Portfolio Selection with Objective Functions
from Cumulative Prospect Theory
Thorsten Hens and János Mayer
13th International Conference on Stochastic Programming
Bergamo, Italy
July 10, 2013
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
How to solve this type of problems numerically?
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
How to solve this type of problems numerically?
Algorithm based on adaptive simplicial grid refinement and its
implementation.
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
How to solve this type of problems numerically?
Algorithm based on adaptive simplicial grid refinement and its
implementation.
Basic setup of the numerical experiments for comparing(C)PT and
MV.
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
How to solve this type of problems numerically?
Algorithm based on adaptive simplicial grid refinement and its
implementation.
Basic setup of the numerical experiments for comparing(C)PT and
MV.
Data–sets used: basic data set, data set with an added European call
option, data–set from a normal distribution.
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
How to solve this type of problems numerically?
Algorithm based on adaptive simplicial grid refinement and its
implementation.
Basic setup of the numerical experiments for comparing(C)PT and
MV.
Data–sets used: basic data set, data set with an added European call
option, data–set from a normal distribution.
Numerical results.
Contents
Portfolio optimization in the Prospect Theory (PT) framework.
Probability distortion and Cumulative Prospect theory (CPT).
How to solve this type of problems numerically?
Algorithm based on adaptive simplicial grid refinement and its
implementation.
Basic setup of the numerical experiments for comparing(C)PT and
MV.
Data–sets used: basic data set, data set with an added European call
option, data–set from a normal distribution.
Numerical results.
Conclusions.
Literature
According to our knowledge the directly relevant literature is rather scarce.
Papers related to the numerical solution of CPT optimization problems.
Assumption: normally distributed resp. elliptically symmetric returns:
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: Multi–Stock Portfolio Optimization under
Prospect Theory, NCCR FINRISK Working Paper 742, January 2012.
Explicit formulas for the two–assets case with one of them being the
risk–free asset:
C. Bernard, M. Ghossoub: Static Portfolio Choice under Cumulative
Prospect Theory, Math. and Fin. Econ. 2(2010)277–306.
X. D. He, X. Y. Zhou: Portfolio Choice Under Cumulative Prospect Theory:
An Analytical Treatment, Management Sci., 57(2011)315–331.
T. Hens, K. Bachmann: Behavioural finance for private banking, Wiley
(2008).
V. Zakamouline, S. Koekebakker: A generalisation of the mean–variance
analysis, European Financial Management 15(2009)934–970.
PT: The Kahneman–Tversky value function
1.0
alpha = 0.6
v(x)
1.0
0.5
0.5
risk seeking
0.5
0.5
losses
risk aversion
x
1.5
2.0
1.0
gains
risk seeking
0.5
risk aversion
1.0
losses
0.5
1.0
1.5
alpha = 0.88
v(x)
1.5
gains
1.0
RP: reference point
RP: reference point
1.5
2.0
2.0
(
(x − RP)α
v (x) =
+
α−
−β (RP − x)
if x ≥ RP
if x < RP
~ Kahneman and Tversky (1979):
risk aversion α := α+ = α− = 0.88;
loss-aversion β = 2.25.
~ Exponential PT value function: De Giorgi, Hens and Levy (2003).
x
2.0
PT: Probability distortion
gamma = 0.65
1.0
w(p)
I Probability distortion function:
0.8
0.6
0.4
w (p; γ) =
pγ
1
[ p γ + (1 − p)γ ] γ
p overweighted
p underweighted
I 0 < γ ≤= 1 (KT: γ = 0.65);
0.2
0.0
0.0
p
0.2
0.4
0.6
0.8
I γ = 1 ⇒ no distortion.
1.0
~ The above Tversky–Kahneman (1992) probability weighting function
is not monotone for γ ≤ 0.278; Rieger and Wang (WP: 2004),
(2008), Ingersoll 2008.
~ Alternative weighting functions were proposed by Lattimore et al.
(1992), Wu and Gonzalez (1996), Prelec (1998), Rieger and Wang
(2004), . . .
;
PT: Static (one–period) portfolio optimization
max VPT (λ) :=
S
X
ws · v ( xs (λ) )




