Solving for Interest or Benefit Amounts From Reserves

Why Reserve Data Can Reveal an Unknown

A retrospective reserve records the value, at a specified duration, of past premiums accumulated with interest less past benefits paid. If the premium, mortality experience, and reserve are known, the balance equation can be used in reverse: it may reveal an unknown interest rate or benefit amount. The key is to preserve the timing of each cash flow. A premium paid at the start of a year earns interest for that year; a death claim paid at the end of the year does not.

This tutorial uses a homogeneous cohort of identical policies to make the cash flows visible. The same method applies when the cohort counts are expected counts from a mortality basis rather than literal integer counts. No expenses are included, and premiums are paid annually in advance while death benefits are paid at the end of the year of death. Other payment timings require a different accumulation equation.

Definition: Let $N_k$ be the number of policies in force at the start of policy year $k+1$, and let $D_k$ be the number of deaths during that year, so $N_{k+1}=N_k-D_k$. Let $P$ be the annual premium per policy, $B$ the death benefit per claim, and $i$ the effective annual interest rate. Write $F_k$ for the cohort’s total fund at the end of year $k$, after that year’s claims are paid. If $N_k>0$, the retrospective reserve per policy in force is ${}_kV_x=F_k/N_k$.

The Retrospective Balance Equation

At the beginning of year $k+1$, the cohort has fund $F_k$ and receives $N_kP$ in premiums. The combined amount earns one year of interest. Then $D_k$ claims of amount $B$ are paid at year-end. Therefore the total fund and the reserve per survivor satisfy

F_{k+1}=(F_k+N_kP)(1+i)-D_kB, \qquad {}_{k+1}V_x=\frac{F_{k+1}}{N_{k+1}}.

For a newly issued cohort with no initial fund, $F_0=0$. Repeated application gives the total fund at duration $t$:

F_t=P\sum_{j=0}^{t-1}N_j(1+i)^{t-j} -B\sum_{j=0}^{t-1}D_j(1+i)^{t-j-1}.

The exponent on each cash flow reflects its timing. The premium collected at the start of year $j+1$ accumulates for $t-j$ years, while the claim paid at the end of that year accumulates for only $t-j-1$ years. The reserve observation supplies the equation $F_t=N_t\,{}_tV_x$. Solving that equation for $i$ generally produces a polynomial; solving it for $B$, when $i$ is known, is linear.

Fix the timing and duration.
State when premiums and claims are paid, and whether the reserve is measured before or after the year’s claims.

Translate the reserve into a fund.
Multiply the reserve per policy in force by the number of survivors at that duration: $F_t=N_t\,{}_tV_x$.

Build the past cash-flow balance.
Use the fund recursion year by year, or accumulate each premium and claim to duration $t$.

Solve and verify.
Keep only economically admissible interest rates or benefit amounts, then substitute the result back into the balance and any additional reserve observations.

Solving for an Interest Rate

When $B$ and $P$ are known, the observed reserve determines the required accumulated fund. The unknown $i$ appears in the growth factors on past premiums and claims. At one duration, the equation may be simple or may have several algebraic roots. A root must be checked against the stated domain, such as $i\geq 0$, and against every other reserve observation supplied. One reserve is not automatically enough to identify a unique rate.

Worked Example: Recovering Interest From a One-Year Reserve

A hypothetical term contract covers 100 policies at issue. Each policy pays a premium of $1,000 at the start of the year. Ten deaths occur during the year, and each claim is $1,000 at year-end. The end-of-year reserve is $1,100 per policy still in force. Find the effective annual interest rate.

Method. Convert the per-survivor reserve to the cohort fund. There are $100-10=90$ survivors, so

F_1=90(1{,}100)=99{,}000.

The 100 premiums total $100(1{,}000)=100{,}000$, and the claims total $10(1{,}000)=10{,}000$. With premiums at the start and claims at the end,

$$ 100{,}000(1+i)-10{,}000=99{,}000. $$

Adding $10,000 to each side gives $100{,}000(1+i)=109{,}000$. Divide by $100,000: $1+i=1.09$, so $i=0.09$, or 9%.

Check. At 9%, the premiums accumulate to $100{,}000(1.09)=109{,}000$. After $10,000 of claims, the fund is $99,000, and $99{,}000/90=$1,100$ per survivor, as required. The timing trap avoided here is charging interest on the end-of-year claims.

Worked Example: Solving a Two-Year Interest Equation

A hypothetical cohort begins with 100 policies. The annual premium is $1,000, and the death benefit is $1,000. There are 10 deaths in each of the first two years, leaving 90 policies at the start of year 2 and 80 at the end of year 2. The reserve at duration 2 is $2,487.50 per survivor. Find the nonnegative effective annual interest rate.

Method. First convert the reserve to the total fund: $F_2=80(2{,}487.50)=199{,}000$. Let $z=1+i$. At the end of year 1, the fund is

F_1=100(1{,}000)z-10(1{,}000)=100{,}000z-10{,}000.

In year 2, 90 premiums of $1,000 are collected and 10 claims of $1,000 are paid. Thus

F_2=(F_1+90{,}000)z-10{,}000 =100{,}000z^2+80{,}000z-10{,}000.

