Adamou, Alexander - Cooperation and Insurance

7/25/2019 Adamou, Alexander - Cooperation and Insurance

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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Cooperation and insurance

Alexander Adamou

European Bond CommissionLondon, 20 October 2015
http://find/


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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Message

Two related questions:

Why should we cooperate?

Why should we insure?One fundamental answer:

Because all parties gain (by growing faster)

These are not zero-sum games
http://find/


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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Agenda

1 Setup

2 Cooperation

3 Insurance
http://find/


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and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Agenda

1 Setup

2 Cooperation

3 Insurance
http://find/


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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Multiplicative growth

Simple null model of how wealth x(t) changes over time isnoisy multiplicative growth

Change in wealth x(t) over time period t is a random

proportion ofx(t):

x(t) =x(t)Z

Z is a random variable with stationary distribution

Reflects notion that wealth invested creates more of itself butthat outcomes are uncertain
http://find/


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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Illustration

0 1 2 3 4 50

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t

x

0 1 20

0.2

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0.6

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Z = Dx / x

p(Z)
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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Illustration

0 1 2 3 4 50

2

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t

x

0 1 20

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0.6

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Z = Dx / x

p(Z)
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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Illustration

0 1 2 3 4 50

2

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t

x

0 1 20

0.2

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0.6

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Z = Dx / x

p(Z)
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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Illustration

0 1 2 3 4 50

2

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t

x

0 1 20

0.2

0.4

0.6

0.8

Z = Dx / x

p(Z)
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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Illustration

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2

4

6

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t

x

0 1 20

0.2

0.4

0.6

0.8

Z = Dx / x

p(Z)
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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Specific model

Most widely-used model in finance is Geometric BrownianMotion (GBM), e.g. stock price dynamics

Specify noise: Zhas normal (Gaussian) distribution

x(t) =x(t)Z
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Cooperation

and insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Specific model



x(t) =x(t)(t+t)
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Cooperation

and insuranceAlexanderAdamou

Setup

Cooperation

Insurance

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Specific model



x(t) =x(t)(t+t)

N(0, 1) is a standard normal variate(. . .)is normal with mean tand stddev

t

In finance, is drift (or expected rate of return) and isstochastic volatility
http://find/


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Cooperation


Setup

Cooperation

Insurance

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Exponential growth

Wealth grows exponentially

x(T) =x(0)exp[g(T)T]

at a noisy growth rate

g(T) 1T

ln

x(T)

x(0)

=

2

2 +

T

which converges in the long-time limit to

g limT

{g(T)} = 2

2
http://find/http://goback/


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Insurance

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Growth rates

g= 22 is the time-average growth rateRate at which each trajectory grows almost surely as T In finance

2

2 term known as volatility drag

Expectation value of wealth grows at a faster rate

x(T) =x(0) exp(g T) where g =

g = is the ensemble-average growth rateRate at which average over all parallel trajectories grows
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Cooperation


Setup

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Insurance

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Example trajectories

0 20 40 60 80 10010

0

101

102

103

t

x

GBM: mu = 0.05, sigma = 0.2

ensembleaveragetimeaverage

E l j i


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0 200 400 600 800 100010

0

10

5

1010

1015

1020

t

x

GBM: mu = 0.05, sigma = 0.2


E l j i
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Cooperation


Setup

Cooperation

Insurance

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0 2000 4000 6000 8000 1000010

0

10

50

10100

10150

10200

t

x

GBM: mu = 0.05, sigma = 0.2


R i
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Cooperation


Setup

Cooperation

Insurance

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Recipe

Assume individual wealth grows multiplicatively as GBM

Intervention favourable when it increases individuals g

Build simple models of cooperation and insurance

Determine effect of intervention on g

should occur when g increased for all parties

A d
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Cooperation


Setup

Cooperation

Insurance

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Agenda

1 Setup

2 Cooperation

3 Insurance

C d
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Cooperation


Setup

Cooperation

Insurance

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Conundrum

Cooperation is the basis for much of the structure in natureand society, from multicellular organisms to nation states

Persistent behavioural pattern in which individual entities pool

and share their resources

Conundrum: at each decision point, more successful entitiesmust share their resources with less successful ones, therebybooking an immediate net loss

How can we explain this apparent altruism?

