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