#
Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities^{}^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Marginal Survival Functions

#### 2.2. Copulas

#### 2.2.1. One-Parameter Copulas

#### 2.2.2. Two-Parameter Copulas

- a product:$${C}_{\alpha ,\theta}(u,v)={u}^{1-\alpha}{v}^{1-\alpha}{C}_{\theta}({u}^{\alpha},{v}^{\alpha})$$
- a linear mix:$${C}_{\alpha ,\theta}(u,v)=(1-\alpha )uv+\alpha {C}_{\theta}(u,v)$$
- a geometrically-weighted average:$$\phantom{\rule{4pt}{0ex}}{C}_{\alpha ,\theta}(u,v)={\left(uv\right)}^{1-\alpha}{\left({C}_{\theta}(u,v)\right)}^{\alpha}$$

## 3. Calibration Methods

#### 3.1. Marginal Survival Functions

#### 3.2. Copulas

## 4. Calibration Results

#### 4.1. Marginal Survival Functions

#### 4.2. Decreasing Spouses’ Dependence over Generations

- We create artificially two observation windows out of the unique one, by distinguishing the period 29 December 1988–30 June 1991 from the period 1 July 1991–31 December 1993.
- For each cohort, we compute Kendall’s tau in the two sub-windows, in which the same individuals have different initial ages.

#### 4.3. Joint Calibration: Archimedean Copulas

- Class 2P: For both generations, Table 6 and Table 8 show that the two-parameter family that performs best is the Gumbel–Hougaard linear mix. We observe that the LH of the 2P-product is very similar (and in many cases, identical) to those of the 2P-geometric. For the old generation, the linear-mixing copula is closely followed by the product and geometric Gumbel–Hougaard mixes (as pointed out before, the latter two entail identical copulas): the 2P mixes with the GH copula have a LH much higher than all of the other copulas. This is not the case for the young generation, where the LH are of similar order of magnitude for all copulas and way of mixing.

- points on or close to the main diagonal; considering these points only gives an impression of very strong spouses’ dependence (evidently stronger for the old generation than for the young generation);
- the remaining points; considering these points only gives an impression of weak or no spouses’ dependence.

## 5. Effects of Spouses’ Dependence on Pricing

#### 5.1. Prices of Combined Joint Life and Reversionary Annuities

- For each R and each model specification (1P, 2P), the young generation shows ratios of cum-spouses’ dependence to independence price that are closer to one than those of the old generation. This is a clear consequence of the decreasing τ from old generation to young generation: the milder spouses’ dependence of the young generation generates prices that deviate less from the independence prices than the old generation prices.
- In both tables and for each model specification (1P, 2P), the ratio cum-spouses’ dependence/independence is decreasing when R increases. This can be explained, as well. Let us recall that for $R=0$, we have the joint life annuity, and for $R=1$, we have the last survivor policy. Then, R measures the weight given to the last-survivor part of the reversionary annuity, with respect to the joint-life part. When $R=0$, positive spouses’ dependence implies that the joint survival probability is higher than in the independence case, leading to a ratio greater than one. At the opposite, when $R=1$, we have the last survivor, for which positive spouses’ dependence implies lower survivorship after the spouse’s death, implying ratios lower than one. The values $0<R<1$ give all of the intermediate situations between these two extremes. In particular, for $R\in (0,1/2)$, we still have ratios greater than one; for $R\in (1/2,1)$, we have ratios lower than one. For $R=1/2$, the ratio is exactly one: the annuity price is unaffected by the level of spouses’ dependence. In fact, due to Equation (13), the joint survival probability does not enter the premium that reduces to:$$\sum _{t=1}^{+\infty}{v}^{t}\left(\frac{{}_{t}{p}_{x}^{m}{+}_{t}{p}_{y}^{f}}{2}\right)$$
- The practical consequence of having ratios greater than one as long as $R<1/2$ and ratios smaller than one when $R>1/2$ is that insurance companies, by assuming independence when pricing the former, are under-pricing contracts, while they are over-pricing, or prudentially pricing, the latter. Consider the joint life case. Insurance companies that assume independence are not “on the safe side”. The previous tables give a measure of the lack of safety so obtained. Consistently with Item 1 above, the lack of safety is greater for the older generation. Consider now the last survivor policy, for which insurance companies that assume independence are overpricing the contract. This can be interpreted as a prudential maneuver from the point of view of insurers, and the previous tables give a measure of the extent of prudence so obtained. Consistently with Item 1 above, prudence decreases when the younger generation is selected.
- For the old generation, the impact on prices and on the ratio cum-spouses’ dependence/independence is smaller for the 2P copulas than for the 1P copula; the opposite happens for the young generation. As a consequence, for the old generation, the width of the range of prices and ratios when R changes is smaller for the 2P copulas than for the 1P copula; the opposite happens for the young generation. The next item illustrates the mispricing implications of this asymmetry.
- Misspecification of the copula produces opposite mispricing effects on the two generations. Indeed, given the assessed superiority of the 2P model with respect to the 1P one, annuity prices show that when the spouses’ dependence is described with a 1P copula rather than with a 2P one, for the old generation, the insurer over-prices annuities with $R<1/2$ and under-prices annuities with $R>1/2$. Opposite results apply to the young generation: when the spouses’ dependence is described with a 1P copula rather than with a 2P one, the insurer under-prices annuities with $R<1/2$, and over-prices annuities with $R>1/2$. The occurrence of opposite mispricing effects for the two generations could represent a potential for the insurer who wants to natural-hedge reversionary annuities written on one cohort with products on a different cohort.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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^{1.}The survival function given by Equation (5) is biologically reasonable (i.e., it is decreasing over time) if and only if the following condition holds: ${e}^{{b}_{i}^{j}t}\left[{\left({\sigma}_{i}^{j}\right)}^{2}+2{\left({d}_{i}^{j}\right)}^{2}\right]>{\left({\sigma}_{i}^{j}\right)}^{2}-2{d}_{i}^{j}{c}_{i}^{j}.$ In our calibrations, this is always true.^{2.}We did fit more common 2P families, such as Student’s copula and several families in [22], to the data, as well, but the fits obtained were not as good as the ones resulting from the three types specified above. In preliminary work, we investigated also the general three-parameter families as proposed in [23,24], built as follows:$${C}_{\alpha ,\beta ,\theta}(u,v)={u}^{1-\alpha}{v}^{1-\beta}{C}_{\theta}({u}^{\alpha},{v}^{\beta})$$^{3.}To be more precise, the males of the older generation were born between 1.1.1900 and 31.12.1913, while the corresponding females between 1 January 1903 and 31 December 1916, and so on.^{4.}This was indeed the method used in [6], where the authors used censored data on the same dataset.^{5.}Let $\ell \left(\theta \u02cb\right)$ be defined as in Equation (12). Then, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are defined as:$$\mathbf{AIC}=-\frac{2}{n}\left(\ell \left(\theta \u02cb\right)-p\right),\phantom{\rule{2.em}{0ex}}\mathbf{BIC}=-\frac{2}{n}\left(\ell \left(\theta \u02cb\right)-\frac{logn}{2}p\right)$$

