK (y) =
tive distribution function, and to invert it, Q(Y,
K
K
nh
i=1
Remark 0.6. Note that those two
Z approaches can be formally by presented through the
following analytical expression,
estimators.
1
λu FbK−1 (u)du, which is a weighted sum of smoothed
0
Transformation based estimates
In order to derive better estimates, [8] pointed out that it could be more interesting,
{Y1 , . . . , Yn }, to estimate the
quantile of g(Y ), based on a sample {g(Y1 ), . . . , g(Yn )}, where g is a nondecreasing
function. Since g(Q(Y, α)) = Q(g(Y ), α), a natural estimator of the quantile is then
b α) = g −1 Q(g(Y
b
b
Q(Y,
), α), where the estimator Q(g(Y
), α) can be any of the previous
instead of estimating quantiles of
Y,
based on a sample
estimator.
[23] suggested to use this technique with
g
being the cumulative distribution function
of some generalized Champernowne distribution.
10
h
Yi
.
From VaR estimation to allocation estimation
The problem here can be written as follows: a estimated optimal allocation is dened
b n (ω)}, droping the constraints convience and understanding, where
b ∗n = argmin{R
as ω
b n (ω) is an estimator of the α-quantile of random variable ω 0 X , estimated using sample
R
b n (ω) converges to R(ω) for every ω , it seems
{X 1 , ..., X n }. If ω ∗ = argmin{R(ω)}, if R
∗
b n will converge to ω ∗ as n → ∞.
reasonable to expect that ω
Theoritically, convergence - in probability - of the estimator of the risk measure is too
weak to insure the convergence of the optimal allocation, and functional convergence is
necessary, strengthening the pointwise convergence (see e.g. Theorem 5.7. [44]). Since
quantile estimators can be obtained as
L-statistics
(wieghted sums of order statistics), it
can be possible to derive conditions such that functional convergence can be satisted.
Application to portfolio optimization: a real-data based
study
Consider the dataset of weekly log-returns, obtained from daily closing prices of
4
Eu-
ropean stock indices, from 1991 to 1998: Deutch DAX (Ibis), Switz SMI, French CAC,
and British FTSE. The data are sampled in business time, i.e., weekends and holidays are
omitted. Table 1 summarizes statistics of the dataset, including the mean-variance optimal allocation, when the target expected weekly return is
0.35%
(this target will remain
unchanged in this study).
[Table 1 about here.]
[Figure 1 about here.]
The algorithm in order to derive properties of the estimators of allocations is the
following. First, we need to generate samples of log-returns,
•
semiparametric monte-carlo based estimation: assuming that sample
{X 1 , ..., X n }
Lθ , θ can be estimated from those observaLθb: a monte carlo based estimator of quantiles
has been generated using distribution
tions, and then generate samples from
can be derived, for all possible allocation,
•
bootstrap (nonparametric) estimation: since this rst approach can induce model
error if the underlying distribution family
(Lθ , θ ∈ Θ)
is mispecied,a fully non-
parametric approach can also be considered. Estimators of quantiles can be derived
using bootstraped samples.
In this paper, we focus on the second approach, since we wish to avoid model errors.
b
b
Given a scenario (a generated sample {X 1 , ..., X n })
11
•
for each possible allocation
ω of a nite grid G , deteermine an estimation of VaR(Z, α),
where Z is the log-return associate to portfolio allocation ω , which can be estimated
b
b
b
b
0
0
from sample {Z1 , ..., Zn } = {ω X 1 , ..., ω X n }
•
the optimal allocation is obtained at the (empirical) minimum Value-at-Risk for all
the points of the allocation of the grid
G.
[Figure 2 about here.]
Optimal allocations as a function of the probability level α
Table 2 and Figure 49 present the output of
10, 000
simulated portfolio returns of the
same period of time, with optimal alllocation based on a Value-at-Risk criteria for several
value for
α.
Note that the higher the quantile level the more we sell asset 3 to buy asset
4.
[Figure 3 about here.]
[Table 2 about here.]
