Portfolio Selection with Objective Functions from Cumulative Prospect Theory

Transcription

1 Portfolio Selection with Objective Functions from Cumulative Prospect Theory Thorsten Hens and János Mayer 13 th International Conference on Stochastic Programming Bergamo, Italy July 10, 2013

2 Contents Portfolio optimization in the Prospect Theory (PT) framework.

3 Contents Portfolio optimization in the Prospect Theory (PT) framework. Probability distortion and Cumulative Prospect theory (CPT).

4 Contents Portfolio optimization in the Prospect Theory (PT) framework. Probability distortion and Cumulative Prospect theory (CPT). How to solve this type of problems numerically?

5 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.

6 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.

7 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.

8 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.

9 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.

10 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) T.A. Pirvu and K. Schulze: Multi Stock Portfolio Optimization under Prospect Theory, NCCR FINRISK Working Paper 742, January 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) X. D. He, X. Y. Zhou: Portfolio Choice Under Cumulative Prospect Theory: An Analytical Treatment, Management Sci., 57(2011) 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)

11 PT: The Kahneman Tversky value function 1.0 v(x) = 1.0 v(x) = risk seeking risk aversion x risk seeking risk aversion x 0.5 losses gains 0.5 losses gains RP: reference point 1.5 RP: reference point v(x) = { (x RP) α+ if x RP β (RP x) α if x < RP Kahneman and Tversky (1979): risk aversion α := α + = α = 0.88; loss-aversion β = Exponential PT value function: De Giorgi, Hens and Levy (2003).

12 PT: Probability distortion 1.0 w(p) gamma= p overweighted p underweighted p Probability distortion function: p γ w(p; γ) = ; [ p γ + (1 p) γ ] 1 γ 0 < γ = 1 (KT: γ = 0.65); γ = 1 no distortion. The above Tversky Kahneman (1992) probability weighting function is not monotone for γ 0.278; Rieger and Wang (WP: 2004), (2008), Ingersoll Alternative weighting functions were proposed by Lattimore et al. (1992), Wu and Gonzalez (1996), Prelec (1998), Rieger and Wang (2004),...

13 PT: Static (one period) portfolio optimization max V PT (λ) := S w s v( x s (λ) ) s=1 λ T w s := w(p s ; γ): distorted probabilities; K x s : prospects, realizations of portfolio return; x s (λ) := Rs k λ k ; K T := {λ λ k = 1, λ k 0, k}, a simplex. k=1 k=1

14 PT: Different probability weighting for gains and for losses In this case the distorted probabilities are computed as: { w + s := w(p s, γ + ) if x s (λ) RP, w s (λ) := ws := w(p s, γ ) if x s (λ) < RP. where now the distorted probabilities themselves also depend on λ. This leads to S V PT (λ) = w s (λ) v( x s (λ) ). s=1

15 PT: Different probability weighting for gains and for losses In this case the distorted probabilities are computed as: { w + s := w(p s, γ + ) if x s (λ) RP, w s (λ) := ws := w(p s, γ ) if x s (λ) < RP. where now the distorted probabilities themselves also depend on λ. This leads to S V PT (λ) = w s (λ) v( x s (λ) ). If γ + γ then s=1 p s = 1, s = the probability weighting matters. S

16 CPT: main idea (I) Prospect Theory (PT): Small probabilities are overweighted; large probabilities are underweighted, regardless of the size of the loss or gain.

17 CPT: main idea (I) Prospect Theory (PT): Small probabilities are overweighted; large probabilities are underweighted, regardless of the size of the loss or gain. 0.2 Cumulative Prospect Theory (CPT): only small probabilities corresponding to extreme losses or gains are overweighted.

18 CPT: main idea (II) cumulative distrib. function F(x) survival function 1-F(x) F(x) cumulative distribution function RP 5 0 RP 5 10 Rank ordered lottery: ( ( x 1(λ) (λ), p 1(λ) ),..., ( x S(λ) (λ), p S(λ) ) ), where ( 1(λ),..., S(λ) ) is a permutation of ( 1,..., S ) such that x 1(λ) (λ)... x r(λ) (λ) RP x r+1(λ) (λ)... x S(λ) (λ) holds. ( r(λ) = 0 gains in all states; r(λ) = S(λ) losses in all states ). Numerical aspect: sorting.

