overview optimization robust portfolio optimization
TRANSCRIPT
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 1 / 37
Robust Portfolio Optimization
November 2007
Prof. Dr. Heinz ZimmermannWirtschaftswissenschaftliches Zentrum (WWZ), Universitat [email protected]
Daniel NierdermayerWirtschaftswissenschaftliches Zentrum (WWZ), Universitat [email protected]
Overview
. Overview
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 2 / 37
Overview
Overview
. Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 3 / 37
1. Estimation Error
2. Robust Optimization
3. Reverse Optimization
1. Estimation Error
Overview
.1. EstimationError
a. Monte Carlo
b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 4 / 37
Estimation Error
Overview
1. Estimation Error
a. Monte Carlo
b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 5 / 37
Traditional Mean-Variance optimization (Markowitz) assumesknown expected returns µ and covariance matrix Σ.
In practice, µ and Σ must be estimated and thereforecontain estimation error.
This section presents a simulation that highlights the impactof estimation error on optimal asset allocation.
As the simulation uses multivariate normal returns, we firstdiscuss a return generating Monte Carlo method.
Estimation Error
Overview
1. Estimation Error
a. Monte Carlo
b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 6 / 37
a. Generating multivariate normal returns (Monte CarloSimulation)
b. Experiment
a. Monte Carlo - Generating multivariate normal returns
Overview
1. Estimation Error
. a. Monte Carlo
b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 7 / 37
Let x be an N × 1 vector with
x ∼ N (0, IN×N) ,
where IN×N is an N ×N identity matrix.
Then E(x) = 0 and E(xx′) = I.
Let y be defined asy = Ax , (1)
where A is an N ×N matrix.
From Eq. (1) follows E(y) = E(Ax) = AE(x) = 0 andΣ ≡ E(yy′) = E(Axx′A′) = AE(xx′)A′ = AA′
For a known positive definite N ×N matrix Σ an N ×N matrixA must be found such that Σ = AA′.
a. Monte Carlo - Cholesky Decomposition
Overview
1. Estimation Error
. a. Monte Carlo
b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 8 / 37
Use the Cholesky decomposition of Σ (in Matlabchol(Sigma)’)
A = chol(Σ) check that A is a lower triangular matrix (as Matlab returns
an upper triangular matrix, it has to be transposed) A has the property of Σ = AA′
The multivariate return series can be generated by
r = y + µ ,
where µ is a vector of constants (here expected returns).
The random vector r satisfies the properties E(r) = µ andCov(r) = E(rr′) − E(r)E(r′) = E(y + µ)(y + µ)′ − µµ′ =E(yy′) + 2µE(y′) + µµ′ − µµ′ = E(yy′) = Σ.
b. Experiment
Overview
1. Estimation Error
a. Monte Carlo
. b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
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The following experiment is based on DeMiguel/Nogales (2007).
Given are the (theoretical) moments of monthly returns
µ =
0.010.010.010.01
, Σ = 0.04 × IN×N .
The Mean-Variance optimal weights for any risk aversioncoefficient are 0.25 for each asset!1
1To verify this, set µ = a1 and Σ = bI in Eq. (3).
b. Experiment
Overview
1. Estimation Error
a. Monte Carlo
. b. Experiment
c. Results
2. RobustOptimization
3. ReverseOptimization
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Generate 140 returns ri ∼ N (µ,Σ), i = 1 . . . 140.
For different values of t calculate sample moments from
µ =1
120
120+t−1∑
i=t
ri , Σ =1
120 − 1
120+t−1∑
i=t
(ri − µ)(ri − µ)′
Rolling window estimation:
Rebalancing date t = 1: Calculate µ and Σ using r1 . . . r120
Rebalancing date t = 2: Calculate µ and Σ using r2 . . . r121
...
..until t = 20.
