# On the Market-Consistent Valuation of Participating Life Insurance Heterogeneous Contracts under Longevity Risk

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Model Setup

Assets | Liabilities |

$W\left(0\right)$ | $E\left(0\right)=(1-{\alpha}_{1}-{\alpha}_{2})W\left(0\right)$ |

${L}_{1}\left(0\right)={\alpha}_{1}W\left(0\right)$ | |

${L}_{2}\left(0\right)={\alpha}_{2}W\left(0\right)$ | |

$W\left(0\right)$ | $W\left(0\right)$ |

#### 2.1. Contract Structure

**Case****1:**- the minimum interest rate ${g}_{i}\ge 0$, $i=1,2$, promised by the insurer;
**Case****2:**- the contract maturity date ${T}_{i}$, $i=1,2$.

**Case 1.**We assess the impact of different minimum interest rate guarantees. To set up this case recall that ${T}_{1}={T}_{2}=T$ and, without loss of generality, assume that ${g}_{1}>{g}_{2}$. As a result, the payoff promised to the policyholders of group i, $i=1,2$, at maturity T is ${G}_{i}\left(T\right)={N}_{i}\left(T\right)l\left(0\right){e}^{{g}_{i}T}$. Such instance is common to many life insurers since older products still in force often have significantly higher guaranteed rates than those sold more recently which were penalized by the ongoing low interest environment. Although in this stylized model all contracts are issued at the same date, our findings provide some guidance on establishing some contractual parameters, in particular the participation coefficients for which there is usually some discretion on the insurer’s side.

**Case 2.**We assume that the policies from the first and second group expire at time ${T}_{1}$, respectively time ${T}_{2}$, and, without loss of generality, let ${T}_{2}>{T}_{1}$. The minimum guaranteed rates are set to be equal, i.e., ${g}_{1}={g}_{2}=:g$. The guaranteed payoffs are now ${G}_{i}\left({T}_{i}\right)={N}_{i}\left({T}_{i}\right)l\left(0\right){e}^{g{T}_{i}}$, $i=1,2$.

#### 2.2. Modelling Insurance and Financial Risk

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Valuation

## 4. Numerical Analysis

- firstly, the fair participation rates ${\delta}_{i}^{*}$, $i=1,2$, under the pricing measure Q;
- secondly, the annual certainty equivalent log-returns of the life insurance contracts under the real world measure P, henceforth just certainty equivalent returns, denoted by ${C}_{i}$, $i=1,2$, based on the fair participation rates ${\delta}_{i}^{*}$.8

**Results for Case 1.**We set $T\in \left\{12,25\right\}$ and consider reasonable choices for the minimum interest rate guarantees, ${g}_{1}=1.75\%$ and ${g}_{2}=1.25\%$. As previously outlined, a potentially desirable goal of an insurer is to provide all policyholders with the same (expected) rate of return regardless of the individual minimum interest rate guarantee. Hence, comparing the certainty equivalent returns ${C}_{1}$ and ${C}_{2}$ for the two groups in the present case seems particularly interesting. Table 2, Table 3 and Table 4 contain our findings for the fair participation rates and the annual certainty equivalent returns.

