Checking Recursive Reserves Against Prospective Values

Why Compare Two Reserve Calculations?

A prospective reserve values the policy’s future cash flows directly. A recursive reserve carries a value forward one policy year at a time. Under the same contract terms, mortality assumptions, interest rate, and timing convention, the two methods must give the same reserve at a given duration. Comparing them is therefore a useful check: a mismatch signals that at least one input, timing choice, or arithmetic step is inconsistent.

The comparison is meaningful only after fixing what “the reserve at time $k$” means. This tutorial uses an annual discrete term insurance with level premiums. The reserve $V_k$ is measured immediately before the premium due at time $k$, conditional on the insured being alive then. A premium is paid at the beginning of each covered year, including at time $k$; a death benefit is paid at the end of the year of death. There is no premium or benefit after the term ends.

Definition: Under the timing convention above, the prospective reserve at duration $k$ is the conditional expected present value, at time $k$, of benefits after time $k$, less the conditional expected present value of premiums due from time $k$ onward. The recursive reserve is obtained from the corresponding one-year relation, using the reserve at duration $k+1$ for policies that remain in force.

The Prospective Value and the Recursion

Consider an $n$-year term insurance issued at age $x$, with benefit $B_j$ paid at the end of policy year $j$ if death occurs in that year, and annual premium $P$ due at the start of each policy year while the insured is alive. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. At duration $k<n$, the future benefits and premiums can be valued conditionally on survival to that duration.

V_k = \sum_{r=0}^{n-k-1} v^{r+1} {}_rp_{x+k}q_{x+k+r}B_{k+r+1} - P\sum_{r=0}^{n-k-1} v^r {}_rp_{x+k}.

In the benefit sum, $r$ years of survival are required before death in the next year; the payment is then discounted $r+1$ years. In the premium sum, the premium at time $k+r$ is paid if the insured has survived $r$ years from time $k$, and it is discounted $r$ years. In particular, the $r=0$ term includes the premium due immediately at time $k$.

For a level benefit $B$, the recursive relation is

(V_k+P)(1+i)=q_{x+k}B+p_{x+k}V_{k+1}, \qquad k<n,

where $p_{x+k}=1-q_{x+k}$. At the end of the term, $V_n=0$, since no premiums or benefits remain under this term contract. The recursion assumes the same premium, benefit, mortality, interest, and timing conventions as the prospective formula.

Why the Recursion Agrees with Prospective Values

The relation follows by separating the future cash flows into the next policy year and what follows it. At time $k$, the premium $P$ is due immediately. The amount $V_k+P$ is therefore the reserve available after that premium is paid. Accumulated for one year, it must equal the expected value at time $k+1$ of the next year’s outcomes: the death benefit if death occurs, or the next reserve if the insured survives.

(V_k+P)(1+i)=q_{x+k}B+p_{x+k}V_{k+1}.

Equivalently, discounting the right side by one year and subtracting the premium due immediately gives

V_k=-P+v\left(q_{x+k}B+p_{x+k}V_{k+1}\right).

Substituting the prospective expression for $V_{k+1}$ into this equation separates out the first year’s death benefit and premium. The remaining terms are exactly the benefit and premium values from later years, each conditional on surviving the first year. Repeating this step gives the full prospective sums. Thus the two methods are not competing definitions: they are two ways of organizing the same expected future cash flows.

Fix the timing.
State whether the reserve is before or after the premium, and when benefits are paid.

Compute a prospective value.
Include only future cash flows, conditional on being alive at the selected duration.

Compute recursively.
Use the one-year relation and the same assumptions at each duration.

Reconcile the results.
Compare unrounded values first; if they differ, check assumptions and timing before blaming rounding.

Worked Comparisons

Worked Example: Matching a Two-Year Term Reserve

A hypothetical two-year term policy pays $B=\$3{,}000$ at the end of the year of death. Annual premiums are $P=\$100$, due at the start of each covered year. The effective annual interest rate is $i=10\%$, with $q_x=0.20$ and $q_{x+1}=0.10$. Calculate $V_1$ and $V_0$ prospectively and recursively, and verify that they agree.

