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Dependence may arise in insurance when the insureds are clustered into groups e.g. joint-life annuities. This dependence may be produced by sharing a common risk acting on mortality of members of the group. Various dependence models have been considered in literature ; however, the focus has been on either the lower-tail dependence alone or upper-tail dependence alone. This article implements the frailty dependence approach to life insurance problems where most applications have been within medical setting. Our strategy is to use the conditional independence assumption given an observed association measure in a positive stable frailty approach to account for both lower and upper-tail dependence. The model is calibrated on the association of Kenyan insurers 2010 male and female published rates. The positive stable model is then proposed to construct dependence life-tables and generate life annuity payment streams in the competitive Kenyan market.

When annuity payments are concerned, the calculation of expected present values (EPVs), needed in pricing and reserving, requires an appropriate mortality model in order to avoid biased valuations [

Our article contributes to the existing literature in several ways. First, we apply a shared frailty approach to life insurance risk problems where most applications have been within medical setting. Second, most dependence models in literature have focused on either the lower-tail dependence alone or upper-tail dependence alone. We apply the conditional independence assumption for an observed association measure in a positive stable frailty to account for both lower and upper-tail dependence. Third, we incorporate a stochastic dependence structure via a dynamically evolving positive stable process to model time-varying shared risk.

To facilitate an easier discussion of frailty dependence modeling, the following notations are used. We consider joint-life annuity contracts where x is the age of the male annuitant whose future lifetime random variable is t i 1 and y the age of the female annuitant whose future lifetime random variable is t i 1 . v t is the present value factor, s x ¯ y ( t ) is the probability of survival until the last of ( x , y ) dies and a x ¯ y the EPV of the benefit. D x y = v x y l x y and N x y = ∑ k = 0 ∞ D x y + k are commutation functions.

For simplicity of notation of the commutation functions, we suppose that the limiting age of our mortality table is infinity.

The force of mortality λ x ( t ) for a life aged x during time t is assumed piece-wise constant across each whole year of age [ t , t + 1 ) . i.e. λ x ( t ) = λ x + z ( t ) ; 0 < z < 1 . Similar assumptions are found in [

A deterministic financial structure is adopted for the present value factor v t for illustration purposes.

Model calibration with reference to standard mortality tables [

The association of Kenyan insurers (AKI) 2010 published mortality tables is based upon data collected by the association of Kenya insurers for an investigation into the mortality of assured lives in the Republic of Kenya. The AKI 2010 mortality rates are used as the baseline hazard rates in the study. Joint-life and last survivor annuitants member data for policies in-force between 2001-2013 will be used in the study to determine the average age difference for insured couples in Kenya. The annuitants data used in this article was obtained from a major Kenyan insurance company. To preserve confidentiality, we took a sub-sample of 178 joint-life policies.

The proposed model can be applied to any type of joint-life annuity business. In this article, we discuss the case of joint-life last survivor annuities. This is a contract that provides level payments to two or more annuitants until the last of them dies. This can be an immediate annuity for a single lump-sum payment whose expected present value (EPV) of say amount b per annum, payable in arrears, until the last of ( x , y ) dies is given by:

b a x ¯ y = b ∑ t = 1 ∞ v t s x ¯ y ( t ) (1)

where the future lifetime random variable T = M a x ( t i 1 , t i 2 ) .

Under the dependence (frailty) assumption:

s x ¯ y ( t ) = s x ( t ) + s y ( t ) − s x y ( t ) (2)

Assuming independence we have:

s x ¯ y ( t ) = s x ( t ) + s y ( t ) − s x ( t ) ⋅ s y ( t ) (3)

The EPV in Equation (1) becomes:

b ∑ t = 1 ∞ v t [ exp { − ∫ 0 t λ x + s d s } + exp { − ∫ 0 t λ y + s d s } − exp { − ∫ 0 t ( λ x y + s ) d s } ] (4)

b ∑ t = 1 ∞ v t [ exp { − ∫ 0 t λ x + s d s } + exp { − ∫ 0 t λ y + s d s } − exp { − ∫ 0 t ( λ x + s + λ y + s ) d s } ] (5)

respectively. Here again for simplicity of notation, we suppose that the limiting age of our mortality table is infinity. If the purchase price for the annuity is p then the future level payment stream b applying the traditional equivalence principle is:

b = p a x ¯ y (6)

The prevailing traditional insurance practice assumes independence in pricing joint-life annuity contracts thereby adopting the EPV shown in Equation (5) whose joint mortality is obtained by summing up the individual mortality rates. Frailty dependence models focus on modeling the EPV as shown in Equation (4) whose joint mortality accounts for heterogeneity and dependence.

Definition 1. The hazard function for a shared frailty model is given by;

λ i j ( t | Ω i ) = Ω i λ 0 ( t ) ; t > 0 (7)

where λ i j ( t | Ω i ) is the conditional hazard function for the j^{th} individual in the i^{th} group and Ω i is the shared random effect associated with the i^{th} group. λ 0 ( t ) is the population’s base force of mortality.

Formally, the expression of the bivariate net survival function is summarized in the following proposition.

Proposition 1. Under the assumption of independent future life-times for a given shared frailty the bivariate net survival function is;

s ( t i 1 , t i 2 ) = L Ω i ( Λ 01 ( t i 1 ) + Λ 02 ( t i 2 ) ) (8)

Proof. The bivariate conditional survival function for a given shared frailty Ω i at time t i 1 > 0 , t i 2 > 0 is given by; s ( t i 1 , t i 2 | Ω i ) = s ( t i 1 | Ω i ) s ( t i 2 | Ω i ) and from s ( t | Ω ) = exp { − ∫ 0 t λ ( u | Ω ) d u } = exp ( − Ω Λ 0 ( t ) ) we have that s ( t i 1 , t i 2 | Ω i ) = exp { − Ω [ Λ 01 ( t i 1 ) + Λ 02 ( t i 2 ) ] } using expectation s ( t i 1 , t i 2 ) = E [ exp { − Ω [ Λ 01 ( t i 1 ) + Λ 02 ( t i 2 ) ] } ] this simplifies to

s ( t i 1 , t i 2 ) = L Ω i ( Λ 01 ( t i 1 ) + Λ 02 ( t i 2 ) ) □

Copulas have been studied in actuarial science to model joint-life survival functions [

C ϕ ( u , w ) = p { q ( u ) + q ( w ) }

where the generator function p ( . ) is any non-negative decreasing function and non-negative second derivative with p ( 0 ) = 1 and q ( . ) its pseudo-inverse function. A special case showing similarity to shared frailty approach is when p ( s ) = L Ω ( s ) where Ω is the frailty random variable and u = s ( t i 1 ) , w = s ( t i 2 ) this leads to:

C ϕ { s ( t i 1 ) , s ( t i 2 ) } = L Ω [ L − 1 ( s ( t i 1 ) ) + L − 1 ( s ( t i 2 ) ) ]

Comparing this with the marginal survival from shared frailty approach i.e. s ( t i 1 ) = L Ω ( Λ 0 ( t i 1 ) ) and therefore L Ω − 1 ( s ( t i 1 ) ) = Λ 0 ( t i 1 ) shows that

C ϕ { s ( t i 1 ) , s ( t i 2 ) } = L Ω [ L − 1 ( s ( t i 1 ) ) + L − 1 ( s ( t i 2 ) ) ] = L Ω ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) = s ( t i 1 , t i 2 )

If p ( . ) is the Laplace transform of a gamma distribution with scale parameter 1, then the Clayton copula model is obtained. Similarly, the Gumbel copula is obtained if p ( . ) has a positive stable Laplace though the estimation strategies and association measures differ with the frailty approach. Whereas in the copula approach, the marginal survivor functions and the dependence structure have to be specified [

The probability density function (p.d.f) for the positive stable distribution in closed-form is given by;

f ( Ω ) = − 1 π Ω ∑ n = 1 ∞ Γ ( n r + 1 ) n ! ( − Ω − r k / r ) n sin ( r n π ) ; k > 0 , Ω > 0 , 0 < r ≤ 1 (9)

For identifiability we assume k = r (see [

f ( Ω ) = − 1 π Ω ∑ n = 1 ∞ Γ ( n r + 1 ) n ! ( − Ω − r ) n sin ( r n π ) ; Ω > 0 , 0 < r ≤ 1

Proposition 2. The Laplace transform is a special case of the Power Variance Family ( r , k , η ) Laplace given by:

L Ω ( s ) = exp { − k r s r } (10)

As indicated earlier for identifiability reasons we let k = r .

L Ω ( s ) = exp ( − s r ) , 0 < r ≤ 1 (11)

The proposed frailty distribution has many advantages. First, it is easy to implement due to the simplified Laplace transform shown in Equation (11). Second, the positive stable has an infinite mean and variance. This allows for a much higher degree of heterogeneity to be accounted for that would not be possible by using a frailty distribution with finite variance. Third, the positive stable distribution is infinitely divisible, allowing the splitting of the shared risk into cause specific risks which may be easier to interpret. The net bivariate survival, density and hazard functions at time t i 1 > 0 , t i 2 > 0 are:

s ( t i 1 , t i 2 ) = exp { − ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) r } (12)

f ( t i 1 , t i 2 ) = s ( t i 1 , t i 2 ) ⋅ λ 0 ( t i 1 ) λ 0 ( t i 2 ) [ r 2 ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) 2 r − 2 − r ( r − 1 ) ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) r − 2 ] (13)

λ ( t i 1 , t i 2 ) = λ 0 ( t i 1 ) λ 0 ( t i 2 ) [ r 2 ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) 2 r − 2 − r ( r − 1 ) ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) r − 2 ] (14)

In dependence frailty models, the frailty distribution is identifiable through the [