Optimal allocations based on dierent quantile estimates
In this section, ve estimators of quantiles are considered,
1. raw estimator
b α) = Y[α·n]:n
Q(Y,
2. mixture estimator
b α) =
Q(Y,
n
X
λi (α)Yi:n ,
which is the standard quantile estimate
i=1
in R (see [26]),
3. Gaussian estimator
b α) = Y + z1−α sd(Y ),, where sd denotes the empirical stanQ(Y,
dard deviation,
4. Hill's estimator, with
(assuming that
b α) = Yn−k:n
k = [n/5], Q(Y,
k
−ξbk
X
Yn+1−i:n
bk = 1
(1 − α)
, where ξ
log
k
k i=1
Yn−k:n
n
ξ > 0),
5. kernel based estimator is obtained as a mixture of smoothed quantiles, derived as
inverse values of a kernel based estimator of the cumulative distribution function,
n
X
b
i.e. Q(Y, α) =
λi (α)Fb−1 (i/n).
i=1
12
From Figure 49 we can observe that the Gaussian approximation is not ecient in the
case the returns are not normally distributed. Nonparametric estimators are interesting
since they are distribution free (and therefore, there should be no model error), and the
uncertainty is rather small, compared with parametric estimators. And from a computational point of view, the raw estimator performs rather well, compared with more robust
estimators (which are time consuming due to multiple inversion and smoothing).
[Figure 4 about here.]
Conclusion
We have seen in that paper that it was possible to obtain from a nite sample an estimation of the optimal mean-VaR portfolio (minimizing Value-at-Risk with a constraint
on expected return). An algorithm to derive bounds on the estimation of optimal allocations. On the simulated dataset, we have seen that if calculation is much longer than
the optimal mean-variance portfolio, condence bounds are quite similar. An other point
is that
n = 250
past observed data is not sucently large to estimate optimal allocation
with a small condence interval.
If estimation was somehow robust and ecient for
in real life with
100
4 risky assets and 250 observations,
(or even more) assets, much more past observations are necessary.
Among possible extensions, we have assumed here that log-returns
X t 's
were inde-
pendent. As pointed out in litterature, this assumption is not validated by stylized fact.
One idea might be to look forward where the
(e.g. ARCH or GARCH process).
13
X i 's
are driven by some stochastic process
References
[1]
Acerbi,
C. and
[2]
Alexander,
[3]
Artzner,
[4]
[5]
Tasche,
D. (2002) - On the coherence of Expected Shortfall.
Banking & Finance, 26, 1487-1503.
Journal of
G. and Baptista, A. (2002). Economic Implications of Using a Mean-VaR
Model for Portfolio Selection: A Comparison With Mean-Variance Analysis. Journal of
Economic Dynamics & Control, 26, 1159-1193.
Risk.
P.,
Delbaen,
Mathematical
Basak,
prices.
S. and
F., Eber, J.M. and
Finance, 9, 203-228.
Heath, D.
(1999). Coherent Measures of
Shapiro,
The Review of
A. (2001). Value-at-Risk Management: optimzl policies and asset
Financial Studies, 14, 371-405.
V.S. (1978). Safety-First, Stochastic Dominance, and Optimal Portfolio Choice. The
Journal of Financial& Quantitative Analysis, 13, 255-271.
Bawa,
[6]
Beirlant,
J., Goegebeur, Y., Segers, J. and Teugel, J. (2006). Statistics of extremes.
Wiley Interscience.
[7]
Campbell,
[8]
Charpentier,
[9]
Cohen, J. B., and Zinbarg, E.D. (1967). Investment Analysis and PortfoZio Management.
Homewood, Ill.: Richard D. Irwin, Inc., 1967.
R., Huisman, R., and Koedijk, K. (2001). Optimal portfolio selection in a
value at risk framework. Journal of Banking and Finance, 25, 1789-1804.
A., and
Oulidi,
A. (2007). Nonparametric quantile estimation.
submitted.
[10]
Coles,
J.L. and Loewenstein, U. (1988). Equilibrium pricing and portfolio composition
in the presence of uncertain parameters. Journal of Financial Economics , 22, 279-303.
[11]
Consigli,
G. (2002). Tail estimation and meanÐVaR portfolio selection in markets subject
to nancial instability. Journal of Banking & Finance, 26,1355-1382.
[12]
Cornish,
[13]
Dowd,
[14]
Dowd,
[15]
Duarte,
E. A. and Fisher, R.A. (1937). Moments and cumulants in the specication of
distributions. Review of the International Statistical Institute, 5, 307-320.
