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The Value of a Probability Forecast from Portfolio Theory

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Abstract

A probability forecast scored ex post using a probability scoring rule (e.g. Brier) is analogous to a risky financial security. With only superficial adaptation, the same economic logic by which securities are valued ex ante – in particular, portfolio theory and the capital asset pricing model (CAPM) – applies to the valuation of probability forecasts. Each available forecast of a given event is valued relative to each other and to the “market” (all available forecasts). A forecast is seen to be more valuable the higher its expected score and the lower the covariance of its score with the market aggregate score. Forecasts that score highly in trials when others do poorly are appreciated more than those with equal success in “easy” trials where most forecasts score well. The CAPM defines economically rational (equilibrium) forecast prices at which forecasters can trade shares in each other’s ex post score – or associated monetary payoff – thereby balancing forecast risk against return and ultimately forming optimally hedged portfolios. Hedging this way offers risk averse forecasters an “honest” alternative to the ruse of reporting conservative probability assessments.

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References

  • Bayarri M.J., DeGroot M.H. (1989), Optimal reporting of predictions. Journal of the American Statistical Association 84: 214–222

    Article  Google Scholar 

  • Bernardo J.M. (1979), Expected information as expected utility. The Annals of Statistics 7: 686–690

    Google Scholar 

  • Bernardo J.M., Smith A.F.M. (1994), Bayesian Theory. Wiley, New York

    Google Scholar 

  • Bernstein P.L. (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press, New York

    Google Scholar 

  • Bodie Z., Kane A., Marcus A.J. (1999), Investments 4th ed. Irwin McGraw-Hill, Boston

    Google Scholar 

  • Borch K. (1969), A note on uncertainty and indifference curves. Review of Economic Studies 36: 1–4

    Article  Google Scholar 

  • Brier G.W. (1950), Verification of forecasts expressed in terms of probability. Monthly Weather Review 78: 1–3

    Article  Google Scholar 

  • Chamberlain G. (1983), A characterization of the distributions that imply mean–variance utility functions. Journal of Economic Theory 29: 185–201

    Article  Google Scholar 

  • Clemen R.T. and Winkler R.L. (1990), Unanimity and compromise among probability forecasters. Management Science 36: 767–779

    Google Scholar 

  • Cochrane J. (2001), Asset Pricing. Princeton University Press, Princeton

    Google Scholar 

  • Dawid A.P. (1982), The well-calibrated Bayesian. Journal of the American Statistical Association 77: 605–613

    Article  Google Scholar 

  • de Finetti, B. (1937) La Prévision; ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré 7, 1–68. Reprinted as ‘Foresight: Its Logical Laws, Its Subjective Sources’ in Kyburg, H.E. and Smokler, H.E. (1980), Studies in Subjective Probability, 2nd ed, Kreiger, New York, pp. 54–118.

  • de Finetti B. (1962), Does it make sense to speak of ‘good probability appraisers’?. In: Good I.J. (eds) The Scientist Speculates: An Anthology of Partly Baked Ideas. Heinemann, London, pp 357–364

    Google Scholar 

  • de Finetti B. (1965), Methods for discriminating levels of partial knowledge concerning a test item. The British Journal of Mathematical and Statistical Psychology 18: 87–123

    Google Scholar 

  • de Finetti B. (1970), Logical foundations and measurement of subjective probability. Acta Psychologica 34: 129–145

    Article  Google Scholar 

  • de Finetti B. (1974), Theory of Probability, vol 1. Wiley, New York

    Google Scholar 

  • de Finetti, B. (1976), Probability: beware of falsifications, Scientia 111, 283–303. Reprinted in Kyburg, H.E. and Smokler, H.E. (1980), Studies in Subjective Probability, 2nd ed, Kreiger, New York, pp. 194–224.

