Portfolio choice under Cumulative Prospect Theory:
sensitivity analysis and an empirical study
Elisa Mastrogiacomo
University of Milano-Bicocca, Italy
(joint work with Asmerilda Hitaj)
XVI WORKSHOP ON QUANTITATIVE FINANCE
Parma, January 29-30, 2015
Summary
Main Goal and motivation of our work
Literature review and basic notion of CPT
Results obtained from Simulation and Numerical analysis
Problem
Static (one period) portfolio optimization
max V (w · X)
w∈W
where
X = (X1 , . . . , Xn ): random vector of asset’s returns,
�
W := {w ∈ Rn : ni=1 wi = 1, wi ≥ 0}: set of all admissible
portfolio’s weights w = (w1 , . . . , wn )
V = V CPT is the objective function in Cumulative Prospect
Theory
Maximizing CPT-portoflios: existing literature
Papers related to CPT optimization and numerical aspects (among
the others):
H. Levy, M. and Levy: Prospect Theory and Mean-Variance
Analysis, The Review of Financial Studies, 17 (2004)
1015–1041.
T.A. Pirvu and K. Schulze: MultiStock Portfolio Optimization
under Prospect Theory, NCCR FINRISK Working Paper 742,
January 2012.
Notice: Assumption of normally distributed resp. elliptically symmetric returns
Maximizing CPT-portoflios: existing literature
Papers related to CPT optimization and numerical aspects (among
the others):
T.Hens and J. Mayer Portfolio Selection with Objective
Functions from Cumulative Prospect Theory, Swiss Finance
Institute Research Paper No. 14-23 (2013)
Notice: use of a real world data set
� no assumption on the asset’s distribution.
Our contribution
Two directions
Impact of
Skewness and Kurtosis of Asset’s returns
parameters of CPT-objective function
on Mean/Standard Dev. and Mean/(CPT)-Certainty equivalent
Efficient frontiers
� Simulation part
Impact of parameters of CPT-objective function on the Asset
Allocation Decision
Comparison with traditional MV and GMV optimization
� Empirical analysis
Cumulative Prospect Theory
Main reason of our interest in CPT
CPT takes into account several phenomena which are not
completely explained through expected utility theory
risk aversion in choices involving sure gains
risk seeking in choices involving sure losses
losses loom larger than corresponding gains
people underweight extreme losses/gains that are merely
probable in comparison with losses/gains that are obtained
with certainty.
(see Kahneman & Tversky, 1979)
CPT “ingredients”
Piece-wise power utility function
concave for gains
convex for losses
v is steeper for losses than
for gains
v (x) =
�
(x − RP)α , x ≥ RP
−λ(RP − x)β , x < RP
CPT “ingredients”
Weighting probability functions
Roll a die
� x ∈ {1, . . . , 6}
c.d.f. FY
You win x, if it is even
You pay x if it is odd
Assuming equiprobable outcomes, our prospect is
�
−5, −3, −1
p = 16
Y =
+2, +4, +6, p = 16
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CPT “ingredients”
Weighting probability functions
small pobabilities for
extreme gains/losses are
overweighted through
nonlinear functions
T + , T − : [0, 1] → [0, 1]:
strictly increasing
functions s.t.
T + (0) = T − (0) = 0 and
T + (1) =, T − (1) = 1.
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CPT “ingredients”
Weighting probability functions
“modified” c.d.f. FY
T + : fix γ ∈ (0, 1)
T + (p) =
pγ
(p γ + (1 − p)γ )1/γ
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T − (p) =
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(p δ + (1 − p)δ )1/δ
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CPT value for a portfolio - discrete case
Given
X = (X1 , . . . , Xn ), vector of asset’s returns
w ∈ W, weights
y := w · X, portfolio return with {yi }i=−�,...,m possible
outcomes
V CPT (w · X) = V − (w · X) + V + (w · X)
�
�
=
v (y−i )π −
v (yi )π +
w·X (y−i ) +
w·X (yi )
−�≤−i≤0
0≤i≤m
How does Skewness, Kurtosis and CPT parameters
affect Efficient Frontiers on CPT framework?
