IAI CM1 flashcards

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

Whole life insurance APV

The expected present value of a benefit paid at the end of the year of death: sum over k of v^(k+1) times the probability of death in year k.

Annuity-due identity

ä_x equals (1 minus A_x) divided by d, where d is the discount rate i/(1+i). This links whole life insurance and annuity values.

Net premium formula

For whole life insurance, the annual net premium equals A_x divided by ä_x, equivalently d times A_x divided by (1 minus A_x).

Prospective reserve

The reserve at duration k equals the APV of future benefits minus the APV of future net premiums, evaluated on the valuation basis.

Retrospective reserve

The accumulated value of past premiums minus accumulated cost of insurance, using the same valuation basis; it equals the prospective reserve for a standard policy.

Force of mortality

The instantaneous death rate μ at age x, defined as minus the derivative of the log of the survival function. It equals f(x)/S(x).

UDD assumption

Under uniform distribution of deaths, deaths are spread evenly within each year of age, so t-year death probability equals t times the annual q_x.

Constant force assumption

When the force of mortality is constant within each year, survival over a fraction t of the year equals the annual survival probability raised to power t.

Complete expectation of life

e_x is the integral from zero to infinity of t-year survival probability, giving the average future lifetime of a life aged x.

Pure endowment

A benefit paid only if the life survives n years; its APV is v^n times n-year survival probability.

Endowment insurance APV

Equal to the term insurance APV plus the pure endowment APV, since exactly one of death within the term or survival occurs.

Gross premium

The premium equating APV of benefits plus expenses to APV of gross premiums, all computed on the pricing basis rather than the valuation basis.

Multiple decrement table

A table tracking lives subject to several competing decrements; dependent rates describe actual exits while independent rates assume other decrements removed.

Associated single decrement

The hypothetical table where only one decrement operates; its rates relate to dependent rates through integration of the multiple decrement survival function.

Select mortality

Mortality that depends on duration since underwriting, so q at a given age is lower for recently selected lives than for those insured longer.

Thiele's differential equation

A continuous-time reserve recursion: the derivative of the reserve equals the reserve times the force of interest plus premium minus benefit outflow.

Maximum likelihood estimator

The parameter value maximizing the likelihood or log-likelihood. It is consistent and asymptotically normal with variance given by the inverse Fisher information.

Fisher information

The negative expected second derivative of the log-likelihood, or equivalently the variance of the score. It quantifies information about a parameter in a sample.

Cramér–Rao lower bound

A lower bound on the variance of any unbiased estimator, equal to the reciprocal of n times the Fisher information per observation.

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.

Consistent estimator

An estimator that converges in probability to the true parameter value as sample size tends to infinity.

Unbiased estimator

An estimator whose expected value equals the true parameter for every possible parameter value; sample variance with n minus one divisor is unbiased.

Method of moments

An estimation method equating sample moments to population moments and solving for parameters; it is consistent but usually less efficient than maximum likelihood.

Type I error

Rejecting a true null hypothesis, with probability equal to the significance level alpha chosen for the test.

Type II error

Failing to reject a false null hypothesis; one minus this probability is the power of the test.

p-value

The probability, computed assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the observed value.

Confidence interval coverage

The long-run proportion of intervals that contain the true parameter; for a normal mean with known variance it is exact.

Poisson distribution

A count distribution with probability mass λ^k e^(−λ)/k!; its mean and variance both equal λ, and counts over disjoint intervals are independent.

Exponential memorylessness

The exponential distribution satisfies that the conditional distribution of additional waiting time given survival to time s is the same as the original distribution.

Bayes' theorem posterior

The posterior density is proportional to the likelihood times the prior density; the normalizing constant makes it integrate to one.

Conjugate prior

A prior family such that the posterior belongs to the same family; for example, a Beta prior with binomial data yields a Beta posterior.

Delta method

A technique approximating the variance of a smooth function of an estimator by the squared derivative times the estimator's variance.