The Three Stages of (Actuarial) Man

Stephen and I recently presented a pair of papers to the Institute and Faculty of Actuaries: Richards & Macdonald (2024) and Macdonald & Richards (2024).  In these papers we encourage actuaries to use continuous-time models in their work. But where does that leave discrete-time?

Not consigned to the bin! Far from it. Even if our world-view is firmly in continuous time, discrete-time models still play their part. In fact I claim that the journey most working actuaries must take passes from discrete-time to continuous-time and back again to discrete-time, but the journey’s end is very different from the journey’s start.

In the beginning the student sets out to acquire new ideas, and the teacher provides the simplest possible examples by way of introduction and motivation. The student is invited to imagine the world ‘as if’ it really were that simple, and see if a mathematical description fits the imagined ‘facts’. So we suppose we have \(N\) individuals, all identical and independent in some sense, and that each may die in the next year with probability \(q\). This set-up is described (exactly, not approximately) by a binomial random variable with parameters \(N\) and \(q\), and off we go. The student is asked genuinely informative questions (mean? variance? estimate?) and learns much from the answers. This is the first stage.

Soon the actuary (no longer a student) faces real-life examples which do not fit this simple set-up. For example, some observations may be bounded within a half-year, or a quarter-year, or a week, or a day. Or, perhaps, monthly cashflows have to be valued. Clearly, we need to understand mortality over any arbitrary fraction of a year. In one sense, the answer is easy. If our problem is sufficiently regular, redefining the time unit is enough, like working in centimetres instead of metres. The ideas are the same, the time unit is unimportant, which provides a first fundamental insight.

But this will not do. Unless time itself is quantized into indivisible half-years, or weeks, or some other unit (an argument that we leave to the physicists) we will never answer all possible questions in this way. We are led ineluctably to consider the limiting case, that someone is alive at some time \(t\) and may die in the succeeding time \(dt\), where \(dt\) may become as small as we choose. In other words, we are led to consider models in continuous time. Thus the actuary enters the second stage.

Can we rescue anything from the discrete-time model? Yes, a great deal! The key concept is still that of the binary event --- something either happens during a given time interval, or it does not. We obtain a probabilistic model by supposing it happens with some probability \(q'\), or not with probability \((1-q')\) --- a Bernoulli model. The clever bit is to extend this reasoning to the arbitrarily small time \(dt\), and see what happens as we let \(dt \to 0^+\). In the case of mortality, the probability \(q'\) becomes \(\mu \, dt\), where \(\mu\) is the force of mortality, and \((1-q')\) is \((1- \mu \, dt)\). The link between discrete and continuous is provided by the integral calculus. Not quite the familiar \(\int f(t) \, dt\)  integral, but the product integral \(\prod (1 + f(t) \, dt)\), the natural way to construct quantities that nature made multiplicative rather than additive --- quantities like the actuary's survival probability \({}_tp_x\). This is a second fundamental insight. For the record, here is the relevant product integral, see Macdonald & Richards (2024) and references therein for more:

\[ \prod_0^t \big( 1 - \mu_{x+s}\, ds \big) = \exp \left( - \int_0^t \mu_{x+s} \, ds \right) = {}_tp_x. \]

The practical actuary is left with a problem though. How can she actually compute such quantities, involving integrals and other objects, whose basic quality is smoothness? Some integrals can be written down explicitly, but not many. And the answer --- discretize! If we must use a digital computer we can do nothing else. No longer the `toy' discrete-time models of the classroom, however, but a discipline in its own right, with huge impact on practice. If a binomial model should reappear (as it might) it now describes the world  approximately, not exactly. We have reached the third stage.

It is perhaps instructive that both of the actuary's traditional domains have in recent decades been re-upholstered in new forms of integral calculus. Mortality analysis, as above, is now understood in terms of product integrals, while financial mathematics is written in the language of Itô calculus and its stochastic integral. In both, the trajectory can clearly be seen: