Report

Portfolio Optimisation for the Anxious Greg B Davies, PhD Head of Behavioural Finance [email protected] June 2010 “The market can stay irrational, longer than you can stay solvent” John Maynard Keynes Harry Markowitz – Nobel Prize 1990 2 Daniel Kahneman – Nobel Prize 2002 Value function in descriptive theories: CPT Utility Reference Point Losses (£) Gains (£) Loss aversion: Steeper for losses And with risk=variance, expected utility theory is equivalent to mean-variance optimisation r E u x f r , Risk f r , 2 Risk/Return Trade-off (Indifference Curve) Optimal Portfolio Risk free rate Market Portfolio Portfolio Efficient Frontier 4 Only works under specific, and largely incorrect assumptions EU E u x f r , 2 Only true if one of two assumptions holds: 1. Log Returns are normally distributed (no fat tails; no black swans; no skewness) OR 2. Individuals have a rational utility function that is quadratic Neither assumption is valid 5 Example Quadratic Utility Function Utility Return This means both the risk measure and the risk-return trade-off are flawed EU E u x f r , 2 r Risk free rate 6 The exponential function shows aversion to left tail events and preference for positive skewness in log returns Utility All display CRRA throughout domain Low Risk Tolerance Medium Risk Tolerance High Risk Tolerance Log returns These result in a remarkably simple rational trade-off between adjusted risk and expected returns D r r f B 2 T Desirability Compensation for risk Expected Excess Returns Total return – risk-free return 8 Risk / Risk Tolerance Illustrating in risk-return space Maximum Desirability r B 2 r rf Accept Reject Desirability Desirability rf Risk compensation All components measured in % log returns Risk free return B 9 T Because risk is mis-specified, the mean-variance ‘efficient’ frontier is not truly risk-return efficient r True Efficient Frontier Desirability rf (In)efficient Frontier B 10 Traditional portfolio theory trades off risk and return of the portfolio in the long run Expected Portfolio Return Efficient Frontier High risk tolerance, high portfolio risk, high return Low risk tolerance, low portfolio risk, low return Portfolio Risk 11 The emotional experience with the investment journey has potentially greater influence on the final result Portfolio Value Which investor is happier? (Green, black or red) Danger of buying high Danger of selling low Time 12 Ulysses 13 Self-control Dual self model Two systems of reasoning Methods for self-control Differences in short-term and long-term distributions The short term investor has a further emotional transformation of returns Composure value function Rational linear function v r sgn r r c Investors with short-term reactions will attribute utility to returns differently to long-term rational investors Loss aversion 2 v r EU E 1 e T v r sgn r r c Investors with short-term reactions will attribute utility to returns differently to long-term rational investors Loss aversion 2 v r EU E 1 e T v r sgn r r c These result in a remarkably simple rational trade-off between adjusted risk and expected returns D r r f B 2 T Desirability Compensation for risk Expected Excess Returns Total return – risk-free return 17 Risk / Risk Tolerance Effect can be completely reflected through a separate Anxiety score for any investment D r r f adj 2 A T Desirability Excess Returns 18 Compensation for risk Compensation for Anxiety (relative to risk free) Introducing the Anxiety measure 2 v r r adj T T A ln E e v r f r f 2 T 2 Total psychological compensation for returns variability 19 Compensation for rational risk Reduction in anxiety from existence of risk free investment Illustrating in risk-return space Maximum Desirability r B 2 r rf Accept Reject Desirability Desirability rf Risk compensation All components measured in % log returns Risk free return B 20 T Representing Anxiety graphically Maximum Desirability r B 2 r rf Accept ANXIETY Reject Desirability Risk compensation rf Risk free return B 21 T Portfolio optimisation for the anxious r Maximum Desirability True Efficient Frontier rf Desirability B 22 Portfolio optimisation for the anxious r True Efficient Frontier rf Desirability B 23 Portfolio optimisation for the anxious r Anxiety True Efficient Frontier rf Desirability B 24 Portfolio optimisation for the anxious r True Efficient Frontier Anxiety Efficient Frontier rf B 25 Portfolio optimisation for the anxious r True Efficient Frontier Anxiety Efficient Frontier rf Desirability B 26 Why would we use this? Pander to short term self Understand costs of Anxiety Bargaining between planner and doer Use small degree of short-term preferences to ‘take off the edge’ Use different time horizons to overcome myopia 27 The effect of time horizon on risk and desirability Excess Returns 1 Month 28 2 Month Risk Compensation 3 Month 6 Month Rational Desirability 1 Year 2 Year 5 Year The effect of time horizon on anxiety Excess Returns 1 Month 29 2 Month Risk Compensation 3 Month 6 Month Anxiety Compensation 1 Year 2 Year 5 Year The effect of time horizon on anxiety Excess Returns 1 Month 30 2 Month Risk Compensation 3 Month 6 Month Anxiety Compensation 1 Year 2 Year Desirability 5 Year Desirability for the rational and the anxious Rational Desirability 1 Month 31 2 Month 3 Month Desirability 6 Month 1 Year 2 Year 5 Year