Actuarial notation

Actuarial notation is a standardized shorthand system of symbols used by actuaries to express mathematical formulas and quantities related to interest rates, mortality probabilities, survival functions, and financial contingencies in insurance, pensions, and annuities.^[1] This notation enables concise representation of present values, expected lifetimes, premiums, and reserves, facilitating efficient computation and communication in actuarial practice.^[2] The system originated from George King’s Key to the Notation in the Institute of Actuaries' Text-Book, Part II (Life Contingencies), and was first adopted as an international standard at the Second International Actuarial Congress in London in 1898.^[3] It underwent revisions proposed by an international committee at the Eleventh Congress in Paris in 1937, with final adoption in 1950 following wartime delays, replacing symbols like j^{(m)} with i^{(m)} for nominal interest rates and introducing notations such as \ddot{a} for annuities-due.^[3] This revised notation was embraced by major bodies including the Society of Actuaries, the Institute of Actuaries in England, and the Faculty of Actuaries in Scotland, promoting uniformity in global actuarial work.^[3] Key components of actuarial notation include symbols for compound interest (e.g., i for effective interest rate, v = 1/(1+i) for discount factor), life tables (e.g., l_x for number of lives at age x, {}_tp_x for probability of surviving t years from age x), annuities (e.g., a_{\overline{n}|} for present value of an immediate annuity of 1 per year for n years, \ddot{a}_x for whole-life annuity-due), and life insurances (e.g., A_x for net single premium of whole-life insurance, A_{x:\overline{n}|}^1 for term insurance).^[1] It supports both discrete and continuous models, often incorporating assumptions like uniform distribution of deaths, and extends to multi-state transitions and multiple decrements in modern applications such as reserves and gross premiums.^[2] Commutation functions like D_x = v^xl_x and N_x = \sum_{k=0}^\infty D_{x+k} further aid practical calculations of premiums and reserves.^[1]

General Conventions

Core Symbols and Usage

Actuarial notation uses base symbols to represent fundamental quantities. For annuities, a denotes the present value of an annuity-immediate (payments at end of period), while \ddot{a} (with double dots) denotes an annuity-due (payments at beginning of period). For life insurances, A represents the present value of a benefit payable at the end of the year of death, and \bar{A} (with bar) for payment at the moment of death in continuous models. Interest-related symbols include i for the effective annual interest rate and v = 1/(1+i) for the discount factor. Accumulation functions use s for future value. These core symbols form the basis for more complex expressions and are modified by indices and accents to specify terms, contingencies, and payment timings.^[3]

Indexing and Modification Rules

In actuarial notation, temporal indices modify base symbols to specify durations or perpetual terms. A subscript n attached to the right of a core symbol, such as a_{\overline{n}|}, denotes an n-year temporary annuity, where payments cease after n years or upon death if applicable.^[3] An overline above the symbol, as in \bar{a}, indicates a continuous perpetuity, representing an annuity with payments extending indefinitely.^[4] Accents and bars further adapt symbols to distinguish payment timing and continuity. A double accent, typically double dots (e.g., \ddot{a}_{x:\overline{n}|}), signifies a temporary life annuity-due payable at the beginning of each period for up to n years contingent on the life aged x.^[3] In contrast, a single bar over the symbol, such as \bar{A}_x, denotes continuous payment or benefit, as in the present value of a whole life assurance payable immediately upon death for a life aged x.^[4] Modifications for payment frequency use a superscripted subscript (m) to indicate m-thly payments, placed to the right of the core symbol; for example, a^{(m)} represents an annuity with payments made m times per year.^[5] This modifier applies to both discrete and continuous base forms, adjusting the valuation for more frequent compounding or disbursements. When combining modifiers, a logical precedence governs their order to ensure unambiguous parsing, typically starting with left subscripts for deferral or duration, followed by the core symbol, right subscripts for age or term, accents or bars for timing, and finally frequency superscripts. For instance, _{n}E_x parses as an n-year pure endowment for life aged x, where the leading subscript n indicates the deferred term before the endowment symbol E qualified by age x.^[3] This hierarchy, rooted in the International Actuarial Notation, prioritizes temporal constraints before contingent or payment details, facilitating complex expressions like temporary continuous insurances without ambiguity.^[4]