K. (1997). Beyond Value at Risk: The Science of Risk Management. Wiley.
K. and Blake, D.. (2006). After VaR : the theory, estimation, and insurance
applications of quantile-based risk measures. Journal of Risk & Insurance, 73, 193-229.
[16]
Journal of Risk, 1, 71-94.
Duffie, D. and Pan, J. (1997). An overview of Value at Risk. Journal of Derivatives, 4,
[17]
Embrechts,
A. (1999). Fast computation of ecient portfolios.
7-49.
P.,Kluppelberg, C. and
Springer Verlag.
Mikosh,
14
T. (1997). Modeling extremal events.
[18]
Feller,
G.W. (1968). Generalized asymptotic expansions of Cornish-Fisher type.
nals of Mathematical Statistics, 39, 1264-1273
The An-
[19]
Fishburn,
[20]
Föllmer,
[21]
Gaivoronski,
A.A. and Pflug, G. (2000). Value-at-Risk in portfolio optimization : properties and computational approach. Working Paper, 00-2, Norwegian University of Sciences
& Technology.
[22]
Gouriéroux, C. and
CREST, 2006-17.
[23]
Gustafsson,
J., Hagmann, J.,Nielsen, J.P., Scaillet, O. (2006). Local Transformation
Kernel Density Estimation of Loss. Cahier de Recherche 2006-10, HEC Genve.
[24]
Harrel,
[25]
Hill,
[26]
Hyndman,
[27]
Ingersoll,
[28]
P.C. (1970). Utility Theory for Decision-Making. Wiley.
H., and Schied, A. (2004). Stochastic Finance: An Introduction in Discrete
Time. Walter de Gruyter.
Liu,
W. (2006). Ecient Portfolio Analysis. Using Distortion Risk.
F.E. and Davis, C.E. (1982) A new distribution free quantile estimator.
Biometrika, 69, 635-670.
type.
G.W. and
Davis,
The Annals of
A.W. (1968). Generalized asymptotic expansions of Cornish-Fisher
Mathematical Statistics, 39, 1264Ð1273.
R. J. and Fan, Y. (1996). Sample quantiles in statistical packages.
Statistician, 50, 361Ð365.
Jorion,
Hill.
American
J.E. (1987). Theory of Financial Decision Making. Rowman & Littleeld.
P. (1997). Value at Risk : the new benchmark for controlling market risk. McGraw-
[29]
Kast,
R., Luciano, E. and Peccati, L. (1998). VaR and optimization : 2nd international
workshop on preferences and decisions. Trento, July 1998.
[30]
Klein,
[31]
Kroll,
Y., Levy, H. and Markowitz, H.M. (1984). Mean-Variance Versus Direct Utility
Maximization. The Journal of Finance, 39, 47-61.
[32]
Levy,
[33]
Larsen, N., Mausser, H. and Uryasev, S. (2002). Algorithms for optimization of Value
at Risk. in Financial engineering, e-commerce and supply-chain, Pardalos and Tsitsiringos
eds., Kluwer Academic Publichers, 129-157.
[34]
Lemus,
R.W. and Bawa, V.S. (1976). The eect of estimation risk on optimal portfolio choice.
Journal of Financial Economics, 3, 215-231.
of
H. and Sarnat, M. (1972). Safety FirstAn Expected Utility Principle. The
Financial & Quantitative Analysis, 7, 1829-1834.
MIT.
Journal
G. (1999). Portfolio Optimization with Quantile-based Risk Measures. Ph.D. Thesis,
15
[36]
Risk, 10, 38-42.
Markowitz, H.M. (1952). Portfolio selection. Journal of Finance, 7, 77-91.
[37]
Merton,
[38]
Padgett,
[39]
Park,
[40]
Quiggin,
[41]
Ogryczak,
W. and Ruszczynski, A. (2002). Dual stochastic dominance and related meanrisk models. SIAM Journal of Optimization, 13, 60-78.
[42]
Rockafellar,
[43]
Roy,
[44]
van der Waart,
[45]
Von Neumann,
[46]
Wand,
M.P. and
[47]
Wang,
S. (1996). Premium Calculation by Transforming the Layer Premium Density.
Bulletin, 26, 71-92.
