IAI CM2 common mistakes
IAI CM2 · Understand the traps and how to avoid them.
When computing survival probabilities from a life table, candidates use the force of mortality or the central death rate directly as if it were the one-year death probability q_x, or they interpolate l_x values and treat the interpolated value as exact, producing survival probabilities that are subtly wrong.
The notation μ_x, m_x and q_x all look like 'the death rate at age x', and the distinction between the force of mortality, the central exposed-to-risk rate and the annual probability of death is easy to blur when under time pressure. Interpolation (e.g. uniform distribution of deaths) is often applied without checking which assumption the question specifies.
What to do instead
Before computing, write the definitional chain explicitly: q_x = 1 - p_x, p_x = l_{x+1}/l_x, m_x = d_x/L_x, and μ_x = -d/dx ln l_x. State which mortality assumption (UDD, constant force, Balducci) applies and use only the formula valid under that assumption. Check that any interpolated quantity is labelled as an approximation and that the question asked for the exact value.
Candidates estimate mortality rates from exposed-to-risk data without correctly matching deaths to the central exposed-to-risk, for example dividing calendar-year deaths by an exposure figure that includes lives who entered or left mid-year, or ignoring the distinction between initial and central exposure.
The standard approximation q_x ≈ d_x/E_x^c only holds under specific assumptions about when deaths occur within the year. Candidates memorise the formula but not its derivation, so they mis-handle deaths in the year of entry or exit, and they confuse E_x^c with the number of lives at the start of the year.
What to do instead
Derive the estimator once from first principles: the central exposed-to-risk is total time lived in the age band, and the Poisson maximum likelihood estimator is d/E^c. Practise computing E^c from individual records by summing exact durations in the band for each life, including partial years. When a question gives initial exposure, convert using the stated assumption (e.g. deaths uniformly spread) rather than substituting directly.
When graduating a mortality table, candidates apply a smoothing or parametric fit and then report the graduated rates without testing goodness of fit, or they test fit but ignore smoothness, treating a close fit as automatically acceptable.
Graduation is presented in courses as a two-stage process (fit, then test), and candidates often stop after the fit because the test statistics look like optional extras. The trade-off between adherence to data and smoothness is the conceptual heart of the topic but is easy to skip under exam timing.
What to do instead
For every graduation question, structure the answer as: (1) fit, (2) test adherence with a chi-squared or deviance statistic and degrees of freedom adjusted for parameters estimated, (3) test smoothness by examining third or higher differences of the graduated rates, and (4) comment on the balance. State the null hypothesis and the degrees of freedom explicitly rather than just quoting a statistic.
Candidates compute annuity and insurance present values using the wrong timing convention, for example applying an annuity-due factor when the payments are in arrears, or discounting a whole-life insurance benefit by v^{k+1} when the benefit is payable at the end of the year of death versus immediately on death.
The notation ä_x, a_x, A_x and \bar{A}_x differs only by a diacritic or bar, and the exam frequently mixes continuous, annual and m-thly payment patterns. Under pressure, candidates default to the formula they remember best rather than the one matching the contract description.
What to do instead
For each question, first draw a timeline marking payment dates and the death benefit timing, then select the symbol that matches the timeline. Memorise the relationships A_x = v q_x + v p_x A_{x+1} and the due/arrears conversion a_x = ä_x - 1, and verify any answer by checking it lies between the continuous and annual bounds.
In premium and reserve calculations, candidates compute the net premium reserve using the gross premium when the question asks for the net premium reserve, or they omit expenses and then compare against a gross premium valuation without reconciling the two bases.
The recursive reserve equation is taught in several equivalent forms, and the distinction between net premium reserves (based on the net premium) and gross premium reserves (based on the gross premium and including expenses) is a common source of sign and basis errors. Candidates also forget that the reserve is the expected present value of future outgo minus future income at the valuation date.
What to do instead
Write the prospective reserve as EPV(future benefits) + EPV(future expenses) - EPV(future premiums) at the valuation time, and use the same premium basis throughout. Cross-check with the retrospective form and with the recursive relation (V + P)(1+i) = q·S + p·V_{next}. If the question specifies a net premium reserve, set expenses to zero explicitly.
Candidates value pension scheme liabilities using the same discount rate as the scheme's expected asset return, or they ignore the difference between defined benefit and defined contribution promises when describing the employer's risk.
