PMER
Lecture 1
The Law of Large Numbers: expanding the number of risks in a pool, the average loss (or gain) in the
pool eventually becomes certain (or predictable).
The Law of Large Numbers is a statement about the average loss (not about the aggregate
loss) of the pool of risks.
It considers the situation in which the pool of risks becomes large, eventually pooling
infinitely many risks.
Khintchine’s (weak version of the) Law of Large Numbers: the sample mean (or average) of an
expanding pool of independent risks with identical probability distribution eventually converges (in
probability) to the true (or population) mean, whenever the risks have finite expectation.
Aggregate vs Average: pooling of risks does not lead to risk reduction on the aggregate level of the
pool. It leads to increased risk on the aggregate level of the pool. The Law of Large Numbers cannot
be interpreted to imply that pooling of risks leads to risk reduction.
Independent vs Completely dependent: the Law of Large Numbers does not work if there is a
common component to all risks. Systematic risk is not annihilated by risk pooling.
Finite vs Infinite: Because the pool of risks contains at most a finite number of risks, the variability of
the average loss per individual induced by risk pooling is not fully annihilated, and the average loss
does not become fully predictable.
Weak Law of Large Numbers
Suppose μ exists. Then, Khintchine’s Weak Law of Large Numbers says that the sample mean
of an expanding pool of such risks, ( X 1+ …+ X n ) / n, converges in probability to the true
(population) mean, μ. That is, for any positive number epsilon, the probability
[
P |( X 1+ …+ X n ) / n−μ|>ϵ ]
goes to zero as n tends to infinity. ( X i independent identical risks.)
Remove the condition of identical (common) probability distribution.
[ ]
li m n →∞ P |( ( X 1−μ 1) + …+ ( X n−μn ) ) / n|> ϵ =0
And
( 1 / n2 ) ( σ 21 +…+ σ 2n )
Goes to 0 as n tends to infinity, with σ 2i , the variance of the risk X i . Proof follows directly by
applying Cebyshev’s inequality.
Strong Law of Large Numbers
If the risks are i.i.d. and the expectation μ exists, no extra condition is required for the Strong
Law of Large Numbers to be valid:
P [ ω ∈ Ω:li mn → ∞ ( X 1+ …+ X n ) / n=μ ]=1
Remove the condition of identical (common) probability distributions, while maintaining the
condition of independence. P ¿
to be valid also when the probability distributions of the X i \ s are allowed to be non-identical
(but still independent) is that, for some 1 ≤ p ≤ 2,
, p p
li mn →∞ E [ (| X 1−μ1| )
1p
+ …+
(| X n−μn| )
np ] <∞
Risk Spreading
The expanding pool of risks is owned by a similarly expanding number of insurer shareholders. The
expanding pool of risks gets spread over an expanding number of owners. The more risks in the pool,
the more owners subdividing the aggregate risk, and the more the variability of the average risk
borne by a single owner gets reduced.
Cooperating and Mutuality
The expanding pool of risks is placed with an expanding number of cooperating insurance companies
and redistribute the aggregate risk. In the case of ‘identical’ insurance companies, upon
redistribution, each individual insurer bears the average loss of the pool of risks. (Mutuality Principle
of Borch). In will be Pareto efficient to allocate risks based on the total and systematic risk of the
pool. The more risk in the pool, the more cooperating insurance companies redistributing (and
subdividing) the aggregate risk, and the more variability of the average loss, allocated to a single
insurer, gets reduced.
Pareto efficiency/optimality: none of the economic agents involved can be made better off without
making at least one of the other agents worse off.
Mutuality Principle of Borch: Consider a pool of economic agents each facing a similar risk. In this
case of risk redistribution in a pool of cooperating agents, a Pareto efficient allocation is dictated
completely and solely by the total (or aggregate) and systematic risk of the pool, with the specific
characteristics of the individual risks being irrelevant. The Mutuality Principle does not necessarily
mean that the individual allocations are identical (dependent on degree o risk aversion). Under a
Pareto optimal allocation among identical cooperating agents, each agent bears the average loss of
the pool of risks.
Risk Solidarity: if insurance premia are entirely based on the risk factors of fully homogeneous pools,
then premium differentiation is maximized and solidarity is minimized.
Risk Exchange Y 1 +…+Y n =X 1 +…+ X n, X is initial risk and the full allocation requirement is
satisfied.
Maximize: w 1 E U 1 ( Y 1 ) +…+ wn EU n ( Y n )
Consider m possible outcomes or states of nature s1 , … , s m making up Ω with probabilities of
occurring given by p1 , … , pm.
m m
L=w1 ∑ p i U 1 ( y 1 ( s i ) ) +…+ wn ∑ pi U n ( y n ( s i ) ) + λ ( x 1 ( s1 ) + …+ x n ( s1 ) −( y 1 ( s 1 ) +…+ y n ( s1 ) ) ) +…+ λm ( x 1 ( s m ) +…+ x
i=1 i=1
FOC: w j p i U 'j ( y j ( si ) ) −λi =0
U 'j ( y j ( s i ) ) / U 'j ( y j ( s k ) )=U 'l ( y l ( si ) ) / U 'l ( y l ( sk ) )
' ' ' '
Observe that :U j ( y j ( s i ) ) / U l ( y l ( s i ) )=U j ( y j ( sk ) ) / U l ( y l ( sk ) )
Lecture 1
The Law of Large Numbers: expanding the number of risks in a pool, the average loss (or gain) in the
pool eventually becomes certain (or predictable).