Name | $C\left(u,v\right)$ |

Clayton | ${\left({u}^{-\theta}+{v}^{-\theta}-1\right)}^{-\frac{1}{\theta}}$ |

Gumbel-Hougaard | $exp\left[-{\left({\left(-lnu\right)}^{\theta}+{\left(-lnv\right)}^{\theta}\right)}^{\frac{1}{\theta}}\right]$ |

Frank | $-\frac{1}{\theta}ln\left[1+\frac{\left({e}^{-\theta u}-1\right)\left({e}^{-\theta v}-1\right)}{{e}^{-\theta}-1}\right]$ |

Nelsen 4.2.20 | ${\left[ln\left(exp\left({u}^{-\theta}\right)+exp\left({v}^{-\theta}\right)-e\right)\right]}^{-\frac{1}{\theta}}$ |

Special | ${\left(\frac{-W+\sqrt{4+{W}^{2}}}{2}\right)}^{\frac{1}{\theta}};$ with $W=\frac{1}{{u}^{\theta}}-{u}^{\theta}+\frac{1}{{v}^{\theta}}-{v}^{\theta}$ |

OG Male | OG Female | YG Male | YG Female | |
---|---|---|---|---|

a | 961.045 | 790.232 | 528.581 | 619.733 |

σ | 0.007 | 0.057 | 0.019 | 0.5 |

OG | YG | |
---|---|---|

Kendall’s τ | 0.440 | 0.279 |

τ | December 1988–June 1991 | July 1991–December 1993 |
---|---|---|

OG | 0.636 | 0.502 |

YG | 0.543 | 0.411 |

OG | LH-1P | AIC-1P | BIC-1P |
---|---|---|---|

Fr | $11.806$ | $-0.327$ | $-0.294$ |

Cl | $6.04$ | $-0.153$ | $-0.120$ |

GH | $\mathbf{14.396}$ | $-\mathbf{0.406}$ | $-\mathbf{0.373}$ |

Ne | $4.950$ | $-0.12$ | $-0.086$ |

Sp | $2.216$ | $-0.037$ | $-0.004$ |

OG | LH-2Ppr | AIC-2Ppr | BIC-2Pr | LH-2Pl | AIC-2Pl | BIC-2PL | LH-2Pg | AIC-2Pg | BIC-2Pg |
---|---|---|---|---|---|---|---|---|---|

Fr | $32.627$ | −0.928 | −0.862 | $32.625$ | −0.928 | −0.862 | $33.731$ | −0.961 | −0.895 |

Cl | $31.583$ | $-0.896$ | −0.830 | $26.607$ | −0.745 | −0.679 | $31.583$ | $-0.896$ | −0.830 |