- (2.1)
- Group 2, endowed with a lower interest rate guarantee, is naturally provided with a higher fair participation rate ${\delta}_{2}^{*}$ and a perceptibly larger implied certainty equivalent return ${C}_{2}$. In order to examine the goodness of the contract design in (4), middle row, to achieve similar rates of return for different groups of customers, we further carry out the analysis under the assumption that the same payout structure of group 1 is applied to group 2 (but still with different guarantees ${g}_{1}>{g}_{2}$) and find that ${\delta}_{2}^{*}$ and ${C}_{2}$ are even higher. Therefore, if the insurance company aimed at treating both groups fairly when the payoff structures are identical, it should assign a much larger fair participation rate to the second group than to the first one, resulting in a greater difference between ${C}_{1}$ and ${C}_{2}$. That is why our attempt at designing contracts that potentially provide the same rate of return to customers endowed with different minimum interest rate guarantees leads to more desirable results.
- (2.2)
- For any portfolio size, an increase in the longevity risk premium, i.e., lower values for ${E}^{Q}[\Delta ]$, leads to smaller fair participation rates. This is because longevity improvements anticipated by the insurer increase the expected number of survivors, and consequently the value of the outstanding liabilities. To offset this effect and simultaneously ensure fairness, lower participation rates are offered. The same observation holds true for the certainty equivalent returns of the policyholders under the physical measure P. The reason for this is due to the smaller participation rates ${\delta}_{i}^{*}$, $i=1,2$, since a greater degree of conservativeness harms the customers’ benefits.
- (2.3)
- For the exceptional case where ${N}_{1}\left(0\right)={N}_{2}\left(0\right)=1$, the fair participation rates are considerably higher than those obtained with larger portfolio sizes due to the sizeable extinction probability of the groups and the fact that the equity holders seize all the assets pertaining to the extinct group(s). As a compensation, the policyholders need to be served with substantially larger participation rates.
- (2.4)
- It seems that, with a very small portfolio size, e.g., ${N}_{i}\left(0\right)=10$, $i=1,2$, the portfolio is already well-diversified, i.e., the expected impact of the systematic part of the biometric risk is the only component still playing a role, since similar results are achieved as, e.g., when ${N}_{i}\left(0\right)=$ 100,000.
- (2.5)
- It is notable that all certainty equivalent returns lie between the risk free rate of return and the expected rate of return of the assets under P, i.e., ${C}_{i}\in (r,\mu )=(0.03,0.05)$, $i=1,2$. It is true that our life insurance policies cannot beat the pure investment into the assets due to the guaranteed interest rate, although the values obtained are much closer to $\mu $ than to r. Nevertheless, the included guarantees of the insurance products make them much less risky and are crucial for many potential customers when comparing different investment opportunities.

- (3.1)
- We observe that policyholders in the larger group obtain relatively higher fair participation rates ${\delta}_{i}^{*}$, $i=1,2$, and consequently mostly also relatively higher certainty equivalent returns ${C}_{i}$ on their investments when comparing them with the corresponding numbers from Table 2.
- (3.2)
- For the first group, an increase in ${N}_{1}\left(0\right)$ leads to an increase in ${\delta}_{1}^{*}$ and ${C}_{1}$ in general, independently of the size of the other group, while, for the second group, the opposite relations apply. We can conclude that, if there are only a few policyholders holding a lower individual guarantee than the rest, their participation in the surplus distribution must be very high to ensure fair contracts, especially if they represent a clear minority. Another related interesting fact is that a sudden spread within the values for ${\delta}_{i}^{*}$ and ${C}_{i}$, $i=1,2$, occurs as soon as the size ratio between the two groups shifts.

**Results for Case 2.**In this case, the maturity dates of the groups’ policies differ. The quantity ${V}_{2}\left({T}_{1}\right)$ in the definitions of the payoff functions (see Equations (9)–(12)) is given by

- (5.1)
- One of the main questions is: why are fair participation coefficients for the second group so much smaller compared to group 1, even though in Case 1 it was seen that a longer duration leads to higher fair participation rates? A possible explanation is the fact that only a portion of the assets ${\alpha}_{1}W\left({T}_{1}\right)$, pertaining to the first group at ${T}_{1}$, is paid out to its policyholders if the insurer is able to achieve a surplus. As a consequence, group 2 profits from the residual amount staying in the company which boosts the probability of gaining relatively high assets’ values in the future. To maintain fairness, a lower ${\delta}_{2}^{*}$ is required.
- (5.2)
- The spread between ${C}_{1}$ and ${C}_{2}$ is always significantly positive, although it decreases as ${E}^{Q}[\Delta ]$ increases. Clearly, the substantially higher fair participation rates for group 1 play a major role here. Nevertheless, these differences are much less relevant than those between ${\delta}_{1}^{*}$ and ${\delta}_{2}^{*}$.

