Why Premium Timing Matters
A prospective reserve is the expected present value of future benefits less the expected present value of future premiums, conditional on the policy being in force at the valuation time. When premiums are paid quarterly, monthly, or at another frequency more often than annually, the premium stream has its own dates and survival conditions. It cannot generally be valued as though one annual premium were due at each policy anniversary.
The key is to build the future cash flows from the valuation date. For each possible benefit payment, identify the period in which death occurs and the date the benefit is paid. For each future premium, identify its due date and whether the policyholder must survive to that date for the installment to be payable. The reserve is the difference between those two values.
This tutorial uses a discrete timing convention: death benefits are paid at the end of the subperiod of death, and premiums are paid on specified installment dates while the policy is in force. Other contracts can use different timings; the dates in the contract determine the valuation. The methods below are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
A General Formula for Frequent Installments
Suppose the contract divides time into periods of length \(h=1/m\) years, where premiums are paid \(m\) times per year. Let \(N\) coverage periods remain at the valuation date. Let \({}_{k-1}p\) denote the conditional probability of surviving the first \(k-1\) periods, and let \(q_k\) be the conditional probability of death in period \(k\), given survival to its start. These probabilities may vary by period. If the annual effective interest rate is \(i\), the discount factor for \(k\) periods is \(v^{kh}\), where \(v=1/(1+i)\). If the quarterly discount factor is supplied directly, use it consistently instead.
Let \(B_k\) be the benefit paid at the end of period \(k\). Let \(P_j\) be the installment due \(j\) periods after valuation, and let \(\mathcal{K}\) be the set of those future installment dates. The installment is payable only if the insured survives to its due date. Then:
Here, \({}_0p=1\). The first sum values death benefits: the probability of death in period \(k\) is the probability of surviving the preceding periods multiplied by the conditional death probability \(q_k\). The second sum values premiums: the probability of receiving a premium at its due date is the probability of surviving to that date. Premiums due at the valuation time are excluded if the reserve is calculated immediately after that payment.
For a level benefit \(B\), constant survival probability \(p\) per period, constant death probability \(q=1-p\), and period discount factor \(v_h\), define \(r=pv_h\). The benefit value for \(N\) remaining periods is \(Bqv_h\sum_{k=0}^{N-1}r^k\). A level installment \(P\) due at the future dates indexed by \(\mathcal K\) has value \(P\sum_{j\in\mathcal K}r^j\). The set \(\mathcal K\) must come from the actual premium schedule; it is not automatically the same as the set of remaining coverage periods.
Why the Formula Works
The prospective value follows by dividing the future into mutually exclusive outcomes and valuing each payment only on outcomes where it occurs. A death benefit at the end of period \(k\) is paid if the insured survives to the start of that period and dies during it. A premium at a future due date is paid if the insured is alive on that date.
For a death benefit in period \(k\), survival through the first \(k-1\) periods followed by death in period \(k\) has probability \({}_{k-1}p\,q_k\). Discount the payment at the end of period \(k\) by \(v^{kh}\).
A premium due \(j\) periods after valuation is paid only if the insured survives to that date. Its expected present value is therefore \({}_jp\,P_jv^{jh}\).
Summing the benefit values and subtracting the installment values gives the prospective reserve conditional on being in force at valuation.
This derivation also explains why survival is treated differently for the two cash-flow types. The death benefit requires survival to the beginning of the death period and death within it; a premium requires survival to its due date. Applying the same probability to both without checking their dates can produce an incorrect reserve.
Worked Calculations
Worked Example: A Quarterly-Premium Term Policy
A hypothetical one-year term policy pays \(B=\$1{,}000\) at the end of the quarter of death. Premiums of \(\$250\) are due at times \(0, 0.25, 0.50,\) and \(0.75\) years while the policy is in force. Each quarter, the conditional survival probability is \(p=0.80\), and the quarterly discount factor is \(v_h=0.90\). Thus the effective annual interest rate is \(i=(0.90)^{-4}-1\), approximately \(52.42\%\). Find the reserve immediately after the premium at time \(0.25\).
Method. Use the prospective formula from time \(0.25\). Three quarterly coverage periods remain: \(0.25\) to \(0.50\), \(0.50\) to \(0.75\), and \(0.75\) to \(1.00\). The next death benefit is discounted one quarter from valuation. Set \(q=1-p=0.20\) and \(r=pv_h=0.80(0.90)=0.72\).
The benefit value is:
Only two premiums remain: \(\$250\) at time \(0.50\), and \(\$250\) at time \(0.75\). Their values, conditional on being in force at time \(0.25\), are:
Therefore the reserve is \(402.912-309.600=\$93.312\), or \(\$93.31\) to the nearest cent. The method avoids counting premiums by counting coverage periods: a premium is due at \(0.50\) and \(0.75\), but not at maturity, \(1.00\). A common wrong calculation adds a third future premium at maturity, with value \(250(0.72^3)=\$93.312\), and reports a zero reserve. That installment is not in the stated schedule, so the correct reserve remains \(\$93.31\).
Worked Example: A Net Premium and a Reserve After an Installment
A hypothetical one-year term policy pays \(\$2{,}000\) at the end of the quarter of death. The conditional survival probability is \(0.90\) each quarter, and the quarterly discount factor is \(0.95\), equivalent to an annual effective rate of \((0.95)^{-4}-1\), approximately \(22.77\%\). A level premium is due at times \(0, 0.25, 0.50,\) and \(0.75\), provided the policy is in force. First find the net level installment by the equivalence principle, then find the reserve immediately after the installment at \(0.25\).
