It’s your ultimate nightmare. You spend every weekend and evening studying for that actuarial exam, but when you come to sit it, you find that the paper is impossible. Questions are either ambiguously worded, or you are asked to prove statements that are unambiguously false.

Example of a poor question

I have picked an example from the April 2005 CT4 paper and the examiners’ solution. You can download both the paper and the examiners’ report from www.actuaries.org.uk.

Here is the first part of the third question:

An able candidate’s answer

A student with a secure mathematical background might argue as follows.

The question defines Y2=Y3/Y1. I am asked to prove in part (i) that Y3 is independent of Y1 and Y2, but that can’t possibly be true because Y3=Y1*Y2. So I’ll leave part (i) blank. There must be a typo somewhere.

Now for part (ii). From part (i), the examiners believe that ‘Yk:k=1, 2,... is a sequence of independent and identically distributed random variables’. This means, for example, that if s

A weaker candidate’s answer

The weaker candidate might not know how to prove that a sequence is independent and identically distributed. However, the hint suggests trying to prove that any pair (Xs, Xt), s Case Definitio– Examiners’ model solution

(a) s, t both odd Independent because the question says so

(b) t=s+1 Proved in the model solution

(c) s, t both even and Asserted as ‘obvious’

t>= s+4

(d) t=s+2, both eve– Asserted as ‘obvious by symmetry’

(e) t >= s+3 with s and t Not considered

of opposite parity

What the examiners wanted

The examiners made the following comments on question 3:

The section in bold above was absent from the original examiners’ report. It was added after I wrote to them. The less able candidate, and apparently the examiners, could easily overlook the fact that pairwise independence does not imply the full independence whose proof is requested in the original question. The examiners’ report demonstrates that most candidates followed the logic of my hypothetical able candidate.

The black hole in the examiners’ proof

The examiners, in their report, recognise two notions of independence mutual and pairwise. The phrase ‘independent and identically distributed random variables’ occurs twice in the question. This leads to four possible interpretations (table 1).

To sum up, the only way to prove the stated result is to suppose the examiners attach a different meaning to ‘independent and identically distributed’ each time the phrase appears. In that case, a full solution involves a laborious examination of five special cases, of which the marking schedule only gives credit for one.

The next question

Discouraged by question 3, many candidates went straight on to question 4. Question 4 is a multiple decrement question. Ambiguous wording requires candidates to judge whether the status ‘unmarried’ includes or excludes divorced people. The examiners’ report highlights the fate of those who guessed wrongly.

What’s a student to do?

What is the appropriate professional response to this? First, don’t be too harsh on a hapless colleague who failed CT4 it may not be their fault. Just over 40% of candidates passed; presumably some did so only by guessing the same wrong answers that were on the marking schedule.

Some colleagues justify a capricious examination process by the argument that the exams may be unpleasant and degrading but they had to sit them and learn to smile, therefore so should I. By that argument I should just put up with it: I’m not so special that they’ll suspend the system just for me.

A second possibility is to disengage from the system. The Institute, at least, allows students to keep their student status indefinitely. Why lose sleep now for the sake of an exam system that is obviously in chaos? Despite their current state of denial, in five years, the examiners must surely get their act together or the entire system will be outsourced to more competent hands. Either of these is more attractive than what we have today. This has been my personal strategy.

A third route is to make greater use of the exemption system. Many universities offer good courses in stochastic processes. Experts in the subject set and mark the examinations, which are also moderated externally. Furthermore, a pass in some of these examinations can result in an exemption from CT4, so you can take a step towards qualification while learning something useful.

05_10_07.pdf