Method. Use the stated before-premium reserve convention. The corresponding survival probabilities are $p_x=0.80$ and $p_{x+1}=0.90$, and $v=1/1.10=10/11$.

Prospective value at duration 1. The only remaining benefit has probability $0.10$ and is paid one year later. The premium due immediately is included:

V_1=-100+\frac{0.10(3{,}000)}{1.10} =-100+\frac{3{,}000}{11} =\frac{1{,}900}{11} =\$172.727\ldots

Prospective value at issue. The benefit in year 1 has present value $0.20(3{,}000)/1.10=6{,}000/11$. The year 2 benefit is payable only if the insured survives year 1, so its present value is $0.80(0.10)(3{,}000)/(1.10)^2=24{,}000/121$. Thus

\text{EPV of benefits} =\frac{6{,}000}{11}+\frac{24{,}000}{121} =\frac{66{,}000+24{,}000}{121} =\frac{90{,}000}{121}.

The premiums are due at time 0 and, if alive, at time 1. Their present value is

\text{EPV of premiums} =100+\frac{0.80(100)}{1.10} =100+\frac{800}{11} =\frac{1{,}900}{11} =\frac{20{,}900}{121}.

Therefore $V_0=90{,}000/121-20{,}900/121=69{,}100/121=\$571.074\ldots$.

Recursive check. At duration 1, the final-year recursion has $V_2=0$:

(V_1+100)(1.10)=0.10(3{,}000)+0.90(0), \qquad V_1=\frac{300}{1.10}-100=\frac{1{,}900}{11}.

Then at duration 0,

(V_0+100)(1.10) =0.20(3{,}000)+0.80\left(\frac{1{,}900}{11}\right) =600+\frac{1{,}520}{11} =\frac{8{,}120}{11}.

Hence $V_0=8{,}120/12.1-100=69{,}100/121=\$571.074\ldots$. Both methods give $V_1=\$172.73$ and $V_0=\$571.07$, rounded to the nearest cent. The check also catches two tempting arithmetic slips: $240/1.21=24{,}000/121$, not $2{,}400/121$, and $100+800/11=1{,}900/11$, not $1{,}920/11$.

Worked Example: Checking an Intermediate Duration

A hypothetical three-year term policy pays $\$1{,}000$ at the end of the year of death. The annual premium is $\$50$, due at the start of each year. Assume $i=5\%$ and $q_{x}=q_{x+1}=q_{x+2}=0.10$. Find the reserve at duration 1 by both methods.

Prospective method. At duration 1, two years remain. The first remaining death benefit has present value $100/1.05=2{,}000/21$. The second-year benefit requires survival for one year, so its present value is $0.90(0.10)(1{,}000)/(1.05)^2=4{,}000/49$. Premiums are due at duration 1 and, if alive, at duration 2:

\text{EPV of benefits} =\frac{2{,}000}{21}+\frac{4{,}000}{49} =\frac{14{,}000+12{,}000}{147} =\frac{26{,}000}{147}, \qquad \text{EPV of premiums} =50+\frac{0.90(50)}{1.05} =50+\frac{300}{7} =\frac{650}{7}.

Thus $V_1=26{,}000/147-650/7=12{,}350/147=\$84.0136\ldots$.

Recursive method. First calculate the reserve at duration 2:

V_2=-50+\frac{0.10(1{,}000)}{1.05} =-50+\frac{2{,}000}{21} =\frac{950}{21}.

Then

V_1=-50+\frac{0.10(1{,}000)+0.90(950/21)}{1.05} =-50+\frac{985/7}{1.05} =-50+\frac{19{,}700}{147} =\frac{12{,}350}{147}.

Both methods give $V_1=\$84.01$, rounded to the nearest cent. The key is to compare at the same duration: comparing a prospective value at duration 1 with a recursive value at issue would not be a valid check.

Worked Example: Locating a Mortality-Input Discrepancy

A hypothetical three-year term policy pays $\$5{,}000$ at the end of the year of death. Its annual premium is $\$300$, payable at the start of each covered year. The interest rate is $i=4\%$, and the mortality assumptions are $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$. A calculation reports $V_1=\$337.20$, while a recursive calculation using all three stated mortality rates gives a different result. Determine whether the discrepancy is consistent with using $q_{x+2}=0.10$ by mistake in the prospective calculation.

