IFoA CM2 common mistakes
IFoA CM2 · Understand the traps and how to avoid them.
When asked to show that a compound Poisson aggregate claim distribution has a particular mean and variance, the candidate writes the mean as E[N]E[X] but then computes the variance as Var(N)Var(X) or as Var(N)E[X]^2, instead of Var(N)E[X]^2 + E[N]Var(X).
The conditional variance identity is memorised as a single product, and under exam pressure the candidate collapses the two terms into one, often because they only check the mean formula and assume variance follows the same product structure.
What to do instead
Before computing, write the two-line derivation: E[S] = E[N]E[X]; Var(S) = E[N]Var(X) + Var(N)(E[X])^2, derived from Var(S) = E[Var(S|N)] + Var(E[S|N]). Then substitute numbers term by term, labelling each term, so a missing term is visible in the working and can earn method marks.
When transforming a normal variable, the candidate computes the mean of a function of X, such as E[e^X] or E[X^2] for a lognormal, by substituting the mean of X into the function rather than using the moment generating function or the lognormal moment formulas.
Linearity of expectation is over-generalised to nonlinear functions, and the lognormal parameterisation (mu and sigma being parameters of the underlying normal, not the mean and standard deviation of the lognormal) is confused.
What to do instead
State explicitly that E[g(X)] is not g(E[X]) unless g is linear. For lognormal, memorise E[Y] = exp(mu + sigma^2/2) and Var(Y) = exp(2mu + sigma^2)(exp(sigma^2) - 1), and derive them once from the normal MGF. In the exam, write the MGF route out in full before substituting.
When constructing a confidence interval or test for a proportion or Poisson rate, the candidate uses the normal approximation with the estimated standard error even when the counts are small, and does not apply a continuity correction or an exact method where the question implies one.
Textbook formulas are applied mechanically without checking the validity conditions (np and n(1-p) large, Poisson mean large), and the difference between an approximate and an exact interval is not internalised.
What to do instead
Before choosing a method, write down the validity condition and check the numbers. If the condition fails, state that an exact binomial or Poisson interval is required and use the appropriate distribution directly. If the normal approximation is used for a discrete count, apply the continuity correction and say so in the answer.
When carrying out a hypothesis test, the candidate states the conclusion as 'the null hypothesis is true' or 'the probability the null is true is 0.03', rather than reporting the p-value and concluding in terms of evidence against the null at the stated significance level.
The p-value is treated as the posterior probability of the null, and the asymmetry of hypothesis testing (fail to reject does not mean accept) is not understood, so the wording drifts into an incorrect probabilistic claim.
What to do instead
Use a fixed sentence template: 'There is [sufficient/insufficient] evidence at the 5% level to reject H0 in favour of H1; the p-value is x.' Explicitly note that failing to reject H0 is not evidence that H0 is true, and that the p-value is computed assuming H0 holds. Practise writing this sentence for every past-style test question.
When fitting a linear regression or GLM, the candidate interprets a coefficient as the effect of a one-unit increase in the predictor on the response mean without accounting for the link function, and reports the coefficient on the linear predictor scale as if it were on the response scale.
The distinction between the linear predictor and the mean response is blurred, particularly for log and logit links where the coefficient is multiplicative on the response scale.
What to do instead
For every coefficient, state the scale: 'on the log scale' or 'the log-odds'. Then translate: for a log link, exp(beta) is the multiplicative change in the mean per unit increase; for a logit link, exp(beta) is an odds ratio. Write both the raw and translated interpretation in the answer, and check the direction against the data.
When assessing a regression model, the candidate reports R-squared or deviance as the sole measure of fit and does not check residual plots, influential points, or the appropriateness of the assumed error distribution and link.
Model checking is treated as an optional afterthought, and the candidate has not internalised that a high R-squared is compatible with a badly misspecified mean structure or non-constant variance.
What to do instead
Adopt a fixed checklist for every model: plot residuals against fitted values and against each predictor, check for curvature and funnel shapes, inspect standardised residuals and Cook's distance for influential points, and for GLMs check the deviance residual plot and the dispersion parameter. Write one sentence per check in the answer.
When fitting an ARIMA or exponential smoothing model, the candidate differences or transforms the series without checking stationarity first, and then reports the order of the model from software output without justifying the choice of d or the differencing.
The order selection is treated as a software output rather than a modelling decision, and the link between the integration order, unit roots, and stationarity is not secure.
What to do instead
Always plot the series and the ACF/PACF before differencing. State the stationarity test used (e.g. augmented Dickey-Fuller) and its conclusion, then justify d from that conclusion. After fitting, check the residual ACF for whiteness and report the AIC alongside any automated order selection, noting that automated selection is a starting point, not a justification.
When computing transition probabilities for a Markov jump process, the candidate uses the discrete-time transition matrix P for a continuous-time process, or computes the generator matrix Q but then exponentiates it entry by entry instead of computing the matrix exponential.
The discrete and continuous cases are studied together and the notation overlaps, so the candidate applies the familiar P^n formula where exp(Qt) is required, and matrix exponentiation is a less practised operation.
What to do instead
Write down which time scale the process lives on before starting. For a continuous-time chain, compute Q from the rates, then obtain P(t) = exp(Qt) by diagonalisation or by solving the Kolmogorov forward equations. Check that rows of exp(Qt) sum to 1 and that the stationary distribution satisfies pi Q = 0, which catches most errors.
When estimating a survival function from grouped or censored data, the candidate treats censored observations as deaths, or computes the Kaplan-Meier estimate but mishandles ties and the risk set at each death time.
The risk set is the conceptually difficult part, and it is easy to drop censored individuals from the denominator entirely rather than removing them only after their censoring time.
What to do instead
Build a table with columns: time, number at risk, deaths, censorings, and the factor (1 - d/n). At each event time, define the risk set as those with observed time at least that time, including those censored later. Handle ties by reducing the risk set by the number of deaths at that time. Check that the product of factors is non-increasing and ends near the expected survival.
When graduating a mortality table or projecting mortality, the candidate applies the method mechanically without checking that the graduated rates are monotone where expected and that the projection basis is stated, and does not comment on the plausibility of the resulting rates.
Graduation and projection are treated as formula-execution tasks, and the candidate does not build the habit of sanity-checking outputs against the crude rates and against known mortality patterns.
What to do instead
After graduating, plot or tabulate graduated against crude rates, check monotonicity and smoothness, and report a test statistic (e.g. chi-squared) with its degrees of freedom and interpretation. For projections, state the model (e.g. Lee-Carter), the data period, the jump-off rates, and the assumed future improvement, and comment on whether the projected rates are plausible at the oldest ages.