1. 1 The Valuation of Stochastic Cash Flows Leigh J. Halliwell, FCAS, MAAA CAS Reinsurance Seminar Philadelphia, PA June 2-3, 2003 Leigh J. Halliwell, FCAS, MAAA CAS Reinsurance Seminar Philadelphia, PA June 2-3, 2003

2. 2 The Coming Financial Revolution Small wonder that accountants and lawyers control the insurance industry under the ruling theory. To borrow from Karl Marx: Actuaries of the world unite; you have nothing to lose but your chains! (cf. Borch quote, p. 34)

3. 3 Down with Capital Allocation! Capital (or money) does not work, despite advertising slogans. Would capital allocated to a one-year hurricane treaty work off season (Dec-May)? Can we make capital moonlight? Some try to have it both ways. Capital allocation inevitably confuses rate of return (% per year) with return (%). Example of short-lived exposure: 25% per year is 0.5% per week Capital (or money) does not work, despite advertising slogans. Would capital allocated to a one-year hurricane treaty work off season (Dec-May)? Can we make capital moonlight? Some try to have it both ways. Capital allocation inevitably confuses rate of return (% per year) with return (%). Example of short-lived exposure: 25% per year is 0.5% per week

4. 4 Author’s Extensive Argumentation “Conjoint Prediction of Paid and Incurred Losses,” 1997 CAS Loss Reserving Discussion Papers, Appendix F. “ROE, Utility, and the Pricing of Risk,” 1999 CAS Spirng Reinsurance Call Papers, 71-136. “A Critique of Risk-Adjusted Discounting,” 2001 ASTIN Colloquium, www.casact.org/coneduc/reinsure/astin/ 2000/halliwell1.doc. “The Valuation of Stochastic Cash Flows,” 2003 Spring Reinsurance Call Papers, Appendix A. “Conjoint Prediction of Paid and Incurred Losses,” 1997 CAS Loss Reserving Discussion Papers, Appendix F. “ROE, Utility, and the Pricing of Risk,” 1999 CAS Spirng Reinsurance Call Papers, 71-136. “A Critique of Risk-Adjusted Discounting,” 2001 ASTIN Colloquium, www.casact.org/coneduc/reinsure/astin/ 2000/halliwell1.doc. “The Valuation of Stochastic Cash Flows,” 2003 Spring Reinsurance Call Papers, Appendix A.

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6. 6 Implications of Exhibit 1 Present value is a random variable. Exhibit 2 (p. 40) graphs the CDF of the PV of this stochastic cash flow. The value of this stochastic cash flow should lie between 50 and 100. If all points lay on the same isobar, the value would be that of the isobar. Objection: What if the coordinate system changes? If two PV random variables are (almost surely) equal, then they must have the same value. PVs of outcomes are like sufficient statistics. Negative PVs as legitimate as positive; solvency and bankruptcy irrelevant (footnotes 25 and 26). Present value is a random variable. Exhibit 2 (p. 40) graphs the CDF of the PV of this stochastic cash flow. The value of this stochastic cash flow should lie between 50 and 100. If all points lay on the same isobar, the value would be that of the isobar. Objection: What if the coordinate system changes? If two PV random variables are (almost surely) equal, then they must have the same value. PVs of outcomes are like sufficient statistics. Negative PVs as legitimate as positive; solvency and bankruptcy irrelevant (footnotes 25 and 26).

7. 7 Question: Which Wealth is Better? Answer: Which is greater, E[u(W 1 )] or E[u(W 2 )]?

8. 8 Valuing Projects by Expected Utility Current stochastic wealth W Project has present value X and cost q. Wealth with project is W  X  q. Compare E[u(W  X  q)] with E[u(W)]. Borch’s insurance formulation, W  L  p, equivalent (p. 16) Instantaneous formulation Value  H[X] frequent in our actuarial literature. (e.g., Bühlmann, Gerber) Some are so future-oriented that they must displace instantaneous results to the end of an accounting period (with interest)! Why is a future time better than the present? Is an instantaneous problem illegitimate? Current stochastic wealth W Project has present value X and cost q. Wealth with project is W  X  q. Compare E[u(W  X  q)] with E[u(W)]. Borch’s insurance formulation, W  L  p, equivalent (p. 16) Instantaneous formulation Value  H[X] frequent in our actuarial literature. (e.g., Bühlmann, Gerber) Some are so future-oriented that they must displace instantaneous results to the end of an accounting period (with interest)! Why is a future time better than the present? Is an instantaneous problem illegitimate?

9. 9 What’s new? Market-Tempering. Comparing E[u(W  X  q)] with E[u(W)] can give you only a ceiling price, not the appropriate price. An economic agent should be free to choose how much of X to purchase at unit price q. Cash flows are scaleable. New objective: Maximize f(  )  E[u(W  X  q)] “Fundamental Theorem” of Appendix B: For a given q, f(  ) has one and only one maximum. (cf. Exhibit 5, p. 43) Two or more agents together find the unique price q at which each maximizes its expected utility and all of X clears. The agents constitute a market, but each agent is entitled to its own beliefs. The market is an epiphenomenon. Comparing E[u(W  X  q)] with E[u(W)] can give you only a ceiling price, not the appropriate price. An economic agent should be free to choose how much of X to purchase at unit price q. Cash flows are scaleable. New objective: Maximize f(  )  E[u(W  X  q)] “Fundamental Theorem” of Appendix B: For a given q, f(  ) has one and only one maximum. (cf. Exhibit 5, p. 43) Two or more agents together find the unique price q at which each maximizes its expected utility and all of X clears. The agents constitute a market, but each agent is entitled to its own beliefs. The market is an epiphenomenon.