GH | $41.015$ | $-1.182$ | −1.116 | $\mathbf{41.714}$ | $-\mathbf{1.203}$ | $-\mathbf{1.137}$ | $41.015$ | $-1.182$ | −1.116 |

Ne | $18.754$ | $-0.508$ | −0.441 | $12.887$ | −0.329 | −0.263 | $18.754$ | −0.507 | −0.441 |

Sp | $12.619$ | −0.321 | −0.255 | $10.722$ | −0.264 | −0.198 | $13.258$ | −0.341 | −0.275 |

YG | LH-1P | AIC-1P | BIC-1P |
---|---|---|---|

Fr | $4.736$ | $-0.113$ | −0.080 |

Cl | $6.696$ | $-0.173$ | −0.139 |

GH | $2.39$ | $-0.042$ | −0.009 |

Ne | $6.444$ | $-0.165$ | −0.132 |

Sp | $\mathbf{6.832}$ | $-\mathbf{0.177}$ | $-\mathbf{0.144}$ |

YG | LH-2Ppr | AIC-2Ppr | BIC-2Ppr | LH-2Pl | AIC-2Pl | BIC-2Ppl | LH-2Pg | AIC-2Pg | BIC-2Ppg |
---|---|---|---|---|---|---|---|---|---|

Fr | $9.31$ | $-0.221$ | −0.155 | $11.055$ | −0.274 | −0.208 | $8.762$ | −0.204 | −0.139 |

Cl | $9.555$ | $-0.229$ | −0.163 | $11.048$ | −0.274 | −0.208 | $9.555$ | $-0.229$ | −0.163 |

GH | $9.057$ | $-0.214$ | −0.148 | $\mathbf{11.516}$ | $-\mathbf{0.288}$ | $-\mathbf{0.222}$ | $9.057$ | $-0.214$ | −0.147 |

Ne | $8.915$ | $-0.209$ | −0.143 | $9.835$ | −0.237 | −0.171 | $8.915$ | $-0.209$ | −0.143 |

Sp | $10.049$ | $-0.244$ | −0.178 | $11.250$ | −0.280 | −0.214 | $10.049$ | −0.243 | −0.178 |

OG | Gumb-Houg-1P | Gumb-Houg-Linear–Mix-2P |
---|---|---|

θ | $1.758$ | $12.134$ |

α | − | $0.550$ |

YG | Special-1P | Gumb-Houg-Linear–Mix-2P |
---|---|---|

θ | $1.116$ | $6.100$ |

α | − | $0.373$ |

**Table 11.**Reversionary annuity price for the old generation under independence and using best-fit 1P, 2P copulas.

R | Indep. | GH-1P | Ratio | GH-2Pl | Ratio |
---|---|---|---|---|---|

0 | $7.72$ | $8.786$ | $1.138$ | $8.755$ | $1.110$ |

$1/4$ | $9.772$ | $10.305$ | $1.054$ | $10.199$ | $1.044$ |

$1/3$ | $10.456$ | $10.811$ | $1.032$ | $10.740$ | $1.027$ |

$1/2$ | $11.823$ | $11.823$ | 1 | $11.823$ | 1 |

$2/3$ | $13.191$ | $12.835$ | $0.975$ | $12.906$ | $0.978$ |

$3/4$ | $13.875$ | $13.342$ | $0.964$ | $13.448$ | $0.969$ |

1 | $15.926$ | $14.860$ | $0.937$ | $15.072$ | $0.946$ |

**Table 12.**Reversionary annuity price for the young generation under independence and using best-fit 1P, 2P copulas.

R | Indep. | Spec.-1P | Ratio | GH-2Pl | Ratio |
---|---|---|---|---|---|

0 | $16.421$ | $17.056$ | $1.039$ | $17.137$ | $1.044$ |

$1/4$ | $19.271$ | $19.589$ | $1.016$ | $19.580$ | $1.019$ |

$1/3$ | $20.221$ | $20.433$ | $1.010$ | $20.394$ | $1.012$ |

$1/2$ | $22.121$ | $22.121$ | 1 | $22.121$ | 1 |

$2/3$ | $24.021$ | $23.810$ | $0.991$ | $23.650$ | $0.990$ |

$3/4$ | $24.971$ | $24.654$ | $0.987$ | $24.464$ | $0.986$ |

1 | $27.822$ | $27.187$ | $0.977$ | $26.907$ | $0.974$ |

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**MDPI and ACS Style**

Luciano, E.; Spreeuw, J.; Vigna, E.
Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities^{}. *Risks* **2016**, *4*, 16.
https://doi.org/10.3390/risks4020016

**AMA Style**

Luciano E, Spreeuw J, Vigna E.
Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities^{}. *Risks*. 2016; 4(2):16.
https://doi.org/10.3390/risks4020016

**Chicago/Turabian Style**

Luciano, Elisa, Jaap Spreeuw, and Elena Vigna.
2016. "Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities^{}" *Risks* 4, no. 2: 16.
https://doi.org/10.3390/risks4020016