Market-consistent valuation of long-term insurance contracts: valuation framework and application to German private health insurance

Abstract

In this paper we derive a market-consistent value for long-term insurance contracts, with a focus on long-term health insurance contracts as found, e.g., in the German private health insurance industry. To this end, we first set up a health insurance company model and, second, conduct a simulation study to calculate the present value of future profits and the time value of financial options and guarantees from a portfolio of private health insurance policies. Our analysis quantifies the impact of investment results and underwriting surpluses on shareholder profits with respect to profit sharing rules and premium adjustment mechanisms. In contrast to the valuation of life insurance contracts with similar calculation techniques the results indicate that the time value of financial options and guarantees of German private health insurance contracts is substantially smaller in typical parameter settings.

Appendix: Remarks on the stochastic environment

In the following we give a short review of the JY-model. The review is similar to the description of the model in [4, 9, 12, 20].

Consider a financial market with finite horizon $T$ described by the probability space $(\Omega ,\mathcal {F},P)$ and filtration $(\mathcal {F}_t)_{0\le t\le T}$. The probability measure $P$ is the real-world measure. The model is based on the assumption that there exist nominal as well as real prices in the financial market. The inflation (i.e. the development of the consumer price index) explains the difference between the corresponding nominal and real economy. The JY-model is an analog to a two-currency interest rate model, whereas the inflation rate in the JY-model corresponds to the spot exchange rate in the two-currency analog.

The following two equations constitute a Heath-Jarrow-Morton framework for the instantaneous forward rates $f_n(t,T)$ (nominal economy) and $f_r(t,T)$ (real economy). The instantaneous forward rates under the real-world probability measure $P$ satisfy the following stochastic differential equations for $t\in [0,T]$:

$$\begin{aligned} df_n(t,T)&=\alpha _n(t,T)dt+\varsigma _n(t,T)dW_n^P(t),\\ df_r(t,T)&=\alpha _r(t,T)dt+\varsigma _r(t,T)dW_r^P(t) \end{aligned}$$

with initial conditions $f_n(0,T)=f_n^M(0,T)$ and $f_r(0,T)=f_r^M(0,T)$. $\alpha _n(t,T)$ and $\alpha _r(t,T)$ are adapted processes; $\varsigma _n(t,T)$ and $\varsigma _r(t,T)$ are deterministic functions; $W_n^P(t)$ and $W_r^P(t)$ are Brownian Motions. $f_n^M(0,T)$ and $f_r^M(0,T)$ denote the observed instantaneous forward rates in the market at time 0 for maturity $T$; i.e.,

$$\begin{aligned} f_n^M(0,T)=-\frac{\partial \log P_n^M(0,T)}{\partial T}\quad {\rm and} \quad f_r^M(0,T)=-\frac{\partial \log P_r^M(0,T)}{\partial T}. \end{aligned}$$

$P_n^M(0,T)$, $P_r^M(0,T)$ are the bond prices in the nominal and real market for maturity $T$.

The development of the consumer price index $I(t)$ is explained in terms of a Geometric Brownian Motion, i.e.

$$\begin{aligned} dI(t)&=I(t)\mu (t)dt+I(t)\sigma _IdW_I^P(t), \end{aligned}$$

with initial condition $I(0)=I_0>0$, an adapted process $\mu (t)$, and a positive constant volatility parameter $\sigma _I$.

The three Brownian motions $W_n^P(t)$, $W_r^P(t)$, and $W_I^P(t)$ are correlated with correlation coefficients $\rho _{n,r}$, $\rho _{n,I}$ and $\rho _{r,I}$. It is

$$\begin{aligned} dW_n^P(t)dW_r^P(t)=\rho _{n,r}dt,\quad dW_n^P(t)dW_I^P(t)=\rho _{n,I}dt,\quad dW_r^P(t)dW_I^P(t)=\rho _{r,I}dt. \end{aligned}$$

Following [20] we assume a decaying volatility structure. For $t\in [0,T]$ we let

$$\begin{aligned} \varsigma _n(t,T)=\sigma _n\exp \left( -a_n(T-t)\right) \quad {\rm and} \quad \varsigma _r(t,T)=\sigma _r\exp \left( -a_r(T-t)\right) , \end{aligned}$$

with positive constants $a_n$, $a_r$, $\sigma _n$ and $\sigma _r$.

A change of measure from the real-world measure $P$ to the risk-neutral measure $Q^n$ (corresponding to the nominal economy) and a restatement of the stochastic differential equations in terms of short rates yields

$$\begin{aligned} dn(t)&=(\vartheta _n(t)-a_nn(t))dt+\sigma _ndW_n(t),\\ dr(t)&=(\vartheta _r(t)-\rho _{r,I}\sigma _r\sigma _I-a_rr(t))dt+\sigma _rdW_r(t),\\ dI(t)&=I(t)(n(t)-r(t))dt+I(t)\sigma _IdW_I(t). \end{aligned}$$

Again the three Brownian motions $W_n$, $W_r$, and $W_I$ are correlated with the parameters $\rho _{n,r}$, $\rho _{n,I}$, and $\rho _{r,I}$, and we have

$$\begin{aligned} \vartheta _n(t)&=\frac{\partial f^M_n(0,t)}{\partial t}+a_nf^M_n(0,t)+\frac{\sigma _n^2}{2a_n}\left( 1-\exp (-2a_nt)\right) \\ \vartheta _r(t)&=\frac{\partial f^M_r(0,t)}{\partial t}+a_rf^M_r(0,t)+\frac{\sigma _r^2}{2a_r}\left( 1-\exp (-2a_rt)\right) , \end{aligned}$$

to fit the observed term structure at the initial date. $\frac{\partial f^M_n(0,t)}{\partial t}$ and $\frac{\partial f^M_n(0,t)}{\partial t}$ denote the partial derivatives of $f^M_n(0,t)$ and $f^M_r(0,t)$ with respect to the second argument. The equations for the nominal and real interest rate under the risk-neutral measure $Q^n$ are referred to in the literature as the “Hull-White Extended Vasicek” model [4]. Note that the drift term of the inflation process after the measure change is described by the difference of the nominal and real short rate. In economic literature, other authors denote this relation as the Fisher equation.