Worst-Case Approach to Strategic Optimal Portfolio
Selection under Transaction Costs and Trading Limits
Andreev N. A.
Higher School of Economics
Laboratory for Financial Engineering and Risk-Management
IFD Kapital
February 3, 2017
Overview
Investment management process
Portfolio analysis, and risk-management in particular, usually relies on the
underlying assumptions about the market (ARIMA/GARCH models for returns
and VaR, Black-Scholes market model for option pricing etc.)
Portfolio manager considers the results based on the market data analysis and
expert analysis.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Overview
Investment management process
A more robust process allows to modify assumptions of the market model
depending on the market state; an expert can also correct the estimated market
parameters if the market data is not representative or insucient.
Still, part of the analysis depends on some a priori assumptions about the market.
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Portfolio selection
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Overview
Model-free decision support system
Can make a model-free decision support system which gives investment
recommendations?
The answer is YES, but some some model-free assumptions about the market
parameters are still required.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Overview
Model-free decision support system
is to formulate a practical portfolio selection problem and provide it with
eective numerical solution. Statement of the problem should:
Our aim
•
Be model-free to avoid model-induced errors;
•
Allow for uncertain values of the known parameters;
•
Consider non-zero transaction costs;
•
Include trading limits;
•
Consider multi-asset portfolios.
The developed framework should be integrated into the decision support system
(DSS) which produces an optimal strategy or a decision map.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Portfolio selection problem
Main blocks
Allowed assets and known at the beginning of the strategy.
investment policy
known at the beginning of each investment period,
limit rules are specied at the beginning of the
Trading limits
strategy.
Initial state of the system exogenous parameter.
the dynamics of the system for a given
species
Market model
strategy.
External ows of omitted in the current research for simplicity.
portfolio assets
a mapping of the system state to the utility value.
Specied by investor, can be considered part of
Optimal criteria
the investment policy.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Portfolio selection problem
Asset prices
Portfolio is formed at and rebalanced at xed moments , . . . , N − (weekly,
quarterly etc.). Strategy is evaluated at N (nite horizon).
n ∈ Rm , n ∈ R prices of -dimensional risky asset and risk-free asset at n .
t0
t1
t
1
t
X
Y
m
t
∆Yn = ξnY Yn−1 ,
ξnY = rn ∆tn
where n is a risk-free rate.
To illustrate the framework, this research considers a general multiplicative
dynamics
X
r
∆Xn = ξn (Θn )Xn−1
where Θn is a vector of market parameters.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
7 / 21
Portfolio selection problem
Asset prices
Black-Scholes model:
Cox-Ross-Rubinstein model:
worst-case approach:
ξnX ∼ N (·, ·)
with probability n ,
with probability 1 −
any distribution with support n
and expectation n .
ξnX =
u
d
n
n
p
p
n
K
E
For any distribution we can write ξnX = |µn{z∆ n} + σ| n {z∆ n}.
t
drift
p
t
volatility
is the expected value of returns, estimated by the analyst.
is the deviation from the expected value, estimated by risk-manager.
Below, we assume that n and µn are estimated by an expert, while σn is the only
stochastic market parameter.
µn
σn
r
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Portfolio selection problem
Asset prices
distribution of σn is unknown but assumed to have the
expected value of n and domain [σn ; σn ].
Parameters of the distribution class can be estimated by an expert, set to reasonable
default values (e.g. n = 0 or σn = −20%, σn = 20% for MOEX stock market etc.)
Parameters can also be estimated via the exogenous market model, which can be
rejected or changed if inconsistent with the market data (untangling of DSS and the
market model):
Worst-case approach:
E
E
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Portfolio selection problem
Portfolio strategy
X Y
n , Hn asset volumes.
X
X T Xn , W Y = H Y Yn
Wn = Hn
n
n
H
positions.
market values of total risky and risk-free
market value of portfolio.
n ( , ) transaction costs function at n .
Budget equation:
W
C
n = WnX + WnY
H X
t
∆HnX
T
Y
X
n−1 + ∆Hn Yn−1 = −Cn−1 (∆Hn , Xn−1 )
X
⇔
−1
Y
n = Yn−1
H
W
X
XT
n−1 − Hn Xn−1 − Cn−1 (∆Hn , Xn−1 ) ,
Our goal is to nd an optimal strategy
Trading limits: nX ∈ n , = 1, .
H
Andreev N. A. (FERM Lab)
D
n
H
n
1
= , N.
X = H X N .
n n=1
N
Portfolio selection
February 3, 2017
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Portfolio selection problem
Optimal criteria
Let Sn be the system state at n , Sn |
S is a known initial state.
t
X
n
H
be the state given the portfolio n .
H
0
Classic approach:
max
where is derived from the specied market model.
ESQ0 J (SN | HNX ) →
H X ∈A
Q
Worst-case approach:
inf ES0 J (SN | HNX ) → Hmax
X ∈A
Q ∈Q Q
where Q is a set of distributions for market parameters (Θ , . . . , ΘN )T with the
expectation n and support n , = 1, .
1
E
Andreev N. A. (FERM Lab)
K
n
N
Portfolio selection
February 3, 2017
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Portfolio selection problem
Dynamic programming principle
The optimization problem can be solved by using the dynamic programming
principle.
For each state S of the system at n− , we dene the value function as
t
V
n−1 (S) =
1
sup inf
H≥X n ∈A≥n Q ∈Q
ESQ J (SN | HN ),
n
0
= ,N −
1
N (S) = J (S).
