PJENL Actuario Aptitude Exam common mistakes
PJENL Actuario Aptitude Exam · Understand the traps and how to avoid them.
Computing a conditional probability such as P(A|B) by dividing P(A∩B) by P(A) instead of by P(B), or substituting the joint probability into the numerator when the conditional probability is what is given.
Candidates memorize the formula P(A|B)=P(A∩B)/P(B) but under time pressure anchor on the first event mentioned in the prompt and use its marginal probability as the denominator. The reversal is easy to miss because the arithmetic still produces a plausible number between 0 and 1.
What to do instead
Before calculating, write the conditioning event explicitly with a vertical bar and label it 'given'. Use Bayes' theorem in the form P(A|B)=P(B|A)P(A)/P(B) and verify by checking that the two conditional probabilities P(A|B) and P(B|A) are not interchangeable. Sanity-check with a small contingency table when the problem gives counts.
Applying the wrong variance formula for a sum of random variables, treating Var(X+Y) as Var(X)+Var(Y) when X and Y are correlated, or adding variances when the question asks for the variance of a mean.
The independent-case identity is taught first and becomes automatic. Candidates stop reading for the covariance term or for the 1/n² scaling in Var(X̄)=σ²/n, especially when the problem is dressed in an actuarial scenario with aggregate claims.
What to do instead
Always write the general identity Var(aX+bY)=a²Var(X)+b²Var(Y)+2abCov(X,Y) and only drop the covariance after confirming independence is stated. For sample means, derive the scaling from first principles rather than recalling it. Check sign and magnitude: a variance of a mean must be smaller than the individual variance.
Using the nominal annual rate directly as the periodic rate, for example discounting monthly cash flows at i^(12) instead of i^(12)/12, or mixing effective and nominal rates in one accumulation.
Notation like i^(m) and d^(m) is visually similar to i, and candidates convert too quickly without writing the accumulation function. The error is invisible when all rates in a problem are the same nominal value, so it survives a casual check.
What to do instead
Convert every rate to an effective rate per payment period before touching a timeline. Memorize 1+i=(1+i^(m)/m)^m and 1-d=(1-d^(m)/m)^m, and write the conversion line explicitly on scratch paper. Match the compounding frequency to the cash-flow frequency, then discount period by period.
Computing the present value of an annuity-immediate when the cash flows actually occur at the beginning of each period, or misaligning the timing of the first payment in a deferred annuity.
The annuity-immediate symbol a-angle-n is the default in most textbooks, so candidates reach for it without drawing the timeline. Deferred annuities compound the confusion because the deferral period and the payment period must be counted separately.
What to do instead
Draw a horizontal timeline for every annuity problem and mark each payment date with an arrow before selecting a symbol. Use a-double-dot for payments at period start, and for deferred cases discount the whole annuity back by v^m where m is the deferral length. Verify with the first two payments valued individually.
Mixing up the force of mortality μ(x) with the probability of death q(x), or using q(x) where the problem requires the survival probability p(x) in a life-contingency calculation.
Both quantities describe mortality and both appear in life-table work, but they differ in units and magnitude. Candidates under exam pressure grab the tabulated q(x) column and plug it into an exponent or a product where p(x)=1-q(x) belongs.
What to do instead
State the definitional relationships before computing: μ(x)=-d/dx ln(l(x)), q(x)=1-p(x), and p(x)=exp(-∫μ). For multi-year survival use the product of annual p(x) values or the ratio l(x+t)/l(x), never a sum of q(x). Cross-check that survival probabilities decrease with t.
Computing a net premium or reserve using the wrong equivalence principle, for example equating expected present value of premiums to the benefit without including expenses, or omitting the annuity-due versus annuity-immediate distinction in the reserve recursion.
The equivalence principle has several variants (net, gross, portfolio) and candidates apply the net version reflexively. Reserves also require careful timing of the premium payment relative to the valuation date, which is easy to shift by one period.
What to do instead
Write the equivalence equation from scratch each time: EPV(premiums)=EPV(benefits)+EPV(expenses), and label whether each annuity is due or immediate. For reserves, use the prospective formula at the valuation time and confirm the recursion V(t)+P = q·b + p·V(t+1) with the premium paid at the start of the year. Test the reserve at issue equals zero for a net premium reserve.
Selecting a loss distribution for aggregate claims that ignores the distinction between frequency and severity, or fitting a Poisson frequency when the data show overdispersion and a negative binomial is required.
Candidates default to the Poisson because it is the simplest count model, without checking the variance-to-mean ratio. Overdispersed claim counts are common in practice, and the exam often supplies a sample whose variance clearly exceeds its mean.
What to do instead
Compute the sample mean and variance of claim counts first and compare them; if variance exceeds mean, consider negative binomial or a mixed Poisson. For aggregate losses, state whether the model is collective (compound) or individual, and confirm the severity distribution is positive and integrates to one. Check that the chosen model reproduces the observed first two moments.
Applying full credibility to a class with insufficient exposure, or using the square-root rule for partial credibility without checking the standard-for-full-credibility threshold.
The credibility factor Z is often presented as a formula to memorize, and candidates plug in exposure without verifying the threshold condition. The square-root rule Z=√(n/n_F) is only valid below the full-credibility standard.
What to do instead
First compute the full-credibility standard n_F from the chosen criterion (for example, the number of exposures needed so that the observed mean is within a stated percentage of the true mean at a given confidence level). If n≥n_F set Z=1; otherwise use Z=√(n/n_F) and cap at 1. State the criterion you are using and show the threshold calculation.
Confusing the spot rate, forward rate, and yield to maturity, for example discounting a multi-year cash flow at a single yield when the term structure is upward sloping, or computing a forward rate as an arithmetic average of spot rates.
All three are expressed as annualized percentages and appear in the same bond-pricing problems, so candidates substitute one for another. The forward rate in particular is often mistaken for a simple average rather than a geometric relationship.
What to do instead
Write the no-arbitrage relationship explicitly: (1+s_n)^n=(1+s_{n-1})^{n-1}(1+f_{n-1,n}), and solve for f. Discount each cash flow at its own spot rate unless the problem states a flat curve. For bonds, distinguish yield to maturity (a single internal rate) from the spot curve used for arbitrage-free pricing, and verify the price equals the sum of discounted cash flows.
Making a sign or transposition error when solving a linear system for portfolio weights or when computing an inverse matrix, and then reporting the resulting weights without checking that they sum to one.
Matrix algebra on the exam is done by hand under time pressure, and a single sign slip propagates silently. Candidates rarely substitute the answer back into the original system because they are rushing to the next question.
What to do instead
After solving any linear system, substitute the solution into every original equation. For portfolio weights, confirm the weights sum to 1 and that any long-only constraint is satisfied. For 2×2 inverses use the adjugate/determinant form and verify A·A⁻¹=I. Prefer elimination with a written pivot order over mental arithmetic.