Managing a Portfolio of Risks

Abstract

Basic ideas concerning risk pooling and risk transfer, presented in Chap. 1, are progressed further in the present chapter, mainly with the following purposes: 1. to discuss key features of premium calculation when non-homogeneous portfolios are concerned, namely portfolios consisting of risks with various claim probabilities; 2. to analyze, more deeply, the riskiness of a portfolio and the tools which can be used to face potential losses, in particular introducing the role of the shareholders’ capital; 3. to illustrate the possibility, for an insurance company, to transfer, in its turn, risk of losses to another insurer, namely the possibility to resort to reinsurance; 4. to address dynamic aspects of the management of insurance portfolios in a multiyear context.

Appendix

As noted in Sect. 2.3.5, various approximations to the (exact) probability distribution of the total random payment $X^{[\mathrm{P}]}$ can be adopted. Whatever the approximating distribution may be, the goodness of the approximation must be carefully assessed, especially with regard to the right tail of the distribution itself, as this tail quantifies the probability of large payments.

Table 2.21 Right tails of Binomial distribution and Normal approximation

Table 2.22 Binomial distribution and Poisson approximation

Table 2.23 Right tails of Binomial distribution and Poisson approximation

Fig. 2.46

Probability distribution of the random payment ($n = 500$). Binomial distribution and normal approximation

The following examples can provide some ideas about the degree of approximation obtained by using the Poisson (see (2.3.22)–(2.3.24)) and the Normal approximation (see (2.3.25)–(2.3.30)) to the binomial distribution (given by (2.3.21)).

Assume the following data:

• individual loss: $x^{(j)} = 1$, for $j=1,\ldots ,n$;

• probability: $p = 0.005$;

• pool sizes: $n=100$, $n=500$, $n=5\,000$.

The (exact) binomial distribution and the normal approximation have been adopted for $n=500$ and $n=5\,000$; the (exact) binomial distribution and the Poisson approximation have been used for $n=100$. Tables 2.21, 2.22 and 2.23 and Figs. 2.46 and 2.47 show numerical results.

Fig. 2.47

Probability distribution of the random payment ($n = 5\,000$). Binomial distribution and normal approximation

The following aspects should be stressed. In relation to portfolio sizes $n=500$ and $n=5\,000$, the normal approximation tends to underestimate the right tail of the payment distribution (see Table 2.21). Conversely, the Poisson distribution provides a good approximation to the exact distribution, also for $n=100$ (see Tables 2.22 and 2.23); unlike the normal approximation, the Poisson model tends to overestimate the right tail, so that a prudential assessment of the payment follows.