Common Errors in Prospective Reserve Calculations

Why Prospective Reserve Errors Happen

A prospective reserve is the value, at a specified time, of future benefits less the value of future premiums. The arithmetic is often straightforward; the difficult part is deciding exactly which future cash flows belong in the calculation. A reserve at duration $t$ is conditional on the insured being alive at that duration, and its premium and benefit terms must respect the contract’s payment dates.

This tutorial focuses on five related checks: condition on survival to the valuation time, include only premiums still due, use the remaining coverage period, discount benefits from their actual payment dates, and distinguish an annuity-due premium from an annuity-immediate premium. Each calculation below states its timing convention explicitly. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A prospective reserve at time $t$ is the conditional expected present value, at time $t$, of future benefits minus the conditional expected present value of future premiums, given that the insured is alive and the policy is in force at time $t$. The precise cash flows included depend on whether the valuation is immediately before or immediately after a premium due at time $t$.

A Timeline Before a Formula

For a basic annual term policy, assume level premiums are paid at the beginnings of policy years while the insured is alive. A death benefit is paid at the end of the policy year of death. Let $B$ be the benefit, $P$ the annual premium, and $v=1/(1+i)$ the annual discount factor. Let $q_{x+k}$ be the probability of death during the next year for a life aged $x+k$, and let ${}_kp_x$ be the probability of surviving $k$ years from age $x$.

For an $n$-year term policy valued at integer duration $t$, immediately after the premium at time $t$, there are $m=n-t$ years of coverage left. The benefit in the $k+1$st remaining year is paid only if the insured survives the first $k$ years and then dies in the next year. A future premium at time $j$ is payable only if the insured survives to that time.

{}_tV_x = B\sum_{k=0}^{m-1}v^{k+1}{}_kp_{x+t}q_{x+t+k} - P\sum_{j=1}^{m-1}v^j{}_jp_{x+t}, \qquad m=n-t.

The premium sum begins at $j=1$ because the premium due at time $t$ has just been paid. It ends at $m-1$ because premiums are scheduled at the starts of covered years; there is no premium at the end of the term. If the reserve is instead valued immediately before a premium due at time $t$, that due premium is still a future cash flow and must be included. This distinction is not a change in mortality assumptions; it is a change in the valuation instant.

Why the formula holds: Conditional on survival to time $t$, the probability of dying in remaining year $k+1$ is ${}_kp_{x+t}q_{x+t+k}$. Its benefit is paid $k+1$ years after valuation, so its present value is discounted by $v^{k+1}$. Summing these expected discounted benefits gives the first term. A premium at time $j$ is paid with conditional probability ${}_jp_{x+t}$, and its present value is $Pv^j{}_jp_{x+t}$. Subtracting the sum of those expected premium values gives the prospective reserve.

Worked Calculations and Their Error Checks

Worked Example: Conditioning on Survival to the Valuation Date

A hypothetical three-year term policy pays $B=\$10{,}000$ at the end of the year of death. Annual premiums are paid at times 0, 1, and 2 if the insured is alive. Assume $i=0$ and $q=0.10$ in each of the three policy years. Find the net annual premium and the reserve at duration 1, immediately after the premium at time 1.

Method. First value the full contract at issue to find $P$; then re-evaluate only cash flows after time 1, conditional on survival to time 1. With $i=0$, discount factors are all 1.

The probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.10)=0.09$, and $0.90^2(0.10)=0.081$. Thus the expected benefit at issue is

10{,}000(0.10+0.09+0.081)=\$2{,}710.

The expected number of premiums is $1+0.90+0.90^2=1+0.90+0.81=2.71$, since the premiums at times 1 and 2 require survival to those times. Therefore,

P=\frac{\$2{,}710}{2.71}=\$1{,}000.

At duration 1, given survival to that time, the remaining death probabilities are $0.10$ in year 2 and $0.90(0.10)=0.09$ in year 3. The conditional expected future benefit is $\$10{,}000(0.10+0.09)=\$1{,}900$. The only future premium is at time 2, payable with conditional probability $0.90$; its expected value is $0.90(\$1{,}000)=\$900$. Hence

{}_1V_x=\$1{,}900-\$900=\$1{,}000.

Check. The valuation uses exactly two remaining years, and the conditional probability of death over those years is $0.19$. A common error is to reuse the full-term death probability $0.271$ after conditioning on survival to year 1. That gives an erroneous benefit value of $\$2{,}710$, and subtracting the correct $\$900$ premium value would produce $\$1{,}810$. The correction is to restart the projection at the attained duration and use the two remaining conditional death probabilities.

Worked Example: Premium Due at the Valuation Time

A hypothetical two-year term policy pays $\$10{,}000$ at the end of the year of death. Premiums are due at times 0 and 1 if the insured is alive. Assume $i=0$ and annual mortality probability $q=0.10$. Find the net premium and the reserve at time 1 immediately before the second premium is paid. Also find the reserve immediately after that payment.

At issue, the expected benefit is $\$10{,}000[0.10+0.90(0.10)]=\$1{,}900$. The expected number of premiums is $1+0.90=1.90$. Thus $P=\$1{,}900/1.90=\$1{,}000$.

Immediately before the time-1 premium, the insured is alive. There is one year of coverage remaining, so the conditional expected benefit is $0.10(\$10{,}000)=\$1{,}000$. The premium of $\$1{,}000$ is due at that same instant. At zero interest its value is $\$1{,}000$, so the reserve just before payment is

{}_1V_x^-=\$1{,}000-\$1{,}000=\$0.