λ ∈ T



s=1
I ws := w (ps ; γ): distorted probabilities;
I xs : prospects, realizations of portfolio return; xs (λ) :=
K
X
k=1
I T := {λ |
K
X
k=1
λk = 1, λk ≥ 0, ∀k}, a simplex.
Rsk λk ;
PT: Different probability weighting for gains and for losses
In this case the distorted probabilities are computed as:
( +
ws := w (ps , γ + ) if xs (λ) ≥ RP,
ws (λ) :=
ws− := w (ps , γ − ) if xs (λ) < RP.
where now the distorted probabilities themselves also depend on λ. This
leads to
S
X
ws (λ) · v ( xs (λ) ).
VPT (λ) =
s=1
PT: Different probability weighting for gains and for losses
In this case the distorted probabilities are computed as:
( +
ws := w (ps , γ + ) if xs (λ) ≥ RP,
ws (λ) :=
ws− := w (ps , γ − ) if xs (λ) < RP.
where now the distorted probabilities themselves also depend on λ. This
leads to
S
X
ws (λ) · v ( xs (λ) ).
VPT (λ) =
s=1
If γ + 6= γ − then
1
~ ps = , ∀s =⇒ the probability weighting matters.
S
CPT: main idea (I)
PT
1.0
0.8
0.6
0.4
0.2
2
2
4
Prospect Theory (PT):
Small probabilities are
overweighted;
large probabilities are
underweighted,
regardless of the size of the loss or
gain.
CPT: main idea (I)
PT
1.0
0.8
CPT
2
0.6
1.0
0.4
0.8
0.2
0.6
2
0.4
4
0.2
Prospect Theory (PT):
Small probabilities are
overweighted;
large probabilities are
underweighted,
regardless of the size of the loss or
gain.
2
2
4
Cumulative Prospect Theory (CPT):
only small probabilities corresponding
to extreme losses or gains are
overweighted.
CPT: main idea (II)
1.0
0.5
0.8
cumulative distrib. function 0.4
F(x)
0.6
0.4
0.2
5
0.2
cumulative distribution function
5
1-F(x)
0.3
F(x)
RP
survival function
10
0.1
�5
0
RP
5
10
Rank–ordered lottery: ( ( x1(λ) (λ), p1(λ) ), . . . , ( xS(λ) (λ), pS(λ) ) ), where
( 1(λ), . . . , S(λ) ) is a permutation of ( 1, . . . , S ) such that
x1(λ) (λ) ≤ . . . ≤ xr (λ) (λ) ≤ RP ≤ xr +1(λ) (λ) ≤ . . . ≤ xS(λ) (λ) holds.
( r (λ) = 0 ⇒ gains in all states; r (λ) = S(λ) ⇒ losses in all states ).
Numerical aspect: sorting.
CPT: main idea (III)
We introduce the notation x̄s (λ) := xs(λ) (λ), p̄s := ps(λ) and r̄ := r (λ).
Computing the weights:
w1− (λ) := w − ( p̄1 ),






−
−
−
ws (λ) := w ( p̄1 + . . . + p̄s ) − w ( p̄1 + . . . + p̄s−1 ), 1 < s < r̄ , 
ws+ (λ) := w + ( p̄s + . . . + p̄S ) − w + ( p̄s+1 + . . . + p̄S ), r̄ ≤ s < S, 





+
+
w (λ) := w ( p̄ ).
S
S
CPT: main idea (III)
We introduce the notation x̄s (λ) := xs(λ) (λ), p̄s := ps(λ) and r̄ := r (λ).
Computing the weights:
w1− (λ) := w − ( p̄1 ),






−
−
−
ws (λ) := w ( p̄1 + . . . + p̄s ) − w ( p̄1 + . . . + p̄s−1 ), 1 < s < r̄ , 
ws+ (λ) := w + ( p̄s + . . . + p̄S ) − w + ( p̄s+1 + . . . + p̄S ), r̄ ≤ s < S, 





+
+
w (λ) := w ( p̄ ).
S
S
~ Both the ordering of the scenarios and r̄ depend on λ;
CPT: main idea (III)
We introduce the notation x̄s (λ) := xs(λ) (λ), p̄s := ps(λ) and r̄ := r (λ).
Computing the weights:
w1− (λ) := w − ( p̄1 ),