- (6.1)
- The fair participation rates ${\delta}_{i}^{*}$, $i=1,2$, grow with ${N}_{i}\left(0\right)$. Therefore, it is surprising that, unlike the certainty equivalent return ${C}_{1}$ of the first group, ${C}_{2}$ declines as the second group size ${N}_{2}\left(0\right)$ increases (as in Table 3 where this also holds for ${\delta}_{2}^{*}$). Yet, this finding reinforces the fact that low values of ${\delta}_{2}^{*}$, when ${N}_{2}\left(0\right)$ is small, are necessary.
- (6.2)
- The most remarkable feature is given by the variation in the values of ${\delta}_{2}^{*}$ for a given mortality pricing assumption when changing the composition of the portfolio (in particular, when comparing cases with ${N}_{1}\left(0\right)<{N}_{2}\left(0\right)$ to cases with ${N}_{1}\left(0\right)>{N}_{2}\left(0\right)$).

- (7.1)
- Compared to Remark (6.2), the fair participation rate of the second group seems to smooth out over time within one longevity pricing assumption since the fluctuations between the varying pooling schemes subside. Specifically, ${\delta}_{2}^{*}$ goes down if ${N}_{2}\left(0\right)/{N}_{1}\left(0\right)$ is large (unlike Table 4 when compared to Table 3) and it goes up if ${N}_{2}\left(0\right)/{N}_{1}\left(0\right)$ is small (as in Table 4 when compared to Table 3). Looking at the values of ${\delta}_{1}^{*}$, the same pattern is observed, i.e., a high ${N}_{2}\left(0\right)/{N}_{1}\left(0\right)$ results in lower fair participation coefficients and a low ${N}_{2}\left(0\right)/{N}_{1}\left(0\right)$ leads to (much) higher ones.
- (7.2)
- Concerning the certainty equivalent returns, those of group 1 behave quite as expected, i.e., for the first two pooling schemes (low ${\delta}_{1}^{*}$), smaller figures of ${C}_{1}$ are obtained and for the last two combinations (high ${\delta}_{1}^{*}$), larger values occur, compared to Table 6. By contrast, the relevant values of ${C}_{2}$ are always significantly higher than their counterparts in the previous table. A possible reason for that is our assumption made in the specific definition of this quantity given in (25), namely that the premature payoff to the second group conditioned by a default event at time ${T}_{1}$ is invested into the riskless asset until ${T}_{2}$.

## 5. Concluding Remarks

- If the insurer decides to heavily load risk premiums for the systematic part of the insurance risk, lower fair participation rates result. This in turn also hits the customers’ returns, particularly if the presumptions on the longevity risk are very prudent.
- Maintaining usual practised participation rates of $\sim 80\u2013100\%$ (often prescribed by law) can give rise to severe financial problems for the insurer, as certain portfolio and parameter combinations actually imply smaller participation coefficients ensuring the fairness of the contracts.
- If the two groups differ exclusively in the promised minimum interest rate guarantee provided by the insurer (Case 1), the group endowed with the lower minimum interest rate guarantee receives a larger fair participation rate. This increase is intensified if the insurance company does not explicitly aim to provide similar returns to all policyholders. Consequently, the difference between the actual returns widens as well. Thus, the proposed definition of the payoff structure for this case turns out to be an option the insurer can exploit in order to protect the customers and advance the desirable goal of achieving similar returns for everyone.
- In Case 1, another observation leads to the insight that, if there are only a few policyholders holding a lower individual guarantee than the rest, their participation in the surplus sharing must be really high to ensure fair contracts, especially if they represent a clear minority.
- If the two groups differ exclusively in the contract maturity date (Case 2), the fair participation rate for the group with the longer contract duration is much lower, and so is the resulting actual return, although on a considerably smaller scale.
- In Case 2, the group with the longer contract duration receives a remarkably low fair participation rate if it outnumbers the members of the other group.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Actually, Hansen and Miltersen (2002) introduce the diversifiable component of mortality. |

