2. 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 minimize
risk.
If diversiļ¬cation eļ¬ects were intuited early, and Markowitz (1952) proposed a
mathematical model.
ā¢ return is measured by the expected value of the portfolio return,
ā¢ risk is quantiļ¬ed by the variance of this return.
Agenda
1. statistical issue in the mean-variance framework
2. portfolio optimization with general risk measures
2
3. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Portfolio optimization (parametric framework)
Consider a risk measure R (variance or Value-at-Risk). Solve
Ļā
=
ļ£±
ļ£²
ļ£³
argmin{R(Ļt
X)},
u.c. Ļt
1 = 1 and E(Ļt
X) ā„ Ī·,
where X ā¼ L(Īø), Īø unknown.
Īø is unknown but can be estimated using a sample {X1, . . . , Xn}.
āThe 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 valueā (Coles
& Loewenstein (1988)).
If Ļā
= Ļ(Īø) (e.g. mean-variance)
Ļā
= Ļ(Īø).
3
4. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Optimization of standard deviation (or variance)
Allocation
in
the
first asset
Allocation in the second asset
Standard deviation of the portfolio
ā200 ā100 0 100 200
ā200ā1000100200 Allocation in the first asset
Allocationinthesecondasset
Figure 1: Portfolio variance optimization problem.
4
5. 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)
Ļā
=
ļ£±
ļ£²
ļ£³
argminR(Ļt
X),
u.c.Ļ ā {(Ļk)kā{1,...,m}}
where X ā¼ L(Īø).
The idea is to generate samples Xiās,
ā¢ either from a parametric distribution L(Īø),
ā¢ or from a nonparametric distribution (bootstrap approach).
5
6. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Optimization of Value-at-Risk
VaR of the portfolio
ā4 ā2 0 2 4 6
ā2ā1012345
ā4
ā2
0
2
4
6
VaR of the portfolio
ā4 ā2 0 2 4 6
012345
ā4
ā2
0
2
4
6
Figure 2: Optimization in the mean-VaR framework.
6
7. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Classical mean-variance allocation problem
Consider d risky assets, with weekly returns X = (X1, . . . , Xd). Denote
Āµ = E(X) and Ī£ = var(X).
Let Ļ = (Ļ1, . . . , Ļd) ā Rd
denote the weights in all risky assets.
ā¢ the expected return of the portfolio is E(Ļt
X) = Ļt
Āµ,
ā¢ the variance of the portfolio is var(Ļt
X) = Ļt
Ī£Ļ.
ļ£±
ļ£²
ļ£³
Ļā
ā argmin{Ļt
Ī£Ļ}
u.c. Ļt
Āµ ā„ Ī· and Ļt
1 = 1
convex
āā
ļ£±
ļ£²
ļ£³
Ļā
ā argmax{Ļt
Āµ}
u.c. Ļt
Ī£Ļ ā¤ Ī· and Ļt
1 = 1
7
8. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Classical mean-variance allocation problem
The solution can be given explicitly (see Markowitz (1952)) as
Ļā
= Ļ(Āµ, Ī£) = p + Ī·q
where Āµ = E(X), Ī£ = var(X),
p =
1
d
bĪ£ā1
1 ā aĪ£ā1
Āµ and q =
1
d
cĪ£ā1
Āµ ā aĪ£ā1
1 ,
and a = 1t
Ī£ā1
Āµ, b = Āµt
Ī£ā1
Āµ, c = 1t
Ī£ā1
1, d = bc ā a2
.
Note that p is an allocation, and q indicates how the original portfolio should
be modiļ¬ed.
8
9. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Eļ¬cient frontier
first asset
ā0.2 0.0 0.2 0.4 ā0.4 0.0 0.4
ā0.20.00.2
ā0.20.00.20.4
second asset
third asset
ā0.20.20.6
ā0.2 0.0 0.2
ā0.40.00.4
ā0.2 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
10. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Inference issues
In practice, Āµ = [Āµi] and Ī£ = [Ī£i,j] are unknown, and should be estimated.
A natural idea is to deļ¬ne
Āµi =
1
n
n
t=1
Xi,t et Ī£i,j =
1
n ā 1
n
t=1
(Xi,t ā Āµi)(Xj,t ā Āµj).
Given n observed observed returns,
Āµ|Ī£ ā¼ N Āµ,
Ī£
n
and nĪ£|Ī£ ā¼ W (n ā 1, Ī£) .
where the two random variables Āµ and Ī£ are independent, given Ī£.
10
11. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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: Eļ¬cient frontiers and estimation.
11
12. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Parametric bootstrap
Assume that X ā¼ L(Īø). estimate Īø by Īøn. The procedure is the following
1. generate n returns X1, . . . , Xn from L(Īøn);
2. estimate Āµ and Ī£, i.e. Āµn and Ī£n,
3. solve the minimization problem, i.e.
Ļā
=
1
d
bĪ£
ā1
1 ā aĪ£
ā1
Āµ + Ī·
1
d
cĪ£
ā1
Āµ ā aĪ£
ā1
1 ,
Using several simulations, the distribution of the Ļā
k and vark(Ļā
k
t
X) can be
obtained.
12
13. 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 Āµ and Ī£, i.e. Āµn and Ī£n,
3. solve the minimization problem, i.e.
