The Doctrine of Probabilities

I rediscovered my Faculty of Actuaries diploma recently, having misplaced it in a house move some years ago. It testifies to my knowledge of ‘the doctrine of probabilities’, which is nice. But not long after I received it, Prof Hans Bühlmann classified actuaries like me as ‘Actuaries of the First Kind’ and said:

Contrary to [the Actuary] of the First Kind in life assurance, whose methods were essentially deterministic, [the Actuary of the Second Kind] had to master the skills of probabilistic thinking.

Bühlmann (1989)

How could I be, at one and the same time, knowledgeable in ‘the doctrine of probabilities’ and ‘essentially deterministic’ ? The answer lies in the deeper nature of those probabilities. I was indeed expert at calculations using probabilities such as \({}_tq_x\) and \({}_tp_x\), defined in International Standard Actuarial Notation as follows, for some arbitrary person \(A\):

\begin{eqnarray*}{}_tq_x &= {\rm Pr}[A {\rm \ dies\ before\ age\ }x+t|A {\rm \ is\ alive\ at\ age\ }x]\\{}_tp_x &= {\rm Pr}[A {\rm \ is\ alive\ at \ age\ }x+t|A {\rm \ is\ alive\ at\ age\ }x].\end{eqnarray*}

Moreover, if we define \(l_x\) to be the life table function with radix \(l_0\) then we can give \({}_tp_x\) another probabilistic interpretation, as an expectation:

\[l_x {}_tp_x = {\rm E}[{\rm Number\ out\ of\ }l_x{\rm \ alive\ at\ age\ }x{\rm\ who\ are\ still\ alive\ at\ age\ }x+t]\qquad(2)\]

However, look at these statements more closely. They are mainly defined in words. As such, they would be recognizable to late Victorian actuaries. The standard notation was proposed at the 2nd International Congress of Actuaries (ICA) in London in 1898; and approved at the 3rd ICA in Paris in 1900.

Coincidentally, the 2nd International Congress of Mathematicians also took place in Paris in 1900. It was immortalized by David Hilbert announcing his list of problems, whose solutions he hoped would propel mathematics into the new century. Part of Hilbert’s 6th Problem was to find an axiomatic basis for probability theory, then viewed as a renegade offshoot of applications, maybe not even proper mathematics at all (so much for the 3rd ICA!)

In fact, probability and statistics is mainly a 20th century construction. It had to await a proper theory of sets (for events) and measures on sets (for probabilities) and integration (for expectations), and Hilbert’s problem was finally answered by Kolmogorov in 1933. Only then could words be replaced by well-defined mathematics. To see the difference, represent the future lifetime of \(A\) at birth by a non-negative random variable \(T\) . Then we can rewrite the definitions (1) as follows:

\begin{eqnarray*}{}_tq_x = {\rm Pr}[T\leq x+t | T > x ]\\{}_tp_x = {\rm Pr}[T > x + t | T > x ].\end{eqnarray*}

The essence of Hilbert’s problem was, how should we represent mathematically the random ‘thing’ whose probability or expectation we calculate? Presented with an expression ${\rm Pr}[X]$ like (1), or ${\rm E}[X]$ like (2) (which actuaries had been using in some form for two centuries) we ought to ask, “what is the mathematical nature of the quantity \(X\) inside the brackets?”

In this example the answer involved a set of points \(\omega\) in a suitable sample space \(\Omega\) and a function \(T:\Omega\mapsto[0,\infty)\) such that it makes sense to define the sets \(\mathcal{A} = \{\omega\in\Omega : T(\omega)\leq x\}\) (in the case of \({}_tq_x\)) or \(\bar{\mathcal A} = \{\omega\in\Omega: T(\omega) > x\}\) (in the case of \({}_tp_x\)). We can write these sets less formally as the events \(\{T\leq x\}\subseteq\Omega\) and \(\{T > x\} \subseteq\Omega\).  The hard bit, that took 300 years from Pascal to Kolmogorov to figure out, lies in the words ‘makes sense’.

Prof Bühlmann was quite right. I could manipulate \({}_tq_x\), \({}_tp_x\) and their more exotic relatives, but had given little or no thought to the random objects of which these were the probabilities. That conceptual leap had in fact been made by actuaries (Hickman 1964, Bühlmann 1976, Gerber 1976) noticing that the random present value of a life assurance benefit could be written in the general form \(v^T\) (and much more, details omitted). It then made sense to ask questions like: what is \({\rm E}[v^T]\)? What is \({\rm Var}[v^T]\)? In other words, to think like an Actuary of the Second Kind. Meanwhile, financial economists had begun to ask, “what is the nature of that \(v\) inside the brackets?” and even more radically, “what is the nature of that \({\rm E}\) outside the brackets?”

Forty years ago this seemed quite futuristic. After all, I worked in an institution 150 years old, for the most part selling souped-up (pre-)Victorian policies, competing with twenty or so equally ancient institutions, all governed by rules originating in 1870. But that is another story.