A ( t i 1 , t i 2 ) = s ( t i 1 , t i 2 ) ∂ 2 ∂ t i 1 ∂ t i 2 s ( t i 1 , t i 2 ) ∂ ∂ t i 1 s ( t i 1 , t i 2 ) ∂ ∂ t i 2 s ( t i 1 , t i 2 ) (15)

Using the positive stable as frailty distribution, the cross ratio function from Equation (15) becomes:

A ( t i 1 , t i 2 ) = 1 − ( 1 − 1 r ) ( Λ 0 ( t i 1 ) + Λ 0 ( t i 2 ) ) − r (16)

From Equation (16), values of r close to zero indicate high association between t i 1 and t i 2 because A ( t i 1 , t i 2 ) takes values greater than 1, r close to one indicate low association between t i 1 and t i 2 since A ( t i 1 , t i 2 ) takes values near 1 while r = 1 corresponds to independence i.e. A ( t i 1 , t i 2 ) = 1 .

We present below four examples with specific baseline distributions to find the frailty dependence hazard functions with explicit expressions.

Example 1.

Let λ 0 ( t ) follow a Weibull( a , τ ) distribution with p.d.f f 0 ( t ) = τ a t τ − 1 exp ( − a t τ ) ; a , τ > 0 ; t ≥ 0 where τ is the shape parameter and a the scale parameter.

Then the survival, hazard and cumulative hazard functions are;

1) s 0 ( t ) = exp ( − a t τ ) , t > 0.

2) λ 0 ( t ) = τ a t τ − 1 , t > 0.

3) Λ 0 ( t ) = a t τ , t > 0.

From Equation (14) the positive stable Weibull (PSW henceforth) frailty dependence hazard is described explicitly as:

λ ( t i 1 , t i 2 ) = a 1 τ 1 t i 1 τ 1 − 1 ⋅ a 2 τ 2 t i 2 τ 2 − 1 ⋅ [ r 2 ( a 1 t i 1 τ 1 + a 2 t i 2 τ 2 ) 2 r − 2 − r ( r − 1 ) ( a 1 t i 1 τ 1 + a 2 t i 2 τ 2 ) r − 2 ] (17)

The Weibull distribution is widely used in the analysis of lifetime data because it is flexible enough to account for an increasing ( τ > 1 ), decreasing ( τ < 1 ) or constant ( τ = 1 ) hazard rate. Further, the law of Weibull is useful in mortality models for annuitants see e.g. [

Example 2.

Let λ 0 ( t ) follow a Lognormal( μ , σ 2 ) distribution with parameters μ , σ the p.d.f is given by;

f 0 ( t ) = 1 σ t 2 π e − ( ln t − μ ) 2 2 σ 2 ; t , σ > 0 , − ∞ < μ < ∞

Then the survival, hazard and cumulative hazard functions are:

1) s 0 ( t ) = 1 − Φ ( ln t − μ σ ) , t > 0.

2) λ 0 ( t ) = f 0 ( t ) 1 − Φ ( ln t − μ σ ) , t > 0.

3) Λ 0 ( t ) = − ln ( 1 − Φ ( ln t − μ σ ) ) , t > 0.

From Equation (14) the positive stable Lognormal frailty dependence hazard is:

λ ( t i 1 , t i 2 ) = f 0 ( t i 1 ) 1 − Φ ( ln t i 1 − μ 1 σ 1 ) ⋅ f 0 ( t i 2 ) 1 − Φ ( ln t i 2 − μ 2 σ 2 ) ⋅ [ r 2 ( − ln ( 1 − Φ ( ln t i 1 − μ 1 σ 1 ) ) − ln ( 1 − Φ ( ln t i 2 − μ 2 σ 2 ) ) ) 2 r − 2 − r ( r − 1 ) ( − ln ( 1 − Φ ( ln t i 1 − μ 1 σ 1 ) ) − ln ( 1 − Φ ( ln t i 2 − μ 2 σ 2 ) ) ) r − 2 ] (18)

The Lognormal is also used in modeling failure time data because it can take various unimodal shapes i.e. bathtub-shaped or hump-shaped.

Example 3.

Let λ 0 ( t ) follow a Gamma( p , φ ) with p.d.f

f 0 ( t ) = φ p t p − 1 exp ( − φ t ) Γ ( p ) ; t > 0 , φ > 0 , p > 0

Then the survival, hazard and cumulative hazard functions are:

1) s 0 ( t ) = γ ( p , φ t ) Γ ( p ) , t > 0.

2) λ 0 ( t ) = φ p t p − 1 exp ( − φ t ) γ ( p , φ t ) , t > 0.

3) Λ 0 ( t ) = − ln ( γ ( p , φ t ) Γ ( p ) ) , t > 0.

The Gamma is widely used in survival analysis to generate mixtures in exponential and Poisson models. It has positive support and is also a good choice for the baseline hazard.

