M7 Optimal Reinsurance with Expected Utility
Topics in Insurance, Risk, and Finance
Introduction
Learning outcomes
- Define the collective risk model and the compound Poisson distribution.
- Know and derive properties of the compound Poisson distribution.
- Simulate from the compound Poisson distribution.
- Define reinsurance, proportion reinsurance, and excess of loss reinsurance.
- Find the optimal reinsurance contract using the expected utility theory.
Insurance portfolio
In many insurance applications, portfolio losses can be simplified to the following two models.
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Denote by \(S\) the aggregate risk in a fixed period. Let \(X_1,X_2,\dots\) be a sequence of iid individual claims. We have
- Individual risk model: \[S=\sum_{i=1}^nX_i,\] where \(n\) is a constant.
- Collective risk model: \[S=\sum_{i=1}^NX_i,\] where \(N\) is a random number.
We will cover some aspects of these models; advanced topics will be covered by another subject.
Reinsurance
- A reinsurance contract is an agreement in which one party (the reinsurer) agrees to indemnify another party (the insurer) for parts of its insurance risk.
- Reinsurance thus can be seen as a particular form of insurance.
- Why reinsurance?
- There are many reasons: prevention of extreme losses, reduction of capital requirements, increasing underwriting capacity and so on.
Two types of reinsurance: proportion reinsurance
Proportion reinsurance: the reinsurer covers a prespecified proportion of each risk in the portfolio and the reinsurance premium is in proportion to the risk ceded.
If the insurer has a retained proportion \(\alpha\), then when a loss \(X\) occurs, the insurer will need to pay \(\alpha X\) and the reinsurer will pay \((1-\alpha)X\).
Two types of reinsurance: excess of loss reinsurance
Excess of loss reinsurance: the reinsurer pays the claim which is beyond a prespecified limit. In other words, the insurer’s liability is capped. The cap is referred to as the retention of the insurer.
If the insurer has a retention limit \(M\), then when a loss \(X\) occurs, the insurer will need to pay \(\min(X,M)\) and the reinsurer will pay \(\max(X-M,0)\).
Optimal reinsurance
- Given the various choices of reinsurance products, which is the optimal one for the insurer?
- We will study the insurer’s aggregate risk after reinsurance in the framework of the expected utility theory.
Compound Poisson distribution
Compound Poisson distribution
Let \(X_1,X_2,\dots\) be a sequence of iid individual claims and \(N\) be the number of claims, independent of all \(X\) in a fixed period. The portfolio loss \(S\) below is called the collective risk model: \[S=\sum_{i=1}^NX_i.\] By convention, \(S=0\) when \(N=0\).
- In the special case where \(N\) follows the Poisson distribution, \(S\) is said to follow the compound Poisson distribution.
- We will assume that the aggregate risk follows the compound Poisson distribution.
Recap: Poisson distribution
A random variable \(X\) is said to follow the Poisson distribution with parameter \(\lambda\), denoted by \(X\sim Poisson(\lambda)\), if \[\p(X=x)=\frac{e^{-\lambda}\lambda^x}{x!},\] where \(x=0,1,2,\dots\).
Properties of the Poisson distribution:
- \(\E(X)=\lambda\)
- \(\var(X)=\lambda\)
- \(M_X(t)=\E(\exp(tX))=\exp(\lambda(e^t-1))\)
Recap: mgf
We have \[\begin{align*} M_X(t)&=\E(\exp(tX))\\ &=\sum_{x=0}^\infty e^{tx}\frac{e^{-\lambda}\lambda^x}{x!}\\ &=e^{-\lambda}\sum_{x=0}^\infty \frac{(\lambda e^{t})^x}{x!}\\ &=e^{-\lambda}\exp(\lambda e^{t})\\ &=\exp(\lambda(e^t-1)). \end{align*}\]
Properties of compound Poisson distributions
Notations:
- \(X_i\sim F\) with density \(f\) such that \(F(0)=0\) (i.e., claims are positive).
