Reserves for Varying Benefits and Premiums

Why Varying Cash Flows Need Careful Timing

A policy's benefit need not stay the same from one policy year to the next. For example, a hypothetical term policy might provide a larger death benefit in later years, while its premiums may change according to a scheduled pattern. To value the policy at an intermediate duration, it is not enough to multiply one benefit by one insurance value or one premium by one annuity value. Each amount must be matched to the years in which it applies, and to the probability that the cash flow occurs.

This tutorial uses a discrete annual model. The reserve is prospective: conditional on the insured being alive at the valuation date, it is the expected present value of future benefits minus the expected present value of future premiums. The earlier material on prospective reserves establishes this viewpoint; here the central task is handling schedules that vary by year.

Model and timing convention: Consider an $n$-year term policy issued at age $x$. A death in policy year $j$ produces a benefit $B_j$ payable at the end of that year, at time $j$. A premium $C_t$ is due at time $t$, at the beginning of policy year $t+1$, and is paid only if the insured is alive then. Premiums are scheduled for $t=0,1,\ldots,n-1$. The reserve at time $k$, where $0\leq k<n$, is evaluated immediately after any premium due at time $k$. Benefits and premiums stop when the policy terminates at death or at the end of year $n$. The model assumes no expenses, lapses, or other cash flows.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let $q_{x+j-1}$ be the probability of death during policy year $j$, conditional on being alive at the start of that year, and let $p_{x+j-1}=1-q_{x+j-1}$. More generally, ${}_tp_y$ is the probability that a life aged $y$ survives $t$ years. These mortality probabilities are conditional on survival to the age shown in the subscript.

The Varying-Cash-Flow Reserve Formula

At time $k$, only cash flows after time $k$ belong in the prospective reserve. For a benefit payable at time $j$, the insured must first survive from time $k$ to the start of policy year $j$, then die during that year. For a premium payable at time $t$, the insured must survive from time $k$ to time $t$.

Theorem: Under the annual term-policy model above, the net prospective reserve at time $k$, conditional on survival to that time and measured immediately after the premium due then, is

{}_kV_x = \sum_{j=k+1}^{n} v^{\,j-k}\,{}_{j-k-1}p_{x+k}\,q_{x+j-1}\,B_j - \sum_{t=k+1}^{n-1} v^{\,t-k}\,{}_{t-k}p_{x+k}\,C_t.

A sum over no payment dates is zero. In particular, the premium $C_k$ is excluded because the reserve is immediately after that premium has been paid; any premium due at time $n$ is also excluded because the contract has ended.

The notation in the benefit sum reflects the timing convention. From time $k$ to the start of year $j$ there are $j-k-1$ full years of survival; the death probability for year $j$ is then $q_{x+j-1}$. The benefit is paid at time $j$, so it is discounted for $j-k$ years. By contrast, a premium due at time $t$ is paid if the insured survives the $t-k$ years from the valuation date to that premium date, and is discounted for $t-k$ years.

Why the Formula Holds

The formula follows by conditioning on when each cash flow occurs and using linearity of expectation. It does not require the amounts $B_j$ or $C_t$ to be constant.

Condition on being alive at time $k$.
The reserve is calculated for a policy known to be in force at the valuation date. Thus, all subsequent survival probabilities are measured from age $x+k$.

Value a benefit in year $j$.
The conditional probability of death in year $j$ is ${}_{j-k-1}p_{x+k}q_{x+j-1}$. If that death occurs, the amount $B_j$ is paid at time $j$; its conditional expected present value is $v^{j-k}{}_{j-k-1}p_{x+k}q_{x+j-1}B_j$.

Value a future premium at time $t$.
The premium is paid only if the insured is alive at time $t$, an event with conditional probability ${}_{t-k}p_{x+k}$. Its expected present value is $v^{t-k}{}_{t-k}p_{x+k}C_t$.

Sum the cash flows with their signs.
Expected present values of benefits are liabilities and enter positively; expected present values of premiums reduce the net liability and are subtracted. Summing all applicable dates gives the stated reserve.

This also explains why a level-benefit or level-premium shortcut may fail. If the benefit amount differs by year, each year has a different amount multiplying its own probability and discount factor. If premiums change, each premium amount likewise needs its own survival probability and discount factor. The amounts should not be averaged first unless a separate argument proves that the averaging preserves the value.

Worked Calculations

Worked Example: A Two-Year Reserve with a Changing Benefit

A hypothetical three-year term policy is valued at time 1, immediately after the premium due then. The effective annual interest rate is $i=5\%$. Conditional death probabilities in policy years 2 and 3 are $q_{x+1}=0.20$ and $q_{x+2}=0.25$. The death benefits are $B_2=\$6{,}000$ and $B_3=\$9{,}000$. A premium of $C_2=\$500$ is due at time 2 if the insured is alive; there are no other future premiums. Find ${}_1V_x$.

Method. Value each remaining death benefit using its conditional death probability, then subtract the expected present value of the one future premium. The survival probability through year 2 is $p_{x+1}=1-0.20=0.80$, and $v=1/1.05$.

The year-2 benefit is paid at time 2 if death occurs in year 2. Its present value at time 1 is $v(0.20)(\$6{,}000)=\$1{,}200/1.05=\$1{,}142.8571$.

To receive the year-3 benefit, the insured must survive year 2 and die in year 3. The conditional probability is $(0.80)(0.25)=0.20$. Its present value at time 1 is $v^2(0.20)(\$9{,}000)=\$1{,}800/1.1025=\$1{,}632.6531$.

The premium at time 2 is paid only if the insured survives year 2. Its expected present value at time 1 is $v(0.80)(\$500)=\$400/1.05=\$380.9524$.

Therefore, ${}_1V_x=\$1{,}142.8571+\$1{,}632.6531-\$380.9524=\$2{,}394.5578$. Rounded to the nearest cent, ${}_1V_x=\$2{,}394.56$. The result is positive, consistent with the expected present value of remaining benefits exceeding that of remaining premiums. A common error avoided here is discounting the year-3 benefit for only one year rather than two years from the valuation date.

Worked Example: Varying Benefits and Premiums from Issue

A hypothetical three-year policy has $i=0\%$. Conditional death probabilities by policy year are $0.10$, $0.20$, and $0.25$. Death benefits payable at the ends of years 1, 2, and 3 are $\$4{,}000$, $\$6{,}000$, and $\$8{,}000$. The issue premium $C_0$ has already been paid. Additional premiums of $C_1=\$300$ at time 1 and $C_2=\$500$ at time 2 are paid only if the insured is alive on the respective dates. Find the reserve at time 0 immediately after the issue premium.

Method. At $i=0$, no discounting is needed, but each benefit and premium still has its own occurrence probability. The year-1 benefit has expected value $0.10(\$4{,}000)=\$400$.

The probability of death in year 2 from issue is $(0.90)(0.20)=0.18$, so the year-2 benefit has expected value $0.18(\$6{,}000)=\$1{,}080$.

The probability of death in year 3 from issue is $(0.90)(0.80)(0.25)=0.18$, so the year-3 benefit has expected value $0.18(\$8{,}000)=\$1{,}440$. Total expected benefits are $\$400+\$1{,}080+\$1{,}440=\$2{,}920$.

The time-1 premium is paid with probability $0.90$, giving expected value $0.90(\$300)=\$270$. The time-2 premium is paid with probability $(0.90)(0.80)=0.72$, giving expected value $0.72(\$500)=\$360$. Total expected future premiums are $\$270+\$360=\$630$.

Thus ${}_0V_x=\$2{,}920-\$630=\$2{,}290$. The calculation is checked by separately totaling all three benefit values and both premium values. The common mistake is to count a future premium as certain: both installments depend on survival to their payment dates.

Worked Example: An Intermediate Reserve with a Changing Schedule

A hypothetical four-year term policy is valued at time 1 immediately after the premium due then. The interest rate is $i=0\%$. Conditional death probabilities in years 2, 3, and 4 are $0.20$, $0.25$, and $0.20$. The benefits for deaths in those years are $\$7{,}000$, $\$9{,}000$, and $\$11{,}000$, respectively. A premium of $\$1{,}200$ is due at time 2 if the insured is alive. There are no later premiums. Calculate the reserve.

Method. Begin at the valuation date and use conditional probabilities from that date. In year 2, death probability is $0.20$, so its benefit value is $0.20(\$7{,}000)=\$1{,}400$.

For year 3, survival through year 2 and death in year 3 have probability $(0.80)(0.25)=0.20$. The benefit value is $0.20(\$9{,}000)=\$1{,}800$.

