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Maths & stats

Mortality: Incoherent rating factors

A pre-requisite for pension plan mortality modelling is that the rating factors used should be coherent, which I’ll define shortly.

The insidious problem with incoherent rating factors is that a model may produce poor or even systemically biased forecasts while simultaneously performing well on standard model fit and selection diagnostics. Yikes!

For large experience datasets in particular, rating factor incoherence can be a bigger issue than model fitting and selection. But too often in practice, I’ve seen this issue trivialised or deemed ‘obvious’.

Mortality: Suddenly AIC

In this article, I’m going to look at choosing between mortality models using Hirotugu Akaike’s information criterion, the AIC.

I’m going to run through – at a very high level – the rationale behind the AIC and its construction because

  • the standard result looks so trivial that people sometimes assumes it’s an arbitrary convention, and
  • I’m going to generalise it (a little).

Mortality: Proportional hazards

A/E diagnostics are important but, if we have any mortality experience data, we should be using it to develop a model that takes account of that data, even if it’s nothing more than a simple how-much-heavier-or-lighter-is-the-mortality-of-this-population-than-average model. Otherwise, we’re not making full use of available information.

There are lots of possible approaches, including complex parametric formulas designed to capture all typically observed effects. But I promised concision and so in this article I’ll expound what I think is simultaneously one of the most powerful general approaches and one of the simplest. And the beauty of it is: we’ve already done most of the work.

Mortality: Log-likelihood

I think it’s a shame that the ‘log’ in ‘log-likelihood’ is so often presented as a technical convenience or a device for avoiding numerical under/overflow. Yes, it is definitely both of these things, but it is much more fundamental.

Expected log-probability, i.e. entropy, lies at the heart of information theory. And the concept of entropy itself is pervasive, having extended beyond thermodynamics, its original home, into quantum physics and general relativity, as well as information theory.

So, without further ado, let’s define log-likelihood for mortality experience data.

Mortality: A over E

Why ‘A over E’?

‘A over E’ literally refers to ‘actual’ deaths divided by ‘expected’ deaths as a measure of how experience data compares with a mortality.

In practice, ‘A over E’ is often interpreted as meaning the whole statistical caboodle, which is how I’ll use it here.

In the previous article we defined experience data, variables and mortality with respect to that data, and the \(\text{A}\) (actual) and \(\text{E}\) (expected) deaths operators.

In this article we’ll put \(\text{A}\) and \(\text{E}\) to work.

Mortality: Measures matter

This is the first in a series of articles outlining mortality experience analysis fundamentals, by which I mean estimating the underlying mortality for individuals in a defined group (e.g. members of a pension plan) from mortality experience data.

This will be fairly technical, but I’ll aim

  • to be concise,
  • to pull out the key insights, including what does and doesn’t matter in practice, and
  • to map concepts one-to-one to the process of actually carrying out a mortality experience analysis or calibrating and selecting mortality models.

On contemporary mortality models for actuarial use

Stephen Richards’ and Angus Macdonald’s paper ‘On Contemporary Mortality Models for Actuarial Use’ (due to be discussed at the Institute of Actuaries on 24 October 2024) makes the case for the following in mortality experience analysis1:

  1. Use individual data if at all possible.

  2. Work in continuous time and use instantaneous mortality rates, i.e. μ rather than q.

  3. Consider mortality experience data as comprising a series of Bernoulli trials over infinitesimally small time periods.

The paper could be read as a polemic against actuaries who can’t help but think in terms of q and whose first instinct is to group all time and age-dependent data on annual grids. Which is fine by me – I agree with the thrust of the paper and, in particular, the above three points.

So, having welcomed the paper, I do have a few observations…