A Critique of Risk-Adjusted Discounting Leigh J. Halliwell, FCAS, MAAA XXXII International ASTIN Colloquium Washington, DC July 8-11, 2001 Leigh J. Halliwell, FCAS, MAAA XXXII International ASTIN Colloquium Washington, DC July 8-11, 2001

2 Risk-Adjusted Discounting Two typical statements: “Value is defined as the present value of a cash-flow. The present value is found by discounting the flow at a rate that reflects the riskiness of the stream.” “Because cash flows are uncertain, they are discounted at a rate that reflects this uncertainty.” Sounds good but … Two typical statements: “Value is defined as the present value of a cash-flow. The present value is found by discounting the flow at a rate that reflects the riskiness of the stream.” “Because cash flows are uncertain, they are discounted at a rate that reflects this uncertainty.” Sounds good but …

3 A Philosophical Question … Fact:Because of risk-aversion, the certainty equivalent value of a risky cash flow is less than its actuarial present value, i.e., CEV[CF] < APV[CF]. Question:Is CEV grounded on discounting at a higher that risk-free interest rate? Grounding means logical priority. We demand more than the existence of an r that solves the equation: Fact:Because of risk-aversion, the certainty equivalent value of a risky cash flow is less than its actuarial present value, i.e., CEV[CF] < APV[CF]. Question:Is CEV grounded on discounting at a higher that risk-free interest rate? Grounding means logical priority. We demand more than the existence of an r that solves the equation:

4 … and some Down-to-Earth Questions “THIS IS THE CHALLENGE: to show that the discount rate is more fundamental than the shape of the distribution, and that it is derivable from anything other that seeing what people use for risk loads and using trivial equivalence.”Rodney E. Kreps Questions: If the YTM of a 5-year STRIP is 5.4 percent per year, and that of a 10-year STRIP is 5.5, is the 10-year STRIP a better investment? Does it have a higher RoE? Is money “working harder” in one than in the other? (see Appendix A of paper) “THIS IS THE CHALLENGE: to show that the discount rate is more fundamental than the shape of the distribution, and that it is derivable from anything other that seeing what people use for risk loads and using trivial equivalence.”Rodney E. Kreps Questions: If the YTM of a 5-year STRIP is 5.4 percent per year, and that of a 10-year STRIP is 5.5, is the 10-year STRIP a better investment? Does it have a higher RoE? Is money “working harder” in one than in the other? (see Appendix A of paper)

5 Seemingly a no-brainer: ‘DCF’ = discounted CASH flow. Therefore, flows of hard cash are discounted. With normal investments cash goes out and later RETURNS. Insurance is backwards. First premium, later losses. Normal order restored by changing the cash flow into an equity flow, usually by allocating capital. Equity goes out and later returns (RoE). Betrayal of DCF, since no longer a hard cash flow (DEF?). Just what flows; what gets discounted? RoE  DCF. Seemingly a no-brainer: ‘DCF’ = discounted CASH flow. Therefore, flows of hard cash are discounted. With normal investments cash goes out and later RETURNS. Insurance is backwards. First premium, later losses. Normal order restored by changing the cash flow into an equity flow, usually by allocating capital. Equity goes out and later returns (RoE). Betrayal of DCF, since no longer a hard cash flow (DEF?). Just what flows; what gets discounted? RoE  DCF. Discount What?

6 No-Arbitrage Pricing Linearity of pricing demands the existence of a state-price function, v(t,  ), the certainty equivalent of 1 dollar at time t in state . And v must be factorable as v(t)  (  ). Linearity of pricing demands the existence of a state-price function, v(t,  ), the certainty equivalent of 1 dollar at time t in state . And v must be factorable as v(t)  (  ).

7 Adjust Probability not Discount

8 Inconsistencies of Discounting Too little discounting of short-term flows (#1) Too much discounting of long-term flows (#2) Too rapid price appreciation (#3) Can always repackage to force risk-neutral price Binomial Asset Example: 50/50 chance of paying $120 or $80 at time T Too little discounting of short-term flows (#1) Too much discounting of long-term flows (#2) Too rapid price appreciation (#3) Can always repackage to force risk-neutral price Binomial Asset Example: 50/50 chance of paying $120 or $80 at time T

9 #2 #1 #3 Inconsistencies 1-3

10 Risk-Neutral Repackaging Binomial Asset = Asset 1 + Asset 2 Asset 1: guaranteed $100 at time T Asset 2: 50/50 chance of ± $20 at time T Prices: Binomial Asset = Asset 1 + Asset 2 Asset 1: guaranteed $100 at time T Asset 2: 50/50 chance of ± $20 at time T Prices:

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12 Implications It is circular to base present price on future price. Capital allocation hinders correct pricing. RoE is a poor measure of performance. Overcapitalized? Since when is too much money a problem? There are risks for which premiums neither can nor should be loaded. Such risks are managed through decisions of quantity, not of price. It is circular to base present price on future price. Capital allocation hinders correct pricing. RoE is a poor measure of performance. Overcapitalized? Since when is too much money a problem? There are risks for which premiums neither can nor should be loaded. Such risks are managed through decisions of quantity, not of price.