In the following, for each µ, Σ optimal weights are calculated.
c. Results
Overview
1. Estimation Error
a. Monte Carlo
b. Experiment
. c. Results
2. RobustOptimization
3. ReverseOptimization
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µ and Σ are unknown:
2 4 6 8 10 12 14 16 18 20−1
−0.5
0
0.5
1
1.5
2
wei
ghts
rebalancing dates
T=120
µ and Σ are estimated using 120 monthly returns ri (rolling window).The dashed line shows optimal portfolio weights without estimationerror (= 0.25) and γ = 1.
c. Results
Overview
1. Estimation Error
a. Monte Carlo
b. Experiment
. c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 12 / 37
µ is estimated and Σ is known:
2 4 6 8 10 12 14 16 18 20−1
−0.5
0
0.5
1
1.5
2
wei
ghts
rebalancing dates
T=120
c. Results
Overview
1. Estimation Error
a. Monte Carlo
b. Experiment
. c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 13 / 37
µ is known and Σ is estimated:
2 4 6 8 10 12 14 16 18 20−1
−0.5
0
0.5
1
1.5
2
wei
ghts
rebalancing dates
T=120
c. Results
Overview
1. Estimation Error
a. Monte Carlo
b. Experiment
. c. Results
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 14 / 37
Remarks:
Estimation error can cause strong deviation of estimatedportfolio weights from theoretically optimal weights.
– In the above example where µ and Σ are unknown themaximal estimated weight is 0.78 and the minimal weightis −0.09 while (theoretically) optimal weights are all 0.25.
Estimation error in expected returns has larger impact onasset allocation than estimation error in the covariancematrix.
Higher risk aversion leads to a portfolio allocation closer tothe global minimum variance portfolio (GMVP) (see nextsection). As the GMVP does not rely on expected returns,higher risk aversion reduces estimation error (besidesreducing portfolio risk)!
2. Robust Optimization
Overview
1. Estimation Error
.2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 15 / 37
Robust Optimization
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 16 / 37
The previous section has shown the effects of estimationerror in expected returns and the covariance matrix onoptimal asset allocation.
Optimal weights using sample estimates (esp. of expectedreturns) can strongly deviate from their theoretical values.
This section shows a robust portfolio optimization methodbased on Scherer (2007) (and Tutuncu and Konig (2004))that mitigates the problem of estimation error.
Traditional Optimization
Overview
1. Estimation Error
2. RobustOptimization
.TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 17 / 37
In traditional mean variance optimization of a portfolio with Nassets, expected utility is maximized using the followingLagrangian function
L = µ′w − 1
2γw′
Σw − λ(w′1 − 1) (2)
where
µ: is an N × 1 vector of expected returns,Σ: is the N ×N covariance matrix of assets’ returns,w: is an N × 1 vector of portfolio weights,γ: is the parameter of relative risk aversion,λ: is a Lagrange multiplier.
Note that (w′1 − 1) is the constraint requiring that the optimal
weight vector’s elements sum up to 1.
Traditional Optimization
Overview
1. Estimation Error
2. RobustOptimization
.TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 18 / 37
The weight vector that maximizes Eq. (2) can be found by
∂L
∂w= µ − γΣw − λ1 = 0
∂L
∂λ= w′
1 − 1 = 0
The solution is given by
w∗
mv =1
γΣ
−1
(
µ − 1′Σ
−1µ
1′Σ
−111
)
︸ ︷︷ ︸
w∗spec
+Σ
−11
1′Σ
−11
︸ ︷︷ ︸
w∗
min
(3)
where w∗
min is the portfolio weight of the global minimumvariance portfolio and w∗
spec is a speculative demand.
Robust Optimization, Scherer(2007)
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
.RobustOptimization
a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 19 / 37
Overview: Scherer(2007) (based on Tutuncu/Konig (2004))
Estimation error in sample means µ has larger effect onportfolio allocation than estimation error in the covariancematrix.
Instead of plugging the mean vector into the Mean-Varianceoptimizer, we will use a ‘worst case’ vector µ∗.
The worst case expected portfolio return w′µ∗ is lower than(or equal to) w′µ; this makes it generally less attractive toinvest into the speculative weight w∗
spec which depends on µ
(or µ∗).
The robust optimization method presented here leads tohigher investment into the global minimum variance portfoliow∗
min than in traditional Mean-Variance optimization.
Robust Optimization – a. Worst Case Expected Returns
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
. a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 20 / 37
The vector µ is drawn from a multivariate normal distribution
µ ∼ N (µ,Ω) ,
where Ω is an N ×N positive definite covariance matrix.
We consider a realization of µ as an extreme event if
(µ − µ)′Ω−1(µ − µ) ≥ κ21−α,N .