−
−
−
ws (λ) := w ( p̄1 + . . . + p̄s ) − w ( p̄1 + . . . + p̄s−1 ), 1 < s < r̄ , 
ws+ (λ) := w + ( p̄s + . . . + p̄S ) − w + ( p̄s+1 + . . . + p̄S ), r̄ ≤ s < S, 





+
+
w (λ) := w ( p̄ ).
S
S
~ Both the ordering of the scenarios and r̄ depend on λ;
~ p̄1 + . . . + p̄s = Fλ ( x̄s (λ) );
~ p̄s+1 + . . . + p̄S = 1 − Fλ ( x̄s (λ) ).
CPT: main idea (III)
We introduce the notation x̄s (λ) := xs(λ) (λ), p̄s := ps(λ) and r̄ := r (λ).
Computing the weights:
w1− (λ) := w − ( p̄1 ),






−
−
−
ws (λ) := w ( p̄1 + . . . + p̄s ) − w ( p̄1 + . . . + p̄s−1 ), 1 < s < r̄ , 
ws+ (λ) := w + ( p̄s + . . . + p̄S ) − w + ( p̄s+1 + . . . + p̄S ), r̄ ≤ s < S, 





+
+
w (λ) := w ( p̄ ).
S
S
~ Both the ordering of the scenarios and r̄ depend on λ;
~ p̄1 + . . . + p̄s = Fλ ( x̄s (λ) );
~ p̄s+1 + . . . + p̄S = 1 − Fλ ( x̄s (λ) ).
The objective function in CPT is
VCPT (λ) :=
r
X
s=1
ws− (λ)
· v ( x̄s (λ)) +
S
X
s=r +1
ws+ (λ) · v ( x̄s (λ)).
PT versus CPT: First impressions (I)
CPTH Λ1, Λ2L
PTH Λ1, Λ2L
-0.060
-0.054
-0.062
-0.056
-0.064
-0.058
-0.066
-0.060
0.2
0.4
0.6
0.8
1.0
Λ1
0.2
0.4
0.6
An artificial example with 2 assets and 3 scenarios
probability
0.5
0.2
0.3
Asset1
-0.035
0.03
-0.025
Asset2
-0.01
-0.07
0.01
0.8
1.0
Λ1
PT versus CPT: First impressions (II)
CPTH Λ1, Λ2L
PTH Λ1, Λ2L
0.030
0.025
0.025
0.020
0.020
0.015
0.015
0.010
0.2
0.4
0.6
0.8
1.0
Λ1
0.2
0.4
0.6
0.8
A further artificial example with 2 assets and 3 scenarios
probability
0.5
0.2
0.3
Asset1
0.035
-0.03
0.025
Asset2
0.005
0.09
-0.01
Parameter settings: RP = 0, α+ = α− = 0.88, β = 2.25, γ = 0.65.
1.0
Λ1
Numerical aspects
The main sources of numerical difficulties:
Both the PT and the CPT portfolio optimization models are
non–convex optimization problems: the objective function is not
concave, in general.
Since the KT value function is not differentiable at the RP, both VPT
and VCPT are non-smooth.
Is the non–smoothness of the KT value function the sole source of
the non–smoothness of VCPT ? Most probably not, because sorting
introduces abrupt changes in ws (λ) when λ crosses a point where the
ordering changes. Is VCPT continuous at all? The answer is probably
yes. These issues will be further explored experimentally and
theoretically.
Numerical aspects
The main sources of numerical difficulties:
Both the PT and the CPT portfolio optimization models are
non–convex optimization problems: the objective function is not
concave, in general.
Since the KT value function is not differentiable at the RP, both VPT
and VCPT are non-smooth.
Is the non–smoothness of the KT value function the sole source of
the non–smoothness of VCPT ? Most probably not, because sorting
introduces abrupt changes in ws (λ) when λ crosses a point where the
ordering changes. Is VCPT continuous at all? The answer is probably
yes. These issues will be further explored experimentally and
theoretically.
To get first ideas on the behavior of VCPT :
Why not begin with the beginning? Let us try grid–search approaches for
CPT–optimization first! We employ simplicial grids.
Simplices
x3
(2) - simplex (triangle),
3
unit simplex in R
(0) - simplex (point)
(3) - simplex (tetrahedron)
1
(1) - simplex (line segment)
1
1
x2
x1
n–dimensional simplex in