2 | We assume that the insurance company issues no further debt, raises no capital and pays no dividends to the equity holders within the time frame of interest. |

3 | Note that we allow for different participation rates for the two groups as the insurance company’s goal is to set these rates so as to achieve fairness for both groups, see Section 3. |

4 | An alternative rule uses the weights ${\alpha}_{i}/\left({\alpha}_{1}+{\alpha}_{2}\right)$, $i=1,2$, so that the splitting rule is decided by the groups’ initial contributions. Choosing this alternative could, if only one group survives until time T, result in the equity holders receiving the remaining assets after the insurer has served the group still existent. |

5 | It may happen in (4) that the payoff in the third row is smaller than that in the middle one, corresponding to a lower assets’ value. However, this fairly rare event does not result in a contradiction since it is down to the insurer to decide to what extent the goal of achieving equal rates of return shall be pursued. |

6 | Through this way of modelling of the outstanding liabilities, it could happen that the payments to the equity holders decrease or even vanish, although the assets’ value increases at the same time. The rationale behind this circumstance is that when the assets pertaining to the policyholders as a whole create some surplus over the minimum guarantees, the insurer’s primary goal is to provide them with their regular bonuses, if possible. Some alternative modelling methods apply if the insurance company wants to calculate the possible bonus payments to the different stakeholders based on their initial contributions. In this case, the definition of ${\zeta}_{i}\left(T\right)$, $i=1,2$, needs to take into account that only $\left({\alpha}_{1}+{\alpha}_{2}\right)W\left(0\right)$ is provided by the policyholders of the two groups at time 0 leading to a modification of (5) and (7). However, for the sake of brevity, we examine only the case described before. |

7 | By assuming that the markets are arbitrage-free, such a probability measure Q exists. As insurance markets are incomplete, the measure Q is chosen among infinitely many equivalent martingale measures. |

8 | To calculate ${\delta}_{i}^{*}$, $i=1,2$, we solve numerically the equations in (16) with ${V}_{i}$, $i=1,2$, given by (17) for Case 1, and (18) and (19) for Case 2. Due to the complicated structure of the outstanding liabilities, the computation of ${V}_{i}$, $i=1,2$, is based on a standard Monte Carlo simulation encompassing 100,000 draws. The calculation of ${C}_{i}$, $i=1,2$, is again based on the Monte Carlo method. |

9 | The parameters of $\Delta \sim \Gamma (\beta ,\theta )$ are calculated via ${E}^{Q}\left[\Delta \right]=\beta \theta $ and $Va{r}_{Q}\left(\Delta \right)=\beta {\theta}^{2}$. |

10 | Under the physical measure P, $R\left({T}_{1},{T}_{2}\right)\sim \mathcal{N}\left(\left(\mu -{\sigma}^{2}/2\right)({T}_{2}-{T}_{1}),{\sigma}^{2}({T}_{2}-{T}_{1})\right)$. |

11 | Under the physical measure P, ${N}_{2}\left({T}_{2}\right)|\left(\Delta ,{N}_{2}\left({T}_{1}\right)\right)\sim Binomial\left({N}_{2}\left({T}_{1}\right),{\left({}_{{T}_{2}-{T}_{1}}{\tilde{p}}_{{x}_{2}+{T}_{1}}^{*}\right)}^{\Delta}\right)$. |

**Figure 1.**Case 1: fair participation rates in percentage when ${E}^{Q}[\Delta ]=0.8$. Triangles show rates from Table 3 for given combinations of group sizes. Circles additionally show rates from Table 2 when group sizes fulfil ${N}_{2}\left(0\right)={N}_{1}\left(0\right)=10,100,1000,$ 100,000 (

**a**) and ${N}_{1}\left(0\right)={N}_{2}\left(0\right)=1000,\mathrm{100,000},100,10$ (

**b**).

**Figure 2.**Case 2: 100 scenarios of the outstanding liability in terms of the participation rate when ${E}^{Q}[\Delta ]=0.8$ (grey dotted if no default at maturity ${T}_{1}$, grey otherwise). Additionally, the mean (black dashed) and discounted mean (black) of the outstanding liability, and the premium (black dotted) are shown.

