Abstract
Mortality models are fundamental to quantify mortality/longevity risks and provide the basis of pricing and reserving. In this article, we consider a family of mortality jump models and propose a new generalized Lee–Carter model with asymmetric double exponential jumps. It is asymmetric in terms of both time periods of impact and frequency/severity profiles between adverse mortality jumps and longevity jumps. It is mathematically tractable and economically intuitive. It degenerates to a transitory exponential jump model when fitting the US mortality data and is the best fit compared with other jump models.
Appendix A: the compensation term for the permanent-effect jumps
The compensation term for the permanent-effect jumps, , can be calculated as
Let . The density of is
Then, we can obtain
Appendix B: derivation of the LLF for the model with asymmetric double exponential jumps
In the following, we derive in each of the nine cases. We denote and the density function and cumulative distribution function of a standard normal variable, respectively.
Case 1: .
Case 2: ,
where and .
By convolution techniques, we obtain
where we use the formula .
Therefore, .
Case 3: ,
where and .
where we use the formula .
Therefore, .
Case 4: .
Therefore, .
Case 5: ,
where and .
Similar to Case 2, we can obtain the density function (conditional on )
Case 6: ,
where and .
Similar to Case 3, we can obtain the density function (conditional on )
Case 7: ,
where , .
Case 8: ,
where and
First, we derive the density function of .
When ,
When ,
Differentiating , we can get the probability function
Next, we derive the density function of
Therefore,
Case 9: ,
where
and , .
As in Case 8, we first derive the density function of .
When , .
When ,
Here, we use
Differentiating , we can get the probability function
We, then, derive the density function of .
Therefore,
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- 1
We refer interested readers to Cairns, Blake, and Dowd (2008) for a review of discrete and continuous stochastic mortality models.
- 2
We can relax this assumption by assuming a Poisson distribution. However, a closed form of the likelihood function cannot be derived in this case, and it complicates the mathematics in parameter estimation.
- 3
©2014 by Walter de Gruyter Berlin / Boston