[48]
Wirch,
[49]
Yaari,
[35]
Litterman,
R. (1997). Hot spots and edges II.
R.C. (1992). Continuous time nance (second ed). Blackwell Publishers.
W.J. (1986). A Kernel-Type Estimator of a Quantile Function From RightCensored Data . Journal of the American Statistical Association, 81, 215-222?
C. (2006). Smooth nonparametric estimation of a quantile function under right censoring using beta kernels. Technical Report (TR 2006-01-CP), Departement of Mathematical
Sciences, Clemson University.
J. (1993). Generalized Expected Utility Theory: The Rank-Dependent Expected
Utility model. Kluwer-Nijho.
Journal of
R.T. and
Risk, 2, 21-41.
Uryasev,
S. (2000). Optimization of Conditional Value-at-Risk.
A.D. (1952). Safety First and the Holding of Assets.
Econometrica, 20, 431-449.
A.W. (1998).Asymptotic statistics. Cambridge University Press.
J. and Morgenstern, O. (1947). Theory of games and economic behaviour. Princeton University Press.
ASTIN
J., and
Jones,
M.C. (1995). Kernel Smoothing. New York: Chapman Hall.
Hardy,
Insurance: Mathematics
M. (1999). A Synthesis of Risk Measures for Capital Adequacy.
and Economics, 25, 337-348.
M. (1987). A Dual Theory of Choice Under Risk.
16
Econometrica, 55, 95-115.
List of Figures
1
2
3
4
Scatterplot of log-returns. . . . . . . . . . . . . . . . . . . . .
Value-at-Risk for all possible allocations on the grid G (surface and level curves), with α = 75% on the left and α = 97.5%
on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal allocations for dierent probability levels (α =
75%, 77.5%, 80%, ..., 95%, 97.5%), with allocation for the rst
asset (top left) up to the fourth asset (bottom right). . . . .
Optimal allocations for dierent 95% quantile estimators,
with allocation for the rst asset (top left) up to the fourth
asset (bottom right). . . . . . . . . . . . . . . . . . . . . . . . .
17
18
19
20
21
● ●
●
●● ●●●
●
●
●●
●●
●
●●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●●
●
●
●●
●
●
●●●●
●
●
●
●
●
●
●
●
●
●
●●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●● ●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●
●
●
●●
●●
●● ●
●●
●●
●
●
●
●
●●●
●
●
●●
● ●●
●
●
●
●
●
●●
●
●●●●● ●
●●
●
●
● ●●
−0.05
0.05
●
SMI (2)
●
●
● ● ●● ●
●
●
●●●●
●●
●
●
●●●
●
●●
●●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●●
●●●
●●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●●
●
●
●
●●●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●● ●
●
●●
●
●
●●●
●
●●
●
●
●
●
●●●●●
●●
●
●
●
●
● ● ● ●● ●●
●●●● ●
● ●●
●●
● ●●
● ●
● ●
●●
●
●●●●●
●
●
●●
●●
●●
● ●●
●●●
●
●
●
●
●
●
●●●●●●●
●
●
●
●●●
●●
●
●
●
●
●
●●
●
●
●
●
●
●●●
●
●
●
●●●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●●
●
●
●●
●
●●
●
●
●
●
●●
●
●
●●●●●
●
●
●●
●
●
●
●
●
●●●●●●
●
●
●
●●●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●● ● ●
● ●●●●
●●● ●
●
●
●●
●●
●
●●●● ●
●●●●
●
●●●● ●
●●●
●
●
●
●
●
●
●
●
●
●
●●
●
●●●●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ● ●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●●●
