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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

1


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

2


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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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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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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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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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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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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Efficient frontier

first asset

−0.2 0.0 0.2 0.4 −0.4 0.0 0.4

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.20

.00

.2

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.4

second asset

third asset

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.20

.20

.6

−0.2 0.0 0.2

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.40

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.4

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fourth asset

Portfolio with 4assets

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00

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01

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Efficient Frontier

Standard deviation

Exp

ect

ed

va

lue

Figure 3: Solving a variance optimization problem.

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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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0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

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.25

Efficient Frontier, with 250 past observations

Standard deviation

Exp

ecte

d v

alu

e

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0.0

00

.05

0.1

00

.15

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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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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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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.

13


0.4 0.6 0.8 1.0

01

23

45

Allocation in the first asset

Allocation weight

Dens

ity

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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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−0.2 0.0 0.2 0.4 0.6 0.8 1.0

−0.2

0.0

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1.0


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catio

n in

sec

ond

asse

t

Joint distribution of optimal allocations (1−2)

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et


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sset


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et


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Figure 6: Joint distributions of optimal allocations ω∗k’s.

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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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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.

17


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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0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

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1.0

Empirical quantile estimation

Value

Pro

ba

bili

ty

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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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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

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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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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].

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(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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−0.2 0.0 0.2 0.4 0.6 0.8 1.0

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Figure 10: Joint distributions of optimal allocations ω∗k’s, smoothed quantileestimator.

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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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Figure 12: Joint distributions of optimal allocations ω∗k’s, raw quantile estima-tor.

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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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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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Some referencesColes, J.L. & Loewenstein, U. (1988). Equilibrium pricing and portfolio composition in the presence of uncertainparameters. Journal of Financial Economics, 22, 279-303.

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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