Stochastic Dominance A Tool for Evaluating Reinsurance Alternatives

How do I decide whether an investment is “profitable”. Is return commensurate with “risk”? Does investment diversify my portfolio or concentrate exposure? Is investment consistent with my preferred operating risk? Is return commensurate with “risk”? Does investment diversify my portfolio or concentrate exposure? Is investment consistent with my preferred operating risk?

Common Measures of Risk and Reward Internal Rate of Return Return on Equity Net Present Value Loss Ratio Return on Capital Expected Policyholder Deficit Internal Rate of Return Return on Equity Net Present Value Loss Ratio Return on Capital Expected Policyholder Deficit

Problems with these measures... Blow up in real life. Can’t compare investments of different size. “Show me the capital!” Don’t consider portfolio-level impact. Blow up in real life. Can’t compare investments of different size. “Show me the capital!” Don’t consider portfolio-level impact.

What to do then? Utility Theory (Please suppress groans) Basic premise is “Tell me how much a return of W is worth to you...” “…then we can see if the investment improves your expected worth.” Utility Theory (Please suppress groans) Basic premise is “Tell me how much a return of W is worth to you...” “…then we can see if the investment improves your expected worth.”

Review of Utility Theory A utility function is a transformation that maps dollars to utility (worth). The shape of this function reflects our investment objectives and preferred operating risks. Common features include Wealth Preference and Risk Aversion A utility function is a transformation that maps dollars to utility (worth). The shape of this function reflects our investment objectives and preferred operating risks. Common features include Wealth Preference and Risk Aversion

Wealth Preference “Greed is good.” A utility function U(w) possesses Wealth Preference if and only if U’(w)  0 for all w with at least one strict inequality. In other words, my utility function is increasing (there are a lot of ways to be increasing, though). “Greed is good.” A utility function U(w) possesses Wealth Preference if and only if U’(w)  0 for all w with at least one strict inequality. In other words, my utility function is increasing (there are a lot of ways to be increasing, though).

Risk Aversion I hate losing more than I like winning. A utility function U(w) possesses Risk Aversion if and only if it satisfies Wealth Preference and U’’(w)  0 for all w with at least one strict inequality. In other words, my utility function is increasing at a decreasing rate (i.e. it’s curved). I hate losing more than I like winning. A utility function U(w) possesses Risk Aversion if and only if it satisfies Wealth Preference and U’’(w)  0 for all w with at least one strict inequality. In other words, my utility function is increasing at a decreasing rate (i.e. it’s curved).

A Less Common Feature: Ruin Aversion Also called Decreasing Absolute Risk Aversion, Skewness Preference, etc. Losing a little is bad, but losing everything is intolerable. Enter reinsurance... Also called Decreasing Absolute Risk Aversion, Skewness Preference, etc. Losing a little is bad, but losing everything is intolerable. Enter reinsurance...

Ruin Aversion A utility function U(w) possesses Ruin Aversion if and only if it satisfies Risk Aversion and U’’’(w)  0 for all w with at least one strict inequality. In other words, my utility is curved but “flattening out” as it goes. A utility function U(w) possesses Ruin Aversion if and only if it satisfies Risk Aversion and U’’’(w)  0 for all w with at least one strict inequality. In other words, my utility is curved but “flattening out” as it goes.

Fine Point These three features of utility functions are nested. Wealth Preference Risk Aversion Ruin Aversion

Great! Now what? “A man who seeks advice about his actions will not be grateful for the suggestion that he maximize expected utility.” A.D. Roy “A man who seeks advice about his actions will not be grateful for the suggestion that he maximize expected utility.” A.D. Roy

Stochastic Dominance Avoids need to select or parameterize a utility function. Instead, select a class of utility functions (e.g. Wealth Preference). Then develop investment selection rules that yield maximum expected utility for all such utility functions. Avoids need to select or parameterize a utility function. Instead, select a class of utility functions (e.g. Wealth Preference). Then develop investment selection rules that yield maximum expected utility for all such utility functions.

Wealth Preference (Broadest Class) My utility function may be linearly increasing, may have Risk Aversion, or Ruin Aversion. If I allow such a broad class of utility functions, I will need an austere selection rule! My utility function may be linearly increasing, may have Risk Aversion, or Ruin Aversion. If I allow such a broad class of utility functions, I will need an austere selection rule!