dimension of simplex:
#vertices:
#edges:
#2-dim facets:
RN : the convex hull of n + 1 affinely independent points in RN .
2
3
3
1
3
4
6
4
4
5
10
10
5
6
15
20
6
7
21
35
7
8
28
56
8
9
36
84
9
10
45
120
10
11
55
165
n+1
k +1
Simplices: barycentric coordinates
x3
U
x(2)
3
unit simplex in R
T
1
1
R
The (n − 1)–dimensional
unit simplex in n :
R
x(1)
1
x(0)
x1
An (n − 1)–dimensional
simplex in n :
x
T = {x | x =
n
X
x2
µk x (k) ,
k=1
U
= {x | x =
= {x |
n
X
n
X
k=1
n
X
µk = 1, µk ≥ 0, ∀k }.
k=1
µk e (k) ,
n
X
µk = 1, µk ≥ 0, ∀k }
k=1
xj = 1, xj ≥ 0, ∀j }.
j=1
x (1) , . . . , x (n) are the vertices of the simplex T ;
µ1 , . . . , µn are the (uniquely determined) barycentric coordinates of x ∈ T .
A simplicial partition of the unit simplex corresponds to a simplicial subdivision of T and
vice versa.
Regular grids (triangulations, meshes) over the unit
simplex U
e3
e3
gc=2
e1
gc=4
e
2
e
e1
2
A regular simplex is a simplex with all edges having the same length.
A regular simplicial grid or mesh with grid constant gc ∈
N++ results by partitioning the unit
simplex U into congruent regular subsimplices, such that the edges of U become partitioned into
gc equal pieces.
Adaptive simplicial grid method
Primary objects: grid–points in the current mesh.
Compute the objective function value for each of the grid–points in the current mesh as
well as at the centroids of subsimplices. Choose one point p with the highest objective
value.
) which contains p. Refine the grid
Look for a subsimplex in a coarser grid (e.g., gc1 < gc
2
in the subsimplex found.
Repeat the procedure with this subsimplex.
I This procedure is embedded into an outer loop for finding starting points via generating
uniformly distributed points over the starting simplex.
e3
e3
7→
p
e1
e
2
barycentric coordinates = spatial coordinates
e1
e
2
barycentric coordinates 7→ spatial coordinates
Authors:
H.W. Kuhn (1960), (1968); T. Hansen and H. Scarf (1969); B.C. Eaves (1970), (1971).
Implementation
Generating a grid:
Regular grid (mesh) over the unit–simplex U, corresponding to the grid–constant gc ∈
Tgc := { x =
n
X
1
(k0 , k1 , . . . , kn )T |
ki = gc, ki ∈
gc
i=0
The grid is computed via compositions
n
X
(n + 1)–tuples (k0 , k1 , . . . , kn ) with
ki = gc, gc, ki ∈
N++ :
N+ , ∀i }
N+ , ∀i. For setting up a list of all
i=0
compositions, a recursive algorithm of D. Knuth has been employed.
Primary data structure:
A list of all compositions for gc;
the spatial coordinates of the current simplex.
Subsimplices are constructed only according to the needs of the algorithm.
Crucial ingredient of the algorithm:
For a given point p in the current simplex, the problem is to find the vertices of a subsimplex of
the partition, which contains p. For this an algorithm of Kuhn and Eaves has been implemented.
Comparing (C)PT and MV
With V = VPT and V = VCPT we compare the solutions of:
max V ( ξ T λ )




λ
1lT λ
=
1 


≥ 0,
λ
Optimal solution:
versus
λ

V ( ξTλ )
λ ∈ Λ∗MV
Optimal solution: λ∗MV .
Λ∗MV : the set of portfolios along
the MV efficient frontier.
λ∗ .
Λ∗MV ⊂ {λ | 1lT λ = 1, λ ≥ 0}

 max
⇒
V ( ξ T λ∗MV ) ≤ V ( ξ T λ∗ ).
Comparing (C)PT and MV
With V = VPT and V = VCPT we compare the solutions of:
max V ( ξ T λ )