**Figure 3.**Case 2: fair participation rate ${\delta}_{2}^{*}$ in percentage (

**left**), assets’ and liability’s values (

**right**) in terms of the size of the first group when ${N}_{2}\left(0\right)=3000$ and ${E}^{Q}[\Delta ]=0.8$. In the right-hand plot, total assets’ value $W\left({T}_{1}\right)$ at ${T}_{1}$ (dark grey), outstanding liability for the first group ${L}_{1}\left({T}_{1}\right)$ at ${T}_{1}$ (light grey) and assets’ value ${W}^{\prime}\left({T}_{2}\right)$ at ${T}_{2}$ (black) are shown.

Symbol | Description | Value |
---|---|---|

$l\left(0\right)$ | Initial contribution of a single policyholder | 35 |

$e\left(0\right)$ | Equity holders’ share of initial assets | 0.3 |

${x}_{1}\left(={x}_{2}\right)$ | Initial age of policyholders | 40 |

${\lambda}_{1}\left(={\lambda}_{2}\right)$ | Age independent Gompertz parameter | $2.6743\xb7{10}^{-5}$ |

${c}_{1}\left(={c}_{2}\right)$ | Age dependent Gompertz parameter | 1.098 |

$\beta $ | Shape parameter of $\Delta $ under Q | $\left\{1.6,6.4,10\right\}$ |

$\theta $ | Scale parameter of $\Delta $ under Q | $\left\{0.25,0.125,0.1\right\}$ |

r | Risk free short rate | 3% |

$\sigma $ | Assets’ volatility | 15% |

$\mu $ | Assets’ expected instantaneous rate of return | 5% |

$\gamma $ | Adjustment factor to force of mortality | 0.9 |

**Table 2.**Fair participation rates and certainty equivalent returns for Case 1 with $T=12$, different portfolio sizes with ${N}_{1}\left(0\right)={N}_{2}\left(0\right)$ and different values of ${E}^{Q}\left[\Delta \right]$. All results are in percentage.

$\mathit{T}=12$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.4$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.8$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=1$ | ||||
---|---|---|---|---|---|---|---|

${N}_{1}\left(0\right)$ | ${N}_{2}\left(0\right)$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ |

${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ||

1 | 1 | 72.42 | 78.01 | 76.38 | 81.60 | 78.37 | 83.41 |

4.22 | 4.32 | 4.35 | 4.44 | 4.42 | 4.51 | ||

10 | 10 | 70.03 | 75.65 | 71.32 | 76.66 | 71.94 | 77.14 |

4.30 | 4.41 | 4.35 | 4.44 | 4.37 | 4.46 | ||

100 | 100 | 70.05 | 75.68 | 71.34 | 76.69 | 71.96 | 77.17 |

4.30 | 4.41 | 4.35 | 4.44 | 4.37 | 4.46 | ||

1000 | 1000 | 70.06 | 75.68 | 71.35 | 76.70 | 71.97 | 77.19 |

4.30 | 4.41 | 4.35 | 4.45 | 4.37 | 4.46 | ||

100,000 | 100,000 | 70.06 | 75.68 | 71.35 | 76.70 | 71.97 | 77.18 |

4.30 | 4.41 | 4.35 | 4.45 | 4.37 | 4.46 |

**Table 3.**Fair participation rates and certainty equivalent returns for Case 1 with $T=12$, different portfolio sizes with ${N}_{1}\left(0\right)\ne {N}_{2}\left(0\right)$ and different values of ${E}^{Q}\left[\Delta \right]$. All results are in percentage.

