SUBJECTIVE VS OBJECTIVE PROBABILITIES

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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
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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

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Perception vs Reality Subject vs Object
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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!

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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

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Possible Design for a Wisebot ...
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Logical and Analogical Model Building
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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)

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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

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Probability as Betting Odds

• Betting odds Q1 means pay (Q1) for 1 stake
  (dividend convention)
• Ramsey Probability after Ramseys critique of
  Keynes

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Pure Arbitrage Pricing of Odds

• H horse in the race punters bet Nk on horse k
• The breakeven odds Q1 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

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Inferring Subjective Probability

• W.W. Snyder, Horse Racing Testing The Efficient
  Market, Journal of Finance 33, 1109 (1978)

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Expectation Pricing of Odds

• W.W. Snyder, Horse Racing Testing The Efficient
  Market, Journal of Finance 33, 1109 (1978)

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Subjective Bias of Bettors

• W.W. Snyder, Horse Racing Testing The Efficient
  Market, Journal of Finance 33, 1109 (1978)

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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

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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

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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!

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Miners vs Industrials (Estimated Total Return
Relative 36 to 04)
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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