Actuarial present value

In actuarial science, an actuarial present value can be defined as the present value of a contingent event. In the field of life insurance, one can think of this as the market value of an insurance policy given some interest rate. The calculation of an actuarial present value draws from the theories of expected values, present values and interest theory, one must have a strong understanding of all concepts in order to calculate an actuarial present value. Implicit in all these factors is the evaluation of risk.

Life insurance

Let $T$ be the future lifetime random variable of an individual age x and $Z$ be the present value random variable of a whole life insurance benefit of 1 payable at the instant of death.

$\,Z=v^{T}=(1+i)^{-T}=e^{-\delta T}$

where i is the interest rate and δ is the equivalent force of interest.

To calculate the actuarial present value we need to calculate the expected value $\,E(Z)=E(v^{T})$ of this random variable Z. For someone aged x this is denoted as $\,{\overline {A}}_{x}\!$ in actuarial notation. It can be calculated as

$\,{\overline {A}}_{x}\!=E(v^{T})=\int _{0}^{\infty }v^{t}f_{T}(t)\,dt=\int _{0}^{\infty }v^{t}\,_{t}p_{x}\mu _{x+t}\,dt$

where $f_{T}$ is the probability density function of T, $\,_{t}p_{x}\!$ is the probability of a life age $x$ surviving to age $x+t$ and μ denotes force of mortality.

The actuarial present value of an n-year term insurance policy 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

$\,_{n}E_{x}=P(T>n)v^{n}=\,_{n}p_{x}v^{n}.$

In practice the best information available about the random variable T is drawn from life tables, which give figures by year. The actuarial present value of a benefit of 1 payable at the birthday after death would be

$\,A_{x}=\sum _{k=0}^{\infty }v^{k+1}P(k<T<k+1)=\sum _{k=0}^{\infty }v^{k+1}\,_{k}p_{x}q_{x+k}$

where $\,q_{x}\!$ is the probability of death between the ages of $x$ and $x+1$.

In practice an insurance policy pays soon after death, which requires an adjustment of the formula.

Life annuity

The actuarial present value of a life annuity of 1 per year paid continuously can be found in two ways:

Aggregate payment technique (taking the expected value of the total present value):

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 the life annuity of 1 per year paid continuously as long as the person is alive, and is given by:

$Y=a_{\overline {T|}}={\frac {1-(1+i)^{-T}}{\delta }}={\frac {1-v^{T}}{\delta }}.$

The expected value of Y is:

$\,{\overline {a}}_{x}=\int _{0}^{\infty }a_{\overline {t|}}f_{T}(t)\,dt=\int _{0}^{\infty }a_{\overline {t|}}\,_{t}p_{x}\mu _{x+t}\,dt$

Current payment technique (taking the total present value of the function of time representing the expected values of payments):

$\,{\overline {a}}_{x}=\int _{0}^{\infty }v^{t}(1-F_{T}(t))\,dt=\int _{0}^{\infty }v^{t}\,_{t}p_{x}\,dt\,$

where F(t) is the cumulative distribution function of the random variable T.

The equivalence follows also from integration by parts.

In practice life annuities are not paid continuously. The actuarial present value calculated as if the payments were made at the end of each year would be given by

$a_{x}=\sum _{k=1}^{\infty }v^{k}(1-F_{T}(k))=\sum _{k=1}^{\infty }v^{k}\,_{k}p_{x}.$

This present value is smaller due to two effects: the payments are made later, and there is no payment in the year of death instead of continuously until the day of death.

See also

• Actuarial science
• Actuarial notation
• Actuarial reserve
• Actuary
• Force of mortality
• Life table
• Present value

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