V
The Bellman-Isaacs equation:
n−1 (S) =
V
If nX ∈ Arg max Qninf∈Qn ESQn
Z ∈Dn (S)
optimal strategy.
∗
H
Andreev N. A. (FERM Lab)
sup
inf
HnX ∈Dn (S) Qn ∈Qn
ESQn Vn (Sn | HnX )
n (Sn | Z ) for n = 1, N , then
V
Portfolio selection
X ∗ N
H
n
n=1
is an
February 3, 2017
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Portfolio selection problem
Solution of the Bellman-Isaacs equation
According to the Bellman-Isaacs equation, to nd the optimal strategy at n one
must nd inmum over a set of possible distributions n of market parameters at
n.
Fortunately, when n is concave in Θn , the extreme measure n∗ is concentrated
in + 1 points of the known support Kn where is the number of the market
parameters. If n is a polyhedron, n∗ is concentrated in + 1
points of
.n
Therefore, n∗ can be found by solving the + 1-dimensional optimization
problem in general case, or simply by iterating over combinations of + 1 extreme
points in polyhedral case.
t
Q
t
V
Q
p
p
K
Q
p
extreme
K
Q
p
p
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
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Portfolio selection problem
Solution of the Bellman-Isaacs equation
Concavity in the unknown parameters holds for many practical cases. For example,
when
• Prices have multiplicative dynamics;
• Transaction costs function depends on the market value of the deal (e. g.
linear costs) and convex (always for order-driven markets);
• Terminal utility function depends on the liquidation value of the portfolio;
• Trading limits at each moment are a combination of linear constraints, e. g.
of the form
: T n ≤ βn n
or
: | |T n ≤ βn n ,
where n is the portfolio market capital at n , n is a matrix.
Z
Z
X
W
W
Andreev N. A. (FERM Lab)
Portfolio selection
Z
Z
t
U
X
W
February 3, 2017
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Portfolio selection problem
Numeric modeling
Let
n+1 = I + µn+1 ∆tn+1 + σn+1 ∆tn+1 ,
r̃n+1 = 1 + rn+1 ∆tn+1 ,
p
s
be a set of combinations of + 1 corners of Kn+ , and let S = (
If n+ is concave in σn+ , we can write
Gn+1
V
m
1
X
Y
n (X , H , W ) =
V
X
1
, HX , W Y )
.
1
sup X
min
G ∈Gn
mX
+1
p
i
n+1 (G )Vn+1 sn+1 (Gi )X , Z ,
+1
Z ∈Dn+1 (X ,H ,W Y )
i =1
Y
X
T
X
W
r̃n+1 − (Z − H ) X r̃n+1 − Cn (Z − H , X )r̃n+1 ,
where
s
n
< N,
n+1 (Gi ) = I + µn+1 ∆tn+1 + diag(Gi ) ∆tn+1 ,
p
and solve the equation numerically.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
15 / 21
Portfolio selection problem
Numeric modeling:
m
= 1,
Cn H X
( ,
) = λ|H |X
Optimal strategy
Portfolio wealth
λ = 0. Due to correct predictions of the
analyst, DSS recommendations (considering
choice between long and short position) are
always correct and portfolio wealth grows during
each period.
←−
Andreev N. A. (FERM Lab)
Prices of risky and risk-free assets.
Portfolio selection
February 3, 2017
16 / 21
Portfolio selection problem
Numeric modeling:
m
= 1,
Cn H X
( ,
) = λ|H |X
Optimal strategy
Portfolio wealth
Under non-zero transaction costs λ = 0.05, trade
sizes decrease, lowering total gain of the strategy.
←−
Andreev N. A. (FERM Lab)
Prices of risky and risk-free assets.
Portfolio selection
February 3, 2017
17 / 21
Portfolio selection problem
Numeric modeling:
m
= 1,
Cn H X
( ,
) = λ|H |X
Optimal strategy
Portfolio wealth
Increasing λ up to 0.12, we see that costs
are so high that any risky investments are
disadvantageous.
←−
Andreev N. A. (FERM Lab)
Prices of risky and risk-free assets.
Portfolio selection
February 3, 2017
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Conclusion
The worst-case approach is a valid alternative to the classic stochastic portfolio
selection approach. Under reasonable conditions the strategy can be found
numerically, parallel computing is quite eective.
The main advantage is the robustness of the framework: at the beginning of an
investment period the required market parameters can be estimated either by an
expert, or straight from market data, or by using an appropriate exogenous market
model (chosen by an expert). Model risk can thus be minimized.
The main drawback is the performance of the strategy since less assumptions
about the market are used. Valid assumptions would provide better results. On the
other hand, incorrect assumptions could lead to additional losses.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
19 / 21
References
Andreev, N. A. (2015). Worst-Case Approach to Strategic Optimal Portfolio Selection Under
Transaction Costs and Trading Limits. Higher School of Economics Research Paper No. WP
BRP 45/FE/2015, page 46.
Andreev, N. A. (2017). Boundedness of the Value Function of the Worst-Case Portfolio
Selection Problem with Linear Constraints. Higher School of Economics Research Paper
WP BRP 59/FE/2017, page 22.
No.
Deng, X.-T., Li, Z.-F., and Wang, S.-Y. (2005). A minimax portfolio selection strategy with
equilibrium. European Journal of Operational Research, 166(1):278292.
Andreev N. A. (FERM Lab)
Portfolio selection
February 3, 2017
20 / 21
E-mail:
[email protected]
[email protected]
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