●
●
●
●●
●●
●
●
●
●●●
●
●
●●
●
●
●●
●
● ●●
●●
●
●
●
●
●
●
●
●
●
● ●
● ● ● ●●● ●
●
●●
●
●
● ●●
●
●
● ●●● ●
●●●
●●●
●●●●●
●
●
●
●●●●
●●
●●
●●
●●●●
●
●
●
●●
●●
●●
●
●
●
●●
●
●
●●●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
● ●●●●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●●
●
●
●●●
●
●
●●
●
●
●●●
●●
●
●●
●●
●
●●
●●
●
●
●●●●●
● ●●●●
●
● ●●
●
●
●
●
● ●
●
●
●
●
●
●●
●●●●●
●●
●
●●●
● ●
●
●●●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●●
●
●●●
● ●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●●●
●
●
●●
●
●●
●●●●
● ●
●●
●●
●
●
●
●
●
●
●
●
● ● ●●
●
●
●
●
CAC (3)
●
●
● ●
●
●
●
●
●●
●●
●
●●●●
●
●●
●
●
●
●●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●●
●
●●●
●
●●
●●
●
●
●
●●
●
●
●
●
●●●
●
●●
●
●●
●
●
●
●
●
●
●●
●●
●
●●
●●●●
●●●●● ●●
● ●
●●
●
●
●
●
●
●●● ●●
●
●●●●
● ● ●●
●●
●●
●●●
●
●●●
●
●
●
●●
●
●
●
●
●
●
●●●●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●●●● ●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●● ●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●●●●
●
●
●
●
●
●
●●●
●●
●
●
●
●●
●
●●●
●
●
●●
●●
●●●●
●● ● ●●
●●
●
●
●
●
● ● ●● ●
●
●●
●
● ●●●
● ●●
●
●● ●
●●
●●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●●●
●●
●
●
●●● ●
●
●● ●
● ●●●● ●●
●
●●
●
●
●
● ●
●
● ●●
●●
●● ●
●●●
●
●
●
●
●
●●
●●
●
●●
●●
●
●
●
●
●
●●●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ● ●●●
●
●
●
●
●
●●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●●
●●
●
●
●
● ●●
●
●
●
●
●● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●●
●
●
●
●
●●
●
●
●
●●
●●
●
●
●●
●●●●
●●
●
● ●● ●●
● ●
●
●●
●
FTSE (4)
−0.05
0.05
−0.10
0.00
0.05
●
● ●● ●
●
●●●●
●●
●
●
●●
●●●
●
●
●●●
●●
●
●●
●
●
●●●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●●●
●
●
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
● ●
●●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●●
●
●●●●
●
●
●
●
●
●●●
●●
●
●
●●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●
●●
●●
●●
●●
● ●●●●
●
●
●
●
●
0.10
−0.05
0.05
DAX (1)
●
● ●●● ●
● ●
●
●
●
●●
●
●
●
●●
●●●●●
●
●
●●●
●●
●
●●
●
●
●
●
●●
●
●
●●
●●
●
●●
●
●
●
●●●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●● ●●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●
●
●
●
●●
●
●
● ● ●●●●
●
●●
●
●
●
●
●●
●
●
●
●
● ●
●●
●●
● ●● ●●
● ●
●
●
● ●
●
● ●
0.05
●
−0.05
−0.05
●
0.00
0.05
−0.10
−0.05
0.10
Figure 1: Scatterplot of log-returns.
18
Value−at−Risk (75%) on the grid
t)
0
−1
−2
oc
th a
sse
−3
ati
n (4
All
Allo
catio
on
(3r
da
ss
et)
Quantile
Allocation (4th asset)
1
2
Value−at−Risk (75%) on the grid
−3
−2
−1
0
1
2
Allocation (3rd asset)
Value−at−Risk (97,5%) on the grid
0
−1
−2
(3r
da
ss
et)
Quantile
−3
t)
oc
sse
All
th a
ati
on
Allo
catio
n (4
Allocation (4th asset)
1
2
Value−at−Risk (97.5%) on the grid
−3
−2
−1
0
1
2
Allocation (3rd asset)
Figure 2: Value-at-Risk for all possible allocations on the grid
with
α = 75%
on the left and
α = 97.5%
on the right.
