Impossible Things

Impossibility has often featured in humourous fiction.  From Lewis Carroll's White Queen, who "believed as many as six impossible things before breakfast", to Douglas Adams' Restaurant at the End of the Universe, there is entertainment value in absurdity.

Actuaries are not widely known for their comedic talents.  However, in the past they have been known to entertain some impossible things when it comes to \(q\)-type models.  This was especially the case when it came to analysing mortality in the presence of other important decrements.  Consider the case of \(n\) lives observed over a year of age where there are two decrements: (i) mortality, and (ii) entry into long-term care (LTC).  We will ignore other sources of exit such as lapse for the moment.  The number of deaths observed during this year of age is \(d^{\rm mort}\), while the corresponding number of entries into long-term care is \(d^{\rm care}\).  If the actuary wants to model the mortality experience, this is obviously not a binomial situation, as there are three possible statuses at the end of the year: (i) alive, (ii) dead, and (iii) entered into care.

However, the historic actuarial mindset was so fixated on \(q\)-type models that actuaries nevertheless sought to shoe-horn competing-risk data into such binomial models.  This was done by artificially reducing the number of lives exposed to risk to try and allow for the fact that there were other types of exit.  Thus, in our example the actuary would estimate \(q^{\rm mort}\), the rate of mortality in the absence of care exits, using:

\[\hat q^{\rm mort} = \frac{d^{\rm mort}}{n^{\rm mort}}\qquad(1)\]

where \(n^{\rm mort}\) is a pseudo-life-count for the purposes of estimating mortality, with a corresponding \(n^{\rm care}\) for the purposes of estimating care-inception rates.  If these were to form part of a statistical model, we would then have:

\[d^{\rm mort}\sim{\rm Binomial}(q^{\rm mort}, n^{\rm mort})\qquad(2)\]

\[d^{\rm care}\sim{\rm Binomial}(q^{\rm care}, n^{\rm care})\qquad(3)\]

How are these pseudo-life-counts calculated?  Adapting the denominator of equation (2.1) in Benjamin & Pollard (1986), and assuming events happen midway through the year of age, the pseudo-life-counts for the binomial \(q\)-type models are as follows:

\[n^{\rm mort} = n-\frac{1}{2}d^{\rm care}\qquad(4)\]

\[n^{\rm care} = n-\frac{1}{2}d^{\rm mort}\qquad(5)\]

However, a problem arises when this approach meets real data.  In Richards & Macdonald (2024) we cite the example of a portfolio of home-reversion plans for the age interval 88-89.  There were \(n=42\) lives over this year of age, with \(d^{\rm mort}=5\) deaths and \(d^{\rm care}=3\) care events.  Using equations (4) and (5), this gives \(n^{\rm mort} = 40.5\) and \(n^{\rm care} = 39.5\).  It might be computationally expedient to do this, but what on earth is 0.5 of a binomial trial?  For a better match to reality, modern actuaries use continuous-time methods based around the mortality hazard, \(\mu^{\rm mort}\), or the LTC inception hazard, \(\mu^{\rm care}\).  Besides their other advantages, continous-time methods don't pretend impossible things like half a person.

On a closing note, it is perhaps not entirely fair to say that actuaries are devoid of all humour.  After all, the absurdity of trying to make every risk look like a \(q\)-type binomial model was lampooned in Macdonald (1994).  He's just not quite as widely read as Carroll or Adams.