Note that (µ − µ)′Ω−1(µ − µ) is χ2N distributed (chi-square
distributed with N degrees of freedom – N is the number ofassets).
Robust Optimization – a. Worst Case Expected Returns
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
. a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 21 / 37
0 5 10 15 200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
chi s
quar
e
κ1−α,N2
N = 4α = 5%
5%
where κ21−α,N = Inv-χ2
N (1 − α) and Inv-χ2N is the inverse
chi-square cumulative density function with N degrees offreedom.Example: κ2
0.95,4 = 9.4877. (use chi2inv(0.95,4) in Matlab).
Robust Optimization – a. Worst Case Expected Returns
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
. a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 22 / 37
Now consider the following set of expected return vectors
(µ − µ)′Ω−1(µ − µ) = κ21−α,N . (4)
Geometrically, Eq. (4) describes an N -dimensional ellipsoid. Wewill restrict our analysis to expected return vectors µ that lie onthis ellipsoid.
On this ellipsoid the value of µ with the lowest expected portfolioreturn is obtained by maximizing the Lagrangian function
L(µ, λ) = w′µ − w′µ − λ
2
((µ − µ)′Ω−1(µ − µ) − κ2
1−α,N
)
where w is any portfolio weight vector.
Robust Optimization – a. Worst Case Expected Returns
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
. a. Worst Case µ
b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 23 / 37
Setting ∂L/∂µ = 0 and ∂L/∂λ = 0 and solving this system oflinear equations with respect to µ yields
µ∗ = µ − κ1−α,N√w′
ΩwΩw
andw′µ∗ = w′µ − κ1−α,N
√
w′Ωw . (5)
From Eq. (5) it is obvious that w′µ∗ ≤ w′µ.
The value w′µ∗ is particularly low for high values of κ1−α,N .
For Ω = 1
TΣ Eq. (5) can be rewritten as
w′µ∗ = w′µ − κ1−α,NT−1/2
√
w′Σw , (6)
where Σ is the covariance matrix of assets’ returns.
Robust Optimization – b. Deriving Optimal Weights
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
. b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 24 / 37
The traditional Mean-Variance objective is to maximize theLagrangian
L = µ′w − 1
2γw′
Σw − λ(w′1 − 1) .
Lrob = µ′
∗w − 1
2γw′
Σw − λ(w′1 − 1)
= µ′w − κ1−α,NT−1/2
√
w′Σw − 1
2γw′
Σw − λ(w′1 − 1) .
Robust Optimization – b. Deriving Optimal Weights
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
. b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 25 / 37
Solving ∂Lrob/∂w = 0 and ∂Lrob/∂λ = 0 gives
w∗
rob =
(
1 − T−1/2κ1−α,N
γσ∗p + T−1/2κ1−α,N
)
1
γΣ
−1
(
µ − 1′Σ
−1µ
1′Σ
−111
)
+Σ
−11
1′Σ
−11
w∗
rob =
(
1 − T−1/2κ1−α,N
γσ∗p + T−1/2κ1−α,N
)
w∗
spec + w∗
min (7)
Note that σ∗p =√
w∗′
robΣw∗
rob which makes Eq. (7) not directly
solvable. However, Eq. (7) can be solved numerically.
Robust Optimization – b. Deriving Optimal Weights
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
. b. Derivation
c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 26 / 37
Moreover, note that
If κ1−α,N → 0 or T → ∞,
w∗
rob =1
γΣ
−1
(
µ − 1′Σ
−1µ
1′Σ
−111
)
+Σ
−11
1′Σ
−11
= w∗
mv
If κ1−α,N → ∞ or T → 0,
w∗
rob =Σ
−11
1′Σ
−11
= w∗
min
Robust Optimization – c. Example
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
. c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 27 / 37
Impact of estimation error in traditional M-V Optimization
2 4 6 8 10 12 14 16 18 20−1
−0.5
0
0.5
1
1.5
2
wei
ghts
rebalancing dates
T=120
(γ = 1)
Robust Optimization – c. Example
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
. c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 28 / 37
Impact of estimation error in robust M-V Optimization
2 4 6 8 10 12 14 16 18 20−1
−0.5
0
0.5
1
1.5
2
wei
ghts
rebalancing dates
T=120
(γ = 1)
Robust Optimization – c. Example
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
. c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 29 / 37
Investment into w∗
spec increases with higher sample size T .(Note again that σ∗
p is not constant in Eq. (7) - the curve below wassimulated numerically.)
