Lee Carter Model - Mortality

Question

Helllo

Althoug not technically a QF question, I was wondering whether someone can help my anyways. The Lee Carter model is a stochastic mortality model.

Usually, one models the central death rates as follows:

$\log(m(x,t)) = a(x) + b(x)\kappa(t) +\varepsilon(x,t)$

In the past, I have also seen that instead of $m(x,t)$ the formula is applied to the probability of dying within one year denoted by $q$:

$\log(q(x,t)) = a(x) + b(x)\kappa(t) +\varepsilon(x,t)$.

Usually, one uses/assumes one of the following relationships:

$q(x,t)=\frac{m(x,t)}{(1+\frac{1}{2}m(x,t))}$ or $q(x,t)=1-\exp(-m(x,t))$.

I am wondering which model approach is more appropriate? That is, to model $m(x,t)$ or $q(x,t)$ with the above approach? And why?

Thanks a lot,

Answer 1

Depending on what the death rate is applied to (e.g. humans or butterflies or whatever...), the assumption that $m(x,t)$ is rather small compared to 1 is more or less valid.

Assuming that this assumption holds, then both of your expressions for $q(x,t)$ would yield to:

• $q(x,t) = \frac{m(x,t)}{1 + 0.5 m(x,t)} \approx m(x,t)$,

• $q(x,t) = 1 - e^{- m(x,t)} \approx m(x,t)$.

Have a look here:

Therefore, if the assumption holds, both approaches are the same.