A market-consistent framework for the fair evaluation of insurance contracts under Solvency II

Abstract

The entry into force of the Solvency II regulatory regime is pushing insurance companies in engaging into market consistence evaluation of their balance sheet, mainly with reference to financial options and guarantees embedded in life with-profit funds. The robustness of these valuations is crucial for insurance companies in order to produce sound estimates and good risk management strategies, in particular, for liability-driven products such as with-profit saving and pension funds. This paper introduces a Monte Carlo simulation approach for evaluation of insurance assets and liabilities, which is more suitable for risk management of liability-driven products than common approaches generally adopted by insurance companies, in particular, with respect to the assessment of valuation risk.

Appendices

A Change of measure derivation

The dynamics of r(t) under the real-world $\mathbb {P}$ measure follows a Vasicek dynamics as in Sect. 3.3 Eq. (14)

$$\begin{aligned} dr(t)= & {} a \, (\bar{r} - r(t)) dt +\sigma \, dW^{\mathbb {P}}(t), \end{aligned}$$

where $W^{\mathbb {P}}$ is a Brownian motion under the measure $\mathbb {P}$. We define the Radon–Nykodim derivatives from the real-world measure $\mathbb {P}$ to the risk-neutral measure $\mathbb {Q}$ as

$$\begin{aligned} \frac{d \, \mathbb {Q}}{d \, \mathbb {P}} (t) = e^{-\frac{1}{2} \int _0^t m^2(s) \mathrm{d}s + \int _0^t m(s) \mathrm{d}W^{\mathbb {P}}(s) } \end{aligned}$$

where m(t) is the market price of interest rate risk, which has the following form

$$\begin{aligned} m(t) = \frac{\theta (t) - a \bar{r}}{\sigma }, \end{aligned}$$

with $\theta (t)$ a deterministic function of time.

By Girsanow theorem, the short rate r(t) has the following dynamics under the risk-neutral $\mathbb {Q}$ measure

$$\begin{aligned} dr(t)= & {} (\theta (t)-a r(t)) dt +\sigma dW(t), \end{aligned}$$

where W(t) is a Brownian motion under $\mathbb {Q}$. The previous process is an Hull–White process, and it can be written as in Eq. (1) for conveniently chosen functions $\alpha (t)$ and $\theta (t)$ (see, for instance, Brigo and Mercurio 2006, pp. 72–73).

As in Sect. 3 the default risk of sovereign issuers is modelled using an intensity model (also called Cox processes or doubling stochastic Poisson processes) with intensities modelled using correlated CIR processes, for $i=1,2,\ldots ,I$,

$$\begin{aligned}&\tau _i = \inf \left\{ t \ge 0: \int _0^t \lambda _i(s) \mathrm{d}s > \xi _i \right\} ,\\&dy_{i}(t) = b_i \left( \bar{s}_i - y_{i}(t) \right) dt + \eta _i \sqrt{y_{i}(t)} \, dZ^\mathbb {P}_i(t), \\&dZ^\mathbb {P}_i(t) \, dZ^\mathbb {P}_j(t) = \rho _{ij} \, dt, \end{aligned}$$

where I is the number of sovereign bond issuers, $\tau _i$ is the stochastic time of default of the ith issuer, $\xi _i$ are i.i.d. unitary exponential random variables and $Z^\mathbb {P}(t)$ ia a I-dimensional Brownian motions under $\mathbb {P}$.

By Girsanov theorem for point processes (Bremaud 1981), defining the Radon–Nykodim derivatives as

$$\begin{aligned} \frac{d \, \mathbb {Q}}{d \, \mathbb {P}} (t) = \prod _{i=1}^I \left( \prod _{n \ge 1} \left( 1+ \frac{\psi _i(t)}{y_i(t)}\right) \text {I}(\tau _i(n) \le t)\right) e^{\int _0^t \psi _i(s) \, \mathrm{d}s}, \end{aligned}$$

with $\psi _i(t)$ deterministic functions of time, we obtain that the credit risk intensity of the i-th issuer under the risk- neutral probability is

$$\begin{aligned} s_i(t) = \psi _i(t) + y_i(t), \end{aligned}$$

where $y_i(t)$ follows the process in Eq. (5).