  • DeGroot M.H. (1970), Optimal Statistical Decisions. McGraw-Hill, New York

    Google Scholar 

  • DeGroot M.H. and Feinberg S.E. (1982), Assessing probability assessors: calibration and refinement. In: Gupta S.S. and Berger J.O. (eds) Statistical Decision Theory and Related Topics III, vol I. Academic Press, New York, pp. 291–314

    Google Scholar 

  • DeGroot M.H. and Feinberg S.E. (1983), The comparison and evaluation of forecasters. The Statistician 32: 12–22

    Article  Google Scholar 

  • DeGroot M.H. and Feinberg S.E. (1986), Comparing probability forecasters: basic binary concepts and multivariate extensions. In: Goel P.K. and Zellner A. (eds) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Elsevier Science Publishers, Amsterdam, pp. 247–264.

    Google Scholar 

  • Elton E.J., Gruber M.J., Brown S.J., Goetzman W.N. (2003), Modern Portfolio Theory and Investment Analysis, 6th ed. Wiley, New York

    Google Scholar 

  • Epstein L. (1985), Decreasing risk aversion and mean-variance analysis. Econometrica 53: 945–962

    Article  Google Scholar 

  • Friedman D. (1983), Effective scoring rules for probabilistic forecasts. Management Science 29: 447–454

    Google Scholar 

  • Good I.J. (1952), Rational decisions. Journal of the Royal Statistical Society, Series B 14: 107–114

    Google Scholar 

  • Good I.J. (1976), Information, rewards, and quasi-utilities. In: Leach J.J. et al (eds) Science, Decision and Value. D. Reidel, Dordrecht, pp. 115–127.

    Google Scholar 

  • Good I.J. (1983), Good Thinking: The Foundations of Probability and Its Applications. University of Minnesota Press, MN

    Google Scholar 

  • Granger C.W.J. and Pesaran M.H. (2000), Economic and statistical measures of forecast accuracy. Journal of Forecasting 19: 537–560

    Article  Google Scholar 

  • Grinblatt M. and Titman S. (1989), Portfolio performance evaluation: old issues and new insights. Review of Financial Studies 2: 393–421

    Article  Google Scholar 

  • Grinblatt M. and Titman S. (1998), Financial Markets and Corporate Strategy. Irwin McGraw-Hill, New York

    Google Scholar 

  • Haugen R.A. (1997), Modern Investment Theory. Prentice Hall, NJ

    Google Scholar 

  • Huang C. and Litzenberger R.H. (1988), Foundations for Financial Economics. Prentice-Hall, NJ

    Google Scholar 

  • Ingersoll, J.E. (1987), Theory of Financial Decision Making, Rowman and Littlefield, Savage, MD.

  • Jurczenko, E. and Maillet, B. (2001), The three-moment CAPM: theoretical foundations and an asset pricing models comparison in a unified framework, Working Paper. TEAM-ESA 8059 du CNRS, University of Paris.

  • Kadane J.B and Winkler R.L (1988). Separating probability elicitation from utilities. Journal of the American Statistical Association 83: 357–363

    Article  Google Scholar 

  • Kilgour D.M. and Gerchak Y. (2004), Elicitation of probabilities using competitive scoring rules. Decision Analysis 2: 108–113

    Article  Google Scholar 

  • Kroll Y., Levy H. and Markowitz H.M. (1984), Mean-variance versus direct utility maximization. Journal of Finance 39: 47–75

    Article  Google Scholar 

  • Kraus A. and Litzenberger R. (1976), Skewness preference and the valuation of risk assets. Journal of Finance 31: 1085–1100

    Article  Google Scholar 

  • Krouse C.G. (1986), Capital Markets and Prices: Valuing Uncertain Income Streams. Elsevier, Amsterdam

    Google Scholar 

  • Lad F. (1996), Operational Subjective Statistical Methods: A Mathematical, Philosophical and Historical Introduction. Wiley, New York

    Google Scholar 

  • LeRoy S.E. and Werner J. (2001), Principles of Financial Economics. Cambridge University Press, Cambridge MA

    Google Scholar 

  • Leland H.E. (1999), Beyond mean-variance: performance measurement in a nonsymmetrical world. Financial Analysts Journal 55: 27–36

    Article  Google Scholar 

  • Levy H. and Markowitz H.M. (1979), Approximating expected utility by a function of mean and variance. American Economic Review 69: 308–317

    Google Scholar 

  • Lewis M. (2003), Moneyball. Norton, New York

    Google Scholar 

  • Lichtendahl, K.C. and Winkler, R.L. (2005), Probability elicitation, scoring rules and competition among forecasters, Working Paper, Fuqua School of Business, Duke University. Durham, NC.