Ļā
=
1
d
bĪ£
ā1
1 ā aĪ£
ā1
Āµ + Ī·
1
d
cĪ£
ā1
Āµ ā aĪ£
ā1
1 ,
Using several simulations, the distribution of the Ļā
k and vark(Ļā
k
t
X) can be
obtained.
13
14. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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 in the second asset
Allocation weight
Density
ā0.3 ā0.1 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 Ļā
kās.
14
15. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinsecondasset
Joint distribution of optimal allocations (1ā2)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinthirdasset
Joint distribution of optimal allocations (1ā3)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinfourthasset
Joint distribution of optimal allocations (1ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the second asset
Allocationinfourthasset
Joint distribution of optimal allocations (2ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the second asset
Allocationinthirdasset
Joint distribution of optimal allocations (2ā3)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the third asset
Allocationinfourthasset
Joint distribution of optimal allocations (3ā4)
Figure 6: Joint distributions of optimal allocations Ļā
kās.
15
16. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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(Ļā
k
t
X)
16
17. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Value-at-Risk minimization
With V aR(X, p) = Fā1
(p) = sup{x, F(x) < p}, the program is
ļ£±
ļ£²
ļ£³
Ļā
ā argmin{VaR(Ļt
X, Ī±)}
u.c. E(Ļt
X) ā„ Ī·,Ļt
1 = 1
nonconvex
ļ£±
ļ£²
ļ£³
Ļā
ā argmax{E(Ļt
X)}
u.c. {VaR(Ļt
X, Ī±)} ā¤ Ī· ,Ļt
1 = 1
In the previous framework (mean-variance), it could be done easily since
ā¢ there are only a few estimates of the variance
ā¢ there exists an analytical expression of the optimal allocation,
In the case of Value-at- Risk minimization,
ā¢ there are several estimators of quantiles (see Charpentier & Oulidi
(2007)),
ā¢ numerical optimization should be considered.
17
18. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Quantile estimation
ā¢ raw estimator of the quantile, X[pn]:n = Xi:n = Fā1
n (i/n) such that
i ā¤ pn < i + 1.
ā¢ weighted average of Fā1
n (p), e.g. Ī±Xi:n + (1 ā Ī±)Xi+1:n,
ā¢ weighted average of Fā1
n (p), e.g.
n
i=1
Ī±iXi:n =
1
0
Ī±uFā1
n (u)du,
ā¢ smoothed version of the cdf, Fā1
K (p) where FK(x) =
1
nh
n
i=1
K
x ā Xi
h
ā¢ semiparametric approach, based on Hillās estimator, Xnāk:n
n
k
(1 ā p)
āĪ¾k
,
where Ī¾k =
1
k
k
i=1
log Xn+1āi:n ā log Xnāk:n (if Ī¾ > 0),
ā¢ fully parametric approach, Xn + u1āpvar(X) (if X ā¼ N(Āµ, Ļ2
))
18
19. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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 Fā1
(Ā·).
19
20. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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 Fā1
K (Ā·).
20
21. 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
Ļ(p)Fā1
X (p)dp,
for some distortion function Ļ : [0, 1] ā [0, 1].
21
22. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Parametric bootstrap
Assume that X ā¼ L(Īø). The procedure is the following
1. generate n returns X1, . . . , Xn from L(Īø);
2. estimate for all Ļ on a ļ¬nite grid, estimate VaR(Ļt
X),
3. solve the minimization problem on the grid to get numerically Ļā
n.
Using several simulations, the distribution of Ļā
n and var(Ļā
nX) can be
obtained.
22
23. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinsecondasset
Joint distribution of optimal allocations (1ā2)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinthirdasset
Joint distribution of optimal allocations (1ā3)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinfourthasset
Joint distribution of optimal allocations (1ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the second asset
Allocationinfourthasset
Joint distribution of optimal allocations (2ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the third asset
Allocationinfourthasset
Joint distribution of optimal allocations (3ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the second asset
Allocationinthirdasset
Joint distribution of optimal allocations (2ā3)
Figure 10: Joint distributions of optimal allocations Ļā
kās, smoothed quantile
estimator.
23
24. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.16 0.18 0.20 0.22 0.24
05101520
Density of estimated optimal 99% quantile
Optimal ValueāatāRisk
Density
Figure 11: Distribution of VaRk(Ļā
k
t
X, 95%).
24
25. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinsecondasset
Joint distribution of optimal allocations (1ā2)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinthirdasset
Joint distribution of optimal allocations (1ā3)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the first asset
Allocationinfourthasset
Joint distribution of optimal allocations (1ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the second asset
Allocationinfourthasset
Joint distribution of optimal allocations (2ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the third asset
Allocationinfourthasset
Joint distribution of optimal allocations (3ā4)
ā0.2 0.0 0.2 0.4 0.6 0.8 1.0
ā0.20.00.20.40.60.81.0
Allocation in the second asset
Allocationinthirdasset
Joint distribution of optimal allocations (2ā3)
Figure 12: Joint distributions of optimal allocations Ļā
kās, raw quantile estima-
tor.
25
26. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
0.16 0.18 0.20 0.22 0.24
010203040
Density of estimated optimal 95% quantile
Optimal ValueāatāRisk
Density
Figure 13: Distribution of VaRk(Ļā
k
t
X, 95%).
26
27. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
Conclusion
Dealing with only 4 assets, it is diļ¬cult 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
28. Arthur CHARPENTIER & Abder OULIDI - Quantile estimation and optimal portfolios
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 eļ¬cient 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 eļ¬ect 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.
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