Prospective value using the stated assumptions. At duration 1, there are two years remaining. The year 2 death benefit has present value $0.10(5{,}000)/1.04=500/1.04$. The year 3 benefit requires one year of survival and then death, so its present value is $0.90(0.20)(5{,}000)/(1.04)^2=900/(1.04)^2$. Premiums are due now and one year later if alive:

V_1 =\frac{500}{1.04}+\frac{900}{(1.04)^2} -\left(300+\frac{0.90(300)}{1.04}\right) =\$753.2544\ldots

Test the suspected error. If the year 3 mortality rate is incorrectly entered as $0.10$, the second benefit term becomes $0.90(0.10)(5{,}000)/(1.04)^2=450/(1.04)^2$. The mistaken prospective value is

\frac{500}{1.04}+\frac{450}{(1.04)^2} -\left(300+\frac{270}{1.04}\right) =\$337.2044\ldots

This rounds to the reported $\$337.20$, so the discrepancy is explained by the mortality input. For a recursive check with the stated rates, first find $V_2=-300+0.20(5{,}000)/1.04=\$661.5385\ldots$. Then $V_1=-300+[0.10(5{,}000)+0.90V_2]/1.04=\$753.2544\ldots$, agreeing with the correct prospective value. This comparison isolates the changed mortality rate rather than attributing the mismatch to rounding.

Common Traps When Reconciling Values

A reserve check works only when both calculations refer to the same contract state and cash-flow assumptions. In practice, the cause of a mismatch is often not the reserve method itself, but a small inconsistency carried into one calculation.

Mixing before-premium and after-premium values. Here $V_k$ is before the premium at time $k$, so the prospective calculation includes that premium. An after-premium value would exclude it, and the two values differ by the premium due at that time.
Using a duration-specific mortality rate incorrectly. The second future year’s death probability is conditional on surviving the first future year. Use the mortality rate for the attained age in that year, not the rate for the current year.
Discounting a benefit or premium for the wrong number of years. A death benefit at the end of the next year is discounted one year; a premium due immediately is not discounted.
Comparing rounded intermediate values. Keep full precision through both calculations. Round only the final reserves, as in the examples.
Changing the policy cash flows between methods. A benefit amount, premium duration, or survival condition omitted from one method will prevent agreement even if each method is otherwise applied correctly.

Key takeaway: A recursive reserve and a prospective reserve at the same duration must agree when they use the same reserve timing, benefit and premium schedule, mortality assumptions, and interest rate. A useful verification reports both values and traces any mismatch back to a specific assumption or cash flow.

Check Your Understanding

For each problem, use the before-premium reserve convention and compare the stated cash flows carefully.

A two-year term policy has $B=\$2{,}000$, $P=\$100$, $i=5\%$, $q_x=0.10$, and $q_{x+1}=0.20$. Calculate $V_1$ and $V_0$ prospectively. (Answer: $V_1=\$280.95$; $V_0=\$331.29$.)
A three-year term policy has $B=\$10{,}000$, $P=\$250$, $i=4\%$, and $q_x=q_{x+1}=q_{x+2}=0.05$. Use the recursion to calculate $V_2$ and then $V_1$. (Answer: $V_2=\$230.77$; $V_1=\$441.57$.)
A two-year term policy has $B=\$4{,}000$, $P=\$300$, $i=5\%$, $q_x=0.10$, and $q_{x+1}=0.15$. Find $V_0$ by first calculating $V_1$ recursively. (Answer: $V_1=\$271.43$; $V_0=\$313.61$.)
For a two-year term policy, $B=\$3{,}000$, $P=\$100$, $i=10\%$, $q_x=0.20$, and $q_{x+1}=0.10$. A prospective calculation at duration 1 uses $q_{x+1}=0.20$. Calculate the correct $V_1$, the mistaken value, and the size of the discrepancy. (Answer: correct $V_1=\$172.73$; mistaken value $=\$445.45$; discrepancy $=\$272.73$.)

What You'll Learn