M9 Ruin Probability with Reinsurance
Topics in Insurance, Risk, and Finance
Ruin and reinsurance
Learning outcomes
- Find optimal reinsurance contracts by minimising ruin probabilities, in the classes of proportion reinsurance and excess of loss reinsurance, respectively.
- Know the optimal type of reinsurance contract given a budget constraint.
Ruin and reinsurance I
We have discussed how reinsurance affects insurers in a fixed period of time using the EUT. Next, we study the case of classical risk process using the ruin theory.
Given a compound Poisson aggregate claims process, the surplus claim process of an insurer without reinsurance is \[U(t)=u+ct-\sum_{i=1}^{N(t)}X_i,\] where
- \(u\) is the initial surplus
- \(c\) is the premium income per unit of time
- \(N(t)\) is the number of claims before or at time \(t\)
- \(X_i\) is the \(i\)th claim without reinsurance.
Ruin and reinsurance II
With reinsurance, the surplus claim process will become \[U^*(t)=u+c^*t-\sum_{i=1}^{N(t)}X_i^*,\] where
- \(u\) is the initial surplus
- \(c^*\) is the premium income per unit of time, net of reinsurance
- \(N(t)\) is the number of claims before time \(t\)
- \(X_i^*\) is the \(i\)th claim, net of reinsurance.
Here, we make an implicit assumption that the premium is paid to reinsurer continuously.
Ruin and reinsurance III
- We are still in the classical risk model with reinsurance so the previous results can be applied here.
- Since it is generally hard to obtain an analytic expression of the ultimate ruin probability, we focus on studying the adjustment coefficient. With reinsurance, the adjustment coefficient \(R^*\) is given by \[\lambda +c^* R^*=\lambda \E[\exp(R^*X_1^*)]\]
- Lundberg’s inequality is used to approximate the ultimate ruin probability.
- We say a reinsurance arrangement is optimal in the sense that it maximizes the adjustment coefficient (hence minimises the approximated ultimate ruin probability).
Ruin and reinsurance IV
For the rest of the lecture, we make the assumptions below:
Let \(X\) be the individual claim with \(X\sim F\), \(F(0)=0\), and \(f\) the density.
The loading factors of insurer and reinsurer are \(\theta\) and \(\theta_R\).
The Poisson parameter is \(\lambda\).
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\(h\) is a reinsurance arrangement, i.e., the insurer pays \(h(X)\le X\) under the reinsurance arrangement. For instance,
- \(h(x)=\alpha x\) gives the proportional reinsurance
- \(h(x)=\min(x,M)\) gives the excess of loss reinsurance.
Then the insurer’s premium rate, net of reinsurance, is given by, \[c^*=(1+\theta)\lambda \E[X]-(1+\theta_R)\lambda \E[X-h(X)]\] with \(c^*>\lambda \E[h(X)]\).
Example: Proportional reinsurance
Proportional reinsurance: example
Suppose that the individual claim amount \(X\) follows the exponential distribution \(F(x)=1-e^{-x}.\) Recall that \(M_X(r)=1/(1-r)\) for \(r<1\). Hence, we have \[c^*=((1+\theta)-(1+\theta_R)(1-\alpha))\lambda.\] Then the adjustment coefficient can be calculated by \[\lambda M_{X^*}(r)-\lambda-c^*r=0,\] where \(M_{X^*}(r)=1/(1-\alpha r)\).
We will consider the following three cases:
- \(\theta=\theta_R=0.2\)
- \(\theta=0.2\) and \(\theta_R=0.25\)
- \(\theta=0.05\) and \(\theta_R=0.25\)
Proportional reinsurance: example
- \(\theta=\theta_R=0.2\)
Proportional reinsurance: example
- \(\theta=0.2\) and \(\theta_R=0.25\)
Proportional reinsurance: example
- \(\theta=0.05\) and \(\theta_R=0.25\)
Example: Excess of loss reinsurance
Excess of loss reinsurance: example
- The adjustment coefficient can be solved by \[\lambda +c^*r=\lambda\left(\int_0^M \exp(rx)f(x)dx+\exp(rM)(1-F(M))\right).\]
- Assume that the individual claim follows \(F(x)=1-e^{-x}\), \(\theta=0.1\), \(\theta_R=0.2\).
Excess of loss reinsurance: example
Constraint one: the premium income per unit time, net of reinsurance, exceeds aggregate claims per unit time: \[\begin{align*} c^*&=(1+\theta)\lambda m_1-(1+\theta_R)\lambda \E[\max(X-M,0)]\\ &=1.1\lambda-1.2\lambda e^{-M}\\ &\ge \lambda \E[\min(X,M)]=\lambda(1-e^{-M}), \end{align*}\] which gives \(M\ge \log 2=0.693\).
Excess of loss reinsurance: example
Constraint two: We assume that the insurer pays for reinsurance out of the premium income it receives: \[\begin{align*} 1.1\lambda \ge 1.2\lambda \int_M^\infty (x-M)e^{-x}dx=1.2\lambda e^{-M}, \end{align*}\] which gives \(M\ge \log(12/11)=0.0870\).
Hence, in this example it is sufficient to consider Constraint one, as in the case of proportional reinsurance.
Excess of loss reinsurance: example
The equation for the adjustment coefficient:
Excess of loss reinsurance: upper bound
- Remember an upper bound of adjustment coefficient without reinsurance is \[R\le \frac{2(c-\lambda m_1)}{\lambda m_2},\] where \(c\), \(m_1\), \(m_2\) are the premium income per unit of time, the first moment and the second moment of the individual claim.
- This formula can be still applied with reinsurance. However, \(c\), \(m_1\) and \(m_2\) changes with reinsurance.
- We continue with the previous example with excess of loss reinsurance and compute an upper bound of \(R\). That is, we need to compute \(c^*\), \(m_1^*\), and \(m_2^*\) which denote the corresponding component with reinsurance.
Excess of loss reinsurance: upper bound
The optimal type of reinsurance
The optimal type of reinsurance
Consider the following question:
- An insurer can choose from different types of reinsurance contracts.
- It has a budget constraint on the reinsurance premium rate \(c^*=C\), \(C\in\R\).
- What type of reinsurance contract should be considered?
The optimal type of reinsurance
Note that the cost of reinsurance is always the same under the above assumption.