From Equation (14) the positive stable Gamma frailty dependence hazard is described explicitly as:

λ ( t i 1 , t i 2 ) = φ p 1 t i 1 p 1 − 1 exp ( − φ 1 t i 1 ) γ ( p 1 , φ 1 t i 1 ) ⋅ φ 2 p 2 t i 2 p 2 − 1 exp ( − φ 2 t i 2 ) γ ( p 2 , φ 2 t i 2 ) ⋅ [ r 2 ( − ln ( γ ( p 1 , φ 1 t i 1 ) Γ ( p 1 ) ) − ln ( γ ( p 2 , φ 2 t ) Γ ( p 2 ) ) ) 2 r − 2 − r ( r − 1 ) ( − ln ( γ ( p 1 , φ 1 t i 1 ) Γ ( p 1 ) ) − ln ( γ ( p 2 , φ 2 t ) Γ ( p 2 ) ) ) r − 2 ] (19)

The performance of the model selection criteria for the Bayesian estimation technique is validated in a comparative study with the traditional MLE method. In the Bayesian method the deviance information criteria (DIC), Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) is applied whereas in the MLE method the Standard Error information is used.

1) DIC = D ¯ + p D where: D ¯ is the posterior mean of − 2 log L measuring the quality of the goodness-of-fit of the considered model to the data.

D ^ = − 2 log L is the posterior mean of stochastic nodes and p D = D ¯ − D ^ is the effective number of parameter. Smaller values of DIC indicate better models and could give negative values.

2) AIC = D ^ + 2 p where: p = number of parameters of the model.

3) BIC = D ^ + p × log ( n ) where: p= number of parameters of the model and n= sample size. The advantage of the BIC is that it includes the BIC penalty for the number of parameters being estimated.

Bayesian Analysis

The Bayesian method treats all unknown parameters as random variables in a statistical model and derives their distribution conditional upon known information. This method has been applied in actuarial modeling e.g. by [

The hyperparameters of initial values are chosen to be MLE estimates determined outside of OpenBUGS using standard techniques e.g. for the Weibull τ 1 = 0.7 , a 1 = 7 . The actual data to be estimated by the model is specified to be the males and females densities obtained from the AKI 2010 mortality data through standard numerical approximations. Parameters are estimated considering only the range of ages [55, 109]. Burn in period is set at 2000 as per the BGR plot to ensure sequence of draws from the posterior distribution have minimal autocorrelation and can be found by taking values from a single run of the Markov chain. This diminishes the effect of the starting distribution. We run 3 chains in parallel and after 10,000 iterations convergence will be monitored and if stationarity has been achieved (implying estimates are not dependent on the prior distributions) the mean posterior distribution will be picked as a point estimate. Models with smaller values of the DIC, BIC or AIC are preferred.

The WinBUGS codes used to analyze the dataset using Weibull are available upon request.

Brooks-Gelman-Rubin Diagnostic and Trace Plots

The BGR convergence diagnostic plots for the monitored nodes are presented in

plots also monitored in

Comparison with MLE

The MLE approach is concerned with obtaining parameter values say, ( τ , a ) that maximizes the probability of observing the dataD given those parameters, p ( D | τ , a ) . The likelihood function gives the probability of the observed sample generated by the model. Generally, maximization of the likelihood function to find the ML estimates is done algebraically, but can be computational intensive. In this article, the MLE algorithm is implemented using MASS package run in R. The output is given in

Discussion

On the basis of Bayesian the Weibull distribution is chosen since the DIC, BIC and AIC are smallest compared to the other distributions. Also using the MLE approach the Weibull is also chosen because the Standard Errors is smallest compared to the other distributions. It is interesting to note from

Baseline Model | Parameter Estimates | DIC | BIC | AIC |
---|---|---|---|---|

Weibull | τ 1 = 0.67 , a 1 = 7.15 (male) | −192.2 | −192.82 | −192.2 |

τ 2 = 0.7555 , a 2 = 10.17 (female) | −208.9 | −209.52 | −209.0 | |

Lognormal | μ 1 = − 3.711 , σ 1 2 = 2.3568 (male) | −97.22 | −95.75 | −95.24 |

μ 2 = − 3.727 , σ 2 2 = 1.6697 (female) | −101.1 | −99.52 | −99.0 | |

Gamma | p 1 = 7.746 , φ 1 = 0.5645 (male) | −188.6 | −189.22 | −188.7 |

p 2 = 11.89 , φ 2 = 0.6853 (female) | −205.5 | −206.02 | −205.5 |

Baseline Model | Parameter Estimates | Standard errors |
---|---|---|

Weibull | τ 1 = 0.6720 , a 1 = 7.1346 (male) | (0.0687, 0.0114) |

τ 2 = 0.7558 , a 2 = 10.0388 (female) | (0.0753, 0.0090) | |

Lognormal | μ 1 = − 3.7097 , σ 1 2 = 2.31248 (male) | (0.2050, 0.1449) |

μ 2 = − 3.726 , σ 2 2 = 1.63809 (female) | (0.1726, 0.1220) | |

Gamma | p 1 = 7.830 , φ 1 = 0.571 (male) | (0.0909, 1.8737) |

p 2 = 12.0341 , φ 2 = 0.6935 (female) | (0.1126, 2.7592) |

informative priors shared by field experts. The parameter estimates used in the study are as shown in