λ
1lT λ
=
1 


≥ 0,
λ
Optimal solution:
versus
λ

V ( ξTλ )
λ ∈ Λ∗MV
Optimal solution: λ∗MV .
Λ∗MV : the set of portfolios along
the MV efficient frontier.
λ∗ .
Λ∗MV ⊂ {λ | 1lT λ = 1, λ ≥ 0}

 max
⇒
V ( ξ T λ∗MV ) ≤ V ( ξ T λ∗ ).
Levy and Levy (2004): if ξ has a multivariate normal distribution and
short–sales are allowed, then the solutions of the above optimization
problems coincide.
Comparing (C)PT and MV
With V = VPT and V = VCPT we compare the solutions of:
max V ( ξ T λ )




λ
1lT λ
=
1 


≥ 0,
λ
Optimal solution:
versus
λ

V ( ξTλ )
λ ∈ Λ∗MV
Optimal solution: λ∗MV .
Λ∗MV : the set of portfolios along
the MV efficient frontier.
λ∗ .
Λ∗MV ⊂ {λ | 1lT λ = 1, λ ≥ 0}

 max
⇒
V ( ξ T λ∗MV ) ≤ V ( ξ T λ∗ ).
Levy and Levy (2004): if ξ has a multivariate normal distribution and
short–sales are allowed, then the solutions of the above optimization
problems coincide.
Central question in our experiments:
Is this so in general? Is it sufficient to optimize V along the MV frontier if
we do not have a normal distribution and/or λ ≥ 0 is imposed?
How to compare the two types of optimal portfolios?
Proximity measures:
I Kroll, Levy and Markowitz (1984)
IOBJR =
V (ξ T λ∗MV ) − V (ξ T λnaive )
V (ξ T λ∗ ) − V (ξ T λnaive )
where λnaive is the “naı̈ve” portfolio with equal weights.
I Kallberg and Ziemba (1983)
ICER =
CEv [ ξ T λ∗ ] − CEv [ ξ T λ∗MV ]
CEv [ ξ T λ∗ ]
with the certainty equivalent defined as CEv [ ξ T λ ] = v −1 V (ξ T λ) .
I DeMiguel, Garlappi and Uppal (2009), De Giorgi and Hens (2009)
ICED := CEv [ ξ T λ∗ ] − CEv [ ξ T λ∗MV ]
Interpretation: the difference as added value in monetary terms.
In our computational results all three quantities are expressed in % terms
with annualized values for ICED .
Data–sets (I)
I Basic monthly–returns data set. Monthly data for 8 indices, 1994
February – 2011 May (208 months); we would like to express our
thanks to Dieter Niggeler of BhFS for providing us with this data–set.
GSCITR
I3M
HFRIFFM
MSEM
MXWO
NAREIT
PE
JPMBD
Goldman Sachs Commodity Index; total return;
3 months US Dollar LIBOR interest rate;
Hedge Fund Research International, Fund of Funds,
market defensive index;
Morgan Stanley Emerging Markets Index, total return,
stocks;
MSCI World Index, total return, stocks
(developed countries);
FTSE, US Real Estate Index, total return;
LPX50, LPX Group Zurich, Listed Private Equities,
total return;
JP Morgan Bond Index, Developed Markets , total return.
I Sample from the corresponding normal distribution. With the
empirical µ and Σ from the basic data–set we have generated a
multivariate normal sample with sample–size 10’000.
I Investors’ data. PT parameter values for 48 subjects from:
Abdellaoui, Bleichrodt and Paraschiv: Loss aversion under prospect
theory: A parameter–free measurement, Management Science, 2007.
Data–sets (II)
Testing for multivariate normality
H0 : Multivariate normality holds for the basic data–set. Results from
Mardia’s tests involving multivariate skewness and kurtosis:
mv. skewness
test stat. χ2 (120)
p–value
mv. kurtosis
test stat. N (0, 1)
p–value
Original Data–Set
N=208
15.88
550.62
0
113.12
18.88
0
Normal sample
N=208
3.08
106.65
0.80
77.57
-1.38
0.17
Normal sample
N=10’000
0.06
104.14
0.85
79.74
-1.02
0.31
p = 0 for both cases ⇒ H0 can be rejected, e.g., on a 99.9% significance
level.
Data–sets (III)
We wish to investigate also the influence of probability weighting =⇒ empirical
distributions needed. These we have generated via k–means clustering of the
basic data–sets. k=???
Additional goal: For achieving a clear deviation from multivariate normality we wish to add
a European call option to the empirical distribution, having positive payoff in only one state. We
solve the LP:

max
ε

π,ε



S

X




πs = 1



s=1
computing state prices with rf = 0.2%

S

X

k

rs πs = rf , k = 1, . . . , K 




s=1



πs ≥ ε, s = 1, . . . , S.
Optimal solution: (π ∗ , ε∗ ). If ε∗ > 0 holds =⇒ we add a call option on the index MXWO such
S
X
that
r̂s πs∗ = rf holds for the added column r̂ =⇒ the new data–set is also arbitrage–free.
s=1
Trial–and–error with increasing k for getting scenarios with ε∗ > 0 in the LP
above =⇒ k = 15 works; the call option can be appended.
Numerical experiments
We have three data sets, each of them consisting of 15 scenarios and
corresponding probabilities:
~ Basic scenario data–set. This is obtained from the basic monthly
returns data–set via k–means clustering.
~ Scenario data–set with an added option. Computed on the basis
of the previous scenarios by adding a European call option on the
index MXWO; having a positive payoff in a single state.
~ Scenario data–set from a normal distribution. Generated via
k–means clustering from the normal sample.
For each of these data–sets and for each one of the 48 investors
X we have computed the PT and CPT optimal portfolios as well as their
counterparts by optimizing along the MV–frontier.
X For the comparisons the indices IOBJR , ICED and ICER have been
computed.
X In addition, for the PT and CPT optimal portfolios their geometrical
distance to the MV–frontier has been computed.
CPT versus MV (I)
call option added
basic data set
CPT optimal portfolios versus MV‐frontier
normal distribution
CPT optimal portfolios versus MV‐frontier
0.6
CPT optimal portfolios versus MV‐frontier
0.6
0.5
0.5
0.4
0.4
0.7
0.6
0.3
0.2
expe
ected value (%)
expe
ected value (%)
expe
ected value (%)
0.5
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0
0.1
0
2
4
6
8
10
12
0
0
standard deviation (%)
4
6
8
10
12
0
standard deviation (%)
CPT versus MV; CED index
45.00
40 00
40.00
Frequen
ncy
30.00
25.00
20.00
15.00
10.00
5 00
5.00
0.00
40.00
30.00
35.00
6
8
10
12
30.00
20 00
20.00
15.00
25.00
20.00
15.00
10.00
10.00
5.00
5.00
0.00
0.00
CED index
4
CPT versus MV; CED index
35.00
25.00
35.00
2
standard deviation (%)
CPT versus MV; CED index
50.00
Frequen
ncy
2
Frequen
ncy
0
CED index
CED index
Observations. (µ, σ) space: in the first two cases most of the points corresponding to CPT optimal portfolios are quite away
from the MV–frontier, whereas in the third case the points are on the frontier. Frequency histograms for the ICED index: there
is a substantial deviation from 0 in the first two cases, contrary to the third case.
CPT versus MV (II)
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
-18.46
76.89
97.07
99.43
38.15
-1.544
3.604
ICED
0.0028
2.1838
0.1734
67.87
9.877
6.265
41.61
ICER
0.0002
0.171
0.014
5.151
0.751
6.207
41.06
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
99.66
99.99
100.0
100.0
0.049
-6.616
44.85
ICED
-0.001
0.0002
0.000
0.006
0.001
4.590
28.16
ICED
0.034
13.04
5.238
83.72
19.26
2.184
7.325
ICER
0.003
1.047
0.435
6.525
1.508
2.104
6.987
Call option added
Basic data–set
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
-21.99
58.48
73.69
97.70
36.40
-0.906
2.396
ICER
-0.00009
0.00001
0.000
0.0005
0.00008
4.594
28.19
Normal approximation
35
30
25
20
15
13.0441
10
5
0
‐5
0.0002
2.1838
normal approx. original with call option
ICED across the three data-sets
CPT versus MV (III): Optimal portfolios
CPT: KT value function, S = 15
K = 9: GSCITR, HFRIFFM, I3M ...
Algorithm: adaptive grid method
1.2
1
assset weights
0.8
0.6
0.4
0.2
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S25
S26
S27
S28
S29
S30
S31
S32
S33
S34
S35
S36
S37
S38
S39
S40
S41
S42
S43
S44
S45
S46
S47
S48
0
GSCITR
HFRIFFM
I3M
JPMBD
MSEM
MXWO
NAREIT
PE
Call_MXWO
Call option: Maximizing CPT; heavy investments into the call option
