Market inconsistencies of market-consistent European life insurance economic valuations: pitfalls and practical solutions

Abstract

The Solvency II directive has introduced a specific so-called risk-neutral framework to valuate economic accounting quantities throughout European life insurance companies. The adaptation of this theoretical notion for regulatory purposes requires the addition of a specific criterion, namely market-consistency, in order to objectify the choice of the valuation probability measure. This paper points out and fixes some of the major risk sources embedded in the current regulatory life insurance valuation scheme. We compare actuarial and financial valuation schemes. We then first address operational issues and potential market manipulation sources in life insurance, induced by both theoretical and regulatory pitfalls. For example, we show that the economic own funds of a representative French life insurance company can vary by almost 140%, as already observed by market practitioners, when the interest rate model is calibrated in October or on the 31st of December. We then propose various modifications of the current implementation, including a first product-specific valuation scheme, to limit the impact of these market-inconsistencies.

Appendix: The LIBOR market model—actuarial version (\(LMM_{ins}\))

The LIBOR Market Model provides forward rates dynamic modeling. The following presentation assumes that the considered swaptions are options on yearly swaps. Therefore, only annual payment dates are used. This assumption is often chosen in practice when calibrations are made on yearly forward rates, which is generally the case.

Let $0 \le j \le J$ be a set of $J+1$ successive dates. We get the 1 year-forward zero-coupon prices from, $\forall j\in [\![1;J]\!] \text {, }0\le t\le j,$

$$\begin{aligned} F_j(t)=\left( \frac{P(t,j)}{P(t,j+1)}-1\right) . \end{aligned}$$

Let n be the ceiling function defined by:

$$\begin{aligned} \left\{ \begin{aligned} n(0)=1&\\ n(t)=\lceil t \rceil&\quad \text { if }\,\,t\ge 0. \end{aligned} \right. \end{aligned}$$

Let $(g_j)_{j\in [\![1;J]\!] }$ and $\Phi (t)$ be the following deterministic functions: $\forall j\in [\![1;J]\!] \text {, }0\le t\le j$,

$$\begin{aligned} g_j(t)=[a+b(j-n(t)+1)]e^{-c(j-n(t)+1)}+d, \end{aligned}$$

and

$$\begin{aligned} \Phi (t)=\theta +(1-\theta )\times e^{-\kappa t}. \end{aligned}$$

In a general framework, the 1 year-forward diffusion is given, under the spot LIBOR measure, by the following equation (see [5]): for $0\le t\le j$,

$$\begin{aligned} dF_j(t)=F_j(t)\left( \Phi (t)^2\sum \limits _{i=n(t)}^j\left[ \frac{F_i(t)}{1+F_i(t)}g_i(t)g_j(t)\rho _{i,j}(t)\right] dt+\Phi (t)g_j(t)dZ_j^M(t)\right) , \end{aligned}$$

with $Z^M$ a M-dimensional Brownian motion, and $\rho _{i,j}(t)dt=d<Z^M_i,Z^M_j>_t$.

Practitioners suppose $Z^M$ can be decomposed through a two-dimensional standard Brownian motion:¹²

$$\begin{aligned} W(t)=\left(\begin{array}{l} W^1(t)\\ W^2(t) \end{array}\right) . \end{aligned}$$

In addition, the specification of the correlation structure is given by the user through a set of parameters

$$\begin{aligned} \beta _j=\left(\begin{array}{l} \beta ^1_j\\ \beta ^2_j \end{array}\right) _{j\in [\![1;M]\!] }, \end{aligned}$$

such that $\rho _{i,j}(t)=\beta _{i-n(t)+1}\beta _{j-n(t)+1}$. The $\beta$ parameters are therefore meta-parameters, often a priori assessed based on a historical forward data set.

Denoting $\eta _j(t)=\Phi (t)g_j(t)\beta _{j-n(t)+1}$, the Forward diffusion is assumed to be, for $0\le t\le j$,

$$\begin{aligned} dF_j(t)=F_j(t)\left( \sum \limits _{i=n(t)}^j\left[ \frac{F_i(t)}{1+F_i(t)}\eta _i(t)\eta _j(t)\right] dt+\eta _j(t).dW(t)\right) . \end{aligned}$$

The calibration therefore aims to estimate, in an optimal fashion, the six parameters $P=\left( a,b,c,d,\kappa ,\theta \right)$.

Suppose a user wants to fit the at-the-money receiver swaption Black implied volatilities of maturities $(m_i)_{i \in \llbracket 1;I\rrbracket }$ and tenors $(n_i)_{i \in [\![1;I]\!] }$, denoted as $(\sigma _i)_{i \in [\![1;I]\!] }$. Using the Rebonato formula (see [32], [5, 21]), the LMM model implied volatility $\sigma ^{LMM}_i(P)$ can be approximated by:

$$\begin{aligned} \sigma _i^{LMM}(P)\approx \sqrt{\frac{1}{m_i}\sum \limits _{k=1}^{m_i-1}\sum \limits _{q=1}^2\left( \sum \limits _{j=m_i}^{m_i+n_i-1}y_j^{m_i,n_i}\frac{F_j(0)}{S^{m_i,n_i}}\eta _j^q(i)\right) ^2}, \end{aligned}$$

with

$$\begin{aligned} y_k^{m_i,n_i}=\frac{P(0,k+1)}{\sum \nolimits _{j=m_i+1}^{m_i+n_i}P(0,j)} \end{aligned}$$

and

$$\begin{aligned} S^{m_i,n_i}=\sum \limits _{k=1}^{n_i-1}y_k^{m_i, n_i}F_k(0). \end{aligned}$$

The parameters are estimated by:

$$\begin{aligned} \hat{P}=\text{Argmin}\left[ \sum \limits _{i=1}^I\left( \frac{\sigma _i-\sigma _i^{LMM}(P)}{\sigma _i}\right) ^2\right] . \end{aligned}$$

This calibration approach is standard in day-to-day trading on the swaption market using the specific daily data. The Black implied volatilities are indeed very sensitive to the yield curve. To project the interest rates in the economic scenarios table, actuaries simply identify the spot LIBOR measure to the risk neutral measure. This $LMM_{ins}$, used for very long term projections, is simply not justified by market standards.