Financial Engineering: Portfolio
Optimization
Daniel P. Palomar
Hong Kong University of Science and Technology (HKUST)
ELEC5470 - Convex Optimization
Fall 2015-16, HKUST, Hong Kong
Outline of Lecture
• Data model
• Return, risk measures, and Sharpe ratio
• Basic mean-variance portfolio formulation
• Max Sharpe ratio portfolio formulation
• CVaR portfolio formulation
• Robust portfolio design
• Index replication with `1-norm minimization
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Data Model
• Consider M financial assets, e.g., stocks of a major index such as
S&P 500 and Hang Seng Index.
• Denote the absolute prices at time t by pm,t, m = 1, . . . , M .
• The simple (linear) and continuously-compounded (logarithmic) returns are
rm,t = (pm,t − pm,t−1) /pm,t−1
ym,t = log (pm,t) − log (pm,t−1) .
• Observe that, for small values of the return, we have ym,t ≈ rm,t.
• We will mainly use linear returns, stacked as rt ∈ RM , and assume
that they are i.i.d. random variables with mean µ ∈ RM and
covariance matrix Σ ∈ RM ×M .
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Return
• Suppose we invest 1$ in the mth asset at time t − 1, then at time t
we will have an amount of 1$× (pm,t/pm,t−1).
• The relative benefit or return will be (pm,t/pm,t−1 − 1) which is
precisely equal to rm,t.
• If now we invest 1$ distributed over the M assets according to the
portfolio weights w (with 1T w = 1), then the overall return will be
w T rt .
• The key quantity is then the random return wT rt.
• The expected or mean return is wT µ and the variance is wT Σw.
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Risk Control
• In finance, the mean return is very relevant as it quantifies the
average benefit.
• However, in practice, the average performance is not good enough
and one needs to control the probability of going bankrupt.
• Risk measures control how risky an investment strategy is.
• The most basic measure of risk is given by the variance: a higher
variance means that there are large peaks in the distribution which
may cause a big loss.
• There are more sophisticated risk measures such as Value-at-Risk
(VaR) and Conditional VaR (CVaR).
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Mean-Variance Tradeoff
• The mean return wT µ and the variance (risk) wT Σw constitute
two important performance measures.
• Usually, the higher the mean return the higher the variance and
vice-versa.
• Thus, we are faced with two objectives to be optimized. This is a
multi-objective optimization problem.
• They define a fundamental mean-variance tradeoff curve (Pareto
curve). The choice of a specific point in this tradeoff curve depends
on how agressive or risk-averse the investor is.
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Sharpe Ratio
• The Sharpe ratio is akin to the SINR in communication systems.
• It is defined as the ratio of the mean return (excess w.r.t. the return
of the risk-free asset rf ) to the risk measured as the square root of
the variance (standard deviation):
wT µ − rf
S = √
.
T
w Σw
• The Sharpe ratio can be understood as the expected return per unit
of risk.
• The portfolio w that maximizes the Sharpe ratio lies on the Pareto
curve.
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Mean-Variance Optimization (Markowitz, 1956)
• There are two obvious formulations for the portfolio optimization.
• Maximization of mean return:
maximize
wT µ
subject to
wT Σw ≤ α
1T w = 1.
w
• Minimization of risk:
minimize
wT Σw
subject to
wT µ ≥ β
1T w = 1.
w
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• Another obvious formulation is the scalarization of the multiobjective optimization problem:
maximize
wT µ − γwT Σw
subject to
1T w = 1.
w
• These three formulations give different points on the Pareto optimal
curve.
• They all require knowledge of one parameter (α, β, and γ).
• If we consider the Sharpe ratio as a good measure of performance,
we could consider instead maximizing it directly.
• Additional constraints can be included such as w ≥ 0 to allow for
long-only transactions (no short-selling).
