PJENL Actuario Aptitude Exam flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Law of total probability

For a partition of the sample space into disjoint events B_i, the probability of A equals the sum over i of P(A given B_i) times P(B_i).

Bayes' theorem

Posterior probability of B given A equals P(A given B) times P(B) divided by the total probability of A, obtained by summing over all causes.

Variance of a sum

The variance of X plus Y equals the sum of their variances plus twice their covariance; the covariance term vanishes only when the variables are uncorrelated.

Poisson process properties

Counts over an interval of length t are Poisson with mean rate times t; interarrival times are exponential with mean equal to the reciprocal of the rate.

Maximum likelihood estimator

The parameter value maximizing the joint density of the observed sample, usually found by setting the derivative of the log-likelihood to zero.

Central limit theorem

For independent identically distributed variables with finite variance, the standardized sample mean converges in distribution to the standard normal as sample size grows without bound.

Compound interest accumulation

A unit invested at effective annual rate i accumulates to (1+i) raised to the power t after t years, producing exponential rather than linear growth.

Annuity-immediate present value

The present value of n level payments of one at each year end equals (1 minus v to the n) divided by i, where v is the one-year discount factor.

Nominal versus effective rate

A nominal rate compounded m times per year converts to an effective annual rate of (1 plus the nominal rate over m) raised to m, minus one.

Force of interest

The continuous compounding rate delta equals the natural logarithm of one plus the effective annual rate; accumulation over t years is e to the delta t.

Macaulay and modified duration

Macaulay duration is the present-value-weighted average payment time; modified duration divides it by one plus the yield and approximates proportional price sensitivity.

Net present value decision rule

Accept an investment when the present value of its cash inflows exceeds the initial outlay; the internal rate of return sets this difference to zero.

Loan outstanding balance

After a scheduled payment, the remaining balance equals the present value of all future payments discounted at the loan rate, or equivalently the original principal accumulated minus payments accumulated.

Survival probability notation

The symbol t p x denotes the probability that a life aged x survives at least t more years; its complement, t q x, is the probability of death within t years.

Whole life insurance APV

The actuarial present value of a unit payable at death integrates the discount factor times the probability of death at each time over all future times.

Insurance-annuity relationship

For a whole life policy with benefits at year end, the annuity-due value equals one minus the insurance value, all divided by the discount rate.

Equivalence principle premium

The net premium is chosen so the actuarial present value of future premiums equals that of future benefits, with no provision for expenses or profit.

Prospective reserve

The reserve at duration t equals the present value of future benefits minus the present value of future net premiums, both evaluated at the attained age.

Recursive reserve relation

The reserve plus the premium, accumulated one year at interest, equals the expected death benefit plus the expected next-year reserve, weighted by death and survival probabilities.

Aggregate loss mean and variance

For a random sum of random claim sizes, the mean is the product of expected counts and expected severity; the variance adds a term for count variability.

Bühlmann credibility factor

The credibility weight equals exposure divided by exposure plus k, where k is the expected process variance divided by the variance of hypothetical means.

Credibility premium formula

The estimate is the credibility weight times the observed sample mean plus one minus that weight times the prior or manual mean.

Full credibility standard

The exposure needed for full credibility grows with the square of the standard normal quantile and with the squared coefficient of variation of severity.

Panjer recursion

A recursion computing the aggregate claim distribution for count distributions in the (a,b,0) class, using the severity probabilities and two parameters.

Stop-loss expected cost

For retention d, expected reinsurer payment equals the integral from d to infinity of the survival function of aggregate losses, valid for nonnegative losses.

Capital asset pricing model

Expected return equals the risk-free rate plus beta times the market risk premium, where beta measures the asset's sensitivity to market movements.

Put-call parity

For European options on a non-dividend stock, call minus put equals stock price minus the present value of the strike, discounted at the risk-free rate.

Bond price convexity adjustment

The percentage price change approximately equals minus modified duration times the yield change plus half the convexity times the squared yield change.

Modigliani-Miller with taxes

With corporate taxes and deductible interest, firm value increases by the tax rate times debt relative to an otherwise identical unlevered firm.

Matrix invertibility conditions

A square matrix is invertible exactly when its determinant is nonzero, its columns are linearly independent, and zero is not one of its eigenvalues.

Eigenvalue trace and determinant

The sum of a matrix's eigenvalues equals its trace, and their product equals its determinant, providing a quick check on computed eigenvalues.

Hessian classification test

At a stationary point, a positive definite Hessian indicates a local minimum, a negative definite Hessian a maximum, and an indefinite one a saddle point.