Interest Theory Notation

Interest and Discount Rates

In actuarial notation for deterministic interest theory, the effective annual interest rate, denoted by i, represents the interest earned over one year on an investment of 1, expressed as a decimal such that the accumulated value at the end of the year is $1 + i. This rate assumes annual compounding and serves as the foundational measure for calculating present and future values in financial contexts.^[6] The nominal interest rate, denoted by i^{(m)}, is the annual rate quoted with compounding m times per year, where m is a positive integer such as 4 for quarterly or 12 for monthly compounding. The equivalent effective annual rate corresponding to this nominal rate is given by (1 + \frac{i^{(m)}}{m})^m - 1, which aligns the nominal rate with the effective rate i. Conversely, to express the effective rate i in nominal terms compounded m times per year, the formula is i^{(m)} = m \left[ (1 + i)^{1/m} - 1 \right].^[6]^[7]^[8] The force of interest, denoted by \delta, is the instantaneous rate of interest, representing the continuous compounding limit as m \to \infty. For a constant force, it relates to the effective annual rate by \delta = \ln(1 + i), and the accumulation function over time t is e^{\delta t}.^[6]^[9] Discount factors in actuarial notation use v = \frac{1}{1 + i} to denote the present value of 1 due in one year under the effective rate i. For multiple periods, this extends to v^t = \frac{1}{(1 + i)^t} for the present value of 1 due in t years, facilitating calculations for deferred payments.^[6] Accumulation notation includes s_{\bar{n}|}, which represents the accumulated value at the end of n years of an annuity-immediate paying 1 at the end of each year, under effective rate i. The formula for this is s_{\bar{n}|} = \frac{(1 + i)^n - 1}{i}.^[6]^[10] As per general conventions, accumulation symbols like s_{\bar{n}|} employ uppercase letters to denote future values.^[6] For example, if the accumulated value factor s_{\bar{5}|} = 5.5256 at some rate i, solving \frac{(1 + i)^5 - 1}{i} = 5.5256 yields i \approx 0.05 or 5%, illustrating how interest rates can be derived from given accumulation factors.^[10]

Accumulation and Present Value Functions

In actuarial notation, accumulation and present value functions quantify the time value of money for fixed payment streams known as annuities certain, which are non-contingent on survival and assume payments occur at regular intervals under a given interest rate i or discount factor v = 1/(1+i). These functions extend the basic interest and discount rates by applying them to multi-period cash flows, enabling the valuation of savings, loans, and retirement plans. The notations distinguish between immediate payments (at the end of each period), due payments (at the beginning), continuous payments, and variations like deferred, increasing, or decreasing streams.^[3]^[7] The present value of an n-year annuity certain immediate, denoted a_{\bar{n}|}, represents the discounted value of n unit payments made at the end of each year. It is calculated as a_{\bar{n}|} = \sum_{k=1}^n v^k = \frac{1 - v^n}{i}. For an annuity due, denoted \ddot{a}_{\bar{n}|}, payments occur at the beginning of each year, yielding \ddot{a}_{\bar{n}|} = \sum_{k=0}^{n-1} v^k = \frac{1 - v^n}{d}, where d = i/(1+i) = 1 - v is the discount rate. In continuous time, the present value \bar{a}_{\bar{n}|} assumes moment-by-moment payments at rate 1, given by \bar{a}_{\bar{n}|} = \int_0^n v^t \, dt = \frac{1 - v^n}{\delta}, where \delta = \ln(1+i) is the force of interest.^[7]^[3] The accumulated value at the end of n years for an annuity immediate, denoted s_{\bar{n}|}, is the future value of n unit payments made at the end of each year, compounded forward: s_{\bar{n}|} = \sum_{k=0}^{n-1} (1+i)^{n-k} = \frac{(1+i)^n - 1}{i}. For an annuity due, \ddot{s}_{\bar{n}|} = (1+i) s_{\bar{n}|} = \frac{(1+i)^{n+1} - (1+i)}{i}. Deferred versions postpone the annuity; for example, the present value of an m-year annuity immediate deferred n years is _{n|}a_{\bar{m}|} = v^n a_{\bar{m}|}, representing no payments for the first n years followed by the standard annuity.^[7]^[3] Increasing and decreasing annuities certain adjust payments arithmetically over time. The present value of an increasing annuity immediate, (Ia)_{\bar{n}|}, involves payments of 1 at the end of year 1, 2 at the end of year 2, up to n at the end of year n, given by (Ia)_{\bar{n}|} = \sum_{k=1}^n k v^k. A closed-form expression is (Ia)_{\bar{n}|} = \frac{\tilde{s}_{\bar{n}|} - n v^n}{i}, where \tilde{s}_{\bar{n}|} is the accumulated value at time n of the increasing payments, \tilde{s}_{\bar{n}|} = \sum_{k=1}^n k (1+i)^{n-k} = \frac{(1+i)^{n+1} - (n+1)(1+i) + n}{i^2}. An equivalent form is (Ia)_{\bar{n}|} = \frac{\ddot{a}_{\bar{n}|} - n v^n}{i}. For a decreasing annuity immediate, (Da)_{\bar{n}|} = \sum_{k=1}^n (n-k+1) v^k = n a_{\bar{n}|} - (Ia)_{\bar{n}|}. These notations support valuations for escalating benefits or amortizing loans.^[7]^[11]^[3] Perpetuities extend annuities indefinitely. The present value of a perpetuity immediate is a_{\infty} = \sum_{k=1}^\infty v^k = \frac{1}{i}, while the continuous version is \bar{a} = \int_0^\infty v^t \, dt = \frac{1}{\delta}. The due perpetuity is \ddot{a}_{\infty} = \frac{1}{d}. These are used for infinite-horizon investments like endowments.^[7]^[3] To illustrate the difference between due and immediate annuities, consider an effective annual interest rate of 9% (i = 0.09, v \approx 0.9174). The present value of a 5-year annuity immediate of $100 per year is

