Portfolio Optimisation for the Anxious Greg B Davies, PhD Head of Behavioural Finance June 2010

2 “The market can stay irrational, longer than you can stay solvent” John Maynard Keynes Harry Markowitz – Nobel Prize 1990Daniel Kahneman – Nobel Prize 2002

Value function in descriptive theories: CPT Losses (£) Utility Loss aversion: Steeper for losses Reference Point Gains (£)

4 And with risk=variance, expected utility theory is equivalent to mean-variance optimisation Portfolio Efficient Frontier Risk free rate Market Portfolio Optimal Portfolio Risk/Return Trade-off (Indifference Curve)

5 Only works under specific, and largely incorrect assumptions  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 Example Quadratic Utility Function Return Utility

6 This means both the risk measure and the risk-return trade-off are flawed Risk free rate

The exponential function shows aversion to left tail events and preference for positive skewness in log returns Low Risk Tolerance Medium Risk Tolerance High Risk Tolerance Utility Log returns All display CRRA throughout domain

8 These result in a remarkably simple rational trade-off between adjusted risk and expected returns Desirability Expected Excess Returns Total return – risk-free return Compensation for risk Risk / Risk Tolerance

9 Illustrating in risk-return space Risk free return All components measured in % log returns Reject Accept Risk compensation Desirability Maximum Desirability Desirability

10 Because risk is mis-specified, the mean-variance ‘efficient’ frontier is not truly risk-return efficient Desirability (In)efficient Frontier True Efficient Frontier

11 Traditional portfolio theory trades off risk and return of the portfolio in the long run Portfolio Risk Expected Portfolio Return Efficient Frontier Low risk tolerance, low portfolio risk, low return High risk tolerance, high portfolio risk, high return

12 The emotional experience with the investment journey has potentially greater influence on the final result Which investor is happier? (Green, black or red) Portfolio Value Time Danger of selling low Danger of buying high

Ulysses  Self-control  Dual self model  Two systems of reasoning  Methods for self-control  Differences in short-term and long-term distributions 13

The short term investor has a further emotional transformation of returns Composure value function Rational linear function

Investors with short-term reactions will attribute utility to returns differently to long-term rational investors Loss aversion

Investors with short-term reactions will attribute utility to returns differently to long-term rational investors Loss aversion

17 These result in a remarkably simple rational trade-off between adjusted risk and expected returns Desirability Expected Excess Returns Total return – risk-free return Compensation for risk Risk / Risk Tolerance

18 Effect can be completely reflected through a separate Anxiety score for any investment Compensation for risk Compensation for Anxiety ( relative to risk free) Excess Returns Desirability

19 Introducing the Anxiety measure Total psychological compensation for returns variability Compensation for rational risk Reduction in anxiety from existence of risk free investment

20 Illustrating in risk-return space Risk free return All components measured in % log returns Reject Accept Risk compensation Desirability Maximum Desirability Desirability

21 Representing Anxiety graphically Risk free return Reject Accept Risk compensation Maximum Desirability Desirability ANXIETY

22 Portfolio optimisation for the anxious Desirability True Efficient Frontier Maximum Desirability

23 Portfolio optimisation for the anxious Desirability True Efficient Frontier

24 Portfolio optimisation for the anxious Desirability True Efficient Frontier Anxiety

25 Portfolio optimisation for the anxious True Efficient Frontier Anxiety Efficient Frontier

26 Portfolio optimisation for the anxious True Efficient Frontier Anxiety Efficient Frontier Desirability

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

28 The effect of time horizon on risk and desirability

29 The effect of time horizon on anxiety

30 The effect of time horizon on anxiety

31 Desirability for the rational and the anxious