Valuing a Life Annuity with Gompertz-Makeham Mortality

# Life annuity and the use of Gompertz-Makeham


# Life annuity and the use of Gompertz-Makeham












The question goes as follows:

> Consider a life annuity contract that pays the holder a yearly fixed amount from a certain time until the death of the holder of the contract. (a) Suppose the value today of the random cash flow $C_K$ in k years is $E[C_k]e^{-r_kk}$, where $r_k$ is the current $k$-year zero rate. All the zero rates are assumed to be known. Determine an expression for the current value of an annuity whose holder is x years old, the annuity pays the yearly amount $c$ starting from $y$ years from today and the current mortality rate is given by the Gompertz-Makeham formula: $\mu_0(x)=A+R^{\alpha x}$

So if I've understood correctly $C_k$ can be seen as a random variable and it's value can be described as mentioned above. This is some what confusing because I feel that it would be more logical to describe $E[C_k]e^{-r_kk}$ as the liability, which could be described a random variable...anyways.

Moreover the holder of age x gets paid a yearly amount $c$ thus the value of the liability can be described as:

$E[L]=\sum^{\infty}_{k=y}E[C_k]e^{-r_kk}=\sum^{\infty}_{k=y}ce^{-r_kk}$.

This is obviously if you assume that the holder survives to infinity which makes no sense. Thus applying a probability that the holder dies after a certain payment time $k$ seems logical. Thus:

$E[L]=\sum^{\infty}_{k=y}ce^{-r_kk}P(\tau>k)$.

Or if one wants to be more correct and thorough one can reason in the following way:

Assume that the liability is the constant pay $c$ if the holder dies after a certain payment $k$ leading to:

$L=\sum^{\infty}_{k=y}C_ke^{-r_kk}I(\tau>k)$, where I is the indicator function. The value of $L$ thus becomes:

$E[L]=\sum^{\infty}_{k=y}ce^{-r_kk}P(\tau>k)$.

Here is where I get stuck. In the solution they define the probability $P(\tau>k)$ as:

$P(\tau>k)=e^{-\int^{k}_0(A+Re^{\alpha(x+u)})du}$.

I don't really follow how they manage to define the probability. There is an example in the book but they jump over the explanation. It's probably something simple but I fail to see what how.

Any form of help is appreciated.