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Max Sharpe Ratio Portfolio Formulation
• Let’s start by formulating the maximization of the Sharpe ratio as
follows:
√
T
T
minimize
w Σw/ w µ − rf
w
1T w = 1
w≥0
subject to
• This is a quasi-convex problem as can be seen in the epigraph form:
minimize
w,t
subject to
t
T
t w µ − rf
1T w = 1
w≥0
1/2 ≥ Σ w
which can be solve by bisection over t.
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Sharpe Ratio Formulation in Convex Form
• Theorem: The maximization of the Sharpe ratio can be rewritten
in convex form as the QP
minimize
w̃T Σw̃
subject to
(µ − rf 1) w̃ = 1
1T w̃ ≥ 0
w̃ ≥ 0.
w̃
T
Proof:
T
• Defining √
w̃ = tw with t = 1/ w µ − rf > 0, the objective
becomes w̃T Σw̃, the sum constraint becomes 1T w̃ = t, and the
problem becomes
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minimize
w,w̃,t
subject to
√
w̃T Σw̃
T
t = 1/ w µ − rf > 0
w̃ = tw
1T w̃ = t > 0
w̃ ≥ 0.
• Observe that the first constraint 1/t = wT µ − rf can be rewritten
T
in terms of w̃ as 1 = (µ − rf 1) w̃. So the problem becomes
√
minimize
w̃T Σw̃
w,w̃,t
subject to
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T
(µ − rf 1) w̃ = 1
w̃ = tw
1T w̃ = t > 0
w̃ ≥ 0.
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• Note that the strict inequality t > 0 is equivalent to t ≥ 0 because
t = 0 can never happen as w̃ would be zero and the first constraint
would not be satisfied.
• We can now get rid of w and t in the formulation as they can be
directly obtained as w = w̃/t and t = 1T w̃:
minimize
w̃T Σw̃
subject to
(µ − rf 1) w̃ = 1
1T w̃ ≥ 0
w̃ ≥ 0.
w̃
T
• QED!
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Value-at-Risk (VaR & CVaR) Models
• Mean-variance model penalizes up-side and down-side risk equally,
whereas most investors don’t mind up-side risk.
• Solution: use alternative risk measures (not variance).
• VaR denotes the maximum loss with a specified confidence level
(e.g., confidence level = 95%, period = 1 day).
• Let ξ be a random variable representing the loss from a portfolio
over some period of time:
VaRα = min {ξ0 : Pr (ξ ≤ ξ0) ≥ α} .
• Undesirable properties of VaR: does not take into account risks
exceeding VaR, is nonconvex, is not sub-additive.
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• The Conditional VaR (CVaR) takes into account the shape of the
losses exceeding the VaR through the average:
CVaRα = E [ξ | ξ ≥ VaRα] .
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CVaR Portfolio Formulation
• Dealing directly with VaR and CVaR quantities is not tractable.
• Let f (w, r) be an arbitrary cost function, where w is the optimization variable (portfolio) and r denotes the random parameters (asset
returns). Example: f (w, r) = −wT r.
• Consider, for example, the maximization of the mean return subject
to a CVaR risk constraint on the loss:
maximize
wT µ
subject to
CVaRα (f (w, r)) ≤ c
1T w = 1
w
where
CVaRα (f (w, r)) = E [f (w, r) | f (w, r) ≥ VaRα (f (w, r))] .
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CVaR in Convex Form
• Theorem: Define the auxiliary function
Z
1
+
Fα (w, ζ) = ζ +
[f (w, r) − ζ] p (r) dr.
1−α
Then, we have the following two results:
(a) VaRα is a minimizer of Fα (w, ζ) with respect to ζ:
VaRα (f (w, r)) = arg min Fα (w, ζ)
ζ
(b) CVaRα equals minimal value (w.r.t. ζ) of Fα (w, ζ):
CVaRα (f (w, r)) = min Fα (w, ζ)
ζ
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Proof CVaR in Convex Form
(a) The minimizer ζ ? of Fα (w, ζ) satisfies: 0 ∈ ∂ζ Fα (w, ζ ?). For
example we choose the following subgradient:
Z
1
?
sζ Fα (w, ζ ) = 1 −
I (f (w, r) ≥ ζ ?) p (r) dr
1−α
1
= 1−
Pr (f (w, r) ≥ ζ ?) = 0
1−α
where I (·) is the indicator function. Consequently,
Pr (f (w, r) ≥ ζ ?) = 1 − α =⇒ ζ ? = VaRα (f (w, r)) .