19 CPT: main idea (III) We introduce the notation x s (λ) := x s(λ) (λ), p s := p s(λ) and r := r(λ). Computing the weights: w 1 (λ) := w ( p 1 ), w s (λ) := w ( p p s ) w ( p p s 1 ), 1 < s < r, w s + (λ) := w + ( p s p S ) w + ( p s p S ), r s < S, w + S (λ) := w + ( p S ).

20 CPT: main idea (III) We introduce the notation x s (λ) := x s(λ) (λ), p s := p s(λ) and r := r(λ). Computing the weights: w 1 (λ) := w ( p 1 ), w s (λ) := w ( p p s ) w ( p p s 1 ), 1 < s < r, w + s (λ) := w + ( p s p S ) w + ( p s p S ), r s < S, w + S (λ) := w + ( p S ). Both the ordering of the scenarios and r depend on λ;

21 CPT: main idea (III) We introduce the notation x s (λ) := x s(λ) (λ), p s := p s(λ) and r := r(λ). Computing the weights: w 1 (λ) := w ( p 1 ), w s (λ) := w ( p p s ) w ( p p s 1 ), 1 < s < r, w + s (λ) := w + ( p s p S ) w + ( p s p S ), r s < S, w + S (λ) := w + ( p S ). Both the ordering of the scenarios and r depend on λ; p p s = F λ ( x s (λ) ); p s p S = 1 F λ ( x s (λ) ).

22 CPT: main idea (III) We introduce the notation x s (λ) := x s(λ) (λ), p s := p s(λ) and r := r(λ). Computing the weights: w 1 (λ) := w ( p 1 ), w s (λ) := w ( p p s ) w ( p p s 1 ), 1 < s < r, w + s (λ) := w + ( p s p S ) w + ( p s p S ), r s < S, w + S (λ) := w + ( p S ). Both the ordering of the scenarios and r depend on λ; p p s = F λ ( x s (λ) ); p s p S = 1 F λ ( x s (λ) ). The objective function in CPT is V CPT (λ) := r ws (λ) v( x s (λ)) + s=1 S s=r+1 w + s (λ) v( x s (λ)).

23 PT versus CPT: First impressions (I) CPT Λ, Λ An artificial example with 2 assets and 3 scenarios probability Asset1 Asset

24 PT versus CPT: First impressions (II) PT Λ, Λ CPT Λ, Λ A further artificial example with 2 assets and 3 scenarios probability Asset1 Asset Parameter settings: RP = 0, α + = α = 0.88, β = 2.25, γ = 0.65.

25 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 V PT and V CPT are non-smooth. Is the non smoothness of the KT value function the sole source of the non smoothness of V CPT? Most probably not, because sorting introduces abrupt changes in w s (λ) when λ crosses a point where the ordering changes. Is V CPT continuous at all? The answer is probably yes. These issues will be further explored experimentally and theoretically.

26 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 V PT and V CPT are non-smooth. Is the non smoothness of the KT value function the sole source of the non smoothness of V CPT? Most probably not, because sorting introduces abrupt changes in w s (λ) when λ crosses a point where the ordering changes. Is V CPT 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 V CPT : Why not begin with the beginning? Let us try grid search approaches for CPT optimization first! We employ simplicial grids.

27 Simplices x 3 (0) - simplex (point) (2) - simplex (triangle), unit simplex in R 3 1 (3) - simplex (tetrahedron) (1) - simplex (line segment) 1 1 x 2 x 1 n dimensional simplex in R N : the convex hull of n + 1 affinely independent points in R N. dimension of simplex: #vertices: #edges: #2-dim facets: ( n + 1 k + 1 )

28 Simplices: barycentric coordinates x 3 U unit simplex in R 3 1 x (2) x T 1 1 x (1) x 1 x (0) x 2 An (n 1) dimensional simplex in R n : n T = { x x = µ k x (k), k=1 n µ k = 1, µ k 0, k }. k=1 The (n 1) dimensional unit simplex in R n : U = { x x = = { x n µ k e (k), k=1 k=1 n x j = 1, x j 0, j }. j=1 n µ k = 1, µ k 0, k } x (1),..., x (n) are the vertices of the simplex T ; µ 1,..., µ n 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.