Approach
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Cooperation


Setup

Cooperation

Insurance

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Approach

Start with population of non-cooperators

Introduce cooperation as a two-stage process:

Grow Share

Compare growth rates of cooperators and non-cooperators

Non cooperation
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Cooperationand insurance

AlexanderAdamou

Setup

Cooperation

Insurance

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Non-cooperation

Population ofNentities with resources xi(t) for i= 1 . . .N

Non-cooperating entities follow independent GBMs

For clarity, we express this in two distinct stages

xi(t) = xi(t)(t+

ti) (Grow)

xi(t+ t) = xi(t) + xi(t) (Dont share)

where i

N(0, 1) are independent standard normal variates

Non-cooperators grow at g= 22

Simulation
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AlexanderAdamou

Setup

Cooperation

Insurance

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Simulation

0 100 200 300 400 500 600 700 800 900 100010

10

100

1010

1020

1030

1040

1050

1060

1070

time t

Resourcesx(t)

slope g

slope gt(x

1x

2)

slope gt(x

i)

((x1x

2)(t))/2

(x1(t)+x

2(t))/2

x1(t)

x2(t)

Cooperation
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Cooperation

Now imagine population cooperates to produce Nentities withresources yi(t) for i= 1 . . .N

Cooperators grow independently then pool and share resourcesat each time step

We treat case of equal sharing: y1=y2 =. . .= yN

yi(t) = yi(t)(t+

ti) (Grow)

yi(t+ t) = yi(t) +yi(t) (Dont share)

Cooperation
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Cooperation

Now imagine population cooperates to produce Nentities withresources yi(t) for i= 1 . . .N

Cooperators grow independently then pool and share resourcesat each time step

We treat case of equal sharing: y1=y2 =. . .= yN

yi(t) = yi(t)(t+

ti) (Grow)

yi(t+ t) = yi(t) +Nj=1yj(t)

N (Share)

Illustration
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Illustration

Noise reduction
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Noise reduction

Noise term is now a sum ofNindependent normal variates

yi(t) = yi(t)(t+

ti)

yi(t+ t) = yi(t) +

Nj=1yj(t)

N

Noise reduction


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AlexanderAdamou

Setup

Cooperation

Insurance

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Noise reduction


yi(t) = yi(t)(t+

ti)

yi(t+ t) = yi(t) +yi(t)

t+

t

Nj=1j

N

Noise reduction


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AlexanderAdamou

Setup

Cooperation

Insurance

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Noise reduction

Noise term is now a sum ofN

independent normal variates

yi(t) = yi(t)(t+

ti)


t+

t

N

Noise reduction


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AlexanderAdamou

Setup

Cooperation

Insurance

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Noise reduction


yi(t) = yi(t)(t+

ti)


t+

t

N

where

N

i=jjN

N(0, 1)

Noise reduction
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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yi(t) = yi(t)(t+

ti)


t+

t

N

where

N

i=jjN

N(0, 1)

In effect replaced by N

Cooperators grow at gN= 22N> g

Simulation
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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0 100 200 300 400 500 600 700 800 900 100010

10

100

1010

1020

1030

1040

1050

1060

1070

time t

Resourcesx(t)

slope g

slope gt(x

1x

2)

slope gt(x

i)

((x1x

2)(t))/2

(x1(t)+x

2(t))/2

x1(t)

x2(t)

Growth story
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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y

Faster growth from cooperation due to noise (risk) reduction

limN {gN} = limN 22N

==g

g is upper bound and represents infinite cooperation

Simple model is easily generalised, e.g.

idiosyncratic parameters i, i conditions for mutuallybeneficial cooperation

partial sharing (c.f. taxation and redistribution)

In finance, portfolios and economies with well-managed risksshould grow faster in long run

Agenda
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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g

1 Setup

2 Cooperation

3 Insurance

Puzzle
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Judged by its effect on expected wealth, buying insurance isonly rational at a price at which it is irrational to sell

Classically insurance contracts are zero-sum games with nomutually beneficial price for buyer and seller

Why, then, do insurance contracts exist?

Classical resolutions appeal to utility theory (i.e. psychology)and/or asymmetric information (i.e. deception)

We will show that insurance contracts exist with a range ofprices that increase gfor both buyer and seller

Model contract
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AlexanderAdamou

Setup

Cooperation

Insurance

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A shipowner sends a cargo from St Petersburg to Amsterdam

owners wealth, Wown= $100, 000 gain on safe arrival of cargo, G = $4, 000

probability ship will be lost, p= 0.05

replacement cost of ship, C= $30, 000 voyage time, t= 1 month

An insurer with wealth Wins = $1, 000, 000 proposes to insure

the voyage for a fee, F = $1, 800

If the ship is lost, the insurer pays the owner L= G+C

Outcomes
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Transfer of uncertainty from owner to insurerWithout insurance

P(Wown=G) = 1 p, P(Wown= C) =p

P(Wins= 0) = 1With insurance

P(Wown=G F) = 1

P(W

ins=F) = 1

p, P

(Wins

=F

L) =p

Should the owner sign the contract? Should the insurer haveproposed it?