$\mathit{T}=12$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.4$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.8$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=1$ | ||||
---|---|---|---|---|---|---|---|

${N}_{1}\left(0\right)$ | ${N}_{2}\left(0\right)$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ |

${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ||

10 | 1000 | 68.51 | 74.59 | 69.80 | 75.66 | 70.44 | 76.18 |

4.26 | 4.38 | 4.31 | 4.42 | 4.33 | 4.44 | ||

100 | 100,000 | 68.55 | 74.57 | 69.91 | 75.64 | 70.56 | 76.15 |

4.26 | 4.37 | 4.31 | 4.41 | 4.33 | 4.43 | ||

1000 | 100 | 71.34 | 76.75 | 72.57 | 77.71 | 73.16 | 78.18 |

4.33 | 4.44 | 4.38 | 4.47 | 4.40 | 4.49 | ||

100,000 | 10 | 71.63 | 77.01 | 72.84 | 77.96 | 73.43 | 78.43 |

4.34 | 4.44 | 4.38 | 4.48 | 4.40 | 4.50 |

**Table 4.**Fair participation rates and certainty equivalent returns for Case 1 with $T=25$, different portfolio sizes with ${N}_{1}\left(0\right)\ne {N}_{2}\left(0\right)$ and different values of ${E}^{Q}\left[\Delta \right]$. All results are in percentage.

$\mathit{T}=25$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.4$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.8$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=1$ | ||||
---|---|---|---|---|---|---|---|

${N}_{1}\left(0\right)$ | ${N}_{2}\left(0\right)$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ |

${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ||

10 | 1000 | 82.45 | 88.11 | 84.59 | 89.59 | 85.56 | 90.24 |

4.61 | 4.72 | 4.67 | 4.77 | 4.69 | 4.79 | ||

100 | 100,000 | 82.56 | 88.09 | 84.80 | 89.57 | 85.81 | 90.22 |

4.61 | 4.72 | 4.67 | 4.77 | 4.70 | 4.79 | ||

1000 | 100 | 85.80 | 90.45 | 87.58 | 91.58 | 88.39 | 92.09 |

4.69 | 4.78 | 4.74 | 4.82 | 4.76 | 4.83 | ||

100,000 | 10 | 86.13 | 90.73 | 87.87 | 91.84 | 88.66 | 92.33 |

4.69 | 4.79 | 4.74 | 4.82 | 4.76 | 4.84 |

**Table 5.**Fair participation rates and certainty equivalent returns for Case 2 with $\left({T}_{1},{T}_{2}\right)=(10,12)$, different portfolio sizes with ${N}_{1}\left(0\right)={N}_{2}\left(0\right)$ and different values of ${E}^{Q}\left[\Delta \right]$. All results are in percentage.

$\left({\mathit{T}}_{1},{\mathit{T}}_{2}\right)=(10,12)$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.4$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.8$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=1$ | ||||
---|---|---|---|---|---|---|---|

${N}_{1}\left(0\right)$ | ${N}_{2}\left(0\right)$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ |

${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ||

1 | 1 | 75.52 | 49.27 | 78.36 | 52.07 | 79.90 | 53.47 |

4.26 | 4.04 | 4.36 | 4.16 | 4.42 | 4.22 | ||

10 | 10 | 73.62 | 47.73 | 74.51 | 48.80 | 74.92 | 49.32 |

4.34 | 4.13 | 4.37 | 4.18 | 4.39 | 4.20 | ||

100 | 100 | 73.63 | 47.74 | 74.53 | 48.82 | 74.95 | 49.34 |

4.34 | 4.13 | 4.37 | 4.18 | 4.39 | 4.20 | ||

1000 | 1000 | 73.64 | 47.74 | 74.53 | 48.82 | 74.96 | 49.34 |

4.34 | 4.13 | 4.37 | 4.18 | 4.39 | 4.20 | ||

100,000 | 100,000 | 73.63 | 47.74 | 74.53 | 48.82 | 74.96 | 49.35 |

4.34 | 4.13 | 4.37 | 4.18 | 4.39 | 4.20 |

**Table 6.**Fair participation rates and certainty equivalent returns for Case 2 with $\left({T}_{1},{T}_{2}\right)=(10,12)$, different portfolio sizes with ${N}_{1}\left(0\right)\ne {N}_{2}\left(0\right)$ and different values of ${E}^{Q}\left[\Delta \right]$. All results are in percentage.