  • Lindley D.V. (1982a), Scoring rules and the inevitability of probability. International Statistical Review 50: 1–26

    Article  Google Scholar 

  • Lindley D.V. (1982b), The improvement of probability judgements. Journal of the Royal Statistical Society Series A 145: 117–126

    Article  Google Scholar 

  • Lindley D.V. (1988), The use of probability statements. In: Lindley D.V. and Clarotti C.A. (eds) Accelerated Life Testing and Experts’ Opinions in Reliability. Elsevier, Amsterdam, pp. 25–57.

    Google Scholar 

  • Lintner J. (1965), The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47: 13–37

    Article  Google Scholar 

  • Luenberger D.G. (1998), Investment Science. Oxford University Press, New York

    Google Scholar 

  • Markowitz H.M. (1952), Portfolio selection. Journal of Finance 7: 77–91

    Article  Google Scholar 

  • Markowitz H.M. (1959), Portfolio Selection: Efficient Diversification of Investments. Wiley, New York

    Google Scholar 

  • Matheson J. and Winkler R.L. (1976), Scoring rules for continuous probability distributions. Management Science 22: 1087–1096

    Google Scholar 

  • Meyer J. (1987), Two moment decision models and expected utility maximization. American Economic Review 77: 421–430

    Google Scholar 

  • Meyer J. and Rasche R. (1992) Sufficient conditions for expected utility to imply mean-standard deviation rankings: empirical evidence concerning the location and scale condition. The Economic Journal 102: 91–106

    Article  Google Scholar 

  • Milne F. (1995) Finance Theory and Asset Pricing. Oxford University Press, Oxford

    Google Scholar 

  • Mossin J. (1966), Equilibrium in a capital asset market. Econometrica 34: 768–783

    Article  Google Scholar 

  • Murphy A.H. (1973), Hedging and skill scores for probability forecasts. Journal of Applied Meteorology 12: 215–223

    Article  Google Scholar 

  • Murphy A.H. (1993), What is a good forecast? An essay on the nature of goodness in weather forecasting. Weather and Forecasting 8: 281–293

    Article  Google Scholar 

  • Murphy A.H. and Daan H. (1985), Forecast evaluation. In: Murphy A.H., Katz R.W. (eds) Probability, Statistics and Decision Making in the Atmospheric Sciences. Westview Press, Oxford, pp. 379–437.

    Google Scholar 

  • Murphy A.H. and Epstein E.S. (1967a), Verification of probabilistic predictions: a brief review. Journal of Applied Meteorology 6: 748–755

    Article  Google Scholar 

  • Murphy A.H. and Epstein E.S. (1967b), A note on probability forecasts and “hedging”. Journal of Applied Meteorology 6: 1002–1004

    Article  Google Scholar 

  • Murphy A.H. and Epstein E.S. (1972a), Scalar and vector partitions of the probability score: Part I The two-state situation. Journal of Applied Meteorology 11: 273–282

    Article  Google Scholar 

  • Murphy A.H. and Epstein E.S. (1972b), Scalar and vector partitions of the probability score: Part II. N-State situation. Journal of Applied Meteorology 11: 1183–1192

    Article  Google Scholar 

  • Murphy A.H. and Winkler R.L. (1970) Scoring rules in probability assessment and evaluation. Acta Psychologica 34: 273–286

    Article  Google Scholar 

  • Murphy A.H. and Winkler R.L. (1971), Forecasters and probability forecasts: some current problems. Bulletin of the American Meteorological Society. 52: 239–247

    Article  Google Scholar 

  • Murphy A.H. and Winkler R.L. (1984), Probability forecasting in meteorology. Journal of the American Statistical Association 79: 489–500

    Article  Google Scholar 

  • Murphy A.H. and Winkler R.L. (1987), A general framework for forecast evaluation. Monthly Weather Review 115: 1330–1338

    Article  Google Scholar 

  • Murphy A.H. and Winkler R.L. (1992), Diagnostic verification of probability forecasts. International Journal of Forecasting 7: 435–455

    Article  Google Scholar 

  • Nau R.F. (1985), Should scoring rules be ‘effective’?. Management Science 31: 527–535

    Google Scholar 

  • Nau R.F. (2001), De Finetti was right: probability does not exist. Theory and Decision 51: 89–124

    Article  Google Scholar 

  • Nau R.F. and McCardle K. (1991), Arbitrage, rationality, and equilibrium. Theory and Decision 31: 199–240

    Article  Google Scholar 

  • Offerman, T., Sonnemans, J., van de Kuilen, G. and Wakker, P.P. (2006), Correcting proper scoring rules for risk attitudes, Working Paper. Department of Economics, University of Amsterdam.