Goodness of Fit Test

A chi-square goodness-of-fit test of the data for Weibull baseline distribution is as shown in

The PSW dependence model given in Equation (17) is shown in

Inspired by [

Name | Value |
---|---|

Chi-squared statistic | 2970 |

Degree of freedom | 2916 |

Chi-squared p-value | 0.2384 |

Name | p-value | Test Statistic |
---|---|---|

Kolmogorov-Smirnov test | 0.1463 | 0.21818 |

consider two dependence measures i.e. Spearman’s correlation ρ = 0.74 (black curve) and Pearson correlation r = 0.68 (red curve) when the age difference of insured couples is greater than four. As shown in

Equation (17) is then applied to generate the dependence λ x y f r a i l t y hazard rates using Pearson correlation r = 0.68 (to be consistent with the parametric model chosen) shown in

a x y f r a i l t y = N x y D x y where D x y = v x y l x y and N x y = ∑ k = 0 45 D x y + k are commutation functions.

The independence λ x y i n d hazard rates in

INDEPENDENCE LIFETABLE CONSTRUCTION | PURCHASE PRICE: 1000 | |||||||
---|---|---|---|---|---|---|---|---|

AGE(y) | I_{xy} | S_{xy} | λ_{xy} ind | D_{xy} | N_{xy} | a_{xy}_ind | PAYMENT STREAM | AGE(x) |

55 | 100,000 | 0.987322 | 0.01276 | 33,650.42 | 748,844.1 | 22.25363 | 44.9364899 | 55 |

56 | 98,732.15 | 0.989001 | 0.01106 | 32,572.34 | 715,193.7 | 21.95708 | 45.54338661 | 56 |

57 | 97,646.23 | 0.989673 | 0.010381 | 31,582.44 | 682,621.3 | 21.61395 | 46.26641343 | 57 |

58 | 96,637.84 | 0.989943 | 0.010108 | 30,643.42 | 651,038.9 | 21.24563 | 47.06849595 | 58 |

59 | 95,665.97 | 0.989986 | 0.010064 | 29,740.44 | 620,395.5 | 20.86033 | 47.93787231 | 59 |

60 | 94,708.01 | 0.989892 | 0.010159 | 28,865.32 | 590,655 | 20.46244 | 48.87001675 | 60 |

61 | 93,750.7 | 0.989705 | 0.010348 | 28,013.28 | 561,789.7 | 20.0544 | 49.86435956 | 61 |

62 | 92,785.53 | 0.989454 | 0.010602 | 27,181.26 | 533,776.4 | 19.63766 | 50.92256147 | 62 |

63 | 91,807.06 | 0.989153 | 0.010906 | 26,367.27 | 506,595.2 | 19.21303 | 52.04801895 | 63 |

64 | 90,811.23 | 0.988808 | 0.011255 | 25,569.87 | 480,227.9 | 18.781 | 53.24528938 | 64 |

65 | 89,794.87 | 0.988414 | 0.011653 | 24,787.94 | 454,658 | 18.34191 | 54.51995834 | 65 |

66 | 88,754.53 | 0.987967 | 0.012106 | 24,020.34 | 429,870.1 | 17.89608 | 55.87814592 | 66 |

67 | 87,686.54 | 0.98746 | 0.012619 | 23,265.99 | 405,849.7 | 17.44391 | 57.32659994 | 67 |

68 | 86,586.97 | 0.986891 | 0.013196 | 22,523.76 | 382,583.7 | 16.98579 | 58.87274996 | 68 |

69 | 85,451.88 | 0.986254 | 0.013841 | 21,792.64 | 360,060 | 16.52209 | 60.52501545 | 69 |

70 | 84,277.27 | 0.985547 | 0.014558 | 21,071.65 | 338,267.4 | 16.0532 | 62.29287286 | 70 |

71 | 83,059.23 | 0.984762 | 0.015355 | 20,359.9 | 317,195.7 | 15.57943 | 64.18719582 | 71 |

72 | 81,793.58 | 0.983891 | 0.01624 | 19,656.53 | 296,835.8 | 15.10113 | 66.22021465 | 72 |

73 | 80,475.98 | 0.982889 | 0.01726 | 18,960.67 | 277,179.3 | 14.61864 | 68.40580518 | 73 |

74 | 79,098.91 | 0.981855 | 0.018312 | 18,270.81 | 258,218.6 | 14.13285 | 70.75714002 | 74 |

75 | 77,663.65 | 0.980663 | 0.019526 | 17,587.53 | 239,947.8 | 13.64306 | 73.29732669 | 75 |

76 | 76,161.88 | 0.979333 | 0.020884 | 16,909.26 | 222,360.3 | 13.15021 | 76.0444313 | 76 |

77 | 74,587.82 | 0.977847 | 0.022402 | 16,235.09 | 205,451 | 12.65475 | 79.02171141 | 77 |

78 | 72,935.47 | 0.976181 | 0.024108 | 15,564.15 | 189,215.9 | 12.15716 | 82.25603251 | 78 |

