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
Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Quantile estimationand optimal portfolios
Arthur Charpentier & Abder OulidiENSAI-ENSAE-CREST & IMA Angers
Journée SFdS, Mai 2007
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Portfolio management and optimal allocations
Idea: allocating capital among a set of assets to maximize return and minimizerisk.
If diversification effects were intuited early, and Markowitz (1952) proposed amathematical 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
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Portfolio optimization (parametric framework)
Consider a risk measure R (variance or Value-at-Risk). Solve
ω∗ =
argmin{R(ωtX)},u.c. ωt1 = 1 and E(ωtX) ≥ 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 areusually not known, however; and are often estimated or guessed at usingobserved returns and other available data. In empirical applications, theestimated parameters are used as if they were the true value” (Coles& Loewenstein (1988)).
If ω∗ = ψ(θ) (e.g. mean-variance)
ω∗ = ψ(θ).
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Optimization of standard deviation (or variance)
Allocation in the first assetAllocation in the second asset
Standard deviation of the portfolio
−200 −100 0 100 200
−2
00
−1
00
01
00
20
0Allocation in the first asset
Allo
catio
n in
th
e s
eco
nd
ass
et
Figure 1: Portfolio variance optimization problem.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Portfolio optimization (parametric framework)
In the case of no explicit expression of the optimum, solve (numerically)
ω∗ =
argminR(ωtX),
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).
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Optimization of Value-at-Risk
VaR of the portfolio
−4 −2 0 2 4 6
−2
−1
0 1
2 3
4 5
−4−2
0 2
4 6
VaR of the portfolio
−4 −2 0 2 4 6
01
23
45
−4−2
0 2
4 6
Figure 2: Optimization in the mean-VaR framework.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Classical mean-variance allocation problem
Consider d risky assets, with weekly returns X = (X1, . . . , Xd). Denoteµ = 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(ωtX) = ωtµ,
• the variance of the portfolio is var(ωtX) = ωtΣω.
ω∗ ∈ argmin{ωtΣω}u.c. ωtµ ≥ η and ωt1 = 1
convex⇐⇒
ω∗ ∈ argmax{ωtµ}u.c. ωtΣω ≤ η′ and ωt1 = 1
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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 =1d
(bΣ−11− aΣ−1µ
)and q =
1d
(cΣ−1µ− aΣ−11
),
and a = 1tΣ−1µ, b = µtΣ−1µ, c = 1tΣ−11, d = bc− a2.
Note that p is an allocation, and q indicates how the original portfolio shouldbe modified.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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
0.0
00
0.0
01
0.0
02
0.0
03
0.0
04
0.0
05
0.0
06
Efficient Frontier
Standard deviation
Exp
ect
ed
va
lue
Figure 3: Solving a variance optimization problem.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Inference issues
In practice, µ = [µi] and Σ = [Σi,j ] are unknown, and should be estimated.
A natural idea is to define
µi =1n
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,Σn
)and nΣ|Σ ∼ W (n− 1,Σ) .
where the two random variables µ and Σ are independent, given Σ.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.05 0.10 0.15 0.20 0.25
0.0
00
.05
0.1
00
.15
0.2
00
.25
Efficient Frontier, with 250 past observations
Standard deviation
Exp
ecte
d v
alu
e
0.05 0.10 0.15 0.20 0.25
0.0
00
.05
0.1
00
.15
0.2
00
.25
Efficient Frontier, with 1000 past observations
Standard deviationE
xp
ecte
d v
alu
e
Figure 4: Efficient frontiers and estimation.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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∗ =1d
(bΣ
−11− aΣ
−1µ
)+ η
1d
(cΣ
−1µ− aΣ
−11)
,
Using several simulations, the distribution of the ω∗k and vark(ω∗ktX) can be
obtained.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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∗ =1d
(bΣ
−11− aΣ
−1µ
)+ η
1d
(cΣ
−1µ− aΣ
−11)
,
Using several simulations, the distribution of the ω∗k and vark(ω∗ktX) can be
obtained.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.4 0.6 0.8 1.0
01
23
45
Allocation in the first asset
Allocation weight
Dens
ity
0.0 0.2 0.4
01
23
45
Allocation in the second asset
Allocation weight
Dens
ity−0.3 −0.1 0.0 0.1
02
46
8
Allocation in the third asset
Allocation weight
Dens
ity
0.05 0.15 0.25
02
46
810
Allocation in the fourth asset
Allocation weightDe
nsity
Figure 5: Distributions of optimal allocations ω∗k’s.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
sec
ond
asse
t
Joint distribution of optimal allocations (1−2)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
third
ass
et
Joint distribution of optimal allocations (1−3)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (1−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the second asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (2−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the second asset
Allo
catio
n in
third
ass
et
Joint distribution of optimal allocations (2−3)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the third asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (3−4)
Figure 6: Joint distributions of optimal allocations ω∗k’s.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.05 0.06 0.07 0.08 0.09 0.10 0.11
020
4060
8010
0
Density of estimated optimal standard deviation
Optimal standard deviation
Dens
ity
Figure 7: Distribution of vark(ω∗ktX)
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Value-at-Risk minimization
With V aR(X, p) = F−1(p) = sup{x, F (x) < p}, the program is
ω∗ ∈ argmin{VaR(ωtX, α)}u.c. E(ωtX) ≥ η,ωt1 = 1
nonconvex<
ω∗ ∈ argmax{E(ωtX)}u.c. {VaR(ωtX, α)} ≤ η′,ωt1 = 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.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Quantile estimation
• raw estimator of the quantile, X[pn]:n = Xi:n = F−1n (i/n) such that
i ≤ pn < i + 1.