  • O’Hagan A. (1994), Bayesian Statistics. Kendall’s Advanced Theory of Statistics Vol. 2B. Cambridge University Press, Cambridge

    Google Scholar 

  • Pliska S.R. (1997), Introduction to Mathematical Finance: Discrete Time Models. Blackwell, Oxford

    Google Scholar 

  • Prelec D. (2004), A Bayesian truth serum for subjective data. Science 306: 462–466

    Article  Google Scholar 

  • Ross S.A. (1989), Finance. In: Eatwell J., Milgate M., Newman P. (eds) The New Palgrave: A Dictionary of Economics. Macmillan, London, pp. 1–34

    Google Scholar 

  • Ross S.A. (1971), Elicitation of personal probabilities and expectations. Journal of the American Statistical Association 66: 783–801

    Article  Google Scholar 

  • Sarin R.K. and Winkler R.L. (1980), Performance-based incentive plans. Management Science 26: 1131–1144

    Article  Google Scholar 

  • Scott W.R. (1979), Scoring rules for probabilistic reporting. Journal of Accounting Research 17: 156–178

    Article  Google Scholar 

  • Sharpe W.F. (1964), Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 19: 425–442

    Article  Google Scholar 

  • Treynor, J.L. (1961), Towards a theory of market value of risky assets, Unpublished manuscript.

  • Winkler R.L. (1967), The quantification of judgment: some methodological suggestions. Journal of the American Statistical Association 62: 1105–1120

    Article  Google Scholar 

  • Winkler R.L. (1969), Scoring rules and the evaluation of probability assessors. Journal of the American Statistical Association 64: 1073–1078

    Article  Google Scholar 

  • Winkler R.L. (1971), Probabilistic prediction: some experimental results. Journal of the American Statistical Association 66: 675–685

    Article  Google Scholar 

  • Winkler R.L. (1986), On “good probability appraisers”. In: Goel P.K, Zellner A. (eds) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Elsevier Science Publishers, Amsterdam, pp. 265–278.

    Google Scholar 

  • Winkler R.L. (1996), Scoring rules and the evaluation of probabilities (with discussion). Test 5: 1–60

    Article  Google Scholar 

  • Winkler R.L. (1999), Evaluation of probabilities: a level playing field?. In: Shanteau J., Mellers B.A., Schum D.A. (eds) Decision Science and Technology: Reflections on the Contributions of Ward Edwards. Kluwer, Boston, pp. 155—170.

    Google Scholar 

  • Winkler R.L. and Murphy A.H. (1968), Good probability assessors. Journal of Applied Meteorology 7: 751–758

    Article  Google Scholar 

  • Winkler R.L. and Murphy A.H. (1970), Nonlinear utility and the probability score. Journal of Applied Meteorology 9: 143–148

    Article  Google Scholar 

  • Yates J.F., Price P.C., Lee J.W. and Ramirez J. (1996), Good probabilistic forecasters: the ‘consumer’s’ perspective. International Journal of Forecasting 12: 41–56

    Article  Google Scholar 

  • Zhao, Y. and Ziemba, W.T. (2002), Mean-variance versus expected utility in dynamic investment analysis, Working Paper. Faculty of Commerce and Business Administration, University of British Columbia.

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  1. National Australia Bank Professor of Finance, School of Business, University of Sydney, Sydney, NSW, 2006, Australia

    D. J. Johnstone

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Johnstone, D.J. The Value of a Probability Forecast from Portfolio Theory. Theor Decis 63, 153–203 (2007). https://doi.org/10.1007/s11238-006-9023-1

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