Quantile Estimation and Optimal Portfolios Optimization

Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Quantile estimation
and optimal portfolios
Arthur Charpentier & Abder Oulidi
ENSAI-ENSAE-CREST & IMA Angers
Journée SFdS, Mai 2007
1

Portfolio management and optimal allocations
Idea: allocating capital among a set of assets to maximize return and minimize
risk.
If diversification effects were intuited early, and Markowitz (1952) proposed a
mathematical model.
• return is measured by the expected value of the portfolio return,
• risk is quantified by the variance of this return.
Agenda
1. statistical issue in the mean-variance framework
2. portfolio optimization with general risk measures
2

Portfolio optimization (parametric framework)
Consider a risk measure R (variance or Value-at-Risk). Solve
ω∗
=



argmin{R(ωt
X)},
u.c. ωt
1 = 1 and E(ωt
X) ≥ η,
where X ∼ L(θ), θ unknown.
θ is unknown but can be estimated using a sample {X1, . . . , Xn}.
“The parameters governing the central tendency and dispersion of returns are
usually 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” (Coles
& Loewenstein (1988)).
If ω∗
= ψ(θ) (e.g. mean-variance)
ω∗
= ψ(θ).
3

Optimization of standard deviation (or variance)
Allocation
in
the
first asset
Allocation in the second asset
Standard deviation of the portfolio
−200 −100 0 100 200
−200−1000100200 Allocation in the first asset
Allocationinthesecondasset
Figure 1: Portfolio variance optimization problem.
4

Portfolio optimization (parametric framework)
In the case of no explicit expression of the optimum, solve (numerically)
ω∗
=



argminR(ωt
X),
u.c.ω ∈ {(ωk)k∈{1,...,m}}
where X ∼ L(θ).
The idea is to generate samples Xi’s,
• either from a parametric distribution L(θ),
• or from a nonparametric distribution (bootstrap approach).
5

Optimization of Value-at-Risk
VaR of the portfolio
−4 −2 0 2 4 6
−2−1012345
−4
−2
0
2
4
6
VaR of the portfolio
−4 −2 0 2 4 6
012345
−4
−2
0
2
4
6
Figure 2: Optimization in the mean-VaR framework.
6

Classical mean-variance allocation problem
Consider d risky assets, with weekly returns X = (X1, . . . , Xd). Denote
µ = E(X) and Σ = var(X).
Let ω = (ω1, . . . , ωd) ∈ Rd
denote the weights in all risky assets.
• the expected return of the portfolio is E(ωt
X) = ωt
µ,
• the variance of the portfolio is var(ωt
X) = ωt
Σω.



ω∗
∈ argmin{ωt
Σω}
u.c. ωt
µ ≥ η and ωt
1 = 1
convex
⇐⇒



ω∗
∈ argmax{ωt
µ}
u.c. ωt
Σω ≤ η and ωt
1 = 1
7

Classical mean-variance allocation problem
The solution can be given explicitly (see Markowitz (1952)) as
ω∗
= ψ(µ, Σ) = p + ηq
where µ = E(X), Σ = var(X),
p =
1
d
bΣ−1
1 − aΣ−1
µ and q =
1
d
cΣ−1
µ − aΣ−1
1 ,
and a = 1t
Σ−1
µ, b = µt
Σ−1
µ, c = 1t
Σ−1
1, d = bc − a2
.
Note that p is an allocation, and q indicates how the original portfolio should
be modiﬁed.
8

Eﬃcient frontier
first asset
−0.2 0.0 0.2 0.4 −0.4 0.0 0.4
−0.20.00.2
−0.20.00.20.4
second asset
third asset
−0.20.20.6
−0.2 0.0 0.2
−0.40.00.4
−0.2 0.2 0.6
fourth asset
Portfolio with 4assets
0.010 0.015 0.020 0.025 0.030
0.0000.0010.0020.0030.0040.0050.006
Efficient Frontier
Standard deviation
Expectedvalue
Figure 3: Solving a variance optimization problem.
9