79 | 71,198.19 | 0.974307 | 0.026029 | 14,895.51 | 173,651.8 | 11.65799 | 85.77805174 | 79 |

80 | 69,368.91 | 0.97219 | 0.028204 | 14,228.24 | 158,756.3 | 11.15783 | 89.6231621 | 80 |

81 | 67,439.76 | 0.969793 | 0.030673 | 13,561.32 | 144,528 | 10.65737 | 93.8318065 | 81 |

82 | 65,402.58 | 0.967065 | 0.03349 | 12,893.8 | 130,966.7 | 10.15734 | 98.4509591 | 82 |

83 | 63,248.52 | 0.963949 | 0.036716 | 12,224.64 | 118,072.9 | 9.658598 | 103.5346902 | 83 |

84 | 60,968.37 | 0.960375 | 0.040431 | 11,552.88 | 105,848.3 | 9.162069 | 109.145658 | 84 |

85 | 58,552.52 | 0.95626 | 0.044726 | 10,877.55 | 94,295.38 | 8.668808 | 115.3561146 | 85 |

86 | 55,991.43 | 0.951503 | 0.049713 | 10,197.81 | 83,417.83 | 8.179977 | 122.2497347 | 86 |

87 | 53,276.01 | 0.945982 | 0.055531 | 9512.983 | 73,220.02 | 7.696852 | 129.9232519 | 87 |

88 | 50,398.16 | 0.939553 | 0.062351 | 8822.66 | 63,707.04 | 7.220842 | 138.4879971 | 88 |

89 | 47,351.75 | 0.932044 | 0.070376 | 8126.821 | 54,884.38 | 6.753487 | 148.0716653 | 89 |

90 | 44,133.89 | 0.923251 | 0.079854 | 7426.03 | 46,757.56 | 6.29644 | 158.8198986 | 90 |

91 | 40,746.67 | 0.912941 | 0.091084 | 6721.659 | 39,331.53 | 5.851461 | 170.8975 | 91 |

92 | 37,199.3 | 0.900837 | 0.104431 | 6016.155 | 32,609.87 | 5.420384 | 184.4888 | 92 |

93 | 33,510.49 | 0.886635 | 0.120322 | 5313.306 | 26,593.71 | 5.005116 | 199.7956 | 93 |

94 | 29,711.59 | 0.869999 | 0.139263 | 4618.593 | 21,280.41 | 4.607552 | 217.035 | 94 |

95 | 25,849.06 | 0.85058 | 0.161837 | 3939.385 | 16,661.81 | 4.229547 | 236.4319 | 95 |

96 | 21,986.68 | 0.827954 | 0.188798 | 3285.059 | 12,722.43 | 3.872816 | 258.2101 | 96 |

97 | 18,203.96 | 0.802014 | 0.22063 | 2666.546 | 9437.37 | 3.539173 | 282.5519 | 97 |

98 | 14,599.82 | 0.772303 | 0.258379 | 2096.674 | 6770.823 | 3.229317 | 309.663 | 98 |

99 | 11,275.48 | 0.738749 | 0.302797 | 1587.516 | 4674.15 | 2.944316 | 339.6374 | 99 |

100 | 8329.751 | 0.701364 | 0.354728 | 1149.78 | 3086.634 | 2.684542 | 372.503 | 100 |

101 | 5842.187 | 0.660341 | 0.414999 | 790.6024 | 1936.854 | 2.449845 | 408.189 | 101 |

102 | 3857.834 | 0.616061 | 0.48441 | 511.8304 | 1146.251 | 2.239514 | 446.5255 | 102 |

103 | 2376.66 | 0.569077 | 0.56374 | 309.1358 | 634.4207 | 2.05224 | 487.2725 | 103 |

104 | 1352.502 | 0.519526 | 0.654838 | 172.4726 | 325.2849 | 1.88601 | 530.2199 | 104 |

105 | 702.6603 | 0.469801 | 0.755445 | 87.84709 | 152.8124 | 1.739527 | 574.869 | 105 |

106 | 330.1108 | 0.419039 | 0.869791 | 40.46146 | 64.96529 | 1.605609 | 622.8165 | 106 |

107 | 138.3293 | 0.368476 | 0.99838 | 16.62247 | 24.50383 | 1.474139 | 678.3623 | 107 |

108 | 50.97098 | 0.318742 | 1.143375 | 6.004881 | 7.881357 | 1.312492 | 761.9096 | 108 |

109 | 16.24657 | 0.27038 | 1.307927 | 1.876476 | 1.876476 | 1 | 1000 | 109 |

INDEPENDENCE LIFETABLE CONSTRUCTION | PURCHASE PRICE: 1000 | |||||||

AGE(y) | I_{xy} | S_{xy} | μ_{xy} frailty | D_{xy} | N_{xy} | a_{xy}_frailty | PAYMENT STREAM | AGE(x) |

55 | 100,000 | 0.956844 | 0.044115 | 33,650.42 | 447,990.3 | 13.31306 | 75.11419 | 55 |

