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 stong 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 life insurance policy. To calculate the actuarial present value of a whole life insurance policy we need to calculate the expected value of the random variable Z. In the context of a whole life insurance policy:


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

where i is the interest rate and &delta; is the equivalent force of interest.

To find the actuarial present value of someone aged x, we need to find $$\,E(Z)=E(v^T)$$; this is denoted as $$\,\overline{A}_x\!$$ in actuarial notation. In actuarial notation the actuarial present value can be calculated as


 * $$\,\overline{A}_x\! = \int_0^\infty v^t\,_tp_x\mu_{x+t}\,dt$$

which, in statistical notation, is equivalent to


 * $$ E(v^T) = \int_0^\infty v^t f_T(t)\,dt$$

Here &mu; 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 policy can be found as


 * $$\,_nE_x = P(T>n)v^n = \,_np_xv^n. $$

In practise the best information available about the random variable T is drawn from life tables, in this case we only know the probabilities of an individual dying in a specific year and need to use the probabilities from the life tables to find the actuarial present value. The actuarial present value of a whole life policy using a life table would be


 * $$\,A_x = \sum_{k=0}^\infty v^{k+1} \,_kp_xq_{x+k}$$

or in statistical notation


 * $$\,A_x = \sum_{k=0}^\infty v^{k+1} P(k<T<k+1).$$

The above calculations give us the actuarial present value of an insurance policy as if the benefit were paid at the end of the year of death. This is not how an insurance policy works in practice, so it is often necessary to convert the actuarial present value paid at the end of the year to an actuarial present value paid at the moment of death by using a fractional age assumption.

Life annuities
The actuarial present value of a life annuity is found similarly to the actuarial present value of a life insurance policy. This time the random variable Y is the present value random variable of the life annuity and is given by:


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

So the present value of a life annuity paid continuously is:


 * $$\,\overline{a}_x = \int_0^\infty a_{\overline{t|}} \,_tp_x\mu_{x+t}\,dt$$

which in statistical notation is equivalent to

$$\,\overline{a}_x = \int_0^\infty a_{\overline{t|}} f_T(t)\,dt.$$

This is called the aggregate payment technique of calculating the actuarial present value of a life annuity. Integrating by parts it can be shown that this statement is equivalent to


 * $$\,\overline{a}_x = \int_0^\infty v^{t} \,_tp_x\,dt\,=\int_0^\infty v^{t} (1-F_T(t))\,dt$$

Where F(t) is the cumulative distribution function of the random variable T. This is called the current payment technique of calculating the actuarial present value.

Since in practise life annuities are not paid continuously it is necessary to calculate the actuarial present value as if the payment were made at the start of the year. This would be given by


 * $$\ddot{a}_x = \sum_{k=0}^\infty v^k \,_kp_x = \sum_{k=0}^\infty v^k (1-F_T(k)). $$