IFoA CM2 flashcards

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

Actuarial control cycle

A loop of problem definition, model construction, parameter estimation, running the model, monitoring actual experience, and revising assumptions, used to keep models fit for purpose.

Deterministic versus stochastic model

A deterministic model gives a single output for given inputs; a stochastic model gives a probability distribution of outputs, allowing quantification of risk and variability.

Principle of parsimony

Among competing models that fit adequately, prefer the one with fewest parameters, since extra parameters add complexity and estimation error without necessarily improving predictive accuracy.

Binomial distribution

Counts successes in n independent trials with constant success probability p; probability of x successes is C(n,x) p^x (1-p)^(n-x), with mean np and variance np(1-p).

Poisson distribution

Models counts of rare independent events with rate lambda; probability of n events is exp(-lambda) lambda^n/n!, and mean equals variance equals lambda.

Poisson limit of binomial

When n is large and p small with np approaching lambda, the binomial distribution converges to the Poisson distribution with mean lambda.

Poisson process waiting time

In a Poisson process with rate lambda, the waiting time to the next event is exponential with mean 1/lambda, and counts in disjoint intervals are independent Poisson variables.

Lognormal distribution

If log Y is normal with mean mu and variance sigma squared, Y is lognormal with mean exp(mu + sigma squared/2) and positive skew, often used for claim sizes.

Central limit theorem

The sum or mean of n independent identically distributed variables with finite variance is approximately normally distributed for large n, whatever the parent distribution.

Maximum likelihood estimator

The parameter value maximising the likelihood of the observed sample; under regularity conditions it is consistent, asymptotically normal, and asymptotically efficient.

Cramer-Rao lower bound

For an unbiased estimator of theta, variance is at least the reciprocal of the Fisher information; reaching the bound means the estimator is fully efficient.

Method of moments

Estimates parameters by setting sample moments equal to the corresponding theoretical moments expressed as functions of the parameters, then solving the resulting equations.

Type I and type II errors

A type I error rejects a true null hypothesis with probability alpha, the significance level; a type II error fails to reject a false null with probability beta.

Power of a test

Power equals one minus beta, the probability of correctly rejecting a false null; it rises with sample size, effect size, and significance level.

Likelihood ratio test

Twice the difference in log-likelihood between unrestricted and restricted models is approximately chi-square distributed, with degrees of freedom equal to the number of restrictions.

Ordinary least squares slope

In simple linear regression the slope estimate equals the covariance of x and y divided by the variance of x, and the intercept makes the fitted line pass through the means.

GLM link function

A monotone function g connecting the expected response to the linear predictor, so g(mu) equals x transpose beta; examples are log, logit, and identity links.

Poisson GLM log link

Models counts with mean mu equal to exp(x transpose beta) and variance equal to mu; coefficients represent log multiplicative effects on the expected count.

Logistic regression coefficients

Each coefficient is the change in the log odds of the outcome per unit increase in its predictor; exponentiating gives the corresponding odds ratio.

Stationarity

A time series is weakly stationary when its mean and variance are constant over time and its autocovariance depends only on the lag between observations.

AR(1) process

X_t equals c plus phi X_{t-1} plus white noise; it is stationary when the absolute value of phi is below one, with mean c/(1-phi).

Random walk

A series whose current value equals the previous value plus independent noise; it is non-stationary with variance increasing linearly in time, and first differences are white noise.

Geometric Brownian motion

A process where dS equals mu S dt plus sigma S dB; the log of S is normal, so prices are lognormal, underpinning the Black-Scholes framework.

Markov property

The conditional distribution of future states given the present state does not depend on earlier history; only the current state matters for prediction.

Stationary distribution

A probability vector pi satisfying pi P equals pi; for an irreducible aperiodic chain it gives the long-run proportion of time spent in each state.

Generator matrix Q

A matrix with non-negative off-diagonal transition rates and rows summing to zero; the transition matrix over time t equals the matrix exponential of Qt.

Kolmogorov forward equations

The derivative of the transition matrix P(t) equals P(t) Q; solving them, or exponentiating Qt, gives transition probabilities for a Markov jump process.

Hazard and survival function

The hazard is minus the derivative of log survival; cumulative hazard integrates the hazard, and survival equals exp(-cumulative hazard).

Kaplan-Meier estimator

A product over event times of one minus deaths divided by those at risk at that time; it estimates the survival function from censored and uncensored data.

Actuarial estimate of q_x

With d deaths among n lives and deaths assumed uniformly spread over the year, q_x is estimated as d divided by (n minus d/2).

Graduation tests

Chi-square tests overall fit, the signs test detects systematic runs of deviations, and cumulative deviation tests detect bias; degrees of freedom fall with fitted parameters.

Gompertz and Makeham laws

Gompertz sets the force of mortality proportional to c to the power x, giving log-linear mortality; Makeham adds a constant term representing age-independent risk.

Lee-Carter model

Log mortality equals an age effect plus an age-sensitivity times a time index; the time index is usually forecast as a random walk with drift for projection.