0 200 400 600 800 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
T
ψ
Shrinkage Intensity
ψ =
„
1 −T−1/2κ1−α,N
γσ∗
p + T−1/2κ1−α,N
«
≤ 1 (8)
Robust Optimization – c. Example
Overview
1. Estimation Error
2. RobustOptimization
TraditionalOptimization
Robust Optimization
a. Worst Case µ
b. Derivation
. c. Example
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 30 / 37
Remarks:
as w′µ∗ ≤ w′µ, robust optimization using µ∗ leads to ‘lessattractive’ expected returns which reduces speculativedemand w∗
spec. This is shown by Eq. (7), where ψ ≤ 1(compare also Eq. (8)).
Thus, the fraction of wealth invested into the globalminimum variance portfolio w∗
min increases with robustoptimization.
As w∗
min does not depend on µ, the resulting portfolioallocation is more robust than in traditional Mean-Varianceoptimization.
This type of robust optimization is equivalent to traditionaloptimization where the risk aversion coefficient is increasedto 1
ψγ. Note that ψ depends on the ‘uncertainty aversion’κ1−α,N and T .
3. Reverse Optimization
Overview
1. Estimation Error
2. RobustOptimization
.3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 31 / 37
Reverse Optimization
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 32 / 37
Reverse optimization (applied in e.g. Black and Litterman(1992)) avoids the problem arising from estimation error inexpected returns.
Traditional Mean-Variance optimization uses expectedreturns (and the covariance matrix) as input and providesportfolio weights as output.
In contrast, reverse optimization uses portfolio weights (andthe covariance matrix) as input and provides expectedreturns as output.
The rationale behind using portfolio weights as input is thatin some cases weights are easier to estimate than expectedreturns.
Reverse Optimization
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 33 / 37
As in Black/Litterman (1992) we confine our analysis toexpected excess returns µe (expected returns minus the riskfreerate).Then in Eq. (2) the constraint λ(w′
1 − 1) can be dropped asweights do not have to sum up to 1 (because implicitly,remaining wealth is invested into the riskfree asset)
L = µ′
ew − 1
2γw′
Σw
This simplifies the solution of the optimal weight. Setting∂L/∂w = 0 gives
µe − γΣw = 0 .
Rearranging yields
w∗ =1
γΣ
−1µe . (9)
Reverse Optimization
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 34 / 37
From Eq. (9) follows the main equation for reverse optimization:
µ∗
e= γΣw∗ .
In e.g. the Black/Litterman model value weighted marketportfolio weights (or strategic asset allocation weights) are usedfor w∗.
Reverse Optimization
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 35 / 37
Remarks:
By definition, plugging µ∗
einto Eq. (9) yields w∗.
However, when changing the value of the risk aversioncoefficient γ, optimal weights obtained in Eq. (9) differ fromthe original reference weights w∗.
A ‘sensible’ value should be chosen for the reference weightsw∗. This improves the economic intuition of the results.
Reverse Optimization
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 36 / 37
Advantages:
The advantage of reverse optimization is that it does not relyon the estimation of expected returns. This leads to portfolioallocations more robust than in traditional Mean-Varianceoptimization.
Resulting weights are often economically easier to interpretthan weights from traditional Mean-Variance optimization(for reasonable reference weights w∗).
Disadvantage:
Reverse optimization may throw away useful informationcontained in (estimated) expected returns.
Literature
Overview
1. Estimation Error
2. RobustOptimization
3. ReverseOptimization
D. Niedermayer / H. Zimmermann : Robust Portfolio Optimization 37 / 37
Black, F., and R. Litterman, 1992 (September-October),Global Portfolio Optimization, Financial Analysts Journal, p.28-43.
DeMiguel, V. and F. J. Nogales (2007), Portfolio Selectionwith Robust Estimation, Working Paper Series, SSRN eLibrary,http://ssrn.com/paper=911596
Scherer, B. (2007): Portfolio Construction and RiskBudgeting, Risk Books, 3rd edition.
Tutuncu, R. H. and M. Konig (2004), Robust AssetAllocation, Annals of Operations Research, 132(1), p. 157-187.