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Licensed Unlicensed Requires Authentication Published by De Gruyter December 3, 2013

A Family of Mortality Jump Models Applied to US Data

  • Hua Chen EMAIL logo

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. 1

    We refer interested readers to Cairns, Blake, and Dowd (2008) for a review of discrete and continuous stochastic mortality models.

  2. 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. 3
Published Online: 2013-12-3

©2014 by Walter de Gruyter Berlin / Boston

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