A sensitivity analysis concept is introduced for prospective reserves of individual life insurance contracts as deterministic mappings of the actuarial assumptions interest rate, mortality probability, disability probability, etc. Upon modeling these assumptions as functions on a real time line, the prospective reserve is here a mapping with infinite dimensional domain. Inspired by the common idea of interpreting partial derivatives of first order as local sensitivities, a generalized gradient vector approach is introduced in order to allow for a sensitivity analysis of the prospective reserves as functionals on a function space. The capability of the concept is demonstrated with an example.
By statute the insurer must currently maintain a reserve to meet future liabilities in respect of its insurance contracts. A major concern of regulators and insurers is the choice of the valuation basis, which is the set of assumptions underlying the calculation of premiums and reserves. In practice the true values of future interest or mortality rates are not perfectly known; especially in recent years, financial markets have experienced increased volatility, and life expectancies have risen in many developed countries with an unforeseen rate.
Therefore, the dependence of the reserve on the elements of the valuation basis was always an important issue. Already Lidstone (1905) studied in a discrete time setting the effect on reserves of changes in valuation basis and contract terms, but dealt only with simple single life policies with payments dependent on survival and death. Norberg (1985) transferred Lidstone’s ideas to a continuous time version, using Thiele’s differential equations. On the same basis, Hoem (1988), Ramlau-Hansen (1988), and Linnemann (1993) obtained a handful of further results. All of these studies yield only qualitative results. They show which direction the prospective reserve or the premium level are shifted to by a parameter change, but do not quantify the magnitude of that effect.
Another approach is to calculate different scenarios and to compare them to each other (see for example Olivieri (2001) or Khalaf-Allah et al. (2006)), but this idea works only for a very small number of parameters.
A third way is to study sensitivities by means of derivatives, which turned out to be a very efficient concept. References using such an approach are Dienst (1995), Bowers et al. (1997), Kalashnikov and Norberg (2003), and Helwich (2003):
Dienst (1995, pp. 66–68, 147–150) uses a finite number of partial derivatives of the net premium with respect to time-discrete disablement probabilities to approximate the relative change of the net premium caused by altered disablement probabilities.
Bowers et al. (1997, pp. 490, 491) calculate the first-order derivative of the expected loss with respect to the interest rate, which they assume to be constant.
Kalashnikov and Norberg (2003) differentiate the prospective reserve and the premium level with respect to one arbitrary real parameter, which also includes parameters such as contract terms. They present a ‘dynamical approach’ that allows one to calculate sensitivities even when the prospective reserve is not a closed form expression but given by a differential equation. In Section 5, Kalashnikov and Norberg generalize their approach to a finite number of real parameters.
Helwich (2003), models the actuarial assumptions as finite dimensional and real-valued vectors, allowing for parameter changes at a finite number of discrete time points. He calculates the gradient of the expected loss of a portfolio of insurance contracts with respect to yearly constant interest and retirement rates.
All of those studies have in common that they only consider a finite number of parameters. The present paper introduces a sensitivity analysis based on some generalized gradient vectors. This allows one to study sensitivities with respect to an infinite number of parameters, which meets, for example, the more realistic idea of actuarial assumptions (e.g., mortality) being functions on a real line rather than on a discrete time grid. Nonetheless, the approach includes also discrete time models and thus generalizes Helwich’s (2003) chapter 5.
Concepts for defining ‘functional sensitivities’ are already known in the literature (for example, see Saltelli et al. (2000), chapter 5.7). They can be applied to prospective reserves as functionals of the valuation basis if intensities exist; however, this has not been done so far. In order to jointly comprise the ‘discrete method’ and the ‘continuous method’ of insurance mathematics as well as intermediate cases, Milbrodt and Stracke (1997) proposed a life insurance modeling framework that is based on cumulative intensities. In order to allow the study of sensitivities of prospective reserves with respect to cumulative interest and transition intensities, the present paper extends in Section 3 the functional sensitivity analysis approaches mentioned above. A generalized gradient vector concept for functionals of right-continuous functions with finite total variation is introduced. Similar notions can be found in nonparametric locally asymptotic statistics; differences and similarities are discussed at the end of Section 3.
In Section 4, the general concept of Section 3 is applied on prospective reserves of life insurance contracts as mappings of the valuation basis. The proofs are quite technical and are therefore placed in the Appendix. Section 5 gives an example in order to demonstrate the introduced sensitivity analysis concept.
In recent years, many researchers have seen interest rates and mortality rates as random processes. Such an approach truly has advantages, but, so far, all authors in the literature make structural assumptions that can hardly be verified, especially in the long term. Therefore, a sensitivity analysis is a valuable complement for today’s risk management in life insurance.
While both ‘discrete’ and ‘continuous’ time methods are widely used in the actuarial literature, it offers only sensitivity analyses with respect to a finite parameter space so far. The present paper fills this gap by offering a continuous method for sensitivity analyses, which–using the approach based on cumulative intensities–contains also the discrete method. The sensitivities obtained are similar to that of Kalashnikov and Norberg (2003) or Helwich (2003).
Finally, note that the sensitivity analysis concept introduced in Section 3 is quite general and can thus be a helpful instrument whenever functionals of cumulative intensities, probability measures, etc. are to be studied