a_{\bar{5}|} = \frac{1 - v^5}{i} \approx 3.8897 \times 100 = \&#36;388.97

. For the same payments as an annuity due,

\ddot{a}_{\bar{5}|} = (1+i) a_{\bar{5}|} \approx 1.09 \times 3.8897 \times 100 = \&#36;424.00

, reflecting the earlier timing of payments. Another example at 5% interest (i = 0.05) for a 15-year annuity immediate of $7,000 per year yields

a_{\bar{15}|} \approx 10.3797 \times 7,000 = \&#36;72,657.61

; for the annuity due, (\ddot{a}{\bar{15}|} = (1+i) a{\bar{15}|} \approx 10.8987 \times 7,000 = $76,290.90.^[7]

Survival and Mortality Notation

Life Table Symbols

Life tables in actuarial science provide a structured representation of mortality and survival patterns within a population, typically based on discrete age intervals. These tables form the foundation for calculating probabilities and expectations related to human longevity, using standardized symbols to denote key quantities derived from observed or projected death rates. The core notations are universally adopted in actuarial practice to ensure consistency across calculations involving life contingencies.^[12] The symbol l_x represents the number of individuals surviving to exact age x in a hypothetical cohort, often starting with a radix such as l_0 = 100,000 for computational convenience. The number of deaths between ages x and x+1, denoted d_x, is calculated as d_x = l_x - l_{x+1}. From these, the one-year survival probability p_x is defined as p_x = \frac{l_{x+1}}{l_x}, while the mortality rate q_x is q_x = 1 - p_x = \frac{d_x}{l_x}. These symbols allow actuaries to construct and interpret survival data directly from mortality inputs.^[12]^[1] The expectation of life at age x, denoted \stackrel{\circ}{e}_x, measures the average remaining lifetime and can be expressed in curtate (whole years) or complete forms. The curtate expectation is