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Proof CVaR in Convex Form (cont’d)
(b)
1
min Fα (w, ζ) = Fα (w, ζ ) = ζ +
ζ
1−α
?
?
Z
+
[f (w, r) − ζ ?] p (r) dr.
Recall that
CVaRα (f (w, r)) = E [f (w, r) | f (w, r) ≥ VaRα (f (w, r))]
Z
1
f (w, r) p (r) dr
=
1 − α r:f (w,r)≥VaRα
Z
1
+
=
[f (w, r) − VaRα] p (r) dr + VaRα
1−α
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CVaR in Convex Form (cont’d)
• Corollary:
min CVaRα (f (w, r)) = min Fα (w, ζ)
w
w,ζ
• In words, minimizing Fα (w, ζ) simultaneously calculates the optimal
CVaR and VaR.
• Corollary: If f (w, r) is convex in w for each r, then Fα (w, ζ) is
convex!
Proof:
1
Fα (w, ζ) = ζ +
1−α
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Z
+
[f (w, r) − ζ] p (r) dr.
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Reduction of CVaR Optimization to LP
• We start by considering discrete distributions or approximating the
continuous one by a discrete one so that:
N
+
1 X k
Fα (w, ζ) = ζ +
pk f w, r − ζ .
1−α
k=1
• Then include dummy variables zk :
+
k
k
zk ≥ f w, r − ζ =⇒ zk ≥ f w, r − ζ, zk ≥ 0
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• The problem of minimizing the CVaR reduces to the following LP:
minimize
w,ζ,z
subject to
ζ+
1
1−α
k
PN
k=1 pk zk
f w, r − ζ ≤ zk ≥ 0
1T w = 1
where the cost function is assumed linear in w: f (w, r) = −wT r.
• The maximization of the mean return subject to a CVaR constraint
becomes:
maximize wT µ
w,ζ,z
PN
1
subject to ζ + 1−α k=1 pk zk ≤ c
k
f w, r − ζ ≤ zk ≥ 0
1T w = 1.
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Robust Porfolio Design
• In practice, as usual, the parameters defining the optimization
problem are not always perfectly known. Hence, the concept of
robust design. We will consider worst-case robust design.
• There are many ways to formulate a worst-case design for the
portfolio design. For example, we could consider a robust meanvariance formulation:
minimize max wT Σw
w
subject to
Σ∈SΣ
min wT µ ≥ β
µ∈Sµ
T
1 w = 1.
• Now, depending how we define the uncertainty sets for the mean
return vector Sµ and for the covariance matrix SΣ, the problem may
be convex or not.
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• We will consider one particular example based on modeling the
returns via a factor model:
r = µ + VT f + where f denotes the random factors distributed according to f ∼
N (0, F) and denotes the a random residual error with uncorrelated
elements distributed according to ∼ N (0, D) with D = diag (d).
• The returns are then distributed according to r ∼
N µ, VT FV + D and the obtained return
using portfolio w has
mean µT w and variance wT VT FV + D w.
• We will consider uncertainty in the knowledge of µ, V, and D,
while F is assumed know; in fact, we consider F = I.
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• We can then formulate the robust mean-variance problem as
T
T
minimize
max
w V V+D w
w
subject to
V∈SV ,D∈SD
min wT µ
µ∈Sµ
T
≥β
1 w = 1.
• We will define the uncertainty as follows:
Sµ = {µ | µ = µ0 + δ, |δi| ≤ γi, i − 1, . . . , M }
SV = {V | V = V0 + ∆, k∆kF ≤ ρ}
SD = D | D = diag (d) , di ∈ di, di , i − 1, . . . , M .
• Let’s now elaborate on each of the inner optimizations...
