IAI CM1 common mistakes
IAI CM1 · Understand the traps and how to avoid them.
When asked for a probability involving a life table, the candidate reads l(x) values directly as probabilities (for example answering l(60)/l(30) when the question asks for the probability that a life aged 30 dies before 60) instead of forming the correct ratio of deaths or survivors and checking that the answer is a probability between 0 and 1.
Life-table notation (l_x, d_x, p_x, q_x, tPx) is dense and visually similar, and under time pressure candidates pattern-match the two numbers in the question to the two l-values in the table without translating the event into survival or death counts first.
What to do instead
Before touching the table, write the event in words and then in symbols (for example 10p40 = l50/l40, or 10|q40 = (l50 - l60)/l40). Compute the ratio, then sanity-check it is a probability and is consistent with the direction of the event (survival vs death). Practise 20 mixed notation-to-ratio translations until the mapping is automatic.
Applying a single aggregate discount factor to expected cash flows when the payment depends on survival, i.e. computing v^n * E[benefit] rather than the sum of v^t * (probability of payment at t) * (amount at t), or vice versa ignoring discounting on a survival-contingent annuity.
Candidates carry over intuition from deterministic compound-interest problems where expected value and discounting commute trivially; they forget that the expectation is over a random payment time, so the discount factor must sit inside the expectation.
What to do instead
Always write the present value as a random variable first: PV = sum over t of v^t * B_t * I(alive at t), then take expectations term by term. Use the identity E[PV] = sum v^t * B_t * tPx as a checklist. Redo any question where the answer came out as a single product of a discount factor and a plain expected value.
Mixing up the force of mortality mu(x) with the mortality rate q_x, for example substituting mu for q in a formula or treating mu as constant over an interval when the question specifies a non-constant force (such as mu(x) = a + bx or Makeham/Gompertz forms).
Both are called 'mortality' and both are small numbers, so the symbols blur; the relationship q_x = 1 - exp(-integral of mu) is easy to state but rarely practised in both directions under exam conditions.
What to do instead
Memorise the bridge q_x = 1 - exp(-∫_x^{x+1} mu(s) ds) and its survival form tPx = exp(-∫_0^t mu(x+s) ds). For any question giving mu, immediately convert to survival probabilities via the integral before doing anything else. Drill both directions (mu to q and q to mu) on at least five parameterised forces.
Using the wrong interest basis for annuities due versus immediate, or applying i instead of d (or the continuous force delta) in annuity formulas, e.g. using a-double-dot = (1-v^n)/d when the payments are in arrears, or forgetting the (1+i) factor that converts an immediate annuity to a due annuity.
The annuity formulas differ only by a small symbol (a_n vs a-double-dot_n, i vs d), and candidates memorise one form and reuse it; the timing of payments (start vs end of year) is easy to skim past in the question stem.
What to do instead
For every annuity question, draw a small timeline marking payment dates and the valuation date before selecting a formula. Build a one-page formula sheet pairing each annuity symbol with its payment timing and its i/d form, and derive a-double-dot_n = (1+i) a_n each time rather than recalling it.
Solving for a premium or reserve but reporting the answer at the wrong time point or on the wrong basis: computing a net premium reserve prospectively at duration 0 when the question asks for the reserve at duration t, or using the gross (expense-loaded) premium where the question specifies net premium reserve, or vice versa.
Reserve questions stack several conventions (net vs gross, prospective vs retrospective, valuation time), and candidates lock onto the first method they recall rather than reading which convention the question fixes.
What to do instead
Before calculating, underline three things in the stem: valuation time t, net vs gross basis, and prospective vs retrospective. Then write the reserve definition (for example tV = A_{x+t} - P * a-double-dot_{x+t} for net prospective) and confirm the time subscripts match t. Check that the reserve at duration 0 equals zero for a net premium reserve.
Confusing the distribution of a sum of independent random variables with the distribution of their average, or applying the central limit theorem to the sum without rescaling, so the variance is off by a factor of n (or n^2).
The algebra Var(sum X_i) = n*Var(X) and Var(mean) = Var(X)/n look almost identical on paper, and candidates under exam pressure drop the n or apply it in the wrong direction.
What to do instead
Write the quantity of interest explicitly as either S_n = X_1 + ... + X_n or X-bar = S_n/n, then state its mean and variance before standardising. Use the standardisation Z = (S_n - n*mu)/(sqrt(n)*sigma) for sums and (X-bar - mu)/(sigma/sqrt(n)) for averages, and verify that increasing n shrinks the spread for an average but widens it for a sum.
Using the normal approximation to a binomial or Poisson count without a continuity correction, or applying it when np or n(1-p) is too small for the approximation to be reasonable, and reporting a tail probability that is materially wrong.
The continuity correction feels like a cosmetic adjustment and is easily forgotten, and candidates rarely check the np >= 5 (or similar) rule before invoking the approximation.
What to do instead
Whenever approximating a discrete count by a continuous normal, convert P(X <= k) to P(X <= k + 0.5) and P(X >= k) to P(X >= k - 0.5) before standardising. State the validity check (np and n(1-p) both comfortably above 5, or lambda large for Poisson) in your working. If the check fails, use the exact distribution.
Computing a confidence interval or hypothesis test with the wrong standard error: using sigma/sqrt(n) when sigma is estimated by s and the sample is small (should be t with s/sqrt(n)), or using the population standard deviation when only the sample standard deviation is given, or mixing one-sided and two-sided critical values.
The formulas for z and t intervals are nearly identical, and the choice hinges on whether sigma is known and the sample size, which candidates skip checking; one-sided versus two-sided is decided by the wording of the alternative hypothesis, which is easy to misread.
What to do instead
Adopt a decision tree: is sigma known? If yes use z; if no and n is small use t with n-1 degrees of freedom. Then read the alternative hypothesis (H1: mu > , < , or !=) to fix one- or two-sided and pick the matching critical value. Write the standard error symbolically before substituting numbers.
Misinterpreting a p-value or confidence interval in the final written conclusion, for example saying the p-value is the probability the null hypothesis is true, or that a 95% confidence interval contains 95% of the data, or that a non-significant result proves the null.
These are the most commonly repeated misinterpretations in statistics teaching, and candidates reproduce them from memory in the interpretation mark of a question even when the calculation is correct.
What to do instead
Memorise three correct sentences and use them verbatim: a p-value is the probability of data at least as extreme as observed, assuming H0 is true; a 95% CI is an interval that would capture the true parameter in 95% of repeated samples; failure to reject H0 is not evidence that H0 is true. Practise writing the conclusion sentence for every past question, not just the number.
In multi-part questions, carrying forward an early arithmetic error into later parts without flagging it, or abandoning a later part because an earlier part was not completed, even though the later part is independent or can be answered using the stated earlier result.
Candidates treat each question as a single chain and either propagate a wrong value silently or freeze when part (a) is unsolved, losing marks that were available for method in parts (b) and (c).
What to do instead
Use the 'own figure' convention: if a later part depends on an earlier answer, state 'using my value from (a)' and proceed with correct method. If (a) is unsolved, look for a later part that is self-contained or that the stem says to answer using a given value, and do that part first. Label each part's answer clearly so an examiner can follow the carry-forward.