e_x = \sum_{k=1}^\infty \, _k p_x

, where _k p_x = \frac{l_{x+k}}{l_x} is the probability of surviving k years from age x. The complete expectation \stackrel{\circ}{e}_x incorporates fractional years, often approximated as e_x + 0.5 under uniform distribution of deaths, though more precise continuous models may be used.^[1] Commutation functions simplify the computation of present values by incorporating both survival data and the discount factor v = \frac{1}{1+i}, where i is the interest rate. The basic function D_x = l_x v^x discounts the survivors at age x to present value. The annuity commutation N_x = \sum_{k=x}^\infty D_k sums these from age x onward. For mortality-related payments, C_x = v^{x+1} d_x discounts deaths in the year following age x, and M_x = \sum_{k=x}^\infty C_k accumulates these values. These functions precompute summations to avoid repetitive calculations in life table applications.^[1] In select and ultimate mortality tables, distinctions are made between recently selected lives (e.g., insured individuals) and the general population. Select notation uses square brackets, such as l_{} for survivors among lives selected at age x, and p_{} = \frac{l_{+1}}{l_{}} for the one-year survival probability post-selection. After a selection period (typically 2–3 years), mortality transitions to ultimate rates denoted without brackets, like p_{x+t} for attained age x+t. This accounts for temporary mortality improvements due to selection effects.^[13] For example, consider constructing a simple life table from given mortality rates q_x. Start with l_0 = 100,000, then l_1 = l_0 (1 - q_0), d_0 = l_0 q_0, and iteratively compute subsequent values using l_{x+1} = l_x (1 - q_x) and p_x = 1 - q_x. If q_0 = 0.005, then l_1 = 99,500, d_0 = 500, and p_0 = 0.995; continuing this process yields the full table for further actuarial use.^[12]

Force of Mortality and Probabilities

The force of mortality, denoted by \mu_x, represents the instantaneous rate at which mortality occurs at exact age x, formally defined as the limit \mu_x = \lim_{h \to 0^+} \frac{{}_h q_x}{h}, where {}_h q_x is the probability of death within h years for a life aged x.^[14] This hazard rate is closely tied to the life table function l_x, the number of survivors to age x, through the differential equation \frac{d}{dx} l_x = -\mu_x l_x, which implies \mu_x = -\frac{d}{dx} \ln l_x.^[14] The survival probability {}_t p_x, the probability that a life aged x survives an additional t years, is derived directly from the force of mortality as {}_t p_x = \exp\left( -\int_0^t \mu_{x+s} \, ds \right).^[14] This expression highlights the force's role in continuous-time mortality modeling, where the integrated force over the interval determines the cumulative hazard. In discrete contexts, such as one-year survival probabilities p_x and death probabilities q_x = 1 - p_x, the force provides a continuous approximation; specifically, the central mortality rate m_x, defined as the average annual death rate over age x to x+1, satisfies m_x \approx \int_0^1 \mu_{x+t} \, dt \approx \mu_{x+0.5}, and q_x = 1 - \exp\left( -\int_0^1 \mu_{x+t} \, dt \right) \approx 1 - \exp(-\mu_x) when \mu_x is small or constant over the year.^[14]^[14] Several parametric models for the force of mortality are commonly used in actuarial practice to fit empirical data and project future survival. The constant force model assumes \mu_x = \mu for all x, leading to exponential survival probabilities {}_t p_x = e^{-\mu t}, which implies memoryless mortality suitable for certain homogeneous populations.^[15] De Moivre's law models uniform distribution of deaths up to a limiting age \omega, yielding \mu_x = \frac{1}{\omega - x} for x < \omega, which simplifies calculations for deterministic lifetimes.^[16] The Gompertz law, a seminal exponential model introduced in 1825, posits \mu_x = B c^x where B > 0 and c > 1, capturing the accelerating mortality observed in adult human populations; parameters are typically estimated from life table data, with c \approx 1.09 in modern fits.^[14]^[17] Extensions of survival probabilities accommodate deferred, temporary, and multi-life scenarios. A deferred survival probability, such as the probability of surviving from age x+n to x+n+t given survival to x, is denoted {}_n|_t p_x = \frac{{}_{n+t} p_x}{{}_n p_x}, representing survival over a future interval conditional on an initial deferral period.^[2] For multiple lives, joint survival notation like p_{xy} denotes the probability that both lives aged x and y survive one year, often under independence assumptions where p_{xy} = p_x p_y, though dependent models adjust for correlations in joint forces of mortality.^[17] To illustrate, consider a life aged 50 with force of mortality \mu_{50+t} = 0.04 + 0.001 t for $0 \leq t \leq 10. The 5-year survival probability is computed as {}_5 p_{50} = \exp\left( -\int_0^5 (0.04 + 0.001 s) \, ds \right) = \exp\left( -[0.04 s + 0.0005 s^2]_0^5 \right) = \exp(-0.2125) \approx 0.808, demonstrating how the integral aggregates varying instantaneous risks over time.

