# Lee Carter Model - Mortality

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,

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.

• Thanks a lot. You are perfectly right that due to the approximations above it is basically equivalent, e.g. for humans in the US up to a certain age as long as mortality is still quite low. However, as the model is used to project future mortality improvements, old ages with higher mortality are also of particular interest. Thus, and due to the fact that most apply it to central death rates $m(x,t)$ and not $q(x,t)$, I am wondering whether there is a deeper reason for this. For example, whether it has do with the fact that $q(x,t)$ cannot be greater than 1, but $m(x,t)$ can. – Strickland Jun 4 '17 at 10:07