- \(m_k=\E(X_i^k)\), \(k=1,2,3,\dots\)
- \(M_{X}(r)=\E(\exp(rX_i))\)
Proof: expectation
Hint: \(\E(S)=\E(\E(S|N))\)
Proof: variance
Hint: \(\var(S)=\E(\var(S|N))+\var(\E(S|N))\)
Proof: moment generating function
Proof:
Proof: moment generating function
Proof:
Example: normal approximation
Suppose that the claim for annual policy \(i\) is \(X_i\sim N(0.7P,4P^2)\), where \(P=5000\) is the annual premium. Claims are independent. Let the initial wealth be 0.1 million. The number of claims follows \(Poisson(100)\), independent of claim sizes. Find the probability that the insurer’s wealth is below 0 at the end of the year assuming 100 policies were sold at the start of the year. Comment on the model assumption.
Example: normal approximation
Optimal reinsurance with expected utility
Optimal reinsurance
- Insurers pay premiums to reinsurers to transfer part of their losses.
- Reinsurance can reduce the variability of the aggregate claims or the probability of ruin.
- A reinsurance contract is said to be optimal if the insurer’s expected utility is maximised or the probability of ruin is minimised.
An expected utility model
- An insurer has utility function \(u\) and wealth random variable \(X\).
- Goal: maximise \(\E(u(X))\)
- The insurer is risk-averse: \(u\) is increasing and concave.
- Risk aversion means: (a) the more wealth the better (b) the marginal utility is decreasing.
- We will study how reinsurance can affect an insurer’s decision making, i.e., how to make the optimal decision in the presence of reinsurance.
- Q: What are possible limitations of this decision model?
Utility function
We make the following assumptions:
The utility function \(u\) is an exponential utility function: \[u(x)=-\exp(-\beta x),\] where \(\beta>0\). This implies that the insurer is risk-averse.
The aggregate risk \(S\) follows the compound Poisson distribution with Poisson parameter \(\lambda\) and the individual claim distribution is \(F\) with density \(f\).
-
The wealth random variable after reinsurance is \[X=w+p-P_R-S_I,\] where
- \(w\): initial wealth
- \(p\): insurance premium
- \(P_R\): reinsurance premium
- \(S_I\): aggregate risk net of reinsurance
Goal
The goal is to maximise the expected utility of the insurer: \[\begin{align*} \max \E[u(X)] &= \max \E[u(w+p-P_R-S_I)]\\ & = \max \exp(-\beta (w+p))(-\exp(\beta P_R))\E[\exp(\beta S_I)]. \end{align*}\] Since \(w\) and \(p\) are constant, the above problem is equivalent to: \[\max (-\exp(\beta P_R))\E[\exp(\beta S_I)]\] We need to decide \(P_R\) and \(S_I\) such that the above expression can be written explicitly.
Proportion reinsurance
Assumes the reinsurer covers \(1-\alpha\) of each claim and the reinsurance premium is calculated by the exponential principle with parameter \(A\) (i.e., for a random loss \(X\), \(P_R=\log \E[\exp(A X)]/A=\log M_X(A)/A\)).
Let us break down the optimisation problem:
- Write down \(S_I\) and \(P_R\)
- Write down the objective function \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
- Find \(\alpha^*\) which maximises \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Proportion reinsurance
Write down \(S_I\) and \(P_R\)
Proportion reinsurance
Write down the objective function \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Proportion reinsurance
Find \(\alpha^*\) which maximises \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Proportion reinsurance
Find \(\alpha^*\) which maximises \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Excess of loss reinsurance
Let us now assume that the insurer effects excess of loss reinsurance with retention level \(M\) and that the reinsurance premium is calculated by the expected value principle with loading \(\theta\) (i.e., for a random loss \(X\), \(P_R=(1+\theta) \E[X]\)).
Let us break down the optimisation problem:
- Write down \(S_I\) and \(P_R\)
- Write down the objective function \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
- Find \(M^*\) which maximises \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Excess of loss reinsurance
Write down \(S_I\) and \(P_R\)
Excess of loss reinsurance
Write down the objective function \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Excess of loss reinsurance
Find \(M^*\) which maximises \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)
Excess of loss reinsurance
Find \(M^*\) which maximises \((-\exp(\beta P_R))\E[\exp(\beta S_I)]\)