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LINKING PSYCHOMETRIC RISK TOLERANCE WITH CHOICE BEHAVIOUR Peter Brooks, Greg B. Davies and Daniel P. Egan FUR Conference – July 2008 Presentation Aims 2 To introduce the Barclays Wealth Risk Tolerance Scale To introduce the effects of an exponential utility function on asset allocation To describe an experiment that provides a link between risk tolerance scores and risk parameters. Are different risk profiles characterised by different risk/utility parameters in choices? Pre-experiment Analysis Overview Examine risk and utility measures using simulated portfolios involving equities and bonds Mix the simulated portfolios with different proportions of cash Holding cash is assumed to be a riskless alternative Calculate the optimal portfolio for different values of the risk parameter 3 Example Utility Measure 5 Year Bond/Equity Mixes Low values of imply risk tolerant behaviour – Optimal portfolio is 100% equities 6 Higher values of imply risk averse behaviour – Optimal portfolio is now a mix of equities and bonds 2 -4 -6 100B 20E-80B -8 -10 40E-60B 50E-50B 60E-40B 80E-20B 100E -12 -14 4 1 e x EU E 0.39 0.37 0.35 0.33 0.31 0.29 0.27 0.25 0.23 0.21 0.19 0.17 0.15 0.13 0.11 0.09 -2 0.07 0 0.05 Expected Utility 4 Optimal Portfolio Mixes with Cash 5 Year Investment Horizon For low – optimal portfolio is 100% Equities For between 0.08 and 0.16, the optimal portfolio is a mix of equities and bonds For greater than 0.17, the optimal asset allocation includes cash. 100 90 Asset Allocation % We have modelled a range of values of the risk parameter for 5 year returns 80 Cash 70 60 50 40 Bonds 30 20 Equities 10 5 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 Pre-experiment Analysis Overview 2 The analysis suggests that the optimal portfolio is sensitive to the value of a risk parameter. Assuming utility maximisation individual choices between portfolios make it possible to calibrate a risk parameter. Choices constrain a risk parameter to a range of values where the portfolio would be preferred by a utility maximising individual. Analysing a number of choices makes it possible to find a “best” value of the risk parameter for each individual. 6 Barclays Wealth Risk Tolerance Scale 8 question psychometric questionnaire Risk Tolerance Scale Distribution Higher=More Risk Tolerant 0.08 Responses given on a 5-point Likert scale 0.07 Produces a score between 8 and 40 Scores bucketed into 5 risk profiles from low up to high. 0.05 Frequency Higher scores signal higher risk tolerance 0.06 0.04 0.03 0.02 Risk Tolerance Score 7 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 0 8 0.01 Experiment Aims 8 To test various risk measures and utility functions using actual choices To estimate risk/utility parameters for individual respondents. To provide a link between the risk tolerance scores and risk parameters. Are different risk profiles characterised by different risk/utility parameters in choices? Experimental Design Create stylised distributions of the final values of an investment. It is difficult to use distributions based upon real data. Increases in volatility cause the tails of the distribution to become long. Long tailed distributions are difficult to display accurately to survey respondents. Take log-normal distribution and set the mean and standard deviation. Generate 120 equally spaced observations across the distribution. Round each of these observations to the nearest integer. Plot the frequency table of the observations to create the distributions for the experiment. 9 Experimental Design Expected utility is increasing in the mean of the distribution. Expected utility is decreasing in the “risk” of the distribution. Create a preference order between two distributions by compensating for an increase in “risk” by increasing the mean. The most risk averse will prefer lower mean and lower “risk” distributions. The least risk averse will prefer higher mean and higher “risk” distributions. 10 Example Distribution Mean = £103,000 11 Example Distribution 2 Mean = £105,000 12 Distribution Comparisons – Example Using Exponential Risk Measures 1 0.8 0.6 Expected Utility 0.4 Mean = 103 Mean = 102 Mean = 104 0.2 Mean = 105 0 -0.2 Mean = 106 -0.4 -0.6 -0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Utility Parameter () 13 0.8 0.9 1.0 Distribution Comparisons – Example Using Exponential Risk Measures 1.2 1 Expected Utility 0.8 Mean = 104 0.6 Mean = 103 0.4 Mean = 102 Mean = 105 0.2 Mean = 106 0 -0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Utility Parameter () 14 0.8 0.9 1.0 Distribution Comparisons – Example Using Exponential Risk Measures 1 0.8 0.6 Expected Utility Mean = 103 0.4 Mean = 102 0.2 Mean = 104 0 Mean = 105 -0.2 Mean = 106 -0.4 -0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Utility Parameter () 15 0.8 0.9 1.0 Experiment Procedures Participants recruited through iPoints 6 section experiment Participants paid in iPoints 1. Demographics All participants reported either gross annual income above £50k or investable wealth above £100k 2. Psychometric Risk Tolerance Delivered through a non-branded external website Respondents had participated in previous surveys but had not participated within the past 6 months Over-sampling of the extreme risk profiles 16 3. Training stage 4. 9 Pairwise choice tasks between distributions 5. Filler Task – maze 6. 9 Pairwise choice tasks between distributions Experimental Results 108 Participants completed all parts of the survey 1 participant removed for inconsistent responses Individuals in higher risk profiles tend to choose higher variance distributions more often Use MLE to estimate the utility risk parameter for individuals - grouped by risk tolerance score 17 0.3 0.25 Frequency Over-sampling of the end points successful 0.35 0.2 0.15 0.1 0.05 0 Low Med-Low Medium Med-High High MLE Fit Results 2 Estimated Utility Parameter 1.5 1 0.5 0 5.00 10.00 15.00 20.00 25.00 -0.5 Mean Risk Tolerance Score 18 30.00 35.00 40.00 Conclusions and Extensions Our psychometric risk tolerance measure is consistent with risky choice There is potential for a behavioural calibration of a risk measure for portfolio optimisation Separate work on whether utility measures are better than variance, VaR or CVaR as risk measures for portfolio optimisation Geographical calibration exercise – current ongoing work 19