Development and Pricing of a New Participating Contract

Abstract

The purpose of this article is to design and price a new type of participating life insurance contract. Participating contracts are popular in France, Denmark, the UK (where they are called with-profit contracts)... They satisfy various covenants and obey various regulations depending on the country where they are issued. The standard literature is Briys and de Varenne [1994, 1997a] and Grosen and Jørgensen [2000], to quote only a few. Bernard, Le Courtois and Quittard-Pinon [2005] study a particular type of participating contract, where an HJM term structure of interest rates is assumed and where the potential default of the issuing company is taken into account. In this paper, we design a new participating contract very similar to the one considered in the above-mentioned study, but where the guaranteed rate corresponds to the return of a portfolio of Government bonds. We show that this new type of contract can be valued in closed-form even when interest rates are stochastic and default of the company is taken into account. Also, this new type of contract is likely to reduce the default risk of the issuing companies, and is less sensitive to interest risk than more standard contracts.

On the regulator–insurer interaction in a structural model

“…Moody's credit rating, a small volatility of 10% leads to a very small default probability and this leads to an Aaa rating of the company, a volatility value of 15% results in Baa. 5 Thanks to its definition, one can easily see that the probability of default is a increasing function with respect to the variable η. The higher the η, the higher the default probability.…”

“…In addition, it is observed that a quite low regulation parameter η which results in a quite low barrier level should be chosen in order to keep the insurance company to at (or below) a reasonable default probability. 5 According to Moody's rating, for a 20-year horizon, the ratings and the respective default probabilities are given as follows: Aaa, 1.55%; Aa, 2.70%, A, 5.24% and Baa, 12.59%.…”

“…Stochastic interest rates have been then introduced in the previous models in [10,18]. Modeling the default of insurance companies in a Black and Cox framework has been first developed by Grosen and Jørgensen [15], and then extended in [4,5]. Finally, the recent work of Chen and Suchanecki [12] shows how to apply the realistic procedure Chapter 11 of the US Bankruptcy code in the insurance field.…”

Protection of a Company Issuing a Certain Class of Participating Policies in a Complete Market Framework

“…In this setting, it is easy to compute this value in closed form, even with stochastic interest rates, when it is assumed that default can occur only at maturity (see, e.g., Bernard et al 2006, the computation of [2.4], p. 182). To eliminate the stochastic discount factor, we move to the T-forward neutral universe.…”

On Surrender and Default Risks

“…It is possible to compute the expectation E Q [ N T ] if the intensity model of N T is specified, while the second expectation can be developed in a closed or semiclosed form (see Bernard, Le Courtois, and Quittard‐Pinon 2006 for simple expressions of ). Next, we note that the following equality can be obtained easily: Therefore, we can express V 3 0 as follows: …”