The Law of Large Numbers is a statement about the average loss (not about the aggregate
loss) of the pool of risks.
It considers the situation in which the pool of risks becomes large, eventually pooling
infinitely many risks.
Khintchine’s (weak version of the) Law of Large Numbers: the sample mean (or average) of an
expanding pool of independent risks with identical probability distribution eventually converges (in
probability) to the true (or population) mean, whenever the risks have finite expectation.
Aggregate vs Average: pooling of risks does not lead to risk reduction on the aggregate level of the
pool. It leads to increased risk on the aggregate level of the pool. The Law of Large Numbers cannot
be interpreted to imply that pooling of risks leads to risk reduction.
Independent vs Completely dependent: the Law of Large Numbers does not work if there is a
common component to all risks. Systematic risk is not annihilated by risk pooling.
Finite vs Infinite: Because the pool of risks contains at most a finite number of risks, the variability of
the average loss per individual induced by risk pooling is not fully annihilated, and the average loss
does not become fully predictable.
Weak Law of Large Numbers
Suppose μ exists. Then, Khintchine’s Weak Law of Large Numbers says that the sample mean
of an expanding pool of such risks, ( X 1+ …+ X n ) / n, converges in probability to the true
(population) mean, μ. That is, for any positive number epsilon, the probability
[
P |( X 1+ …+ X n ) / n−μ|>ϵ ]
goes to zero as n tends to infinity. ( X i independent identical risks.)
Remove the condition of identical (common) probability distribution.
[ ]
li m n →∞ P |( ( X 1−μ 1) + …+ ( X n−μn ) ) / n|> ϵ =0
And
( 1 / n2 ) ( σ 21 +…+ σ 2n )
Goes to 0 as n tends to infinity, with σ 2i , the variance of the risk X i . Proof follows directly by
applying Cebyshev’s inequality.
Strong Law of Large Numbers
If the risks are i.i.d. and the expectation μ exists, no extra condition is required for the Strong
Law of Large Numbers to be valid:
P [ ω ∈ Ω:li mn → ∞ ( X 1+ …+ X n ) / n=μ ]=1
Remove the condition of identical (common) probability distributions, while maintaining the
condition of independence. P ¿
to be valid also when the probability distributions of the X i \ s are allowed to be non-identical
(but still independent) is that, for some 1 ≤ p ≤ 2,
, p p
li mn →∞ E [ (| X 1−μ1| )
1p
+ …+
(| X n−μn| )
np ] <∞
Risk Spreading
The expanding pool of risks is owned by a similarly expanding number of insurer shareholders. The
expanding pool of risks gets spread over an expanding number of owners. The more risks in the pool,
the more owners subdividing the aggregate risk, and the more the variability of the average risk
borne by a single owner gets reduced.
Cooperating and Mutuality
The expanding pool of risks is placed with an expanding number of cooperating insurance companies
and redistribute the aggregate risk. In the case of ‘identical’ insurance companies, upon
redistribution, each individual insurer bears the average loss of the pool of risks. (Mutuality Principle
of Borch). In will be Pareto efficient to allocate risks based on the total and systematic risk of the
pool. The more risk in the pool, the more cooperating insurance companies redistributing (and
subdividing) the aggregate risk, and the more variability of the average loss, allocated to a single
insurer, gets reduced.
Pareto efficiency/optimality: none of the economic agents involved can be made better off without
making at least one of the other agents worse off.
Mutuality Principle of Borch: Consider a pool of economic agents each facing a similar risk. In this
case of risk redistribution in a pool of cooperating agents, a Pareto efficient allocation is dictated
completely and solely by the total (or aggregate) and systematic risk of the pool, with the specific
characteristics of the individual risks being irrelevant. The Mutuality Principle does not necessarily
mean that the individual allocations are identical (dependent on degree o risk aversion). Under a
Pareto optimal allocation among identical cooperating agents, each agent bears the average loss of
the pool of risks.
Risk Solidarity: if insurance premia are entirely based on the risk factors of fully homogeneous pools,
then premium differentiation is maximized and solidarity is minimized.
Risk Exchange Y 1 +…+Y n =X 1 +…+ X n, X is initial risk and the full allocation requirement is
satisfied.
Maximize: w 1 E U 1 ( Y 1 ) +…+ wn EU n ( Y n )
Consider m possible outcomes or states of nature s1 , … , s m making up Ω with probabilities of
occurring given by p1 , … , pm.
m m
L=w1 ∑ p i U 1 ( y 1 ( s i ) ) +…+ wn ∑ pi U n ( y n ( s i ) ) + λ ( x 1 ( s1 ) + …+ x n ( s1 ) −( y 1 ( s 1 ) +…+ y n ( s1 ) ) ) +…+ λm ( x 1 ( s m ) +…+ x
i=1 i=1
FOC: w j p i U 'j ( y j ( si ) ) −λi =0
U 'j ( y j ( s i ) ) / U 'j ( y j ( s k ) )=U 'l ( y l ( si ) ) / U 'l ( y l ( sk ) )
' ' ' '
Observe that :U j ( y j ( s i ) ) / U l ( y l ( s i ) )=U j ( y j ( sk ) ) / U l ( y l ( sk ) )