56 | 95,684.42 | 0.956038 | 0.044957 | 31,566.88 | 414,339.8 | 13.12578 | 76.18596 | 56 |

57 | 91,477.96 | 0.955202 | 0.045832 | 29,587.39 | 382,773 | 12.93703 | 77.2975 | 57 |

58 | 87,379.94 | 0.954334 | 0.046742 | 27,707.78 | 353,185.6 | 12.7468 | 78.45107 | 58 |

59 | 83,389.65 | 0.953432 | 0.047687 | 25,924 | 325,477.8 | 12.55507 | 79.64907 | 59 |

60 | 79,506.39 | 0.952495 | 0.048671 | 24,232.14 | 299,553.8 | 12.36184 | 80.89411 | 60 |

61 | 75,729.42 | 0.951519 | 0.049695 | 22,628.42 | 275,321.6 | 12.16708 | 82.18902 | 61 |

62 | 72,058.01 | 0.950504 | 0.050763 | 21,109.19 | 252,693.2 | 11.97077 | 83.53683 | 62 |

63 | 68,491.4 | 0.949445 | 0.051877 | 19,670.95 | 231,584 | 11.7729 | 84.94086 | 63 |

64 | 65,028.85 | 0.948341 | 0.053041 | 18,310.28 | 211,913.1 | 11.57345 | 86.40468 | 64 |

65 | 61,669.55 | 0.947189 | 0.054256 | 17,023.92 | 193,602.8 | 11.3724 | 87.93221 | 65 |

66 | 58,412.73 | 0.945985 | 0.055528 | 15,808.7 | 176,578.9 | 11.16973 | 89.5277 | 66 |

67 | 55,257.57 | 0.944726 | 0.056861 | 14,661.56 | 160,770.2 | 10.96542 | 91.1958 | 67 |

68 | 52,203.26 | 0.943407 | 0.058257 | 13,579.57 | 146,108.6 | 10.75945 | 92.94159 | 68 |

69 | 49,248.94 | 0.942026 | 0.059723 | 12,559.87 | 132,529 | 10.55179 | 94.77068 | 69 |

70 | 46,393.77 | 0.940576 | 0.061263 | 11,599.72 | 119,969.2 | 10.34242 | 96.6892 | 70 |

71 | 43,636.86 | 0.939053 | 0.062883 | 10,696.49 | 108,369.5 | 10.13131 | 98.70393 | 71 |

72 | 40,977.33 | 0.937451 | 0.06459 | 9847.62 | 97,672.96 | 9.918434 | 100.8224 | 72 |

73 | 38,414.25 | 0.935765 | 0.066391 | 9050.652 | 87,825.34 | 9.703759 | 103.0528 | 73 |

74 | 35,946.71 | 0.933986 | 0.068294 | 8303.217 | 78,774.69 | 9.48725 | 105.4046 | 74 |

75 | 33,573.73 | 0.932108 | 0.070306 | 7603.03 | 70,471.47 | 9.268867 | 107.8881 | 75 |

76 | 31,294.35 | 0.930122 | 0.07244 | 6947.889 | 62,868.44 | 9.048568 | 110.5147 | 76 |

77 | 29,107.56 | 0.928018 | 0.074704 | 6335.671 | 55,920.56 | 8.826304 | 113.2977 | 77 |

78 | 27,012.35 | 0.925786 | 0.077112 | 5764.331 | 49,584.89 | 8.602019 | 116.2518 | 78 |

79 | 25,007.65 | 0.923413 | 0.079678 | 5231.898 | 43,820.55 | 8.375651 | 119.3937 | 79 |

80 | 23,092.4 | 0.920887 | 0.082418 | 4736.475 | 38,588.66 | 8.147125 | 122.7427 | 30 |

81 | 21,265.48 | 0.918191 | 0.08535 | 4276.232 | 33,852.18 | 7.916357 | 126.3207 | 81 |

82 | 19,525.77 | 0.915308 | 0.088495 | 3849.409 | 29,575.95 | 7.683245 | 130.1533 | 82 |

83 | 17,872.09 | 0.912218 | 0.091876 | 3454.308 | 25,726.54 | 7.447669 | 134.2702 | 83 |

84 | 16,303.24 | 0.908899 | 0.095522 | 3089.296 | 22,272.23 | 7.209484 | 138.7062 | 84 |

85 | 14,817.99 | 0.905323 | 0.099464 | 2752.801 | 19,182.94 | 6.968515 | 143.5026 | 85 |

86 | 13,415.07 | 0.90146 | 0.103739 | 2443.308 | 16,430.14 | 6.724546 | 148.7089 | 36 |

87 | 12,093.15 | 0.897275 | 0.108393 | 2159.358 | 13,986.83 | 6.477309 | 154.3851 | 87 |

88 | 10,850.89 | 0.892726 | 0.113476 | 1899.547 | 11,827.47 | 6.226468 | 160.6047 | 88 |

