Decision Making Under Risk: A Prescriptive Approach

1. Decision Making Under Risk:A Prescriptive Approach Martin Sewell mvs25@cam.ac.uk Behavioral Finance & Economics Research Symposium – 2009 23–25 September 2009 Chicago

2. What is risk? Risk is uncertainty We can measure people’s risk tolerance by providing options of the following form, and asking which option they would prefer. A) $100 B) 50% chance of $0 50% chance of $200

3. How best to deal with uncertainty? • A Dutch book is a gambling term for a set of odds and bets which guarantees a profit, regardless of the outcome of the gamble. • If an individual is not susceptible to a Dutch book, their previsions are said to be coherent. • A set of betting quotients is coherent if (Ramsey 1926; de Finetti 1937; Shimony 1955) and only if (Kemeny 1955; Lehman 1955) they satisfy the axioms of probability. • Probability is necessary when dealing with uncertainty. • Probability is sufficient when dealing with uncertainty. • Everyone should be a Bayesian.

4. Philosophical terminology • Normative – ‘ought’ (what we should do according to our model) • Descriptive – ‘is’ (what we actually do in the real world) • Prescriptive – ‘best compromise’ (what we should do in practice)

5. Evolution of risk aversion • Evolution is essentially ‘survivorship bias’. • Although it is the genes that have evolved to survive, and the individuals that have evolved to reproduce, individuals must survive in order to reproduce. • The most fundamental bias is the status quo bias (also known as conservatism). • In general, we prefer to avoid uncertainty. • We also exhibit a preference for known risks over unknown risks, that is, we prefer probabilities that are easy to calculate. This is known as ambiguity aversion.

6. Loss aversion • The idea of loss aversion is that losses and disadvantages have a greater impact on preferences than gains and advantages. • Our ancestors lived in groups, which meant that respect for private property would have likely evolved as a Nash equilibrium. • It could be that apparent ‘loss aversion’ is actually due to a preference for the status quo. • Gal (2006) and Gintis (2007)

7. Growth optimal strategy • A financial market is a multiplicative process • loge(wealth) is additive • Risk neutral in terms of loge(wealth) • Implies a small degree of risk aversion regarding wealth • Because the log utility function is concave • To maximize growth of wealth, maximize the expected value of the logarithm of wealth after each period (Kelly 1956; Breiman 1961).

8. Expected utility theory • Also called von-Neumann Morgenstern utility • Bernoulli (1738) and von Neumann and Morgenstern (1944) • Four axioms define a rational decision maker: completeness, transitivity, independence and continuity • Let U(Oi) be the utility of outcome Oi, and P(Oi) be the probability of occurrence of outcome Oi, then Utility = ∑U(Oi)P(Oi). • Expected utility theory states that one should choose the option with the highest utility. • In this instance, an outcome would be final wealth.

9. Prospect theory • Kahneman and Tversky (1979) • Investors don’t consider expected levels of final wealth, but gains and losses relative to some reference point. • Investors are loss averse. • People tend to overreact to small probability events, but underreact to medium and large probabilities. • In cumulative prospect theory (Tversky and Kahneman 1992), cumulative probabilities are transformed, rather than the probabilities. • Fourfold pattern of risk attitudes: risk aversion for gains and risk seeking for losses of moderate to high probability; risk seeking for gains and risk aversion for losses of low probability.

10. (Cumulative) Prospect theory

11. Expected utility theory vs Prospect theory There are two fundamental reasons why prospect theory (which calculates value) is inconsistent with expected utility theory. • Whilst utility is necessarily linear in the probabilities, value is not. • Whereas utility is dependent on final wealth, value is defined in terms of gains and losses (deviations from current wealth).

12. Sharpe ratio • Most prevalent performance metric in use by the financial industry • Sharpe ratio = E[rp – rf]/σ • Sharpe ratio assumes either 1) normally distributed returns or 2) mean-variance preferences. • Sharpe ratio ignores all moments higher than the second • The third moment is skewness • People like positive skew

13. Positive skew

14. A high Sharpe ratio strategy • Goetzmann, et al. (2002)

15. Risk is a personal thing • Risk preferences vary across individuals. • Risk preferences vary across different groups of people. • Barsky, et al. (1997) found that measured risk tolerance is related to risk-taking behaviour, but also uncovered some perhaps surprising results.

16. Risk seeking • the highly educated • very wealthy • heavy drinkers • those without health insurance • immigrants • Jewish • Hispanics • (especially) Asians

17. Risk averse • those with an average education • average wealth • average income • employees with health insurance • people in their sixties

18. Relative risk • Is your risk profile absolute, or dependent on the risk profile of others around you? • For example, one is presumably more likely to gamble if taken to a casino. • We are competing with our peers, so it is our relative wealth that counts. • Therefore, a relative risk profile makes sense for Homo sapiens, if not for Homo economicus.

19. Risk and time • The value of money in the future is determined by interest rates. • But how do we determine the social discount rate? • How much do you care about the future in 1, 10, 100 or 1000 years time? • This is a moral judgement, and morality is a product of the gene-centred evolutionary forces which shape human social psychology. • We have evolved on a ‘need to know’ basis, that is, we are only required to be aware of our proximal motivations (e.g. sex and caring for kin), rather than our ultimate motivation (reproduction). So most people care only about their living relatives, not the future of mankind.

20. Climate change • Human-induced climate change is a classic case of Hardin’s tragedy of the commons — the benefits of burning fossil fuels accrue to individuals, companies and nations, whilst the costs accrue to the planet as a whole. • Anthropogenic climate change is ‘unequivocal’ (over 90% certain). • Global warming will largely affect other people’s genes in future generations, so we need a risk profile that is global in nature, and considers the present and discounts the future. • We cannot transcend our genes, so policy makers must reconcile ‘is’ with ‘ought’.

21. Economics – the dismal science? • Economics is amenable to quantitative techniques. • Economics is an approximation of psychology. • This is why economics suffers from physics envy. • In economics, why are normative (‘ought’) and descriptive (‘is’) different? • Humans have evolved ‘as if’ reproduction was the sole goal for which they were ‘designed’ and everything they do is a means to that end (Williams 1966). • The neoclassical economist’s canon that individuals are motivated to maximize utility is accurate if and only if utility corresponds to the number of their genes in future generations.

22. Martin Sewell mvs25@cam.ac.uk