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Financial Engineering: PortfolioOptimization
Daniel P. Palomar
Hong Kong University of Science and Technology (HKUST)
ELEC5470 - Convex Optimization
Fall 2017-18, 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) re-
turns arerm,t = (pm,t − pm,t−1) /pm,t−1ym,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 1Tw = 1), then the overall return will be
wTrt.
• The key quantity is then the random return wTrt.
• The expected or mean return is wTµ and the variance is wTΣw.
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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 wTµ and the variance (risk) wTΣw 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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Daniel P. Palomar 6
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):
S =wTµ− rf√
wTΣw.
• 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:
maximizew
wTµ
subject to wTΣw ≤ α1Tw = 1.
• Minimization of risk:
minimizew
wTΣw
subject to wTµ ≥ β1Tw = 1.
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• Another obvious formulation is the scalarization of the multi-
objective optimization problem:
maximizew
wTµ− γwTΣw
subject to 1Tw = 1.
• 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:minimize
w
√wTΣw/
(wTµ− rf
)subject to 1Tw = 1
w ≥ 0
• This is a quasi-convex problem as can be seen in the epigraph form:
minimizew,t
t
subject to t(wTµ− rf
)≥∥∥Σ1/2w
∥∥1Tw = 1
w ≥ 0
which can be solve by bisection over t.
Daniel P. Palomar 10
Sharpe Ratio Formulation in Convex Form
• Theorem: The maximization of the Sharpe ratio can be rewritten
in convex form as the QP
minimizew̃
w̃TΣw̃
subject to (µ− rf1)T
w̃ = 1
1T w̃ ≥ 0
w̃ ≥ 0.
Proof:
• Defining w̃ = tw with t = 1/(wTµ− rf
)> 0, the objective
becomes√
w̃TΣw̃, the sum constraint becomes 1T w̃ = t, and the
problem becomes
Daniel P. Palomar 11
minimizew,w̃,t
√w̃TΣw̃
subject to t = 1/(wTµ− rf
)> 0
w̃ = tw
1T w̃ = t > 0
w̃ ≥ 0.
• Observe that the first constraint 1/t = wTµ− rf can be rewritten
in terms of w̃ as 1 = (µ− rf1)T
w̃. So the problem becomes
minimizew,w̃,t
√w̃TΣw̃
subject to (µ− rf1)T
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:
minimizew̃
w̃TΣw̃
subject to (µ− rf1)T
w̃ = 1
1T w̃ ≥ 0
w̃ ≥ 0.
• QED!
Daniel P. Palomar 13
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 optimiza-
tion variable (portfolio) and r denotes the random parameters (asset
returns). Example: f (w, r) = −wTr.
• Consider, for example, the maximization of the mean return subject
to a CVaR risk constraint on the loss:
maximizew
wTµ
subject to CVaRα (f (w, r)) ≤ c1Tw = 1
where
CVaRα (f (w, r)) = E [f (w, r) | f (w, r) ≥ VaRα (f (w, r))] .
Daniel P. Palomar 16
CVaR in Convex Form
• Theorem: Define the auxiliary function
Fα (w, ζ) = ζ +1
1− α
∫[f (w, r)− ζ]
+p (r) dr.
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 minζFα (w, ζ)
(b) CVaRα equals minimal value (w.r.t. ζ) of Fα (w, ζ):
CVaRα (f (w, r)) = minζFα (w, ζ)
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Proof CVaR in Convex Form
(a) The minimizer ζ? of Fα (w, ζ) satisfies: 0 ∈ ∂ζFα (w, ζ?). For
example we choose the following subgradient:
sζFα (w, ζ?) = 1− 1
1− α
∫I (f (w, r) ≥ ζ?) p (r) dr
= 1− 1
1− αPr (f (w, r) ≥ ζ?) = 0
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)
minζFα (w, ζ) = Fα (w, ζ?) = ζ?+
1
1− α
∫[f (w, r)− ζ?]+ p (r) dr.
Recall that
CVaRα (f (w, r)) = E [f (w, r) | f (w, r) ≥ VaRα (f (w, r))]
=1
1− α
∫r:f(w,r)≥VaRα
f (w, r) p (r) dr
=1
1− α
∫[f (w, r)− VaRα]
+p (r) dr + VaRα
Daniel P. Palomar 19
CVaR in Convex Form (cont’d)
• Corollary:
minw
CVaRα (f (w, r)) = minw,ζ
Fα (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:
Fα (w, ζ) = ζ +1
1− α
∫[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:
Fα (w, ζ) = ζ +1
1− α
N∑k=1
pk[f(w, rk
)− ζ]+.
