Risk Factor Contributions and Capital Allocation in Life Insurance in the Solvency II Framework

Abstract

Quantification of capital requirements is a critical issue for any insurer. Solvency is assessed through the regulatory capital, but, in practice, insurance companies usually hold higher levels of economic capital assessed using risk-based models that allow for any type of risk the institution deals with. Once the economic capital of a company is determined, it should be allocated down to lower levels, such as business units, lines and products for a number of purposes and this allocation of capital has crucial importance. This chapter deals with some key aspects related to risk quantification and capital allocation in life insurance. Portfolios composed by different insurance contracts, with both life and death benefits, are investigated. The numerical results obtained explain that the features and benefits of a contract influence not only the assessment of the total risk but also its allocation to single factors, showing that such a risk measurement methodology could be a useful tool for new products improvement and management, in adherence to the principles stated by the Solvency II directive, specifically with respect to the own risk and solvency assessment (ORSA) and the implementation of internal models.

Appendix

If the total loss $L$ is assumed to be a function of two random variables, $L=g\left({F}_1,{F}_2\right)$, ${F}_1$ being the mortality risk and ${F}_2$ the investment risk, then the total variance of the loss can be expressed as the sum of two components, the average expected value of the variance of the loss conditioned by the mortality scenario, $E\left[ Var\left[L|{F}_1\right]\right]$, and the variance of the average expected value conditioned by the mortality scenario,$Var\left[E\left[L|{F}_1\right]\right]$:

$$ Var\left[L\right]=E\left[ Var\left[L|{F}_1\right]\right]+ Var\left[E\left[L|{F}_1\right]\right]. $$

(11.1)

As an alternative it can be considered as the sum of the average expected value of the variance of the loss conditioned by the financial scenario, $E\left[ Var\left[L|{F}_2\right]\right]$, and the variance of the average expected value conditioned by the financial scenario, $Var\left[E\left[L|{F}_2\right]\right]$:

$$ Var\left[L\right]=E\left[ Var\left[L|{F}_2\right]\right]+ Var\left[E\left[L|{F}_2\right]\right]. $$

(11.2)

The value at time T of an annuity for M years is given by the following formula:

$$ {V}_{\mathbb{Q}}^M(T)=\sum_{i=1}^MP\left(T,T+i\right)\kern0.28em {\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i,x\right)|\ {\mathcal{M}}_t\right] $$

(11.3)

where $P\left(T,T+i\right)$ is the price at time $T$ of a zero-coupon bond with maturity $T+i$ conditional on the information in $T$ on the term structure of interest rate and ${\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i,x\right)|\ {\mathcal{M}}_t\right]$ is the risk-adjusted expectation of the survivor index at time $T+i$ of a cohort aged x at time T conditional on the information available in T on the term structure of mortality.

Considering the models chosen the value at time $T$ of an annuity for $M$ years is given by the following formula:

$$ {V}_{\mathbb{Q}}^M(T)=\sum_{i=1}^MP\left(T,T+i,r(t)\right)\kern0.28em {\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i,x\right)|\ k(t)\right] $$

(11.4)

where $P\left(T,T+i,r(T)\right)$ is the price at time $T$ of a zero-coupon bond with maturity $T+i$ conditional on the value assumed by the spot interest rate $r(t)$ in $T$ and ${\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i,x\right)|\ k(T)\right]$ is the risk-adjusted expectation of the survivor index at time $T+i$ of a cohort aged x at time T conditional on the value assumed in T by the mortality trend $k(t)$ of the Cairns-Blake-Dowd model.

In a similar way, the value at time $T$ for a term life insurance with term $M$ is given by:

$$ {\displaystyle \begin{array}{l}{W}_{\mathbb{Q}}^M(T)=\sum \limits_{i=1}^MP\left(T,T+i,r(T)\right)\hfill \\ {}\kern3.64em \left({\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i,x\right)|\ k(T)\right]-{\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i-1,x\right)|\ k(T)\right]\right)\hfill \end{array}} $$

(11.5)

where the difference ${\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i,x\right)|\ k(T)\right]-{\mathbb{E}}_{\mathbb{Q}}\left[S\left(T+i-1,x\right)|\ k(T)\right]$ represents the risk-adjusted expectation of the mortality at time $T+i$ of a cohort aged x at time T conditional on the value assumed by $k(t)$ in $T.$