89 | 9686.864 | 0.887762 | 0.119051 | 1662.524 | 9927.922 | 5.971597 | 167.4594 | 89 |

90 | 8599.633 | 0.882327 | 0.125193 | 1446.986 | 8265.398 | 5.712147 | 175.0655 | 90 |

91 | 7587.685 | 0.876348 | 0.131992 | 1251.681 | 6818.412 | 5.447404 | 183.5737 | 91 |

92 | 6649.455 | 0.869743 | 0.139558 | 1075.401 | 5566.731 | 5.176426 | 193.1835 | 92 |

93 | 5783.314 | 0.862406 | 0.148029 | 916.982 | 4491.33 | 4.897948 | 204.1671 | 93 |

94 | 4987.565 | 0.845001 | 0.157576 | 775.3048 | 3574.348 | 4.61025 | 216.908 | 94 |

95 | 4214.5 | 0.834576 | 0.168417 | 642.2878 | 2799.044 | 4.357927 | 229.4669 | 95 |

96 | 3517.318 | 0.82268 | 0.180832 | 525.5272 | 2156.756 | 4.103985 | 243.6656 | 96 |

97 | 2893.626 | 0.808983 | 0.195189 | 423.8632 | 1631.229 | 3.848479 | 259.8429 | 97 |

98 | 2340.894 | 0.793049 | 0.211977 | 336.1746 | 1207.366 | 3.591484 | 278.4365 | 98 |

99 | 1856.444 | 0.774291 | 0.23187 | 261.3755 | 871.1909 | 3.3331 | 300.021 | 99 |

100 | 1437.429 | 0.751898 | 0.255807 | 198.4125 | 609.8154 | 3.073472 | 325.3649 | 100 |

101 | 1080.8 | 0.724725 | 0.285154 | 146.2608 | 411.4029 | 2.812803 | 355.5173 | 101 |

102 | 783.2826 | 0.6911 | 0.321963 | 103.9204 | 265.142 | 2.551394 | 391.9425 | 102 |

103 | 541.3263 | 0.648499 | 0.369471 | 70.41114 | 161.2216 | 2.289717 | 436.7352 | 103 |

104 | 351.0496 | 0.592961 | 0.433095 | 44.76624 | 90.81045 | 2.028548 | 492.9635 | 104 |

105 | 208.1588 | 0.518012 | 0.522626 | 26.02416 | 46.04421 | 1.769287 | 565.1995 | 105 |

106 | 107.8287 | 0.41282 | 0.657758 | 13.21649 | 20.02004 | 1.514778 | 660.1627 | 106 |

107 | 44.5138 | 0.260909 | 0.884744 | 5.349046 | 6.803557 | 1.27192 | 786.2131 | 107 |

108 | 11.61405 | 0.064307 | 1.343584 | 1.368249 | 1.454511 | 1.063046 | 940.6933 | 108 |

109 | 0.746861 | 1 | 2.744092 | 0.086262 | 0.086262 | 1 | 1000 | 109 |

Impact on Mortality Rates

As shown in

Impact on Annuity EPVs

Consequently, comparing the annuity EPVs and payment streams shows that the independence assumptions lead to an overestimation of the insurer’s liability at the initial stages of the contract thereafter there is an underestimation of liability due to deceleration in the mortality increase at very old ages (longevity risk).

Although there is rich literature in frailty dependence modeling, most applications have been in medical field. Various dependence models have been considered in actuarial literature; however, the focus has been on either the lower-tail dependence or upper-tail dependence.

This article presents the frailty dependence approach calibrated on the AKI 2010 male and female published mortality rates due to limited joint-life mortality data-set available in the Kenyan market. This methodology offers greater flexibility than the lower-tail or upper-tail dependence models while preserving closed-form expressions for the net survival functions. Our strategy is to apply the conditional independence assumption in a positive stable frailty approach to account for lower and upper-tail dependence. A positive stable frailty approach is then applied to construct dependence life-tables. The frailty joint-life mortality rates are proposed to generate life annuity payment streams in the competitive Kenyan market.

The conclusion reached is that comparing the independence mortality assumption with the dependence frailty model shows a decrease in the insurer’s expected liability at the early annuitant’s ages followed by an increase at later ages when dependence is accounted for. This can be explained by the fact that the frail couples shall have died during the early stages of the contract, thereafter deceleration in the mortality increase at very old ages (longevity risk), underscoring the importance of dependence modeling in pricing insurance products. Thus, assuming the joint lives to be independent could lead to biased annuity valuations.

We would like to acknowledge the Association of Kenya Insurer for the study conducted on assured lives during 2010 whose mortality rates are currently being used by insurance companies in Kenya and have been used in the study.

All authors have contributed equally in the development of this article.

The authors declare no conflicts of interest regarding the publication of this paper.

Walter, O., Patrick, W., Otieno, J. and Carolyne, O. (2021) Positive Stable Frailty Approach in the Construction of Dependence Life-Tables. Open Journal of Statistics, 11, 506-523. https://doi.org/10.4236/ojs.2021.114032