7/25/2019 Adamou, Alexander - Cooperation and Insurance
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Cooperation
and insurance
AlexanderAdamou
Setup
Cooperation
Insurance
1/35
Cooperation and insurance
Alexander Adamou
European Bond CommissionLondon, 20 October 2015
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2/54
Cooperation
and insurance
AlexanderAdamou
Setup
Cooperation
Insurance
2/35
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
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Cooperation
and insurance
AlexanderAdamou
Setup
Cooperation
Insurance
3/35
Agenda
1 Setup
2 Cooperation
3 Insurance
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Cooperation
and insurance
AlexanderAdamou
Setup
Cooperation
Insurance
4/35
Agenda
1 Setup
2 Cooperation
3 Insurance
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Cooperation
and insurance
AlexanderAdamou
Setup
Cooperation
Insurance
5/35
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
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Cooperation
and insurance
AlexanderAdamou
Setup
Cooperation
Insurance
6/35
Illustration
0 1 2 3 4 50
2
4
6
8
10
12
14
16
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
6/35
Illustration
0 1 2 3 4 50
2
4
6
8
10
12
14
16
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
6/35
Illustration
0 1 2 3 4 50
2
4
6
8
10
12
14
16
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
6/35
Illustration
0 1 2 3 4 50
2
4
6
8
10
12
14
16
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
6/35
Illustration
0 1 2 3 4 50
2
4
6
8
10
12
14
16
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
7/35
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
7/35
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)(t+t)
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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
7/35
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)(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
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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
8/35
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
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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
9/35
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
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
10/35
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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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
10/35
Example trajectories
0 200 400 600 800 100010
0
10
5
1010
1015
1020
t
x
GBM: mu = 0.05, sigma = 0.2
ensembleaveragetimeaverage
E l j i
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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
10/35
Example trajectories
0 2000 4000 6000 8000 1000010
0
10
50
10100
10150
10200
t
x
GBM: mu = 0.05, sigma = 0.2
ensembleaveragetimeaverage
R i
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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
11/35
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
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
12/35
Agenda
1 Setup
2 Cooperation
3 Insurance
C d
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Cooperation
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
13/35
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
and insuranceAlexanderAdamou
Setup
Cooperation
Insurance
14/35
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
15/35
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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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
16/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
17/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
17/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
18/35
Illustration
Noise reduction
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
19/35
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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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
19/35
Noise reduction
Noise term is now a sum ofNindependent normal variates
yi(t) = yi(t)(t+
ti)
yi(t+ t) = yi(t) +yi(t)
t+
t
Nj=1j
N
Noise reduction
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
19/35
Noise reduction
Noise term is now a sum ofN
independent normal variates
yi(t) = yi(t)(t+
ti)
yi(t+ t) = yi(t) +yi(t)
t+
t
N
Noise reduction
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
19/35
Noise reduction
Noise term is now a sum ofNindependent normal variates
yi(t) = yi(t)(t+
ti)
yi(t+ t) = yi(t) +yi(t)
t+
t
N
where
N
i=jjN
N(0, 1)
Noise reduction
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
19/35
Noise term is now a sum ofNindependent normal variates
yi(t) = yi(t)(t+
ti)
yi(t+ t) = yi(t) +yi(t)
t+
t
N
where
N
i=jjN
N(0, 1)
In effect replaced by N
Cooperators grow at gN= 22N> g
Simulation
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
20/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
21/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
22/35
g
1 Setup
2 Cooperation
3 Insurance
Puzzle
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
23/35
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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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
24/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
25/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
26/35
Seek to maximise rate of change of expectation value of wealth
r Wt
Expected-wealth paradigm
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
26/35
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
Expected-wealth paradigm
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
26/35
Insurers perspective
Uninsured
runins= 0t= $0 pm
Insured
rinins= FpLt = $100 pmChange inr
rins= F
pL
t = $100 pm = rownInsurer should sign
No price range
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
27/35
1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100150
100
50
0
50
100
150
insurance fee ($)
changeinrate($/month)
Zero-sum game
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
28/35
Antisymmetry rown= rins makes insurance zero-sumOne party wins at expense of other an unsavoury business
Existence of insurance contracts requires asymmetry between
contracting parties, e.g. different access to information different subjective assessments of risk one party deceives, coerces, gulls the other
Is this truly the basis for the entire insurance market?
Expected-utility paradigm
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
29/35
Empirical evidence suggests maximising expected wealth is nota good model of human rationality
Introduceutility: nonlinear function U(W) supposed to reflectthe value assigned by humans to an amount of wealth
Encodes psychological preferences, e.g. risk aversion
Now seek to maximise rate of change of expected utility
ru
U(W)
t
Example: use U(W) =W (Cramer 1728) for both parties
Expected-utility paradigm
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
29/35
Owners perspective
Uninsured
ruunown= (1p)U(Wown+G)+pU(WownC)U(Wown)t = 3.37 upm
Insured
ruinown= U(Wown+GF)U(Wown)t = 3.46 upmChange inru
ruown= ruin
own ruun
own= 0.094 upmOwner should sign
Expected-utility paradigm
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
29/35
Insurers perspective
Uninsured
ruunins= 0t= 0 upm
Insured
ruinins= (1p)U(Wins+F)+pU(Wins+FL)U(Wins)t = 0.043 upmChange inru
ruins= ruin
ins ruun
ins= 0.043 upmInsurer should sign
Price range
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
30/35
1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 21000.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
insurance fee ($)
changeinrate($
/month)
Classical resolution
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
31/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
32/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
32/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
32/35
Insurers perspective
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
33/35
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
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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
34/35
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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Cooperationand insurance
AlexanderAdamou
Setup
Cooperation
Insurance
35/35
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
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