29 Regular grids (triangulations, meshes) over the unit simplex U e 3 e 3 gc=2 gc=4 e 1 e 2 e 1 e 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.

30 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. Look for a subsimplex in a coarser grid (e.g., gc1 < gc ) which contains p. Refine the grid 2 in the subsimplex found. Repeat the procedure with this subsimplex. This procedure is embedded into an outer loop for finding starting points via generating uniformly distributed points over the starting simplex. e 3 e 3 p e 1 e 2 e 1 e 2 barycentric coordinates = spatial coordinates barycentric coordinates spatial coordinates Authors: H.W. Kuhn (1960), (1968); T. Hansen and H. Scarf (1969); B.C. Eaves (1970), (1971).

31 Implementation Generating a grid: Regular grid (mesh) over the unit simplex U, corresponding to the grid constant gc N ++: T gc := { x = 1 gc (k 0, k 1,..., k n) T n k i = gc, k i N +, i } The grid is computed via compositions n (n + 1) tuples (k 0, k 1,..., k n) with k i = gc, gc, k 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. i=0

32 Comparing (C)PT and MV With V = V PT and V = V CPT we compare the solutions of: max λ V ( ξ T λ ) 1l T λ = 1 Optimal solution: λ. λ 0, versus max λ V ( ξ T λ ) λ Λ MV Optimal solution: λ MV. Λ MV : the set of portfolios along the MV efficient frontier. Λ MV {λ 1lT λ = 1, λ 0} V ( ξ T λ MV ) V ( ξt λ ).

33 Comparing (C)PT and MV With V = V PT and V = V CPT we compare the solutions of: max λ V ( ξ T λ ) 1l T λ = 1 Optimal solution: λ. λ 0, versus max λ V ( ξ T λ ) λ Λ MV Optimal solution: λ MV. Λ MV : the set of portfolios along the MV efficient frontier. Λ MV {λ 1lT λ = 1, λ 0} V ( ξ T λ MV ) V ( ξt 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.

34 Comparing (C)PT and MV With V = V PT and V = V CPT we compare the solutions of: max λ V ( ξ T λ ) 1l T λ = 1 Optimal solution: λ. λ 0, versus max λ V ( ξ T λ ) λ Λ MV Optimal solution: λ MV. Λ MV : the set of portfolios along the MV efficient frontier. Λ MV {λ 1lT λ = 1, λ 0} V ( ξ T λ MV ) V ( ξt 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?

35 How to compare the two types of optimal portfolios? Proximity measures: Kroll, Levy and Markowitz (1984) I OBJR = V (ξt λ MV ) V (ξt λ naive ) V (ξ T λ ) V (ξ T λ naive ) where λ naive is the naïve portfolio with equal weights. Kallberg and Ziemba (1983) I CER = CE v [ ξ T λ ] CE v [ ξ T λ MV ] CE v [ ξ T λ ] with the certainty equivalent defined as CE v [ ξ T λ ] = v 1 ( V (ξ T λ) ). DeMiguel, Garlappi and Uppal (2009), De Giorgi and Hens (2009) I CED := CE v [ ξ T λ ] CE v [ 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 I CED.

36 Data sets (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. 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 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.

37 Data sets (II) Testing for multivariate normality H 0 : Multivariate normality holds for the basic data set. Results from Mardia s tests involving multivariate skewness and kurtosis: Original Data Set Normal sample Normal sample N=208 N=208 N= mv. skewness test stat. χ 2 (120) p value mv. kurtosis test stat. N (0, 1) p value p = 0 for both cases H 0 can be rejected, e.g., on a 99.9% significance level.