By similar reasoning, using the point process Girsanov theorem, the dynamics of the corporate credit risk intensity changes from the real-world process as in Eq. (15) to the risk- neutral measure process as in Eq. (12).

B Maximum likelihood for default probabilities

In this section, we detail the procedure used to calibrate the CIR parameters via maximum likelihood method for the historical series of corporate default probabilities. The same procedure is applied to historical series of sovereign probabilities bootstrapped from CDS spreads. Let $\theta$ be the set of the model parameters. We define

$$\begin{aligned} f(\lambda ; \theta ) := \varPi (t,T)_{iK}, \end{aligned}$$

(17)

and we assume that the function f is invertible with respect to $\lambda = \lambda (t)$. Once we fix a maturity $\tau$ and an initial rate i, then we can build the historical series of default probabilities $\{\varPi _1,\ldots ,\varPi _N\}$ where $\varPi _n = \varPi (t_n, t_n+\tau )_{iK}$. The likelihood function of the observed default probabilities is

$$\begin{aligned} L(\varPi _1,\ldots ,\varPi _N | \theta ) = \prod _{n=1}^N h_\varPi (\varPi _n;\theta ) \end{aligned}$$

(18)

where $h_\varPi (x;\theta )$ is the conditional density function of the default probabilities (we assume that $\varPi _1,\ldots ,\varPi _N$ are i.i.d.). The default probability density function $h_\varPi$ can be obtained from the density function of $\lambda$, which is a non-central Chi-squared distribution, using the monotonic transformation of random variables, as suggested in Pearson and Sun (1994), in the following way

$$\begin{aligned} h_\varPi (\varPi _n; \theta )= & {} h_{\lambda }(\lambda ^*_n ; \theta ) \, \frac{\partial f^{-1}}{\partial \varPi }(\varPi _n; \theta ) \\= & {} h_{\lambda }(\lambda ^*_n; \theta ) \, \left( \frac{\partial f}{\partial \lambda }(\lambda ^*_n; \theta )\right) ^{-1} \end{aligned}$$

where $\lambda ^*_n = f^{-1}(\varPi _n,\theta )$. Hence, a maximum likelihood estimator of the parameters is

$$\begin{aligned} \hat{\theta } = \text {arg}\max _{\theta } L(\varPi _1,\ldots ,\varPi _N | \theta ) = \text {arg}\max _{\theta } \prod _{n=1}^N h_{\lambda }(\lambda ^*_n; \theta ) \, \left( \frac{\partial f}{\partial \lambda }(\lambda ^*_n; \theta )\right) ^{-1}. \end{aligned}$$

C Invariant probabilistic sensitivity analysis

Let y be a function of the model parameters (for instance, y can be the VOG)

$$\begin{aligned} y(\theta ) : D \rightarrow \mathbb {R}. \end{aligned}$$

with $\theta \in D \subseteq \mathbb {R}^n$ and n is the number of the model parameters.

Let $\varTheta$ be a random vector and $\theta$ is realization, hence $Y= y(\varTheta )$ is the corresponding random model output and $F_Y$ is the CDF of Y.

We define the importance measure of $\theta _i$ with respect to Y as

$$\begin{aligned} \beta _i = \mathbb {E}\left[ d(F_Y, F_{Y | \varTheta _i = \theta _i}) \right] , \end{aligned}$$

where $0<d<1$ is a distance between the unconditional and the conditional CDF. We choose to use the Kuiper’s metric, then

$$\begin{aligned} d(F_Y, F_{Y | \varTheta _i = \theta _i}) = \sup _y \left( F_Y(y) - F_{Y | \varTheta _i = \theta _i}(y) \right) + \sup _y \left( F_{Y | \varTheta _i = \theta _i}(y) - F_Y(y) \right) . \end{aligned}$$

In Table 5, we report the values of the importance measures $\beta _i$. A value near to one of $\beta _i$ signals an important difference between $F_Y$ and $F_{Y | \varTheta _i = \theta _i}$, i.e. the parameter $\theta _i$ is a “critical” parameter of the model.