Fair valuation of cliquet-style return guarantees in (homogeneous and) heterogeneous life insurance portfolios

ABSTRACT

Participating life insurance contracts allow the policyholder to participate in the annual return of a reference portfolio. Additionally, they are often equipped with an annual (cliquet-style) return guarantee. The current low interest rate environment has again refreshed the discussion on risk management and fair valuation of such embedded options. While this problem is typically discussed from the viewpoint of a single contract or a homogeneous* insurance portfolio, contracts are, in practice, managed within a heterogeneous insurance portfolio. Their valuation must then – unlike the case of asset portfolios – take account of portfolio effects: Their premiums are invested in the same reference portfolio; the contracts interact by a joint reserve, individual surrender options and joint default risk of the policy sponsor. Here, we discuss the impact of portfolio effects on the fair valuation of insurance contracts jointly managed in (homogeneous and) heterogeneous life insurance portfolios. First, in a rather general setting, including stochastic interest rates, we consider the case that otherwise homogeneous contracts interact due to the default risk of the policy sponsor. Second, and more importantly, we then also consider the case when policies are allowed to differ in further aspects like the guaranteed rate or time to maturity. We also provide an extensive numerical example for further analysis.

KEYWORDS:

• Life insurance
• fair valuation
• embedded options
• heterogeneous portfolios
• participating contracts
• return guarantee

CLASSIFICATION:

• G13
• G22

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Peter Hieber http://orcid.org/0000-0001-5189-5308

Ralf Werner http://orcid.org/0000-0001-8204-3757

Notes

2 Following ideas for the construction of equivalent martingale measures as for example described by Rogers (1994), it can reasonably be assumed that one specific martingale measure is fixed corresponding to the market equilibrium.

3 These numbers are taken from the annual report of Allianz Life 2017, p. 34, see https://www.allianz.com/.

4 According to Quantitative Impact Study 5 (QIS5) by the German supervisory authority BaFin, the main risk driver of European life insurance companies around 80% is market risk. The biggest share in market risk is interest rate risk (64%).

5 Mortality risk can, with some effort, be included in the current analysis. Under the assumption that, upon death, a fair market-consistent value is paid to the heirs of the policyholder, results are only slightly affected. Surrender risk is discussed in Section 4.5.

6 In this setup, we assume that the return credited to the policyholders is directly linked to a reference account A (the dynamics of A are specified later). The reference account might be a portfolio of (liquidly) traded securities (i.e. a portfolio of equity and bond indices, see, for example, the quite general setup by Barbarin and Devolder 2005).

7 Note that this assumption is relevant if and only if the guaranteed return $g^{\left(i\right)}$ is negative.

8 Working on discrete instead of continuous returns, related articles also use the surplus participation scheme

For realistic parameter sets, the use of either (1) or (2(2) $\begin{array}{rl}{P}_{t+1}^{\left(i\right)}& =P_t^{\left(i\right)}\cdot \exp \left(max\left(g^{\left(i\right)},{\alpha }_{t+1}^{\left(i\right)}\cdot \ln \left(\frac{A_{t+1}}{A_t}\right)\right)\right)\\ & =P_t^{\left(i\right)}\cdot max\left(\exp \left(g^{\left(i\right)}\right),{\left(\frac{A_{t+1}}{A_t}\right)}^{{\alpha }_{t+1}^{\left(i\right)}}\right),\end{array}$(2) ) only very marginally affect the payoff streams of the insurance contracts.

9 As already indicated, we can assume that a specific martingale measure has been chosen by the market in accordance with the market equilibrium.

10 Examples of regulatory restrictions in Germany: (1) The default probability, respectively the amount of equity/reserves, has to be in accordance with Solvency II regulation. (2) At least 90% of the surplus has to be credited to the policyholders. (3) Both the technical interest rate g and the share of risky assets γ are subject to upper bounds.

11 This assumption guarantees that the Vasicek-type structure is preserved under the measure change from $\mathbb{P}$ to $\mathbb{Q}$. Further, the Novikov condition is satisfied for a constant market price of risk (see, e.g. (Brigo and Mercurio 2007)).

12 We use annualized default probabilities as they are easier to compare and interpret and correspond to the one year horizon that is usually the time frame used by regulators. Note, however, that a structural model implies a term structure of default probabilities that cannot be summarized by just one number.

13 Note that the terminal bonus accounts ${\mathcal{B}}_0^{\left(i\right)}$, $i=1,2,\dots ,K$, are part of the insurance company's equity capital $E_0$. This is due to the fact that terminal bonuses are lost in case of a default event. If contract i matures at time $T^{\left(i\right)}$ and the insurance company is still solvent ($\tau >T^{\left(i\right)}$), the contract's surplus ${\mathcal{S}}_{T^{\left(i\right)}}^{\left(i\right)}:=P_0^{\left(i\right)}\cdot max(A_{t+1}/A_0-P_{t+1}^{\left(i\right)}/P_0^{\left(i\right)},0)$ is shared between the policyholders (share ${\beta }^{\left(i\right)}$) and insurance company (share $1-{\beta }^{\left(i\right)}$). The policyholders receive the terminal bonus ${\mathcal{B}}_{T^{\left(i\right)}}^{\left(i\right)}={\beta }^{\left(i\right)}\cdot {\mathcal{S}}_{T^{\left(i\right)}}^{\left(i\right)}$ at time $T^{\left(i\right)}$ (see (3(3) ${\mathcal{B}}_{t+1}^{\left(i\right)}:={\beta }^{\left(i\right)}\cdot P_0^{\left(i\right)}\cdot max\left(\frac{A_{t+1}}{A_0}-\frac{P_{t+1}^{\left(i\right)}}{P_0^{\left(i\right)}},0\right).$(3) )).