10. 10 A New Example Four friends each have $500,000 in cash and exponential utilities with a  5.0E  06. All agree that there is a 0.01 chance of a fire in Abel’s house, whose replacement cost is $250,000. Charlie believes that a partial loss is as likely as a total loss; to the others it’s all or nothing. Baker, Charlie, and Dale ignore their own exposures, except that Dale lives next door to Abel, and reckons that his $250,000 house might catch fire from Abel’s (0.001 probability). How should this “mutual market” insure Abel’s house? Four friends each have $500,000 in cash and exponential utilities with a  5.0E  06. All agree that there is a 0.01 chance of a fire in Abel’s house, whose replacement cost is $250,000. Charlie believes that a partial loss is as likely as a total loss; to the others it’s all or nothing. Baker, Charlie, and Dale ignore their own exposures, except that Dale lives next door to Abel, and reckons that his $250,000 house might catch fire from Abel’s (0.001 probability). How should this “mutual market” insure Abel’s house?

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12. 12 Insurance Interpretation of the Example Charlie is the most optimistic, and takes the largest positive share (52.82%). Baker and Dale agree on the pure premium, but Dale is afraid of positive correlation. The whole insurance premium is $3,267. In effect, Abel pays $707, $1,726, and $126 to his friends, and retains 21.66% of risk (not accidentally the same as Baker’s share). Charlie is the most optimistic, and takes the largest positive share (52.82%). Baker and Dale agree on the pure premium, but Dale is afraid of positive correlation. The whole insurance premium is $3,267. In effect, Abel pays $707, $1,726, and $126 to his friends, and retains 21.66% of risk (not accidentally the same as Baker’s share).

13. 13 Exponential Utility Just about the only game in town, as argued by Hans Gerber and in Appendix C. Has all the desirable properties, including absolute risk aversion (ARA) Some argue for power-curve utility and relative risk aversion (RRA). Mistake of RRA like mistaking movement along a curve for a shifting of the curve. True, as one’s wealth grows, one becomes less risk averse; but one can do this by shifting to less-averse exponential utilities. Only exponential utility is defined for all real numbers. Just about the only game in town, as argued by Hans Gerber and in Appendix C. Has all the desirable properties, including absolute risk aversion (ARA) Some argue for power-curve utility and relative risk aversion (RRA). Mistake of RRA like mistaking movement along a curve for a shifting of the curve. True, as one’s wealth grows, one becomes less risk averse; but one can do this by shifting to less-averse exponential utilities. Only exponential utility is defined for all real numbers.

14. 14 The Role of Solvency Value of a project independent of the level of an agent’s stochastic wealth “Asking a valuation formula to depend on [level] is like asking a shopkeeper to charge lower prices to the poor than to the rich.” (p. 63) Solvency is a constraint, not a criterion. (p. 31) One does not expect optimization to run up against a constraint. Solvency better tested on cash flows, than on reasonable one-period changes in capitalization. “Cash, not capital, is the prima materia” (p. 32) Basle Accords not even workable for banking, much less for insurance. Value of a project independent of the level of an agent’s stochastic wealth “Asking a valuation formula to depend on [level] is like asking a shopkeeper to charge lower prices to the poor than to the rich.” (p. 63) Solvency is a constraint, not a criterion. (p. 31) One does not expect optimization to run up against a constraint. Solvency better tested on cash flows, than on reasonable one-period changes in capitalization. “Cash, not capital, is the prima materia” (p. 32) Basle Accords not even workable for banking, much less for insurance.

15. 15 Stochastic Cash Flows, Stocks, and Investing Why can the price of a closed-end mutual fund differ from its net asset value? Because investors are buying the fund manager, not stocks. A stock is at one remove from stochastic cash flows; it is is the human management of a stochastic perpetuity. (p. 34) Debatable whether risk-adjusted discounting is right for stocks; but insurance projects are nearly perfect examples of stochastic cash flows. “Investment neither piggybacks upon nor supercharges underwriting.” “The business of of insurance should be to underwrite well, not to underwrite to generate funds to invest well.” (p. 34) Why can the price of a closed-end mutual fund differ from its net asset value? Because investors are buying the fund manager, not stocks. A stock is at one remove from stochastic cash flows; it is is the human management of a stochastic perpetuity. (p. 34) Debatable whether risk-adjusted discounting is right for stocks; but insurance projects are nearly perfect examples of stochastic cash flows. “Investment neither piggybacks upon nor supercharges underwriting.” “The business of of insurance should be to underwrite well, not to underwrite to generate funds to invest well.” (p. 34)

16. 16 Errata Because a print font was lacking, several equations (especially in Appendix D) show a ‘K’, which with the screen font is an ellipsis ‘ … ’. Pages 17 and 18: Five instances of u(W) should be E[u(W)]. Only for deterministic wealth is the expectation operator superfluous. Because a print font was lacking, several equations (especially in Appendix D) show a ‘K’, which with the screen font is an ellipsis ‘ … ’. Pages 17 and 18: Five instances of u(W) should be E[u(W)]. Only for deterministic wealth is the expectation operator superfluous.