19
G
(surface and level curves),
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
97.5% 95% 92.5% 90% 87.5% 85% 82.5% 80% 77.5% 75%
1.5
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
3:
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
97.5% 95% 92.5% 90% 87.5% 85% 82.5% 80% 77.5% 75%
Optimal allocation (asset 4)
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● 92.5% 90% 87.5% 85% 82.5% 80% 77.5%
●
97.5%
95%
75%
●
●
●
●
●
1.0
0.5
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.0
weight of allocation
●
1.5
2.0
Optimal allocation (asset 3)
●
●
●
●
●
●
●
●
●
● ● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−1.0
●
Optimal
allocations
75%, 77.5%, 80%, ..., 95%, 97.5%),
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
97.5% 95% 92.5% 90% 87.5% 85% 82.5% 80% 77.5% 75%
Probability level (97.5%−75%)
Figure
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Probability level (97.5%−75%)
1.5
1.0
0.5
0.0
weight of allocation
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Probability level (97.5%−75%)
●
●
−1.0
1.0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.5
●
●
●
●
●
●
●
●
●
●
●
●
0.0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−1.0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
weight of allocation
1.5
1.0
0.5
0.0
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
2.0
−1.0
weight of allocation
2.0
Optimal allocation (asset 2)
2.0
Optimal allocation (asset 1)
Probability level (97.5%−75%)
for
dierent
probability
levels
(α
=
with allocation for the rst asset (top left) up to
the fourth asset (bottom right).
20
●
●
Optimal allocation
(asset 2)
Optimal allocation (asset 1)
●
0
−1
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Est. 1
raw
2
Est. 5
Kernel
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Est.
● 3
Gaussian
●
●
Est. 4
Hill
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Est. 5
Kernel
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2
●
●
Est. 2
mixture
●
1
●
0
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Est. 4
Hill
−1
●
●
●
●
●
● 3
Est.
●
Gaussian
weight of allocation
●
●
●
●
−2
weight of allocation
Est. 2
mixture
●
1
2
Est. 1
raw
●
−3
−3
●
●
Quantile estimator
Quantile estimator
Optimal allocation
(asset 3)
●
Optimal allocation (asset 4)
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
2
Est. 2
mixture
Est.
● 3
Gaussian
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
1
●
●
●
●
●
●
●
●
●
●
●
Est. 1
raw
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
Est. 4
Hill
Est. 5
Kernel
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2
1
0
−1
weight of allocation
−2
Est. 5
Kernel
●
●
●
Est. 4
Hill
0
Est. 3
Gaussian
−1
Est. 2
mixture
weight of allocation
2
Est. 1
raw
●
−3
−3
●
●
Quantile estimator
Figure 4: Optimal allocations for dierent
Quantile estimator
95%
quantile estimators, with allocation for
the rst asset (top left) up to the fourth asset (bottom right).
21
List of Tables
1
2
Descriptive statistics of Eurostocks dataset. . . . . . . . . . .
Mean and standard deviation of estimated optimal allocation, for dierent quantile levels. . . . . . . . . . . . . . . . .
22
23
24
average returns
standard deviation
1
2
3
4
Asset 1
Asset 2
Asset 3
Asset 4
-0.0032
-0.0040
-0.0021
-0.0022
0.024
0.023
0.026
0.019
1.00
0.71
0.75
0.61
1.00
0.64
0.60
Asset 2
1.00
0.63
Asset 3
1.00
Asset 4
-0.2516
0.4846
Pearson's
correlation
optimal allocation
0.2277
0.5393
Asset 1
Table 1: Descriptive statistics of Eurostocks dataset.
23
asset 1
mean
variance
0.2277
asset 2
0.5393
asset 3
−0.2516
asset 4
0.4846
75%
77.5%
80%
82.5%
85%
87.5%
90%
92.5%
95%
97.5%
0.222
0.206
0.215
0.251
0.307
0.377
0.404
0.394
0.402
0.339
(0.253)
(0.244)
(0.259)
(0.275)
(0.276)
(0.241)
(0.243)
(0.224)
(0.214)
(0.268)
0.550
0.558
0.552
0.530
0.500
0.460
0.444
0.448
0.441
0.467
(0.141)
(0.136)
(0.144)
(0.152)
(0.154)
(0.134)
(0.135)
(0.124)
(0.121)
(0.151)
−0.062
−0.083
−0.106
−0.139
−0.163
−0.196
−0.228
−0.253
−0.310
−0.532
(0.161)
(0.176)
(0.184)
(0.187)
(0.215)
(0.203)
(0.163)
(0.141)
(0.184)
(0.219)
0.289
0.319
0.339
0.357
0.357
0.359
0.380
0.410
0.466
0.726
(0.162)
(0.179)
(0.191)
(0.204)
(0.221)
(0.205)
(0.175)
(0.153)
(0.170)
(0.200)
Table 2: Mean and standard deviation of estimated optimal allocation, for dierent quantile levels.
24