Inference issues
In practice, µ = [µi] and Σ = [Σi,j] are unknown, and should be estimated.
A natural idea is to deﬁne
µi =
1
n
n
t=1
Xi,t et Σi,j =
1
n − 1
n
t=1
(Xi,t − µi)(Xj,t − µj).
Given n observed observed returns,
µ|Σ ∼ N µ,
Σ
n
and nΣ|Σ ∼ W (n − 1, Σ) .
where the two random variables µ and Σ are independent, given Σ.
10

0.05 0.10 0.15 0.20 0.25
0.000.050.100.150.200.25
Efficient Frontier, with 250 past observations
Standard deviation
Expectedvalue
0.05 0.10 0.15 0.20 0.25
0.000.050.100.150.200.25
Efficient Frontier, with 1000 past observations
Standard deviation
Expectedvalue
Figure 4: Eﬃcient frontiers and estimation.
11

Parametric bootstrap
Assume that X ∼ L(θ). estimate θ by θn. The procedure is the following
1. generate n returns X1, . . . , Xn from L(θn);
2. estimate µ and Σ, i.e. µn and Σn,
3. solve the minimization problem, i.e.
ω∗
=
1
d
bΣ
−1
1 − aΣ
−1
µ + η
1
d
cΣ
−1
µ − aΣ
−1
1 ,
Using several simulations, the distribution of the ω∗
k and vark(ω∗
k
t
X) can be
obtained.
12

Nonparametric bootstrap
A nonparametric procedure can also be considered. Consider a n sample
{X1, . . . , Xn}
1. generate a bootstrap sample from {X1, . . . , Xn}
2. estimate µ and Σ, i.e. µn and Σn,
3. solve the minimization problem, i.e.
ω∗
=
1
d
bΣ
−1
1 − aΣ
−1
µ + η
1
d
cΣ
−1
µ − aΣ
−1
1 ,
Using several simulations, the distribution of the ω∗
k and vark(ω∗
k
t
X) can be
obtained.
13

0.4 0.6 0.8 1.0
012345
Allocation in the first asset
Allocation weight
Density
0.0 0.2 0.4
012345
Allocation weight
Density
−0.3 −0.1 0.0 0.1
02468
Allocation in the third asset
Allocation weight
Density
0.05 0.15 0.25
0246810
Allocation in the fourth asset
Allocation weightDensity
Figure 5: Distributions of optimal allocations ω∗
k’s.
14

−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
Allocationinsecondasset
Joint distribution of optimal allocations (1−2)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
Allocationinthirdasset
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
Allocationinfourthasset
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
Figure 6: Joint distributions of optimal allocations ω∗
k’s.
15

0.05 0.06 0.07 0.08 0.09 0.10 0.11
020406080100
Density of estimated optimal standard deviation
Optimal standard deviation
Density
Figure 7: Distribution of vark(ω∗
k
t
X)
16

Value-at-Risk minimization
With V aR(X, p) = F−1
(p) = sup{x, F(x) < p}, the program is



ω∗
∈ argmin{VaR(ωt
X, α)}
u.c. E(ωt
X) ≥ η,ωt
1 = 1
nonconvex



ω∗
∈ argmax{E(ωt
X)}
u.c. {VaR(ωt
X, α)} ≤ η ,ωt
1 = 1
In the previous framework (mean-variance), it could be done easily since
• there are only a few estimates of the variance
• there exists an analytical expression of the optimal allocation,
In the case of Value-at- Risk minimization,
• there are several estimators of quantiles (see Charpentier & Oulidi
(2007)),
• numerical optimization should be considered.
17

Quantile estimation
• raw estimator of the quantile, X[pn]:n = Xi:n = F−1
n (i/n) such that
i ≤ pn < i + 1.
• weighted average of F−1
n (p), e.g. αXi:n + (1 − α)Xi+1:n,
• weighted average of F−1
n (p), e.g.
n
i=1
αiXi:n =
1
0
αuF−1
n (u)du,
• smoothed version of the cdf, F−1
K (p) where FK(x) =
1
nh
n
i=1
K
x − Xi
h
• semiparametric approach, based on Hill’s estimator, Xn−k:n
n
k
(1 − p)
−ξk
,
where ξk =
1
k
k
i=1
log Xn+1−i:n − log Xn−k:n (if ξ > 0),
• fully parametric approach, Xn + u1−pvar(X) (if X ∼ N(µ, σ2
))
18