“Controlled experiments”
Main points
Simulation of four hypothetical asset, say A,B,C,D, where
mean, variance, covariance are fixed once for all
skewness and kurtosis: 9 different scenarios, which are the
combination of
Skewness
Kurtosis
Zero
Uniform
Positive
Mixed
Negative
Fat
Numerical optimization of the CPT-portfolio obtained with the
simulated asset returns and different values of the CPT
parameters.
“Controlled experiments”
Main points
Numerical computation of Mean/CPT Certainty Equivalent
efficient frontiers; comparison with the Mean/Stand. Dev.
CPT Certainty Equivalent, considered as a risk measure is
ρCEv (X ) := −v −1 (E[v (X )]),
X r .v .
and Mean/CPT Certainty Equivalent efficient frontier is
E[X ∗ (τ )] �→ ρ(X ∗ (τ )),
with X ∗ (τ ) := X · w∗ being the solution of
�
minw∈W ρ(X · w)
s.t.E[w · X] ≥ τ
Mean/Cert. Equiv. efficient frontier: Zero Skew and
Uniform Tails
S= [0 0 0 0 ]
CPT utility function with _= 0.8 `= 0.88 a=0.61 b=0.9
and
K= [3.5 3.5 3.5 3.5 ]
CPT utility function with h=2.25 `= 0.88 a=0.61 b=0.9
0.12
expected return
expected return
0.12
0.1
0.08
h=1.5
h=2
h=2.25
0.06
0.04
0
0.02
0.04
0.06
0.08
risk C.E.
0.1
0.12
0.1
0.08
0.04
0.14
_=0.2
_=0.6
_=0.88
0.06
0
CPT utility function with h=2.25 _= 0.8 a=0.61 b=0.9
expected return
expected return
0.04
0.1
`=0.3
`=0.7
`=0.88
0.06
0.1
0.12
0.14
0
0.02
0.04
0.06
0.08
risk C.E.
0.1
0.12
0.1
0.08
0.14
a=0.6
a=0.75
a=0.9
0.06
0.04
0.06
0.07
0.08
0.09
CPT utility function with h=2.25 _= 0.8 `= 0.88 a=0.61
0.12
0.13
0.14
0.15
expected return
0.12
0.1
0.08
b=0.6
b=0.75
b=0.9
0.06
0.04
0.02
0.1
0.11
risk C.E.
CARA utility function
0.12
expected return
0.06
0.08
risk C.E.
0.12
0.08
0.04
0.02
CPT utility function with h=2.25 _= 0.8 `= 0.88 b=0.9
0.12
0.04
0.06
0.08
0.1
risk C.E.
0.12
0.14
0.16
0.1
h=1
h=1.25
h=1.5
h=1.75
h=2
0.08
0.06
0.04
0
0.1
0.2
0.3
0.4
risk C.E.
0.5
0.6
0.7
Mean-Variance and Mean/Certainty Equivalent Eff. Front.
S=[0 0 0 0],
0.065
K=[6.5
6.5
6.5
S=[0 0 0 0],
6.5]
0.065
Markowitz
CPT(h=2.25, _=0.8, `=0.88, a=0.61, b=0.69,)
CARA(h=2.25)
0.06
0.055
0.055
0.05
0.05
0.045
0.045
0.04
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.04
0.05
0.75
m "Markowitz"
S=[0 0 ï0.5 ï0.4],
0.08
K=[6.5
6.5
6.5]
0.07
0.07
0.065
0.06
0.06
0.055
0.055
0.5
0.08
S=[0
0.075
0.45
0.07
ï0.4
0.55
ï0.7
ï0.5
0.09
ï0.4],
K=[6.5
0.1
6.5
0.11
6.5
0.12
0.13
6.5]
0.6
0.65
0.7
CPT(h=2.25, _=0.8, `=0.88, a=0.61, b=0.69,)
Markowitz
CARA(h=2.25)
0.05
0.05
0.75
0.06
0.07
m "Markowitz"
S=[ï0.5
0
0.08
0.065
0.05
0.4
0.06
6.5]
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6.5
certainty equivalent "CPT"
6.5
Markowitz
CPT(h=2.25, _=0.8, `=0.88, a=0.61, b=0.69,)
CARA(h=2.25)
0.075
ï0.3],
K=[10.5
10.5
10.5
10.5]
S=[ï0.5
ï0.4
0.09
0.1
0.08
0.09
certainty equivalent "CPT"
ï0.7
ï0.3],
K=[10.5
10.5
0.1
10.5
0.11
10.5]
0.085
0.085
0.08
0.08
0.075
0.075
0.07
0.07
0.065
0.065
!