Expected-wealth paradigm
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Seek to maximise rate of change of expectation value of wealth

r Wt

http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Owners perspective

Uninsured

runown= (1p)GpCt = GpLt = $2, 300 pm

Insured

rinown= GFt = $2, 200 pmChange inr

rown= rinown r

unown=

pL

F

t = $100 pmOwner should not sign

http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Insurers perspective

Uninsured

runins= 0t= $0 pm

Insured

rinins= FpLt = $100 pmChange inr

rins= F

pL

t = $100 pm = rownInsurer should sign

No price range
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100150

100

50

0

50

100

150

insurance fee ($)

changeinrate($/month)

Zero-sum game
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Antisymmetry rown= rins makes insurance zero-sumOne party wins at expense of other an unsavoury business

Existence of insurance contracts requires asymmetry between

contracting parties, e.g. different access to information different subjective assessments of risk one party deceives, coerces, gulls the other

Is this truly the basis for the entire insurance market?

Expected-utility paradigm
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Empirical evidence suggests maximising expected wealth is nota good model of human rationality

Introduceutility: nonlinear function U(W) supposed to reflectthe value assigned by humans to an amount of wealth

Encodes psychological preferences, e.g. risk aversion

Now seek to maximise rate of change of expected utility

ru

U(W)

t

Example: use U(W) =W (Cramer 1728) for both parties

http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Owners perspective

Uninsured

ruunown= (1p)U(Wown+G)+pU(WownC)U(Wown)t = 3.37 upm

Insured

ruinown= U(Wown+GF)U(Wown)t = 3.46 upmChange inru

ruown= ruin

own ruun

own= 0.094 upmOwner should sign

http://find/


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AlexanderAdamou

Setup

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Insurance

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Uninsured

ruunins= 0t= 0 upm

Insured

ruinins= (1p)U(Wins+F)+pU(Wins+FL)U(Wins)t = 0.043 upmChange inru

ruins= ruin

ins ruun

ins= 0.043 upmInsurer should sign

Price range
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 21000.1

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

insurance fee ($)

changeinrate($

/month)

Classical resolution
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Symmetry broken by the different wealths Wown and Wins

Wealths now appear inru since utility function is nonlinearCertain combinations ofWown, Wins and Uadmit a range of

mutually beneficial prices F

So utility theory does not rule out insurance contracts butdoes not rule them in either

Invoking arbitrary and unobservable utility functions hardly asatisfying resolution

Time paradigm
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Decision criterion: maximise gunder multiplicative growth

Reject a priori irrelevant expectation values (averages overparallel universes)

Multiplicative repetition over n voyages

g= limn

1

nt ln

W(nt)

W(0)

= lnW

t

gis rate of change of expected logarithmic wealth

In effect U= lnWis utility function for multiplicative growth

Time paradigm
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Owners perspective

Uninsured

gunown= (1p)ln(Wown+G)+pln(WownC)ln(Wown)

t = 1.9% pm

Insured

ginown= ln(Wown+GF)ln(Wown)

t = 2.2% pm

Change in g

gown= ginown g

unown= 0.24% pm

Owner should sign

Time paradigm
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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Uninsured

gunins = 0t= 0% pm

Insured

ginins= (1p)ln(Wins+F)+pln(Wins+FL)ln(Wins)

t = 0.0071% pm

Change in g

gins= gin

ins gun

ins = 0.0071% pmInsurer should sign

Price range
http://find/


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AlexanderAdamou

Setup

Cooperation

Insurance

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1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 21000.5

0

0.5

1

1.5

2

2.5

3x 10

3

insurance fee ($)

changeinrate($

/month)

Fundamental resolution
http://find/


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AlexanderAdamou

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Cooperation

Insurance

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Fundamental resolution of insurance puzzle:

Owner and insurer should both sign the contract because

it increases the time-average growth rates of their wealths

No appeal to arbitrary utility functions or asymmetriccircumstances

Business happens when both parties gain

Model predicts range of mutually beneficial fees

simple

mathematical basis for insurance pricing

References
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AlexanderAdamou

Setup

Cooperation

Insurance

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O. Peters and A. Adamou (2015)The evolutionary advantage of cooperation

http://arxiv.org/abs/1506.03414

O. Peters and A. Adamou (2015)Rational insurance with linear utility and perfect information

http://arxiv.org/abs/1507.04655

Links also on LML website

http://www.lml.org.uk/alex-adamou.php
http://find/

Adamou, Alexander - Cooperation and Insurance

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