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• First, consider the worst case mean return:
min µT w = µT0 w + min δ T w = µT0 w − γ T |w|
µ∈Sµ
|δi|≤γi
which is a concave function.
• Second, let’s turn to the second term of the worst-case variance:
T
max w Dw =
D∈SD
max
di∈[di,di]
PM
2
d
w
i
i
i=1
=
PM
2 = wT Dw
d
w
i=1 i i
• Finally, let’s focus on the first term of the worst-case variance:
max wT VT Vw ≡ max
V∈SV
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k∆kF ≤ρ
kV0w + ∆wk = kV0wk + ρ kwk
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• Proof: From the triangle inequality we have
kV0w + ∆wk ≤ kV0wk + k∆wk
√
≤ kV0wk + wT ∆T ∆w
≤ kV0wk + kwk k∆kF
≤ kV0wk + kwk ρ
but this upper bound is achievable by the worst-case variable
wT
∆=u
ρ
kwk
where
(
u=
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V0 w
kV0wk
any unitary vector
if V0w 6= 0
otherwise.
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• Finally, the robust portfolio formulation is
2
minimize
(kV0wk + ρ kwk) + wT Dw
subject to
µT0 w − γ T |w| ≥ β
1T w = 1.
w
or, better, as the SOCP
minimize
t2 + wT Dw
subject to
t ≥ kV0wk + ρ kwk
µT0 w ≥ β + γ T |w|
1T w = 1.
w,t
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Index Replication
• Index tracking or benchmark replication is an strategy investment
aimed at mimicking the risk/return profile of a financial instrument.
• For practical reasons, the strategy focuses on a reduced basket of
representative securities.
• This problem is also regarded as portfolio compression and it is
intimately related to compressed sensing and `1-norm minimization
techniques.
• One example is the replication of an index, e.g., Hang Seng index,
based on a reduced set of assets.
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Tracking Error
• Let c ∈ RM represent the actual benchmark weight vector and let
w ∈ RM denote the replicating portfolio.
• Investment managers seek to minimize the following tracking error
performance measure:
q
T
T E1 (w) = (c − w) Σ (c − w).
• In practice, however, the benchmark weight vector c is unknown and
the error measure is defined in terms of market observations.
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Empirical Tracking Error
• Let rc ∈ RN contain N temporal observations
of the returns
of the
benchmark or index and let matrix R = r1 · · · rN ∈ RM ×N
contain column-wise the returns of the individual assets over time.
• The empirical tracking error can be defined as
T
T E2 (w) = rc − R w2 .
• We can then formulate the sparse index replication problem as
minimize
T E2 (w) + γcard (w)
subject to
w≥0
1T w = 1.
w
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Index Replication with `1-Norm Minimization
• The cardinality operator is nonconvex and can be approximated in
a number of ways.
• The simplest approximation is based on the `1-norm:
T
rc − R w + γ kwk
minimize
1
2
w
subject to
w≥0
1T w = 1.
• Of course, there are more sophisticated methods such as the
reweighted `1-norm approximation based on a successive convex
approximation (see lecture on `1-norm minimization for details).
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Simulations of Sparse Index Replication
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Summary
• We have introduced basic concepts and data model for portfolio
optimization.
• Then, we have considered different formulations for the portfolio
optimization problem:
–
–
–
–
basic mean-variance formulations
Sharpe ratio maximization in convex form
CVaR optimization in convex form
worst-case robust designs in convex form
• Finally, we have studied related problem such as the index replication
based on the `1-norm.
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References
• D. Luenberger, Investment Science, Oxford Univ. Press, 1997.
• David Ruppert, Statistics and Data Analysis for Financial Engineering, Springer Texts in Statistics, 2010.
• G. Cornuejols and R. Tutuncu, Optimization Methods in Finance,
Cambridge Univ. Press, 2007.
• F. Fabozzi, P. Kolm, D. Pachamanova, S. Focardi, Robust Portfolio
Optimization and Management, Wiley Finance, 2007.
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