38 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: ε max π,ε S π s = 1 s=1 S rs k π s = r f, k = 1,..., K s=1 π s ε, s = 1,..., S. computing state prices with r f = 0.2% Optimal solution: (π, ε ). If ε > 0 holds = we add a call option on the index MXWO such S that ˆr sπs = r f holds for the added column ˆr = 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.

39 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 we have computed the PT and CPT optimal portfolios as well as their counterparts by optimizing along the MV frontier. For the comparisons the indices I OBJR, I CED and I CER have been computed. In addition, for the PT and CPT optimal portfolios their geometrical distance to the MV frontier has been computed.

40 CPT versus MV (I) basic data set call option added normal distribution 0.6 CPT optimal portfolios versus MV frontier 0.6 CPT optimal portfolios versus MV frontier 0.7 CPT optimal portfolios versus MV frontier expe ected value (%) expe ected value (%) expe ected value (%) standard deviation (%) standard deviation (%) standard deviation (%) Frequen ncy CPT versus MV; CED index Frequen ncy CPT versus MV; CED index Frequen ncy CPT versus MV; CED index CED index 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 I CED index: there is a substantial deviation from 0 in the first two cases, contrary to the third case.

41 CPT versus MV (II) I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Basic data set I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Call option added I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis normal approx. original with call option Normal approximation I CED across the three data-sets

42 CPT versus MV (III): Optimal portfolios 1.2 CPT: KT value function, S = 15 K = 9: GSCITR, HFRIFFM, I3M... Algorithm: adaptive grid method ass set weights 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 GSCITR HFRIFFM I3M JPMBD MSEM MXWO NAREIT PE Call_MXWO Call option: Maximizing CPT; heavy investments into the call option 1.2 Maximizing CPT (Pw. power) along MV frontier K = 9: GSCITR, HFRIFFM, I3M as set weights 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 S GSCITR HFRIFFM I3M JPMBD MSEM MXWO NAREIT PE Call_MXWO Call option: Maximizing CPT along the MV frontier; practically no investment into the call

43 CPT versus MV (IV): Skewness of optimal portfolios folios of optim mal portf kewness sk investors CPT CPT(MV) Original data set: skewness of the optimal portfolios 12 ios al portfoli s of optim skewness CPT CPT(MV) 4 investors Call option added: skewness of the optimal portfolios (cf. Barberis and Huang (2008))

44 PT versus MV (I) basic data set call option added normal distribution 0.6 PT optimal portfolios versus MV frontier 0.6 PT optimal portfolios versus MV frontier 0.7 PT optimal portfolios versus MV frontier expe ected value (%) expe ected value (%) expe ected value (%) standard deviation (%) standard deviation (%) standard deviation (%) Frequen ncy PT versus MV; CED index Frequen ncy PT versus MV; CED index Frequen ncy PT versus MV; CED index 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 I CED index: there is a substantial deviation from 0 in the first two cases, contrary to the third case.

45 PT versus MV (II) I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Basic data set I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Call option added I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Normal approximation normal approx. original with call option I CED across the three data-sets

46 PT versus CPT (I) basic data set call option added normal distribution PT versus CPT; CED index PT versus CPT; CED index PT versus CPT; CED index Frequen ncy Frequenc cy Frequen ncy CED index CED index CED index Observations. Frequency histograms for the I CED 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.

47 PT versus CPT (II) I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Basic data set I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis Call option added I OBJR I CED I CER Min Mean Median Max Std. Dev Skewness Kurtosis normal approx. original with call option Normal approximation I CED across the three data-sets

48 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).

49 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.

50 Conclusions (III) Limitations For normally distributed returns, maximizing a (C)PT objective function along the MV frontier produces more accurate optimal solutions (cf. negative I CED 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 assets (asset classes). Running times for 48 investors, in seconds, under Windows XP, 3.16 GHz processor and 3.26 GB RAM: PT CPT PT(MV) CPT(MV) original data set associated normal distr call option added

51 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: t: In contrast, optimizing CPT along the MV frontier took less than 0.1 seconds for all cases.

52 For the details see: T. Hens and J. Mayer: Cumulative Prospect Theory and Mean Variance Analysis. A Rigorous Comparison, NCCR FINRISK Working Paper 792, November 2012.