IAI CM2 key terms

The probability that a newborn survives to age x, written S(x) = P(T0 > x). It decreases from 1 at age 0 toward 0 and underpins all life-contingent probabilities.

The instantaneous rate of death at age x given survival to that age, equal to minus the derivative of log S(x). It is the continuous-age analogue of qx and drives survival probabilities over short intervals.

The whole number of complete future years lived by a person aged x before death, so K(x) = floor(T(x)). Its distribution gives probabilities of dying in each future year, used to value annual payment contracts.

The exact future lifetime of a life aged x measured continuously. Probabilities such as tpx and tqx are derived from its distribution, and it is used for continuous-time insurance and annuity valuations.

qx is the probability a life aged x dies before age x+1; px = 1 - qx is the probability of surviving that year. Columns lx, dx and ex are built from these one-year rates.

The mean remaining lifetime of a person aged x, equal to the sum over future years of tpx (curtate version) or the integral of tpx over time (complete version). Reported as ex in life tables.

Deaths in an age interval divided by the average population exposed during that interval. It approximates the force of mortality and converts to qx through an exposure assumption such as mx/(1 + 0.5 mx).

The total time a group is observed and alive during a study period, allowing for entries and exits. Deaths divided by exposure gives an occurrence/exposure rate that estimates mortality.

Observed deaths in a study population divided by the deaths expected if standard age-specific rates applied. It summarises mortality relative to a reference, adjusting for differing age structures.

The process of smoothing crude age-specific mortality rates into a smooth, consistent set of rates suitable for actuarial use, balancing fidelity to the data against smoothness and plausibility.

A graduation method that minimises a weighted sum of squared deviations from crude rates plus a penalty on the roughness of fitted rates. A parameter controls the smoothness-versus-fit trade-off.

A parametric mortality model where the force of mortality grows exponentially with age, μ(x) = Bc^x. It fits adult mortality well and underlies several graduation and projection methods.

The expected present value of the death benefit, computed by discounting the benefit at the time of death and weighting by the probability of death at that time, using an assumed interest rate.

A contract paying a sum on death whenever it occurs. Its expected present value Ax equals the sum over all future years of discounted death probabilities, and it can be related to annuity values.

A life annuity paying 1 at the start of each year while the person aged x is alive. Its expected present value sums discounted survival probabilities and links to Ax via the identity 1 = d äx + Ax.

The level premium whose expected present value equals the expected present value of benefits, ignoring expenses and profit. It follows from the equivalence principle applied to the contract at outset.

The rule that premiums are set so the expected present value of premiums equals the expected present value of benefits plus expenses. It is the standard basis for calculating net and gross premiums.

The actual premium charged, set so expected premium income covers expected benefits plus expenses and a profit margin. It exceeds the net premium by the expense and contingency loadings.

The amount an insurer should hold at a given duration to meet future obligations, usually the expected present value of future benefits minus future net premiums. It reflects the retrospective accumulation of the contract.

A differential equation describing how the policy value changes continuously with duration, driven by the interest spread, the mortality cost of the sum at risk, and premium income. It links reserves across time.

A retirement arrangement where the benefit is set by a formula, typically linked to salary and years of service. The employer bears investment and longevity risk, and actuarial valuations assess funding.

A retirement arrangement where contributions are fixed and the final benefit depends on accumulated fund value at retirement. The member bears investment risk, and no promise of a benefit level is made.

The portion of a member's pension entitlement earned to date, based on service and salary up to the valuation date. It is the key liability measure in pension scheme funding valuations.

An actuarial cost method that values pension liabilities by projecting final salary, then allocating benefit accrual to each year of service. It produces a service cost and an accrued liability.

A model in which the same inputs always produce the same output, with no random variation. It is useful for projecting central estimates but cannot quantify the spread of possible outcomes.

A model that incorporates random variation, so outputs are probability distributions rather than single values. It allows quantification of risk and uncertainty, for example via simulation.

A technique that generates many random scenarios from assumed distributions and runs the model on each, producing an empirical distribution of outcomes for risk assessment and pricing.

Evaluating a model under a set of specified adverse or plausible scenarios to see how results respond. It tests sensitivity and robustness rather than producing probability-weighted outcomes.

A continuous-time process W(t) with independent, stationary normal increments, W(0)=0, mean zero and variance t. It is the fundamental building block of continuous-time stochastic models.

A process whose logarithm follows Brownian motion with drift, so the process itself stays positive. It is the standard model for asset prices in the Black-Scholes framework.

A chain rule for stochastic processes that corrects the ordinary calculus expansion by a second-order term involving the quadratic variation. It is used to derive dynamics of functions of diffusions.

An equation expressing the change in a process as a drift term plus a diffusion term driven by Brownian motion. Its solution describes the evolution of quantities such as asset prices or interest rates.

A process whose expected future value, given all information up to now, equals its current value. It formalises fair-game behaviour and is central to no-arbitrage pricing.

An integral of a process with respect to Brownian motion, defined as a limit of sums and generally having mean zero. It allows construction of martingales and models of accumulated noise.

A probability measure under which discounted asset prices are martingales and which is equivalent to the real-world measure. Its existence rules out arbitrage and gives unique prices in complete markets.

The principle that an asset's price equals the expected discounted payoff under the equivalent martingale measure. It avoids estimating real-world expected returns and underlies derivative pricing.

A closed-form expression for the price of a European call or put on a non-dividend-paying stock, depending on stock price, strike, volatility, time to expiry and risk-free rate.

A no-arbitrage relationship linking European call and put prices with the same strike and expiry: call minus put equals the stock price minus the discounted strike. It holds independently of any pricing model.

Sensitivities of an option price to underlying price, its change, volatility and time. Delta and gamma measure price exposure, vega measures volatility sensitivity and theta measures time decay.

A model describing the instantaneous risk-free interest rate as a stochastic process, used to price interest-rate derivatives and to generate the term structure of yields.

A short rate model in which the rate mean-reverts toward a long-term level with constant volatility, so rates can become negative. It yields closed-form bond prices and is analytically tractable.

The relationship between yield and time to maturity for default-free bonds, shown as a yield curve. It reflects expectations, risk premia and market conditions and is central to valuation.