Actuarial reserves

An actuarial reserve is a liability equal to the present value of the future expected cash flows of a contingent event. In the insurance context an actuarial reserve is the present value of the future cash flows of an insurance policy and the total liability of the insurer is the sum of the actuarial reserves for every individual policy. An insurer must keep offsetting assets to pay off this future liability.

The Loss Random Variable
The loss random variable is the starting point in the determination of any type of actuarial reserve calculation. Define $$K(x)$$ to be the future curtate lifetime random variable of a person aged x. Then, for a death benefit of one dollar and premium $$P$$, the loss random variable, $$L$$, can be written in actuarial notation as a function of $$K(x)$$


 * $$ L = v^{K(x)+1} - P\ddot{a}_{\overline{K(x)+1}|}$$

From this we can see that the present value of the loss to the insurance company now if the person dies in t years, is equal to the present value of the death benefit minus the present value of the premiums.

The loss random variable described above only defines the loss at issue. For K(x)>t, the loss random variable at time t can be defined as:


 * $$ {}_{t}L = v^{K(x)+1-t} - P\ddot{a}_{\overline{K(x)+1-t|}}$$

Net Level Premium Reserves
Net level premium reserves, also called benefit reserves, only involve two cash flows and are used for some USGAAP reporting purposes. The valuation premium in a NLP reserve is a premium such that the value of the reserve at time zero is equal to zero. The net level premium reserve is found by taking the expected value of the loss random variable defined above. They can be formulated prospectively or retrospectively. The amount of prospective reserves at a point in time is derived by subtracting the actuarial present value of future valuation premiums from the actuarial present value of the future insurance benefits. Retrospective reserving subtracts accumulated value of benefits from accumulated value of valuation premiums as of a point in time. The two methods yield identical results.

As an example, consider a whole life insurance policy of one dollar issues on (x) with yearly premiums paid at the start of the year and death benefit paid at the end of the year. In actuarial notation, a benefit reserve is denoted as V. Our objective is to find the value of the net level premium reserve at time t. First we define the loss random variable at time zero for this policy. Hence


 * $$L = v^{K(x)+1} - P\ddot{a}_{\overline{K(x)+1|}}$$

Then, taking expected values we have:
 * $$\operatorname{E}[L] = \operatorname{E}[v^{K(x)+1} - P\ddot{a}_{\overline{K(x)+1|}}]$$


 * $$\operatorname{E}[L] = \operatorname{E}[v^{K(x)+1}] - P\operatorname{E}[\ddot{a}_{\overline{K(x)+1|}}]$$


 * $${}_0\!V_x=A_{x}-P\cdot\ddot{a}_{x}$$

Setting the reserve equal to zero and solving for P yields:


 * $$P=\frac{A_{x}}{\ddot{a}_{x}}$$

For a whole life policy as defined above the premium is denoted as $$P_{x}$$ in actuarial notation. The NLP reserve at time t is the expected value of the loss random variable at time t given K(x)>t


 * $$ {}_{t}L = v^{K(x)+1-t} - P_{x}\ddot{a}_{\overline{K(x)+1-t|}}$$
 * $$ \operatorname{E}[{}_{t}L|K(x)>t] = \operatorname{E}[v^{K(x)+1-t}|K(x)>t] - P_{x}\operatorname{E}[\ddot{a}_{\overline{K(x)+1-t|}}|K(x)>t]$$


 * $${}_t\!V_x=A_{x+t}-P_x\cdot\ddot{a}_{x+t}$$ Where $${ }P_x=\frac{A_{x}}{\ddot{a}_{x}}$$