!
K=[6.5 6.5
CPT(h=2.25, _=0.8, `=0.88, a=0.61, b=0.69,)
Markowitz
CARA(h=2.25)
!
!
0.06
0.06
0.06
0.055
0.055
0.05
0.05
0.045
0.045
Markowitz
CPT(h=2.25, _=0.8, `=0.9, a=0.5, b=0.6,)
CARA(h=2.25)
0.04
0.4
0.5
0.6
0.7
standard deviation "Markowitz"
0.8
0.9
CPT(h=2.25, _=0.8, `=0.9, a=0.5, b=0.6,)
Markowitz
CARA(h=2.25)
0.04
1
0.08
0.11
0.12
0.13
certainty equivalent "CPT"
0.14
0.15
0.16
0.17
What are the differences between real portfolios
obtained with CPT, MV and GMV models?
Data set
Basic scenario data-set
Daily data for 12 indices, January 2006 - December 2014
Equities within different industries, such as financial, utilities,
communications, information technology, consumer staples and
energy (Citigroup Inc, Microsoft Corp, Royal Bank of Scotland
Group PLC, Unilever PLC, Volkswagen AG, Deutsche Bank
AG, Total SA, BNP Paribas, Banco Santander, Telefonica,
Intesa Sanpaolo SPA and Enel SPA)
Comments on empirical results
We compared the different models by considering
different rolling window strategies (e.g. in-sample period: 6
month and 1 year, out-of-sample period: 1 day, 1 week, 1
month);
a measure of diversification for each year in the period
2006-2014, different combination of the CPT parameters and
also for MV, GMV and CARA
a risk adjusted performance measure for each year in the
period 2006-2014, different combination of the CPT
parameters and also for MV, MGV and CARA
Two concepts for the comparison with MV and CARA
Measuring the diversification: the modified Herfindahl index.
HI =
�N
1
2
i=1 wi − N
1 − N1
Risk adjusted performance measure: Omega Ratio with τ = 0.
Ω=
E (RP,t − τ )+
E (τ − RP,t )+
Numerical Experiment: Diversification
2006 ï 2014; Modified Herfindahl a=0.6, b =0.6 (175ï1) CPT
1
0.9
0.8
0.7
MV
1
1
1.5
1
1
0.6
0.5
1.5
1.5
2.25
1.5
2.25
0.4
1
1
1.5
2.25
1
1.5
1.5
2.25
2.25
2.25
2.25
0.3 GMV
0.2
h=1
0.7
0.6
8
0.9
0.6
0.6
0.8
h=1.5
h=2.25
8
0.9
8
0.8
0.9
`
0.8
0.8
8
0.8
8
0.8
_
0.8
Numerical Experiment: Risk adjusted performance measure
2006ï2014; Omega Ratio a=0.6, b=0.6 (175 ï1)
2.25
1.2
2.25
2.25
1.5
1.5
2.25
GMV
2.25
2.25
1
2.25
1.5
1
1.5
1.5
1.5
MV
1.15
1.5
1
1
1.1
1
1.05
1
0.7
0.6
h=1
8
0.9
0.6
0.6
0.8
8
0.8
8
.
0
0.9
0.8
h=1.5
8
0.8
8
0.8
h=2.25
0.9
`
8
0.8
_
Further developments
Changing the dataset: what happens if we consider different
portfolio (i.e. with different equities)?
� We worked with a dataset which exhibits “anomalous”
behaviour (very high skenewess and kurtosis for some equities)
CPT in dynamic framework
Thank you for your attention!