Maximizing CPT (Pw. power) along MV frontier
K = 9: GSCITR, HFRIFFM, I3M ...
1.2
1
asset weights
0.8
0.6
0.4
04
0.2
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
S19
S20
S21
S22
S23
S24
S25
S26
S27
S28
S29
S30
S31
S32
S33
S34
S35
S36
S37
S38
S39
S40
S41
S42
S43
S44
S45
S46
S47
S48
0
‐0.2
GSCITR
HFRIFFM
I3M
JPMBD
MSEM
MXWO
NAREIT
PE
Call_MXWO
Call option: Maximizing CPT along the MV–frontier; practically no investment into the call
CPT versus MV (IV): Skewness of optimal portfolios
skkewness of optim
mal portffolios
2
1.5
1
0.5
0
‐0.5
CPT
‐1
CPT(MV)
‐1.5
‐2
25
‐2.5
‐3
investors
Original data–set: skewness of the optimal portfolios
skewnesss of optimal portfoliios
12
10
8
6
CPT
4
CPT(MV)
2
0
‐2
‐4
investors
Call option added: skewness of the optimal portfolios (cf. Barberis and Huang (2008))
PT versus MV (I)
call option added
basic data set
PT optimal portfolios versus MV‐frontier
normal distribution
PT optimal portfolios versus MV‐frontier
0.6
0.6
0.5
0.5
0.4
0.4
PT optimal portfolios versus MV‐frontier
0.7
0.6
0.2
expe
ected value (%)
expe
ected value (%)
expe
ected value (%)
0.5
0.3
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0
0.1
0
2
4
6
8
10
12
0
0
standard deviation (%)
4
6
8
10
12
0
standard deviation (%)
PT versus MV; CED index
45.00
40 00
40.00
Frequen
ncy
30.00
25.00
20.00
15.00
4
6
8
10
PT versus MV; CED index
35.00
30.00
30.00
25.00
25.00
35.00
2
standard deviation (%)
PT versus MV; CED index
50.00
Frequen
ncy
2
20.00
Frequen
ncy
0
20 00
20.00
15.00
15.00
10.00
10.00
10.00
5.00
5.00
5 00
5.00
0.00
0.00
0.00
CED index
CED index
CED index
Observations (similar as in the CPT case). (µ, σ) space: in the first two cases most of the points corresponding to CPT
optimal portfolios are quite away from the MV–frontier, whereas in the third case the points are on the frontier. Frequency
histograms for the ICED index: there is a substantial deviation from 0 in the first two cases, contrary to the third case.
12
PT versus MV (II)
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
-13.80
81.24
98.02
99.88
34.49
-1.806
4.539
ICED
0.002
3.536
0.429
83.85
13.30
5.158
29.67
ICER
0.0002
0.273
0.036
6.206
0.994
5.057
28.61
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
ICED
-0.009
0.00002
0.000
0.005
0.002
-1.895
11.73
ICED
0.002
13.75
5.215
83.85
20.79
2.135
7.024
ICER
0.0002
1.096
0.433
6.532
1.610
2.033
6.559
Call option added
Basic data–set
IOBJR
99.98
99.99
100.0
100.0
0.006
2.379
17.13
IOBJR
-0.917
75.74
88.58
99.87
29.50
-1.492
3.681
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
ICER
-0.0007
0.000002
0.000
0.0004
0.0001
-1.894
11.72
Normal approximation
35
30
25
20
15
13.75413
10
5
0
‐5
3.53643
0.00002
normal approx. original with call option
ICED across the three data-sets
PT versus CPT (I)
call option added
basic data set
PT versus CPT; CED index
normal distribution
PT versus CPT; CED index
45.00
40.00
PT versus CPT; CED index
35.00
35.00
30.00
30.00
25.00
25.00
Frequenccy
Frequen
ncy
30.00
25 00
25.00
20.00
15 00
15.00
Frequen
ncy
35 00
35.00
20 00
20.00
15.00
20 00
20.00
15.00
10.00
10.00
5 00
5.00
5 00
5.00
5.00
0.00
0.00
10.00
CED index
0.00
CED index
CED index
Observations. Frequency histograms for the ICED index: the variation across the three cases is much smaller now as in the
previous CPT–MV and PT–MV comparisons. Interestingly, the difference between PT and CPT is noticeably smaller than the
difference with respect to MV for both PT and CPT.
PT versus CPT (II)
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
-92.65
89.50
99.25
100.9
32.23
-4.349
22.99
ICED
-0.356
0.562
0.089
11.13
1.695
5.249
32.24
ICER
-0.030
0.046
0.007
0.921
0.140
5.254
32.30
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