First-Order Stochastic Dominance Assuming Wealth Preference, A is uniformly preferred to B if and only if F B (w)-F A (w)  0 for all w with at least one strict inequality. In other words, investment A yields greater wealth at every probability. Nice if you can get it! Assuming Wealth Preference, A is uniformly preferred to B if and only if F B (w)-F A (w)  0 for all w with at least one strict inequality. In other words, investment A yields greater wealth at every probability. Nice if you can get it!

Curves may never cross.

Risk Aversion (Narrower Class) My utility function may have Risk Aversion or Ruin Aversion. With a narrower class of utility functions, I can relax my selection rule. My utility function may have Risk Aversion or Ruin Aversion. With a narrower class of utility functions, I can relax my selection rule.

Second-Order Stochastic Dominance Under Risk Aversion, A is uniformly preferred to B if and only if for all w with at least one strict inequality. In other words, investment A has uniformly less down-side risk at every probability. Under Risk Aversion, A is uniformly preferred to B if and only if for all w with at least one strict inequality. In other words, investment A has uniformly less down-side risk at every probability.

Curves may cross but not “too soon”.

Ruin Aversion (Narrowest Class) My utility function must have Ruin Aversion. With an even narrower class of utility functions, I can relax my selection rule even further. My utility function must have Ruin Aversion. With an even narrower class of utility functions, I can relax my selection rule even further.

Third-Order Stochastic Dominance Under Ruin Aversion, A is uniformly preferred to B if and only if for all w with at least at least one strict inequality. Small, probable loss is preferable to remote, possible ruin Under Ruin Aversion, A is uniformly preferred to B if and only if for all w with at least at least one strict inequality. Small, probable loss is preferable to remote, possible ruin

Curves may cross sooner than SSD.

Fine Point Revisited The stochastic dominance orders are nested in reverse order. The stochastic dominance orders are nested in reverse order. Third-Order Second-Order First Order Wealth Preference Risk Aversion Ruin Aversion

Stochastic Dominance Properties Stochastic Dominance assumes little so the comparison is weak. If you don’t see dominance, it may still be a good investment. (Select specific utility function or narrower class.) Dominance is transitive. Dominance is not commutative. Stochastic Dominance assumes little so the comparison is weak. If you don’t see dominance, it may still be a good investment. (Select specific utility function or narrower class.) Dominance is transitive. Dominance is not commutative.

What do I do with it? Investment Decision: Does the portfolio with the investment dominate the portfolio without it? Contract Pricing: What risk loads ensure that each of my contract proposals is not dominated by any of the others? Investment Decision: Does the portfolio with the investment dominate the portfolio without it? Contract Pricing: What risk loads ensure that each of my contract proposals is not dominated by any of the others?

You can do this at home! Generate the same number of simulated NPVs for each investment alternative. Sort results of each simulation in ascending order to approximate F(x) Now let’s test whether alternative A dominates alternative B. Generate the same number of simulated NPVs for each investment alternative. Sort results of each simulation in ascending order to approximate F(x) Now let’s test whether alternative A dominates alternative B.

You can do this at home! If E[A]<E[B] then there is NO dominance of ANY order! STOP. If all A i  B i then FSD, SSD, and TSD all apply. If all CumSum(A) i  CumSum(B) i then SSD, and TSD both apply. If E[A]<E[B] then there is NO dominance of ANY order! STOP. If all A i  B i then FSD, SSD, and TSD all apply. If all CumSum(A) i  CumSum(B) i then SSD, and TSD both apply.

You can do this at home! (But TSD is trickier.) Compute the 2-period running avg. V A,i = (CumSum(A) i-1 + CumSum(A) i )/2 V B,i = (CumSum(B) i-1 + CumSum(B) i )/2 If all CumSum(V A ) i  CumSum(V B ) i then TSD applies. Compute the 2-period running avg. V A,i = (CumSum(A) i-1 + CumSum(A) i )/2 V B,i = (CumSum(B) i-1 + CumSum(B) i )/2 If all CumSum(V A ) i  CumSum(V B ) i then TSD applies.

You can do this at home! Option A wins on an “every-day” basis but has large catastrophe exposure. Option B tends to have a larger limited expected value except at largest limits. POTENTIAL TRAP!!!

For more info... Levy, Stochastic Dominance, Investment Decision Making Under Uncertainty Wolfstetter, Stochastic Dominance: Theory and Applications Elton and Gruber, Modern Portfolio Theory and Investment Analysis. This paper may be down-loaded at... f Levy, Stochastic Dominance, Investment Decision Making Under Uncertainty Wolfstetter, Stochastic Dominance: Theory and Applications Elton and Gruber, Modern Portfolio Theory and Investment Analysis. This paper may be down-loaded at... f