Set this equal to the required $199,000 and divide by $1,000:

$$ 100z^2+80z-209=0. $$

The discriminant is $80^2-4(100)(-209)=90{,}000$, whose square root is 300. The roots are

z=\frac{-80\pm300}{200}=1.1\ \text{or}\ -1.9.

Since $z=1+i$ and the rate must be nonnegative, $z=1.1$ gives $i=10\%$; the negative root is inadmissible. To check, $F_1=100{,}000(1.1)-10{,}000=100{,}000$, and $F_2=(100{,}000+90{,}000)(1.1)-10{,}000=199{,}000$. Dividing by 80 gives $2,487.50. The method avoids treating the duration-2 reserve as the total fund: it is a per-survivor amount and must be multiplied by 80.

Solving for a Benefit Amount

If the interest rate is known, each past claim can be accumulated to the reserve duration. The total fund is a linear function of $B$. Provided at least one claim has occurred by duration $t$, the accumulated-claim coefficient is positive, so the equation can be rearranged directly:

B= \frac{ P\displaystyle\sum_{j=0}^{t-1}N_j(1+i)^{t-j} -N_t\,{}_tV_x }{ \displaystyle\sum_{j=0}^{t-1}D_j(1+i)^{t-j-1} }.

This expression applies under the stated annual timing assumptions. If no claims have occurred by duration $t$, the denominator is zero and that reserve observation contains no information about the benefit amount.

Worked Example: Finding the Death Benefit From a Two-Year Reserve

A hypothetical cohort has 100 policies at issue, 10 deaths in each of the first two years, and 80 survivors at duration 2. The premium is $1,000 per policy at the start of each year, and the effective annual interest rate is 10%. The reserve at duration 2 is $2,225 per survivor. Find the unknown level death benefit $B$.

Method. The required fund is $80(2{,}225)=178{,}000$. Let $B$ be the benefit per claim. At the end of year 1,

F_1=100(1{,}000)(1.1)-10B=110{,}000-10B.

The year-2 premiums total $90(1{,}000)=90{,}000$, which is added before interest; 10 claims are paid at year-end. Therefore

F_2=(F_1+90{,}000)(1.1)-10B =(200{,}000-10B)(1.1)-10B =220{,}000-21B.

Set $F_2=178{,}000$: $220{,}000-21B=178{,}000$. Hence $21B=42{,}000$, and $B=$2{,}000$.

Check. With $B=$2,000$, $F_1=110{,}000-20{,}000=$90,000$. Then $F_2=(90{,}000+90{,}000)(1.1)-20{,}000=$178,000$, or $2,225 for each of 80 survivors. The year-1 claims must also be accumulated through year 2; omitting that accumulation would give the wrong coefficient of $B$.

Using Multiple Reserve Observations

A reserve at another duration provides an important consistency check. If the same $P$, $B$, and $i$ apply throughout, one solved parameter should reproduce all supplied reserves. With unknown interest and unknown benefit simultaneously, a single reserve equation will generally not determine both; additional independent information is needed. Even with several duration observations, solve the equations together and verify the original cash-flow balances rather than relying only on rounded intermediate values.

The balance equations also make the role of the cohort explicit. A reserve per policy in force is conditional on survival to the valuation duration, while the accumulated past premiums and claims are cohort totals. Mixing these bases—using a per-policy premium with a total fund, for example—will produce a scale error. A useful check is that $F_t$ is in dollars for the whole cohort and ${}_tV_x$ is dollars per survivor.

Key takeaway: To infer a contract parameter from a retrospective reserve, first convert the per-survivor reserve to the aggregate fund, then accumulate every past premium and claim according to its payment date. Solve within the stated parameter domain and check all available durations.

Check Your Understanding

Use annual premiums at the start of each year and death claims at year-end unless a question says otherwise. Each problem is hypothetical.

A cohort of 100 policies pays $1,000 each at the start of year 1. Ten claims of $1,000 are paid at year-end, leaving 90 policies. The duration-1 reserve is $1,100 per survivor. Find $i$. (Answer: $9\%$)
A cohort has 20 policies, each paying a $500 premium at the start of the year. Two deaths occur, 18 policies survive, and the duration-1 reserve is $500 per survivor. At $i=5\%$, find the death benefit. (Answer: $750)
A cohort starts with 50 policies, pays $200 per policy at the start of each of two years, and has five deaths in each year. The interest rate is 10%, and the duration-2 reserve is $287.50 per survivor. Find the level death benefit. (Answer: $1,000)
A cohort starts with 40 policies, has four deaths in each of two years, and pays $500 per policy at each year’s start. The death benefit is $500, and the duration-2 reserve is $1,151.5625 per survivor. Find the nonnegative interest rate. (Answer: $5\%$)
A cohort begins with 30 policies, has three deaths in each of two years, and pays $400 per policy at each year’s start. The interest rate is zero. The duration-1 reserve is $300 per survivor. Find the benefit, then verify the duration-2 reserve. (Answer: benefit $1,300; duration-2 reserve $625 per survivor)

What You'll Learn