Here the premium stream remaining just before payment has the annuity-due convention: its first payment is at the valuation time. With one certain payment and $i=0$, its value is $\ddot{a}_{\overline{1}|}=1$. Immediately after the premium has been paid, it is no longer a future premium. The reserve is then $\$1{,}000-\$0=\$1{,}000$.

Check. The two reserves differ by exactly the premium paid at time 1. The common mistake is to use the after-premium convention while calling the result a before-premium reserve, omitting the due premium and reporting $\$1{,}000$ instead of $\$0$. State the instant of valuation before deciding whether the annuity-due payment is included.

Worked Example: Discount the Benefit from Its Contractual Payment Date

A hypothetical two-year term policy pays $\$10{,}000$ at the end of the year of death. Annual premiums are due at times 0 and 1 while the insured is alive. Assume $i=0.10$ and annual mortality probability $q=0.10$ in each year. Find the net annual premium and the reserve at duration 1 immediately after the premium at time 1.

Method. Price the policy at issue using end-of-year benefit payments, then calculate the remaining benefit and premium values from time 1 using the same payment-time convention. The discount factor is $v=1/1.10=0.9090909$.

The issue-date expected present value of benefits is

\$10{,}000[0.10v+0.90(0.10)v^2] = \$10{,}000[0.0909091+0.0743802] = \$1{,}652.8926.

The issue-date expected present value of premiums per dollar of premium is $1+0.90v=1+0.8181818=1.8181818$. Therefore,

P=\frac{\$1{,}652.8926}{1.8181818}=\$909.0909.

At duration 1, the year-2 death benefit is paid one year after the valuation date. Its conditional expected present value is $0.10(\$10{,}000)v=\$909.0909$. The premium at time 2 is payable if the insured survives the next year, so its expected present value is $0.90(\$909.0909)v=\$743.8017$. Thus,

{}_1V_x=\$909.0909-\$743.8017=\$165.2893.

The premium was calculated without rounding; dollar amounts are shown to four decimal places here to make the check visible. Rounded to the nearest cent, $P=\$909.09$ and ${}_1V_x=\$165.29$. A timing error would value the remaining benefit as if paid immediately, giving $0.10(\$10{,}000)=\$1{,}000$, and a false reserve of $\$1{,}000-\$743.8017=\$256.1983$. The benefit is payable at the end of the year, so it must be discounted for one year.

A Practical Audit for Prospective Reserves

A reliable calculation starts with a timeline, not a memorized annuity label. The following sequence separates the most common sources of error.

Pin down the valuation instant.
Record the duration and whether the instant is just before or just after any premium due at that time.

Condition on being in force.
Use survival probabilities from the attained age or duration. Do not carry unconditional probabilities from policy issue into a reserve calculation.

Count the remaining benefit periods.
For a term ending at duration $n$, a reserve at duration $t$ has $n-t$ years of coverage remaining, not $n$.

Place every cash flow on the timeline.
Discount a benefit from its contractual payment date. Include a premium only if it is still due and only with the probability that it is payable.

Reconcile benefits and premiums.
Check that the reserve equals conditional future benefit value less conditional future premium value, using consistent timing and units.

The premium convention and the benefit convention are independent choices. Premiums paid at the beginnings of years form a life annuity-due pattern; death benefits paid at the ends of years are discounted from those later dates. An annuity-due factor does not make a year-end benefit immediate, and a benefit discounted for a year does not imply that the premium is also paid at year-end.

A final boundary check is useful: once coverage has ended and no further benefit or premium is due, there are no future policy cash flows to value. At earlier durations, however, a zero reserve should not be inferred merely from a premium date; it follows only if the remaining expected benefit and premium values actually offset.

Key takeaway: Before accepting a prospective reserve, verify five items: the insured is conditioned on survival to valuation, the remaining term is correct, each premium is placed on its actual due date, each benefit is discounted from its actual payment date, and the annuity convention matches whether the valuation is before or after a due premium.

Check Your Understanding

For each problem, identify the valuation timing before calculating. Answers are rounded to the nearest cent when necessary.

A hypothetical two-year term policy pays $\$5{,}000$ at the end of the year of death. Premiums are due at times 0 and 1 while alive. Assume $i=0$ and $q=0.20$ each year. Find the net premium and the reserve at time 1 immediately after the premium. (Answer: $P=\$1{,}000$; ${}_1V_x=\$1{,}000$.)
A hypothetical three-year term policy pays $\$6{,}000$ at the end of the year of death and has premiums due at times 0, 1, and 2. Assume $i=0$ and $q=0.25$ each year. Find the net premium and reserve at time 1 immediately after the premium. (Answer: $P=\$1{,}500$; ${}_1V_x=\$1{,}500$.)
A hypothetical two-year term policy pays $\$8{,}000$ at the end of the year of death, with premiums due at times 0 and 1 while alive. Assume $i=0$ and $q=0.10$ each year. Find the net premium and reserve at time 1 immediately before the premium due then. (Answer: $P=\$800$; before-premium reserve $=\$0$.)
A hypothetical two-year term policy pays $\$10{,}000$ at the end of the year of death. Premiums are due at times 0 and 1 while alive. Assume $i=0.25$ and $q=0.20$ each year. Find the net premium and reserve at time 1 immediately after the premium. (Answer: $P=\$1{,}600$; ${}_1V_x=\$576.00$.)
A hypothetical four-year term policy pays $\$20{,}000$ at the end of the year of death, with premiums due at times 0, 1, 2, and 3 while alive. Assume $i=0$ and $q=0.20$ each year. Find the net premium and reserve at time 2 immediately after the premium. (Answer: $P=\$4{,}000$; ${}_2V_x=\$4{,}000$.)

What You'll Learn