87.07
98.75
99.99
100.0
2.934
-2.595
8.813
ICED
-0.015
0.160
0.0004
1.122
0.288
2.134
6.625
ICED
0
0.501
0.034
2.853
0.780
1.621
4.598
ICER
0
0.041
0.003
0.235
0.064
1.610
4.551
Call option added
Basic data–set
Min.
Mean
Median
Max.
Std. Dev.
Skewness
Kurtosis
IOBJR
-56.98
88.32
99.42
100
28.53
-3.507
16.12
ICER
-0.001
0.013
0.00004
0.091
0.024
2.11
6.510
Normal approximation
2.5
2
1.5
1
0.56
0.5
0.50
0.16
0
normal approx. original with call option
ICED across the three data-sets
Conclusions (I)
To the best of our knowledge we are the first to propose and test a
general algorithm for computing asset allocations for CPT. This is a
numerically hard problem that is of high relevance for finance.
We compare the results of maximizing CPT along the mean–variance
efficient frontier and maximizing it without that restriction. We find:
With normally distributed returns the difference is negligible, in
accordance with Levy and Levy (2004).
Using standard asset allocation data of pension funds the difference is
considerable.
With derivatives like call options the restriction to the mean-variance
efficient frontier results in a sizable loss; the results from the original
data–set are magnified.
Our computational results indicate that assumption of normally
distributed returns is an essential presupposition for the application of
the method proposed by Levy and Levy (2004).
Conclusions (II)
Our numerical results fully support the theoretical findings of Barberis
and Huang (2008) concerning positive skewness preference of (C)PT
investors. This is despite the fact that in our case none of the
assumptions in that paper hold.
Can the observed phenomena in fact be mainly attributed to
deviations from normality? Recall that we exclude short–sales in our
computations! Most probably yes, since in the normally distributed
case we still have that constraint, nevertheless the optimal (C)PT
portfolios are located along the MV frontier!
Considering the relation between optimal PT and CPT portfolios we
observed that the difference between PT and CPT is noticeably
smaller than the difference with respect to MV for both PT and CPT.
We presume that this is due to the unimodal nature of the distribution
of our basic data–set, but verifying this requires further investigations.
Conclusions (III)
Limitations
For normally distributed returns, maximizing a (C)PT objective function
along the MV frontier produces more accurate optimal solutions (cf.
negative ICED values for some investors).
The suggested algorithm clearly has its practical limitations regarding the
number of assets in the portfolio; it can be used up to 15–20 assets (asset
classes).
Running times for 48 investors, in seconds, under Windows XP, 3.16 GHz
processor and 3.26 GB RAM:
original data–set
associated normal distr.
call option added
PT
91
90
180
CPT
432
430
845
PT(MV)
3
3.4
3.6
CPT(MV)
5
4.9
5.4
Conclusions (IV)
Limitations: Computational costs
Average elapsed time in seconds for solving a CPT optimization problem, with n
denoting the number of assets. Computing environment: Windows XP, 3.16 GHz
processor and 3.26 GB RAM.
For the dimensions from 10 to 20, we generated discrete distributions with 15
realizations for monthly assets–returns of assets included in the DJIA index, just
for recording the computational times. From 21 assets upwards, the
computational time exceeds 1 hour.
n:
8
9
10
11
12
13
14
15
16
17
18
19
20
t:
9
18
31
55
96
164
268
430
673
1029
1575
2287
3319
In contrast, optimizing CPT along the MV frontier took less than 0.1 seconds for
all cases.
For the details see:
I T. Hens and J. Mayer: “Cumulative Prospect Theory and Mean
Variance Analysis. A Rigorous Comparison”, NCCR FINRISK
Working Paper 792, November 2012.