The actuarial present value (APV) is the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of the
present value
In economics and finance, present value (PV), also known as present discounted value (PDV), is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money ha ...
of a contingent
cash flow
Cash flow, in general, refers to payments made into or out of a business, project, or financial product. It can also refer more specifically to a real or virtual movement of money.
*Cash flow, in its narrow sense, is a payment (in a currency), es ...
stream (i.e. a series of payments which may or may not be made). Actuarial present values are typically calculated for the benefit-payment or series of payments associated with
life insurance
Life insurance (or life assurance, especially in the Commonwealth of Nations) is a contract
A contract is an agreement that specifies certain legally enforceable rights and obligations pertaining to two or more parties. A contract typical ...
and
life annuities. The probability of a future payment is based on assumptions about the person's future mortality which is typically estimated using a life table.
Life insurance
Whole life insurance pays a pre-determined benefit either at or soon after the insured's death. The symbol ''(x)'' is used to denote "a life aged ''x''" where ''x'' is a non-random parameter that is assumed to be greater than zero. The actuarial present value of one unit of whole life insurance issued to ''(x)'' is denoted by the symbol
or
in
actuarial notation
Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.
Traditional notation uses a halo system, where symbols are placed as superscript or subscript before ...
. Let ''G>0'' (the "age at death") be the
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
that models the age at which an individual, such as ''(x)'', will die. And let ''T'' (the future lifetime random variable) be the time elapsed between age-''x'' and whatever age ''(x)'' is at the time the benefit is paid (even though ''(x)'' is most likely dead at that time). Since ''T'' is a function of G and x we will write ''T=T(G,x)''. Finally, let ''Z'' be the present value random variable of a whole life insurance benefit of 1 payable at time ''T''. Then:
:
where ''i'' is the effective annual interest rate and δ is the equivalent
force of interest.
To determine the actuarial present value of the benefit we need to calculate the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of this random variable ''Z''. Suppose the death benefit is payable at the end of year of death. Then ''T(G, x) :=
ceiling(G - x)'' is the number of "whole years" (rounded upwards) lived by ''(x)'' beyond age ''x'', so that the actuarial present value of one unit of insurance is given by:
:
where
is the probability that ''(x)'' survives to age ''x+t'', and
is the probability that ''(x+t)'' dies within one year.
If the benefit is payable at the moment of death, then ''T(G,x): = G - x'' and the actuarial present value of one unit of whole life insurance is calculated as
:
where
is the
probability density function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
of ''T'',
is the probability of a life age
surviving to age
and
denotes
force of mortality at time
for a life aged
.
The actuarial present value of one unit of an ''n''-year term insurance policy payable at the moment of death can be found similarly by integrating from 0 to ''n''.
The actuarial present value of an n year pure
endowment insurance benefit of 1 payable after n years if alive, can be found as
:
In practice the information available about the random variable ''G'' (and in turn ''T'') may be drawn from life tables, which give figures by year. For example, a three year term life insurance of $100,000 payable at the end of year of death has actuarial present value
:
For example, suppose that there is a 90% chance of an individual surviving any given year (i.e. ''T'' has a
geometric distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
* The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \;
* T ...
with parameter ''p = 0.9'' and the set ' for its support). Then
:
and at interest rate 6% the actuarial present value of one unit of the three year term insurance is
:
so the actuarial present value of the $100,000 insurance is $24,244.85.
In practice the benefit may be payable at the end of a shorter period than a year, which requires an adjustment of the formula.
Life annuity
The actuarial present value of a
life annuity
A life annuity is an annuity, or series of payments at fixed intervals, paid while the purchaser (or annuitant) is alive. The majority of life annuities are insurance products sold or issued by life insurance companies. However, substantial cas ...
of 1 per year paid continuously can be found in two ways:
Aggregate payment technique (taking the expected value of the total
present value
In economics and finance, present value (PV), also known as present discounted value (PDV), is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money ha ...
):
This is similar to the method for a life insurance policy. This time the random variable ''Y'' is the total present value random variable of an annuity of 1 per year, issued to a life aged ''x'', paid continuously as long as the person is alive, and is given by:
:
where ''T=T(x)'' is the future lifetime random variable for a person age ''x''. The expected value of ''Y'' is:
:
Current payment technique (taking the total present value of the function of time representing the expected values of payments):
:
where ''F''(''t'') is the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ever ...
of the random variable ''T''.
The equivalence follows also from integration by parts.
In practice life annuities are not paid continuously. If the payments are made at the end of each period the actuarial present value is given by
:
Keeping the total payment per year equal to 1, the longer the period, the smaller the present value is due to two effects:
*The payments are made on average half a period later than in the continuous case.
*There is no proportional payment for the time in the period of death, i.e. a "loss" of payment for on average half a period.
Conversely, for contracts costing an equal lumpsum and having the same
internal rate of return
Internal rate of return (IRR) is a method of calculating an investment's rate of return. The term ''internal'' refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or fin ...
, the longer the period between payments, the larger the total payment per year.
Life assurance as a function of the life annuity
The APV of whole-life assurance can be derived from the APV of a whole-life annuity-due this way:
:
This is also commonly written as:
:
In the continuous case,
:
In the case where the annuity and life assurance are not whole life, one should replace the assurance with an n-year endowment assurance (which can be expressed as the sum of an n-year term assurance and an n-year pure endowment), and the annuity with an n-year annuity due.
See also
*
Actuarial science
Actuarial science is the discipline that applies mathematics, mathematical and statistics, statistical methods to Risk assessment, assess risk in insurance, pension, finance, investment and other industries and professions.
Actuary, Actuaries a ...
*
Actuarial notation
Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.
Traditional notation uses a halo system, where symbols are placed as superscript or subscript before ...
*
Actuarial reserve
*
Actuary
An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require investment management, asset management, ...
*
Life table
In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, the probability that a person of that age will die before their next birthday ("probability of death"). In ...
*
Present value
In economics and finance, present value (PV), also known as present discounted value (PDV), is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money ha ...
References
* Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J., Chapter 4-5
* Models for Quantifying Risk (Fourth Edition), 2011, By Robin J. Cunningham, Thomas N. Herzog, Richard L. London, Chapter 7-8
Applied mathematics
Actuarial science