For year 4, the probability is $(0.80)(0.75)(0.20)=0.12$. The benefit value is $0.12(\$11{,}000)=\$1{,}320$. The total value of benefits is $\$1{,}400+\$1{,}800+\$1{,}320=\$4{,}520$.

The premium due at time 2 is paid with probability $0.80$, so its expected value is $0.80(\$1{,}200)=\$960$. The reserve is ${}_1V_x=\$4{,}520-\$960=\$3{,}560$. As a check, the three death probabilities sum to $0.20+0.20+0.12=0.52$, which is the probability of death during years 2 through 4 conditional on being alive at time 1; the remaining probability $0.48$ survives to the end of the term.

Common Timing and Survival Traps

The cash-flow schedule should be written down before substituting numbers. A benefit amount is associated with a death year; a premium amount is associated with a payment date. The probability attached to each amount depends on that specific event, not simply on its policy-year label.

Using the wrong survival period for a benefit. For death in year $j$ from valuation time $k$, use ${}_{j-k-1}p_{x+k}q_{x+j-1}$, not just $q_{x+j-1}$ unless $j=k+1$.
Treating a future premium as certain. Under this model, a premium at time $t$ is payable only if the insured reaches time $t$, so it must be multiplied by ${}_{t-k}p_{x+k}$.
Discounting from issue instead of from the valuation date. At time $k$, a payment at time $j$ is discounted by $v^{j-k}$, not $v^j$.
Including a premium already paid. A reserve immediately after the premium at time $k$ excludes $C_k$. The convention would differ for a value immediately before that premium.
Applying a level schedule to changing amounts. Do not replace year-specific $B_j$ or $C_t$ with a single amount without proving that the substitution gives the same expected present value.

For example, in the second worked example, incorrectly treating both future premiums as certain would subtract $\$300+\$500=\$800$, rather than the correct $\$630$. That wrong calculation gives $\$2{,}920-\$800=\$2{,}120$. The correction is to weight the time-1 installment by $0.90$ and the time-2 installment by $0.72$, producing the correct reserve of $\$2{,}290$.

Key takeaway: For a varying schedule, value each benefit and premium separately. Use the probability of its specific payment event, discount from the valuation date to its payment date, and then subtract the total expected present value of future premiums from that of future benefits.

Check Your Understanding

For each problem, use the annual timing convention stated above. Each answer is a reserve immediately after any premium due at the valuation date.

A two-year policy has $i=0\%$, $q_x=0.10$, and $q_{x+1}=0.20$. Its death benefits are $\$5{,}000$ in year 1 and $\$7{,}000$ in year 2. The issue premium has been paid, and $C_1=\$400$ is payable at time 1 if alive. Find ${}_0V_x$. (Answer: $\$1{,}400$)
A three-year policy has $i=0\%$, with annual death probabilities $0.10, 0.20, 0.25$, and death benefits of $\$3{,}000,\$5{,}000,\$7{,}000$. After the issue premium, future premiums are $\$200$ at time 1 and $\$300$ at time 2, conditional on being alive. Find ${}_0V_x$. (Answer: $\$2{,}064$)
At time 1, a three-year policy has $i=5\%$, $q_{x+1}=0.20$, $q_{x+2}=0.10$, benefits of $\$4{,}000$ and $\$8{,}000$ in years 2 and 3, and one future premium of $\$600$ due at time 2 if alive. Find ${}_1V_x$, rounded to the nearest cent. (Answer: $\$885.26$)
A three-year policy has $i=0\%$, death probabilities $0.10,0.20,0.25$, and benefits $\$1{,}000,\$3{,}000,\$5{,}000$. With the issue premium paid, future premiums are $\$100$ at time 1 and $\$200$ at time 2 if alive. Find ${}_0V_x$. (Answer: $\$1{,}306$)
At time 1, a three-year policy has $i=0\%$, with conditional death probabilities in years 2 and 3 of $0.20$ and $0.25$. Benefits in those years are $\$6{,}000$ and $\$8{,}000$. A future premium of $\$700$ is payable at time 2 if alive. Find ${}_1V_x$. (Answer: $\$2{,}240$)

Accurate reserve calculations depend as much on defining payment dates as on performing arithmetic. This year-by-year approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus and applies whenever benefit or premium amounts vary over the term under the stated annual model.

What You'll Learn