SUBJECTIVE VS OBJECTIVE PROBABILITIES Reflections on the Pricing of Financial Claims

1. SUBJECTIVE VS OBJECTIVE PROBABILITIES Reflections on the Pricing of Financial Claims Kingsley Jones Quantitative Analyst Bernstein Value Equities AllianceBernstein Australia Limited Q-Group Colloquium, Manly 14 Sep 2006

2. Agenda • Perception vs Reality • Logic vs Analogy in Thinking and Model Building • Bayesian Inference as Plausible Reasoning • Horse Racing and Ramsey Probability (Betting Odds) • Arbitrage vs Expectation Pricing • Index vs Active Investing • Fundamental vs Technical Analysis • History does not Repeat but it Rhymes • Some Research Questions

3. Perception vs Reality: Subject vs Object

4. Logic vs Analogy in Thinking and Model Building • Consider a race of robots with two opposing philosophies ... • Zealot: Everything is True or False – Certainty – “IS A” • Zenbot: Nothing is both True and Universal – Grey – “IS LIKE” • Difference lies in the degree of rational belief accorded to a model of the world – recognizing that the robot internal representation is not the same as the world and that the premise that today X is more plausible in the world may later be replaced by its negation not X tomorrow • Particularly relevant for participatory thinking agents • Today “is like” yesterday therefore it “is a” day to wear a coat!

5. Bi-Cameral Robot • WiseBot ... part Zealot part Zenbot • Zealot: Abstract rules – “If A then B” – provide model • Zenbot: Generate categories – things that are B and not B – are examples of the newly invented category (C, not C)! • Rules can be axiomatic facts: “All swans are white” or deductions “A black bird is not a swan” or inductions “X% of swans are white”. • Categories can be adduced from observation: “I found a black swan, so the category swans must be subdivided and the former rule refined.” It is tricky to describe what this thought process is – creativity? • "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality." A. Einstein

6. Possible Design for a Wisebot ...

7. Logical and Analogical Model Building

8. Bayesian Inference as Plausible Reasoning • Bayes rule can be thought of as a consistency rule for plausible inference • J.M. Keynes, “A Treatise on Probability” (Macmillan, 1921) • R.T. Cox, “Algebra of Probable Inference” (Johns Hopkins U, 1961) • E.T. Jaynes, “Probability – The Logic of Science” (Cambridge, 2003)

9. Mixed Subjective and Objective Probabilities • Analysts may have strong (useful/useless) prior premises or rules • “Subjective” inputs: non-repeating situations analogous to previous experience but maybe not identical – “Looks like a credit bubble but it seems a bit different this time because of clear demographic factors.” • “Objective” inputs: repeating situations with a clear mechanism or behavior at work which provides plausible inference rules – “Usually credit bubbles end when the need to purge excess debt leads to a spike in short term liquidity demands and forced asset sales.” • “The past does not repeat itself, but it rhymes” Mark Twain

10. Probability as Betting Odds • Betting odds Q:1 – means pay $(Q+1) for $1 stake (dividend convention) • Ramsey Probability – after Ramsey’s critique of Keynes

11. Pure Arbitrage Pricing of Odds • H horse in the race – punters bet Nk on horse k • The breakeven odds Q:1 are set by equating net payout with money raised from the punters who did not win in that race • In practice the track will take a margin for costs and profit

12. Inferring Subjective Probability • W.W. Snyder, “Horse Racing: Testing The Efficient Market”, Journal of Finance 33, 1109 (1978)

13. Expectation Pricing of Odds • W.W. Snyder, “Horse Racing: Testing The Efficient Market”, Journal of Finance 33, 1109 (1978)

14. Subjective Bias of Bettors • W.W. Snyder, “Horse Racing: Testing The Efficient Market”, Journal of Finance 33, 1109 (1978)

15. Tote vs Bookies • The Totaliser is set up to offer odds so the track always wins • Bookies offer odds based on handicapping, form and punter foibles • Bookies can make money by arbitraging their superior knowledge of racing form and punter behaviour while the tote makes money by “shaving the coin” i.e. pays out less than was staked in any race • Tote = “SUBJECTIVE” derived from “OBJECTIVE” market • Bookie = “OBJECTIVE” derived from “SUBJECTIVE” form

16. Index Fund vs Active Fund Analogy • Staking money according to market capitalization is a flow based algorithm which assures one of the market return • However, provided conditions for small companies are not adverse compared with large companies this will hindsight bias towards large prior winners and be short small prior losers • In that sense, index fund investing is SUBJECTIVE based on the OBJECTIVELY offered market weights • Conversely, active investing is OBJECTIVE in paying attention to future prospects but these must be assessed SUBJECTIVELY

17. Fundamental vs Technical Analogy • Technical methods study the market for the securities of a company recognizing that they are part interests in the firm with fluctuating demand and supply conditions for purchase and sale • Fundamental methods study the market for the company activities to assess its future earning potential and thus the prospect for higher wealth through accumulated dividends or retained earnings • In this sense, technical methods weigh SUBJECTIVE sentiment based on OBJECTIVELY measured prices and volumes • Conversely, fundamental methods weigh OBJECTIVE earnings prospect based on SUBJECTIVELY constructed models • In practice, both value and momentum figure in setting market prices!

18. Miners vs Industrials (Estimated Total Return Relative – ‘36 to ‘04)

19. Some Research Questions • Greater application of probability estimation models to analysis of investment markets – historical examples like Altman credit score models – but there is more that can be done along these lines • More systematic exploration of how subjective (qualitative) and objective (quantitative) information can be blended and on robust models for deciding which is the weak/strong component • Consideration to the difference between the subjective and objective pricing of derivative claims according to either dynamic replication or static replication (synthesis of forwards via put-call parity) • Recognition that behavioral biases can impact the processing of information due to inattention to alternative hypotheses or the general weakness of truth standards for social propositions