Life Contingent Payments

Annuities and Their Values

In actuarial notation, life-contingent annuities represent the present value of periodic payments made only while the annuitant survives, incorporating both interest discounting and survival probabilities. These notations extend the basic annuity symbols from interest theory by appending life contingencies, such as the age x of the annuitant and survival terms like _t p_x, the probability that a life aged x survives t years.^[2] The whole life annuity-immediate, denoted a_x, provides payments of 1 at the end of each year indefinitely, contingent on survival, with its actuarial present value given by

a_x = \sum_{k=1}^\infty v^k \, _k p_x,

where v = 1/(1+i) is the discount factor and i is the effective annual interest rate.^[2] The corresponding continuous whole life annuity, \bar{a}_x, assumes payments at a continuous rate of 1 per year and is expressed as

\bar{a}_x = \int_0^\infty v^t \, _t p_x \, dt,

using continuous discounting v^t = e^{-\delta t} where \delta = \ln(1+i) is the force of interest.^[2] For a limited duration, the temporary life annuity-immediate a_{x:\bar{n}|} pays 1 at the end of each of the next n years if the annuitant survives, with present value

a_{x:\bar{n}|} = \sum_{k=1}^n v^k \, _k p_x.

^[2] The temporary life annuity-due \ddot{a}_{x:\bar{n}|} is the present value of payments of 1 at the beginning of each of the next n years, contingent on survival:

\ddot{a}_{x:\bar{n}|} = \sum_{k=0}^{n-1} v^k \, _k p_x.

^[2] Increasing annuities scale payments with time to account for inflation or other growth; the whole life increasing annuity-immediate (Ia)_x pays k at the end of year k if alive, with present value

(Ia)_x = \sum_{k=1}^\infty k \, v^k \, _k p_x.

^[2] This can be computed efficiently using commutation functions, where D_x = v^xl_x, N_x = \sum_{j=x}^\omega D_j, and M_x = \sum_{j=x}^\omega C_j with C_j = v^{j+1} d_j and d_j the deaths in year j, giving

(Ia)_x = \frac{N_x - M_x}{D_x i}.

^[18] Joint life annuities involve multiple lives; the joint life annuity-immediate a_{xy} for lives aged x and y pays 1 annually at year-end while both survive, with present value

a_{xy} = \sum_{k=1}^\infty v^k \, _k p_{xy}

, where _k p_{xy} is the joint survival probability.^[2] The last survivor annuity \bar{a}_{xy}, paying while at least one survives, satisfies

\bar{a}_{xy} = a_x + a_y - a_{xy}.

^[2] To illustrate, consider valuing a 10-year temporary life annuity-immediate for a life aged 50 using illustrative life table values where i = 0.05 so v \approx 0.9524, and _k p_{50} from a standard table: 0.995 (k=1), 0.989 (k=2), ..., decreasing to 0.850 (k=10). The present value is computed as

a_{50:\bar{10}|} = \sum_{k=1}^{10} v^k \, _k p_{50} \approx 7.85

, reflecting the discounted expected payments.^[2]

Life Insurances and Benefits

Life insurance benefits in actuarial notation primarily describe the present value of lump-sum payments contingent on the timing of death, distinct from survival-based annuities. These notations facilitate the calculation of expected present values (EPVs) for policies such as whole life, term, and endowment insurances, incorporating discounting via the factor v = (1+i)^{-1} and mortality probabilities like _k p_x (probability of survival from age x to x+k) and _k| q_x (deferred mortality probability). Standard symbols follow the International Actuarial Notation, approved by actuarial societies in 1949 and widely adopted since.^[3] The whole life assurance, denoted A_x, represents the EPV of a unit benefit payable at the end of the year of death for a life aged x. In discrete form, it is given by

A_x = \sum_{k=0}^{\infty} v^{k+1} \, _k| q_x,

where the sum accounts for possible payment timings weighted by death probabilities and discounted values. The continuous analog, \bar{A}_x, assumes payment at the moment of death and integrates over time:

\bar{A}_x = \int_0^{\infty} v^t \, _t p_x \mu_{x+t} \, dt,

with \mu_{x+t} as the force of mortality; this form is useful for models assuming constant force, yielding \bar{A}_x = \mu / (\delta + \mu) where \delta = -\ln v is the force of interest.^[8]^[19] Term assurance, A_{x:\bar{n}|^1}, covers death within n years, paying at the end of the year of death if it occurs before age x+n. Its EPV is

A_{x:\bar{n}|^1 = \sum_{k=0}^{n-1} v^{k+1} \, _k| q_x,

truncating the whole life sum at n. The pure endowment, _{n}E_x, provides a unit benefit only if survival to n years occurs, with

_{n}E_x = v^n \, _n p_x.