Pension mathematics borrows discounting language from life insurance, and candidates conflate the funding discount rate (often a prescribed valuation rate) with the investment return assumption. The DB/DC distinction is conceptually simple but is frequently answered in generic terms.
What to do instead
State the valuation basis separately from the investment strategy: the liability discount rate is a valuation assumption, not a forecast of returns. For DB questions, identify who bears investment and longevity risk (the employer/sponsor) and value the promised benefit accrual; for DC, identify the member as bearing the risk and value the accumulated fund. Show the accrued benefit formula and the salary projection assumption explicitly.
When asked to critique or construct an actuarial model, candidates list model features without stating the purpose of the model or the criteria (accuracy, simplicity, transparency, consistency) against which it should be judged, and they fail to distinguish deterministic from stochastic models when the question hinges on that distinction.
The principles-of-modelling topic is discursive, and candidates treat it as an invitation to write everything they know. Without an explicit purpose statement, the critique drifts and misses the specific trade-off the examiner is testing.
What to do instead
Adopt a fixed structure: purpose of the model, inputs and assumptions, outputs, then evaluation against the criteria of accuracy, simplicity, transparency, consistency and communicability. Explicitly classify the model as deterministic or stochastic, identify which risks are modelled and which are not, and state how results would be validated and communicated.
Candidates manipulate stochastic integrals as if they were ordinary Riemann integrals, for example writing ∫ f dB = f B - ∫ B df without the correction term, or applying the chain rule to functions of Brownian motion without Itô's lemma.
Classical calculus is deeply practised, and the Itô correction term is a small but decisive addition that is easy to drop. Candidates also forget that Brownian paths are of unbounded variation, so the ordinary chain rule fails.
What to do instead
Always write Itô's lemma in full: df(t,B_t) = f_t dt + f_x dB_t + ½ f_xx dt, and check the quadratic variation term is present. For integration by parts, use ∫_0^T f dB = f(T)B_T - f(0)B_0 - ∫_0^T B df - [f,B]_T. Practise with f = B² and f = e^{σB - ½σ²t} until the correction terms are automatic.
Candidates use a martingale result without verifying the integrability or adaptedness conditions, for example applying the optional stopping theorem to a stopped Brownian motion at an unbounded stopping time, or treating a discounted price as a martingale under the wrong measure.
The martingale property is often stated as a slogan ('discounted prices are martingales'), and the technical conditions are glossed over in revision. The measure under which the martingale property holds is frequently left implicit.
What to do instead
State the measure explicitly every time: under the risk-neutral measure Q, the discounted price process is a martingale. Before applying optional stopping, confirm the stopping time is bounded or that the stopped process is uniformly integrable. For problems involving hitting times, use the bounded version first and then take limits with justification.
Candidates derive the Black-Scholes price by plugging into the closed-form formula without checking which measure and which discounting convention applies, and they confuse the risk-neutral drift with the real-world drift, or misuse put-call parity with the wrong discount factor.
The formula is memorised as a black box, so the assumptions (no arbitrage, continuous trading, constant volatility, lognormal prices) and the role of the risk-neutral measure are not connected to the mechanics of the answer. Parity is quoted as C - P = S - K without the discounting on K.
What to do instead
Derive the price from the risk-neutral expectation e^{-rT} E^Q[(S_T - K)^+], state that S_T is lognormal under Q with drift r - ½σ², and then quote the formula. Write put-call parity as C - P = S_0 - K e^{-rT} and verify it holds in your answer. Comment on which assumptions are violated in practice (e.g. constant volatility) when the question asks for a critique.
In interest rate modelling, candidates assume a single constant rate applies to all maturities, or they calibrate a short-rate model and then price a bond without imposing the no-arbitrage condition that reproduces the initial term structure.
Term structure models are taught after the constant-interest framework, and candidates revert to a single rate because it simplifies the arithmetic. The distinction between the real-world measure and the risk-neutral measure, and the role of market price of risk, is often not internalised.
What to do instead
Work with the full yield curve: state the instantaneous short rate r_t, the bond price P(t,T) = E^Q[exp(-∫_t^T r_s ds)], and check that the model reproduces observed bond prices at calibration. When a question involves the market price of risk, write the change of measure explicitly and note that the drift changes while the volatility is preserved.