IAI CM1 key terms

The total amount an investment has grown to at a given time, including all interest credited up to that point. It is the future value of an initial capital under a stated interest basis.

The amount that must be invested today, at a stated interest rate, to produce a specified future payment. It discounts future cash flows back to the valuation date.

The rate actually earned or charged over a unit period, reflecting compounding once per period. It converts a nominal rate into the true growth over the period.

A quoted annual rate convertible m times per year, so interest is credited in m equal instalments. It is not the actual annual growth unless m equals one.

The instantaneous rate of growth of a fund, equal to the derivative of the log of the accumulation function. It links continuous compounding to effective and nominal rates.

The factor, usually written v, that converts a payment due one period ahead into its value today. It equals the reciprocal of one plus the effective interest rate.

The number of times per year interest is added to the fund balance. Higher frequency raises the effective rate for a given nominal rate.

A series of payments made for a fixed, known number of periods with no dependence on survival. Its value depends only on the payment pattern and the interest basis.

A level annuity whose payments fall at the end of each period. Its present value symbol is a-angle-n, and each payment is discounted one period less than the previous.

A level annuity whose payments fall at the start of each period. Each payment is discounted one period more than under an annuity-immediate, so its value is that annuity multiplied by one plus interest.

An annuity whose first payment is postponed for a stated number of periods. Its value equals the value of the corresponding immediate annuity discounted over the deferment period.

An annuity paid in infinitesimal instalments at a constant rate over a period. Its value is obtained by integrating the discount function, and it is the limit of ever more frequent payments.

A payment stream whose amounts rise by a fixed increment each period, often by one unit. Its value is derived by differentiating or summing the level annuity expression.

Any annuity whose payment amounts change over time according to a stated pattern. Increasing, decreasing and geometric patterns are common special cases.

An equation setting the present or accumulated value of payments received equal to that of payments made, at a common date. It is solved for an unknown rate, time or amount.

The interest rate at which the present value of a cash flow's inflows equals that of its outflows. It summarises the return on an investment as a single rate.

The present value of inflows minus the present value of outflows at a chosen interest rate. A positive value indicates the investment adds value at that rate.

Interest credited only on the original capital, never on interest already earned. The accumulation grows linearly in time and is common for short-term transactions.

Interest credited on both the original capital and previously credited interest. The accumulation grows geometrically, and the effective rate is constant over each period.

A period-by-period table splitting each loan payment into interest on the outstanding balance and capital repayment. It tracks the declining balance until the loan is repaid.

The amount of loan capital still owed at a given time. It can be found prospectively as the present value of remaining payments or retrospectively as the accumulated past payments less accumulated capital.

A fund into which regular deposits are made so that, with interest, it equals a target amount at a future date. It is often used to repay a loan's principal at maturity.

The pattern of spot rates across different terms to maturity at a given time. It underlies pricing of cash flows whose timing differs.

The annual yield on a zero-coupon investment maturing at a specified future date, applicable to a single payment at that date. It is used to discount cash flows occurring exactly then.

The rate agreed today for borrowing or lending over a future period. It is implied by the ratio of spot accumulation factors for the start and end of that period.

A function assigning a numerical value to each outcome of a random experiment. Its distribution describes the probabilities of the values it can take.

A function giving the probability that a random variable is at most a given value. It is non-decreasing, right-continuous, and tends to zero and one at the extremes.

For a continuous random variable, a non-negative function whose integral over an interval gives the probability of falling in that interval. Its total integral over the real line equals one.

The probability-weighted average of a random variable's possible values. For a continuous variable it is the integral of the value times its density.

The expected squared deviation of a random variable from its mean, measuring spread. It equals the second moment minus the square of the first moment.

The positive square root of the variance, expressed in the same units as the variable. It is often easier to interpret than the variance.

The ratio of the standard deviation to the mean, giving relative dispersion. It allows comparison of variability across variables with different scales.

The expected value of the exponential of a variable times a parameter, which generates moments through derivatives at zero. It uniquely determines a distribution when it exists in a neighbourhood of zero.

For a non-negative integer-valued variable, the expected value of a parameter raised to the power of the variable. Its derivatives at one yield factorial moments.

A result stating that the standardised sum of many independent, identically distributed variables tends to a standard normal distribution as the count grows. It justifies normal approximations for aggregate claims.

A result stating that the sample mean of independent, identically distributed variables converges to the population mean as the sample size grows. It underpins pooling of independent risks.

A symmetric bell-shaped continuous distribution defined by a mean and variance. Standardising by subtracting the mean and dividing by the standard deviation gives the standard normal.

A discrete distribution for counts of events occurring independently at a constant average rate. Its mean and variance are equal, and it is often used to model claim numbers.

A model for aggregate claims in which the number of claims follows a Poisson distribution and individual claim sizes are independent of it. Its mean and variance follow from the frequency and severity moments.

A framework modelling total claims from a portfolio as a random sum of individual claim amounts, with claim counts independent of severities. It is used to study aggregate loss distributions.

A recursive formula computing the distribution of aggregate claims when the claim-count distribution satisfies a linear recurrence. It avoids direct convolution of severity distributions.