Stochastic DominanceStochastic Dominance

A Tool for Evaluating Reinsurance Alternatives

A Tool for Evaluating Reinsurance Alternatives

How do I decide whether an investment is “profitable”.

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

Common Measures of Risk and Reward

Internal Rate of ReturnReturn on EquityNet Present Value

Loss RatioReturn on CapitalExpected Policyholder Deficit

Internal Rate of ReturnReturn on EquityNet Present Value

Loss RatioReturn on CapitalExpected Policyholder Deficit

Problems with these measures...

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?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 TheoryReview 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 PreferenceWealth 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 AversionRisk 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

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 AversionRuin 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.

Utility Function Examples

-1

-0.5

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Wealth(Dollars, NPV, etc.)

Uti

lity

of

We

alt

h(u

tils

)

Example of Wealth Preference(Linear Utility)The Risk-Neutral Investor,Only Expected Return Matters

Example of Risk AversionNo matter how much I have,losing a dollar always hurts.

Example of Ruin AversionI hate to lose everything but if I have enough money I don't mind losing a dollar; then I looklike the Risk-Neutral Investor.

Fine PointFine PointThese three features of utility

functions are nested.These three features of utility

functions are nested.

Wealth Preference

Risk Aversion

Ruin Aversion

Great! Now what?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 DominanceStochastic 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)

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-OrderStochastic Dominance

First-OrderStochastic Dominance

Assuming Wealth Preference, A is uniformly preferred to B if and only if FB(w)-FA(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 FB(w)-FA(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!

First-Order Stochastic Dominance

(Uniformly Higher Wealth at Every Level of Probability)

-2,000

-1,000

0

1,000

2,000

3,000

4,000

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

Cumulative Distribution Function

F(w)

Term

inal W

ealth

w

Investment A

Investment B

0)()( wFwF AB

BA ww

Curves may never cross.

Risk Aversion(Narrower Class)

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-OrderStochastic Dominance

Second-OrderStochastic 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.

w

AB duuFuF 0)()(

Second-Order Stochastic Dominance

(Uniformly Less Down-Side Risk at Every Level of Probability)

-2,500

-1,500

-500

500

1,500

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

Cumulative Distribution Function

F(w)

Term

inal W

ealth

w

Investment A

Investment B

w

I II

Curves may cross but

not “too soon”.

Ruin Aversion(Narrowest Class)

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-OrderStochastic Dominance

Third-OrderStochastic 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

w v

AB dudvuFuF 0)()(

Third-Order Stochastic Dominance

(Uniformly Less Ruin Risk at Every Level of Probability)

-25,000

-20,000

-15,000

-10,000

-5,000

0

5,000

10,000

0.0% 20.0% 40.0% 60.0% 80.0% 100.0%

Cumulative Distribution Function

F(w)

Te

rmin

al

We

alt

h

w

Investment A

Investment B

Investment B

Investment A

+

-

Curves may cross sooner than SSD.

Fine Point RevisitedFine Point RevisitedThe 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 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?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!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!You can do this at home! If E[A]<E[B] then there is NO

dominance of ANY order! STOP. If all Ai Bi 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 Ai Bi 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.)

You can do this at home!(But TSD is trickier.)

Compute the 2-period running avg. VA,i = (CumSum(A)i-1 + CumSum(A)i)/2VB,i = (CumSum(B)i-1 + CumSum(B)i)/2

If all CumSum(VA)i CumSum(VB)i then TSD applies.

Compute the 2-period running avg. VA,i = (CumSum(A)i-1 + CumSum(A)i)/2VB,i = (CumSum(B)i-1 + CumSum(B)i)/2

If all CumSum(VA)i CumSum(VB)i then TSD applies.

You can do this at home!You can do this at home!

Option A Option B

Mean NPV 0 Mean NPV 0StDev NPV 159 StDev NPV 93

F(x) NPV CumSum VA

CumSum

VB NPV CumSum VB

CumSum

VB

4% -663 -663 -150 -1509% -129 -792 -727 -727 -139 -289 -220 -220

13% -103 -895 -843 -1571 -134 -424 -356 -57617% -59 -954 -924 -2495 -125 -549 -486 -106222% -30 -985 -969 -3465 -110 -659 -604 -166626% -14 -998 -992 -4456 -97 -756 -707 -237430% -10 -1009 -1004 -5460 -84 -840 -798 -317135% -2 -1011 -1010 -6469 -47 -886 -863 -403439% 2 -1009 -1010 -7479 -26 -912 -899 -493343% 5 -1004 -1006 -8485 -9 -921 -917 -585048% 8 -996 -1000 -9485 6 -915 -918 -676852% 46 -950 -973 -10458 30 -886 -901 -766957% 71 -879 -915 -11373 40 -846 -866 -853561% 75 -804 -842 -12215 42 -805 -825 -936065% 79 -725 -765 -12979 47 -758 -781 -1014170% 84 -641 -683 -13663 51 -706 -732 -1087374% 89 -553 -597 -14260 69 -638 -672 -1154578% 90 -463 -508 -14767 76 -561 -599 -1214583% 91 -371 -417 -15184 86 -475 -518 -1266387% 93 -279 -325 -15509 106 -370 -422 -1308691% 95 -184 -231 -15741 118 -252 -311 -1339696% 96 -88 -136 -15877 126 -126 -189 -13585100% 98 10 -39 -15916 128 1 -62 -13648

Option A wins on an “every-day” basisbut has large catastrophe exposure.

Option B tends to have a larger limitedexpected value except at largest limits.

POTENTIAL TRAP!!!

For more info...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...

www.casact.org/pubs/forum/01sforum/01sf095.pdf

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... www.casact.org/pubs/forum/01sforum/01sf095.pdf