Portfolio choice under Cumulative Prospect Theory: sensitivity analysis and an empirical study

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 where max V (w X) w W X =(X 1,...,X n ): random vector of asset s returns, W := {w R n : n i=1 w i = 1, w i 0}: setofalladmissible portfolio s weights w =(w 1,...,w n ) 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. elliptically sym- Notice: Assumption of normally distributed resp. metric 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 v(x) = concave for gains convex for losses vissteeperforlossesthan for gains (x RP) α, x RP λ(rp x) β, x < RP

CPT ingredients Weighting probability functions Roll a die x {1,...,6} You win x, ifitiseven You pay x if it is odd.,"(- c.d.f. F Y ' '$+,"(- '$+,"#- Assuming equiprobable outcomes, our prospect is 5, 3, 1 p = 1 Y = 6 +2, +4, +6, p = 1 6 '*) +,$%- +,$&- $% $& $' "# "( ").,$&-

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) =0and T + (1) =, T (1) =1. % $,")- % $,"(- modified c.d.f. F Y $ "'$+,"#-# ' '*) + "'$+,"(-# & "+,$%-# & "+,$%-# $% $& $' "# "( ") % &,$&- % &,$%-

CPT ingredients Weighting probability functions T + :fixγ (0, 1) modified c.d.f. F Y T + (p) = p γ (p γ +(1 p) γ ) 1/γ % $,")- % $,"(- $ "'$+,"#-# ' + "'$+,"(-# T :fixδ (0, 1) T (p) = p δ (p δ +(1 p) δ ) 1/δ '*) & "+,$%-# & "+,$%-# $% $& $' "# "( ") % &,$&- % &,$%-

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CPT value for a portfolio - discrete case Given X =(X 1,...,X n ), vector of asset s returns w W, weights y := w X, portfolioreturnwith{y i } i=,...,m possible outcomes V CPT (w X) =V (w X)+V + (w X) = v(y i )π w X (y i)+ v(y i )π + w X (y i) 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 Zero Positive Negative Kurtosis Uniform Mixed 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 ρ CEv (X ):= v 1 (E[v(X )]), Xr.v. and Mean/CPT Certainty Equivalent efficient frontier is E[X (τ)] ρ(x (τ)), with X (τ) :=X w being the solution of min w W ρ(x w) s.t.e[w X] τ

Mean/Cert. Equiv. efficient frontier: Zero Skew and Uniform Tails 0.12 S= [0 0 0 0 ] and K= [3.5 3.5 3.5 3.5 ] CPT utility function with = 0.8 = 0.88 =0.61 =0.9 CPT utility function with =2.25 = 0.88 =0.61 =0.9 0.12 expected return 0.1 0.08 =1.5 0.06 =2 =2.25 0.04 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 risk C.E. CPT utility function with =2.25 = 0.8 =0.61 =0.9 0.12 expected return 0.1 0.08 0.06 =0.2 =0.6 =0.88 0.04 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 risk C.E. CPT utility function with =2.25 = 0.8 = 0.88 =0.9 0.12 expected return 0.1 0.08 0.06 =0.3 =0.7 =0.88 0.04 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 risk C.E. CPT utility function with =2.25 = 0.8 = 0.88 =0.61 0.12 expected return 0.1 0.08 =0.6 0.06 =0.75 =0.9 0.04 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 risk C.E. CARA utility function 0.12 expected return 0.1 0.08 0.06 =0.6 =0.75 =0.9 0.04 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 risk C.E. expected return 0.1 =1 0.08 =1.25 =1.5 0.06 =1.75 =2 0.04 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 risk C.E.

Mean-Variance and Mean/Certainty Equivalent Eff. Front. 0.065 S=[0 0 0 0], K=[6.5 6.5 6.5 6.5] Markowitz CPT( =2.25, =0.8, =0.88, =0.61, =0.69,) CARA( =2.25) 0.065 S=[0 0 0 0], K=[6.5 6.5 6.5 6.5] CPT( =2.25, =0.8, =0.88, =0.61, =0.69,) Markowitz CARA( =2.25) 0.06 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 0.75 "Markowitz" 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 certainty equivalent "CPT" 0.08 0.075 S=[0 0 0.5 0.4], K=[6.5 6.5 6.5 6.5] Markowitz CPT( =2.25, =0.8, =0.88, =0.61, =0.69,) CARA( =2.25) 0.08 0.075 S=[0 0 0.5 0.4], K=[6.5 6.5 6.5 6.5] CPT( =2.25, =0.8, =0.88, =0.61, =0.69,) Markowitz CARA( =2.25) 0.07 0.07 0.065 0.065 0.06 0.06 0.055 0.055 0.05 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 "Markowitz" 0.05 0.05 0.06 0.07 0.08 0.09 0.1 0.11 certainty equivalent "CPT" 0.085 S=[ 0.5 0.4 0.7 0.3], K=[10.5 10.5 10.5 10.5] 0.085 S=[ 0.5 0.4 0.7 0.3], K=[10.5 10.5 10.5 10.5] 0.08 0.08 0.075 0.075 0.07 0.07 0.065 0.065 0.06 0.06 0.055 0.055 0.05 0.05 0.045 0.045 0.04 Markowitz CPT( =2.25, =0.8, =0.9, =0.5, =0.6,) CARA( =2.25) 0.4 0.5 0.6 0.7 0.8 0.9 1 standard deviation "Markowitz" 0.04 CPT( =2.25, =0.8, =0.9, =0.5, =0.6,) Markowitz CARA( =2.25) 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 certainty equivalent "CPT"

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 ariskadjustedperformance 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. H I = N i=1 w i 2 1 1 N 1 N Risk adjusted performance measure: Omega Ratio with τ = 0. Ω= E (R P,t τ) + E (τ R P,t ) +

Numerical Experiment: Diversification 1 2006 2014; Modified Herfindahl =0.6, =0.6 (175 1) CPT 0.9 0.8 MV 0.7 1 1 1.5 1 1.5 1 1 1 1 0.6 0.5 0.4 1.5 2.25 2.25 2.25 1.5 2.25 1.5 2.25 1.5 2.25 1.5 2.25 0.3 0.2 GMV 0.7 0.6 0.88 0.6 0.9 0.6 0.88 0.8 0.9 0.8 =1 =1.5 =2.25 0.88 0.88 0.9 0.88

Numerical Experiment: Risk adjusted performance measure 2006 2014; Omega Ratio =0.6, =0.6 (175 1) 1.2 GMV 1.5 2.25 2.25 1.5 2.25 2.25 1.5 2.25 1.5 12.25 2.25 1 1.5 1.15 MV 1.5 1 1.5 1.1 1 1 1.05 1 =1 =1.5 =2.25 0.7 0.6 0.88 0.6 0.9 0.6 0.88 0.8 0.9 0.8 0.88 0.88 0.9 0.88

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

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