$\left({\mathit{T}}_{1},{\mathit{T}}_{2}\right)=(10,12)$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.4$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.8$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=1$ | ||||
---|---|---|---|---|---|---|---|

${N}_{1}\left(0\right)$ | ${N}_{2}\left(0\right)$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ |

${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ||

10 | 1000 | 72.82 | 53.45 | 73.65 | 54.45 | 74.04 | 54.94 |

4.32 | 4.11 | 4.35 | 4.16 | 4.37 | 4.19 | ||

100 | 100,000 | 72.84 | 53.52 | 73.71 | 54.53 | 74.14 | 55.02 |

4.32 | 4.11 | 4.35 | 4.16 | 4.37 | 4.19 | ||

1000 | 100 | 74.30 | 38.77 | 75.21 | 39.88 | 75.65 | 40.43 |

4.36 | 4.15 | 4.39 | 4.20 | 4.41 | 4.22 | ||

100,000 | 10 | 74.45 | 35.60 | 75.36 | 36.68 | 75.80 | 37.21 |

4.36 | 4.16 | 4.40 | 4.20 | 4.41 | 4.23 |

**Table 7.**Fair participation rates and certainty equivalent returns for Case 2 with $\left({T}_{1},{T}_{2}\right)=(12,25)$, different portfolio sizes with ${N}_{1}\left(0\right)\ne {N}_{2}\left(0\right)$ and different values of ${E}^{Q}\left[\Delta \right]$. All results are in percentage.

$\left({\mathit{T}}_{1},{\mathit{T}}_{2}\right)=(12,25)$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.4$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=0.8$ | ${\mathit{E}}^{\mathit{Q}}\left[\mathbf{\Delta}\right]=1$ | ||||
---|---|---|---|---|---|---|---|

${N}_{1}\left(0\right)$ | ${N}_{2}\left(0\right)$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ | ${\delta}_{1}^{*}$ | ${\delta}_{2}^{*}$ |

${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ||

10 | 1000 | 70.61 | 51.90 | 71.29 | 53.75 | 71.66 | 54.62 |

4.25 | 4.39 | 4.27 | 4.48 | 4.29 | 4.52 | ||

100 | 100,000 | 70.62 | 51.99 | 71.35 | 53.83 | 71.75 | 54.71 |

4.25 | 4.39 | 4.28 | 4.48 | 4.29 | 4.52 | ||

1000 | 100 | 77.35 | 39.07 | 78.24 | 41.06 | 78.67 | 42.01 |

4.44 | 4.33 | 4.47 | 4.43 | 4.48 | 4.48 | ||

100,000 | 10 | 78.19 | 36.59 | 79.13 | 38.59 | 79.58 | 39.52 |

4.46 | 4.33 | 4.49 | 4.43 | 4.51 | 4.47 |

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## Share and Cite

**MDPI and ACS Style**

Bacinello, A.R.; Chen, A.; Sehner, T.; Millossovich, P.
On the Market-Consistent Valuation of Participating Life Insurance Heterogeneous Contracts under Longevity Risk. *Risks* **2021**, *9*, 20.
https://doi.org/10.3390/risks9010020

**AMA Style**

Bacinello AR, Chen A, Sehner T, Millossovich P.
On the Market-Consistent Valuation of Participating Life Insurance Heterogeneous Contracts under Longevity Risk. *Risks*. 2021; 9(1):20.
https://doi.org/10.3390/risks9010020

**Chicago/Turabian Style**

Bacinello, Anna Rita, An Chen, Thorsten Sehner, and Pietro Millossovich.
2021. "On the Market-Consistent Valuation of Participating Life Insurance Heterogeneous Contracts under Longevity Risk" *Risks* 9, no. 1: 20.
https://doi.org/10.3390/risks9010020