0.0 0.5 1.0 1.5 2.0
0.00.20.40.60.81.0 Empirical quantile estimation
Value
Probability
0.0 0.5 1.0 1.5 2.0
0.00.20.40.60.81.0
Empirical quantile estimation
Value
Probability
Figure 8: Classical estimation of the quantile, based on F−1
(·).
19

0.0 0.5 1.0 1.5 2.0
0.00.51.01.5
Smoothed empirical quantile estimation
Value
Smootheddensity
0.0 0.5 1.0 1.5 2.0
0.00.20.40.60.81.0
Smoothed empirical quantile estimation
ValueProbability
Figure 9: Smoothed estimation of the quantile, based on F−1
K (·).
20

A short extention to general risk measures
In a much more general setting, spectral risk measures can be considered, i.e.
R(X) =
1
0
φ(p)F−1
X (p)dp,
for some distortion function φ : [0, 1] → [0, 1].
21

Parametric bootstrap
Assume that X ∼ L(θ). The procedure is the following
1. generate n returns X1, . . . , Xn from L(θ);
2. estimate for all ω on a ﬁnite grid, estimate VaR(ωt
X),
3. solve the minimization problem on the grid to get numerically ω∗
n.
Using several simulations, the distribution of ω∗
n and var(ω∗
nX) can be
obtained.
22

−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
k’s, smoothed quantile
estimator.
23

0.16 0.18 0.20 0.22 0.24
05101520
Density of estimated optimal 99% quantile
Optimal Value−at−Risk
Density
Figure 11: Distribution of VaRk(ω∗
k
t
X, 95%).
24

−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.20.00.20.40.60.81.0
k’s, raw quantile estima-
tor.
25

0.16 0.18 0.20 0.22 0.24
010203040
Density of estimated optimal 95% quantile
Optimal Value−at−Risk
Density
Figure 13: Distribution of VaRk(ω∗
k
t
X, 95%).
26

Conclusion
Dealing with only 4 assets, it is diﬃcult to get robust optimal allocation, only
because of statistical uncertainty of classical estimators. Remark: this was
mentioned in Liu (2003) on high frequency data (every 5 minutes, i.e.
n = 10, 000) with 100 assets.
27

Some references
Coles, J.L. & Loewenstein, U. (1988). Equilibrium pricing and portfolio composition in the presence of uncertain
parameters. Journal of Financial Economics, 22, 279-303.
Dowd, K. & Blake, D.. (2006). After VaR: the theory, estimation, and insurance applications of quantile-based risk
measures. Journal of Risk & Insurance, 73, 193-229.
Duarte, A. (1999). Fast computation of eﬃcient portfolios. Journal of Risk, 1, 71-94.
Duffie, D. & Pan, J. (1997). An overview of Value at Risk. Journal of Derivatives, 4, 7-49.
Gaivoronski, A.A. & Pflug, G. (2000). Value-at-Risk in portfolio optimization: properties and computational
approach. Working Paper 00-2, Norwegian University of Sciences & Technology.
Jorion, P. (1997). Value at Risk: the new benchmark for controlling market risk. McGraw-Hill.
Kast, R., Luciano, E. & Peccati, L. (1998). VaR and optimization: 2nd international workshop on preferences and
decisions. Trento, July 1998.
Klein, R.W. & Bawa, V.S. (1976). The eﬀect of estimation risk on optimal portfolio choice. Journal of Financial
Economics, 3, 215-231.
Larsen, N., Mausser, H. & 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.
Litterman, R. (1997). Hot spots and edges II. Risk, 10, 38-42.
Rockafellar, R.T. & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2, 21-41.
28

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