Method. At issue, the present value of benefits is \(2000(0.10)(0.95)\sum_{j=0}^{3}r^j\), where \(r=0.90(0.95)=0.855\). The premium due at time zero is certain, while later premiums depend on survival to their due dates. Their expected present value is \(P\sum_{j=0}^{3}r^j\). Equating the two values gives:
At time \(0.25\), immediately after that installment has been paid, three coverage quarters remain. Their benefit value is:
There are only two future premiums, due at \(0.50\) and \(0.75\). Their value is:
The reserve is \(491.34475-301.34475=\$190.00\). The calculation avoids including the installment just paid at \(0.25\), or inventing a premium at the maturity date. It also checks against the net premium: with constant mortality and interest, the remaining benefit sum and the future premium sum have the same terms except for the first benefit-period term, leaving \(190\).
Worked Example: Varying Mortality and Installment Amounts
A hypothetical policy has three quarterly coverage periods remaining. The benefit is \(\$3{,}000\), payable at the end of the quarter of death. The conditional death probabilities in the three successive quarters are \(0.10\), \(0.20\), and \(0.30\). The quarterly discount factor is \(0.90\), equivalent to an annual effective rate of approximately \(52.42\%\). After the valuation date, premiums of \(\$120\) and \(\$80\) are due one and two quarters later, respectively, if the policy is in force. Find the prospective reserve.
Method. Use period-specific death probabilities for benefits and survival probabilities to each premium date. The conditional survival probabilities are \(0.90\), \(0.80\), and \(0.70\) in the three periods. The benefit value is:
The \(\$120\) premium is due after one quarter. The policyholder must survive the first quarter, so its value is \((0.90)(0.90)(120)=\$97.20\). The \(\$80\) premium is due after two quarters. Survival probability is \((0.90)(0.80)=0.72\), and the two-quarter discount factor is \(0.90^2=0.81\), so its value is \((0.72)(0.81)(80)=\$46.656\). Total future premium value is \(97.20+46.656=\$143.856\).
The reserve is \(1{,}179.792-143.856=\$1{,}035.936\), or \(\$1{,}035.94\) to the nearest cent. The method avoids using one constant mortality probability for all three periods. It also distinguishes survival to a premium date from death during a coverage period.
Timing Checks and Common Traps
Frequent-payment reserves are especially sensitive to one-period shifts. A useful check is to write the remaining dates before substituting numbers. If a policy has \(N\) coverage periods left, that does not mean it has \(N\) future premiums left. For example, when installments are due at the start of each quarter and there is no premium at maturity, a valuation immediately after an installment will generally have fewer future premium dates than remaining coverage periods.
| Cash flow | Question to ask | Valuation detail |
|---|---|---|
| Death benefit | When is death covered and when is the benefit paid? | Use the probability of death in the applicable period and discount to the payment date. |
| Future installment | When is it due, and must the insured survive to that date? | Use survival to the due date and discount to that date. |
| Installment at valuation time | Has it already been paid? | If valuation is immediately after payment, exclude it from future premiums. |
| Installment at maturity | Does the contract actually require it? | Do not include one merely because coverage remains up to maturity. |
A further common error is mixing annual and subannual rates. If the annual effective rate is \(i\), the discount factor for \(1/m\) of a year is \((1+i)^{-1/m}\), not \(1/(1+i)\). Conversely, when a subperiod discount factor is given, use its \(k\)-th power for a payment \(k\) subperiods away. State the timing convention and use it for both benefits and premiums.
Check Your Understanding
For each problem, identify the remaining coverage periods and future premium dates before calculating.
- A one-year quarterly term policy pays \(\$1{,}000\) at the end of the quarter of death. Quarterly survival is \(0.90\), the quarterly discount factor is \(0.95\), and premiums of \(\$100\) are due at \(0, 0.25, 0.50,\) and \(0.75\) years while in force. Find the reserve just after the \(0.25\)-year premium. (Answer: \(\$87.07\))
- A two-year quarterly term policy pays \(\$4{,}000\) at the end of the quarter of death. Quarterly survival is \(0.90\), the quarterly discount factor is \(0.95\), and premiums of \(\$250\) are due at \(0, 0.25,\ldots,1.75\) years while in force. Find the reserve just after the premium at \(0.25\). There is no premium at maturity. (Answer: \(\$847.09\))
- A policy has three quarters remaining and pays \(\$3{,}000\) at the end of the quarter of death. The successive quarterly death probabilities are \(0.10, 0.20,\) and \(0.30\); the quarterly discount factor is \(0.90\). Future premiums of \(\$120\) and \(\$80\) are due one and two quarters from now, if in force. Find the reserve. (Answer: \(\$1{,}035.94\))
- A two-year quarterly term policy has seven coverage quarters remaining at valuation. The conditional survival probability is \(0.90\) each quarter, and the quarterly discount factor is \(0.95\). It pays \(\$4{,}000\) at the end of the quarter of death. Premiums of \(\$250\) are due at the next six quarter dates, with none at maturity. Find the reserve. (Answer: \(\$847.09\))