Given the yearly benefit of 10 000 x 12, the rate assumption of 4,2 % and the mortality assumption of :
the guaranteed amount F be determined through equation (26).
t0 = 40 and z = 65
I need help to get the guaranteed amount F with the variables given.. Please contact me if there is a problem solving this! Thanks for your time! /Ida
3 Pricing model
Below you find a theoretical explanation of the components used in longevity insurance, relaying on the discussion made above. Those will later be used to drive the in price of an longevity insurance which in this paper will be regarded as the fair price in a consumer perspective.
3.1 Life-span and mortality
If the life-span is described as an stochastic variable T, where the probability that a individual at age t, dies before reaching the age of t + h (h > 0) is illustrated with the relation:
(1)
µ = death intensity and, o(h)/h ? 0 when, h ? 0
The probability to die during a future time period is thereby proportional to µ and the length of the period. It will be obvious later on that µ will increase, thus the probability to die the older a person becomes. The distribution function for T is:
(2)
Differentiation of equation gives the probability density function for T;
(3)
The expected value for T becomes:
(4)
When pricing longevity insurance the main concern is the probability that a person at a certain age s, are alive at the age of t. This function is denoted with ps (t) and describe:
(5)
In appendix it is shown that:
(6)
The expected remaining lifetime of an individual at a specific age is also an interesting parameter, which is given by; (7)
Derivation of the equation can be found in the appendix.
Here the death intensity will be assumed to be a strictly increasing function on,
(8)
Where a, ß>0 are constants and t are the individual age. Which is an approximation of reality where an empirical derived function would give higher values at very low ages. In reality it is not very common…