• weighted average of F−1n (p), e.g. αXi:n + (1− α)Xi+1:n,
• weighted average of F−1n (p), e.g.
n∑
i=1
αiXi:n =∫ 1
0
αuF−1n (u)du,
• smoothed version of the cdf, F−1K (p) where FK(x) =
1nh
n∑
i=1
K
(x−Xi
h
)
• semiparametric approach, based on Hill’s estimator, Xn−k:n
(n
k(1− p)
)−ξk
,
where ξk =1k
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))
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Empirical quantile estimation
Value
Pro
ba
bili
ty
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Empirical quantile estimation
ValueP
rob
ab
ility
Figure 8: Classical estimation of the quantile, based on F−1(·).
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
Smoothed empirical quantile estimation
Value
Sm
oo
the
d d
en
sity
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
Smoothed empirical quantile estimation
ValueP
rob
ab
ility
Figure 9: Smoothed estimation of the quantile, based on F−1K (·).
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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−1X (p)dp,
for some distortion function φ : [0, 1] → [0, 1].
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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(ωtX),
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 beobtained.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
sec
ond
asse
t
Joint distribution of optimal allocations (1−2)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
third
ass
et
Joint distribution of optimal allocations (1−3)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (1−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the second asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (2−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the third asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (3−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the second asset
Allo
catio
n in
third
ass
et
Joint distribution of optimal allocations (2−3)
Figure 10: Joint distributions of optimal allocations ω∗k’s, smoothed quantileestimator.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.16 0.18 0.20 0.22 0.24
05
1015
20
Density of estimated optimal 99% quantile
Optimal Value−at−Risk
Dens
ity
Figure 11: Distribution of VaRk(ω∗ktX, 95%).
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
sec
ond
asse
t
Joint distribution of optimal allocations (1−2)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
third
ass
et
Joint distribution of optimal allocations (1−3)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the first asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (1−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the second asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (2−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the third asset
Allo
catio
n in
four
th a
sset
Joint distribution of optimal allocations (3−4)
−0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Allocation in the second asset
Allo
catio
n in
third
ass
et
Joint distribution of optimal allocations (2−3)
Figure 12: Joint distributions of optimal allocations ω∗k’s, raw quantile estima-tor.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.16 0.18 0.20 0.22 0.24
010
2030
40
Density of estimated optimal 95% quantile
Optimal Value−at−Risk
Dens
ity
Figure 13: Distribution of VaRk(ω∗ktX, 95%).
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Conclusion
Dealing with only 4 assets, it is difficult to get robust optimal allocation, onlybecause of statistical uncertainty of classical estimators. Remark: this wasmentioned in Liu (2003) on high frequency data (every 5 minutes, i.e.n = 10, 000) with 100 assets.
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Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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Dowd, K. & Blake, D.. (2006). After VaR: the theory, estimation, and insurance applications of quantile-based riskmeasures. 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 computationalapproach. 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 anddecisions. Trento, July 1998.
Klein, R.W. & Bawa, V.S. (1976). The effect of estimation risk on optimal portfolio choice. Journal of FinancialEconomics, 3, 215-231.
Larsen, N., Mausser, H. & Uryasev, S. (2002). Algorithms for optimization of Value at Risk. in Financialengineering, 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.
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