• Then include dummy variables zk:
zk ≥[f(w, rk
)− ζ]+
=⇒ zk ≥ f(w, rk
)− ζ, zk ≥ 0
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• The problem of minimizing the CVaR reduces to the following LP:
minimizew,ζ,z
ζ + 11−α
∑Nk=1 pkzk
subject to f(w, rk
)− ζ ≤ zk ≥ 0
1Tw = 1
where the cost function is assumed linear in w: f (w, r) = −wTr.
• The maximization of the mean return subject to a CVaR constraint
becomes:maximize
w,ζ,zwTµ
subject to ζ + 11−α
∑Nk=1 pkzk ≤ c
f(w, rk
)− ζ ≤ zk ≥ 0
1Tw = 1.
Daniel P. Palomar 22
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 mean-
variance formulation:minimize
wmaxΣ∈SΣ
wTΣw
subject to minµ∈Sµ
wTµ ≥ β
1Tw = 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 + ε
where f denotes the random factors distributed according to f ∼N (0,F) and ε denotes the a random residual error with uncorrelated
elements distributed according to ε ∼ N (0,D) with D = diag (d).
• The returns are then distributed according to r ∼N(µ,VTFV + D
)and the obtained return using portfolio w has
mean µTw and variance wT(VTFV + 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
minimizew
maxV∈SV,D∈SD
wT(VTV + D
)w
subject to minµ∈Sµ
wTµ ≥ β
1Tw = 1.
• We will define the uncertainty as follows:
Sµ = {µ | µ = µ0 + δ, |δi| ≤ γi, i− 1, . . . ,M}SV = {V | V = V0 + ∆, ‖∆‖F ≤ ρ}SD =
{D | D = diag (d) , di ∈
[di, di
], i− 1, . . . ,M
}.
• Let’s now elaborate on each of the inner optimizations...
Daniel P. Palomar 25
• First, consider the worst case mean return:
minµ∈Sµ
µTw = µT0 w + min|δi|≤γi
δTw = µT0 w − γT |w|
which is a concave function.
• Second, let’s turn to the second term of the worst-case variance:
maxD∈SD
wTDw = maxdi∈[di,di]
∑Mi=1 diw
2i =
∑Mi=1 diw
2i = wTDw
• Finally, let’s focus on the first term of the worst-case variance:
maxV∈SV
wTVTVw ≡ max‖∆‖F≤ρ
‖V0w + ∆w‖ = ‖V0w‖+ ρ ‖w‖
Daniel P. Palomar 26
• Proof: From the triangle inequality we have
‖V0w + ∆w‖ ≤ ‖V0w‖+ ‖∆w‖≤ ‖V0w‖+
√wT∆T∆w
≤ ‖V0w‖+ ‖w‖ ‖∆‖F≤ ‖V0w‖+ ‖w‖ ρ
but this upper bound is achievable by the worst-case variable
∆ = uwT
‖w‖ρ
where
u =
{V0w‖V0w‖ if V0w 6= 0
any unitary vector otherwise.
Daniel P. Palomar 27
• Finally, the robust portfolio formulation is
minimizew
(‖V0w‖+ ρ ‖w‖)2 + wTDw
subject to µT0 w − γT |w| ≥ β1Tw = 1.
or, better, as the SOCP
minimizew,t
t2 + wTDw
subject to t ≥ ‖V0w‖+ ρ ‖w‖µT0 w ≥ β + γT |w|1Tw = 1.
Daniel P. Palomar 28
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 errorperformance measure:
TE1 (w) =
√(c−w)
TΣ (c−w).
• In practice, however, the benchmark weight vector c is unknown and
the error measure is defined in terms of market observations.
Daniel P. Palomar 30
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
TE2 (w) =∥∥rc −RTw
∥∥2.
• We can then formulate the sparse index replication problem as
minimizew
TE2 (w) + γcard (w)
subject to w ≥ 0
1Tw = 1.
Daniel P. Palomar 31
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:
minimizew
∥∥rc −RTw∥∥2
+ γ ‖w‖1subject to w ≥ 0
1Tw = 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.
Daniel P. Palomar 34
References
• Yiyong Feng and Daniel P. Palomar, A Signal Processing Perspec-
tive on Financial Engineering, Foundations and Trends® in Signal
Processing, Now Publishers, vol. 9, no. 1-2, 2016.
• D. Luenberger, Investment Science, Oxford Univ. Press, 1997.
• D. Ruppert, Statistics and Data Analysis for Financial Engineer-
ing, 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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