Lee-Carter Mortality Models: Central Death Rates or Death Probabilities

Summary

The document considers whether the Lee-Carter stochastic mortality model should be fitted to central death rates or one-year probabilities of death. It gives the model’s log-linear form for both quantities and describes two common conversions from a central death rate to a death probability: a fractional adjustment and an exponential relationship.

The answer explains that when the central death rate is small relative to one, both conversions make the death probability approximately equal to the rate. Under that condition, modelling either quantity leads to approximately the same result. The response does not address cases where mortality rates are not small, compare estimation performance, or provide evidence beyond this approximation; the appropriate choice can therefore depend on whether the assumption holds for the population and age range being modelled.

Key ideas

• The Lee-Carter model can be expressed using central death rates or one-year death probabilities.
• Common formulas convert central death rates into annual death probabilities.
• When the central death rate is small, both conversions approximate the death probability by the rate.
• The equivalence may not hold when death rates are not small.

Tags

# Lee Carter Model - Mortality


# 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,

## Answer by JejeBelfort (score 2)

https://quant.stackexchange.com/a/34524

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.

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