These components combine in endowment insurance, A_{x:\bar{n}|}, which pays on death within n years or at n if alive:

A_{x:\bar{n}|} = A_{x:\bar{n}|^1} + _{n}E_x.

Commutation functions simplify computations using life tables; for whole life, A_x = M_x / D_x, where D_x = v^x l_x (discounted survivors) and M_x = \sum_{k=x}^{\omega-1} C_{k+1} with C_k = v^{k} d_{k-1} (discounted deaths), \omega the limiting age, l_x survivors, and d_x deaths. For example, using a table with i=0.05, l_{30}=10000, and derived values, A_{30} \approx 0.162 illustrates policy valuation scale.^[8]^[3] Joint life insurances extend to multiple lives. The symbol A_{xy} denotes the EPV for a unit benefit on the first death of lives aged x and y, payable at year-end:

A_{xy} = \sum_{k=0}^{\infty} v^{k+1} \, _k p_{xy} q_{xy+k},

using joint survival _k p_{xy} and decrement q_{xy+k}. For last survivor status, payable on the second death, the notation \bar{A}_{xy} (or A_{\bar{xy}} in some texts) captures the EPV, often derived as A_{\bar{xy}} = A_x + A_y - A_{xy}. These are applied in policies like joint-and-survivor benefits.^[8]^[19]

Premium Calculations

In actuarial notation, premium calculations for life contingent payments involve determining the net premiums required to fund insurance benefits and annuities on an equivalence principle basis, where the expected present value of premiums equals that of benefits. The net single premium (NSP), denoted as a single upfront payment, represents the actuarial present value of the expected benefits. For a whole life insurance policy paying a benefit of 1 upon death for a life aged x, the NSP is given by P = A_x, where A_x is the present value of the whole life insurance benefit.^[1] Similarly, for a temporary life annuity-due, the NSP (single premium to fund payments of 1 per year) is \ddot{a}_{x:\bar{n}|}. For an n-year endowment insurance (benefit 1 on death within n years or at n if surviving), the NSP is A_{x:\bar{n}|}.^[4] Net annual premiums (NAP) extend this to level periodic payments, typically annual, over the policy term or life. For whole life insurance, the NAP is \Pi = \frac{A_x}{\ddot{a}_x}, where \ddot{a}_x is the present value of a whole life annuity-due of 1 per year; this divides the NSP by the annuity factor to spread the cost.^[20] For policies with limited payment periods, such as an m-year limited payment whole life insurance, the NAP is denoted \Pi^{(m)} = \frac{A_x}{\ddot{a}_{x:\bar{m}|}}, concentrating higher annual payments over m years to fully fund the lifetime benefit.^[1] Gross premiums incorporate loadings for expenses and profit, expressed as G = \Pi + loading, though net premiums \Pi_x form the core of pure risk funding without such additions.^[4] Reserves, or policy values, ensure solvency by quantifying the insurer's liability at intermediate durations. The prospective reserve at duration t for a whole life policy, denoted _{t}V_x, is the present value of future benefits minus future premiums:

_{t}V_x = A_{x+t} - \Pi \ddot{a}_{x+t}.

This forward-looking measure balances ongoing obligations against remaining premium inflows.^[20] The retrospective reserve, alternatively, accumulates past premiums net of past benefits and equals the prospective under equivalence; for an n-year term policy at duration t, it is ^{t}V_x = A_{x+t:\bar{n-t}|}^1 - \Pi \ddot{a}_{x+t:\bar{n-t}|}.^[4] As a representative example, consider calculating the NAP for a 20-payment whole life policy on a life aged x. The premium \Pi^{(20)} is computed as \frac{A_x}{\ddot{a}_{x:\bar{20}|}}, using mortality assumptions from a life table and an interest rate to evaluate A_x (future death benefits) and \ddot{a}_{x:\bar{20}|} (20-year temporary annuity-due); this yields a level annual payment sufficient to fund the lifetime insurance without further contributions after year 20.^[1]