Interest Sensitivity of One-Year Reserve Growth

Why Interest Sensitivity Needs Careful Comparison

A reserve recursion links the reserve at the start of a policy year to the reserve for a policy that survives to the next year. Interest affects the amount accumulated during that year, but a comparison across interest rates can involve two different questions: how the next reserve changes from the same starting reserve, and how reserves differ after the starting reserve has itself developed under different rates. Keeping those comparisons separate is essential when mortality, premiums, and benefits are held fixed.

The results below use an annual discrete model. Premiums are paid at the beginning of each year, a death benefit is paid at the end of the year of death, and the reserve at the end of the year applies to a policy that survives. The interest rate is an effective annual rate. For a clean interest comparison, mortality probabilities, premium amounts, and benefits do not change with the rate.

Definition: Let ${}_kV_x$ be the reserve immediately before the premium at duration $k$, for a policy issued at age $x$. Let $P_k$ be the premium paid then, $b_{k+1}$ the death benefit paid at the end of the year if death occurs during that year, and $i$ the effective annual interest rate. Write ${}_1p_{x+k}$ and ${}_1q_{x+k}$ for the one-year survival and death probabilities, with ${}_1p_{x+k}+{}_1q_{x+k}=1$. Under these timing assumptions, $$ {}_1p_{x+k}\,{}_{k+1}V_x = (1+i)({}_kV_x+P_k)-{}_1q_{x+k}b_{k+1}. $$

The recursion is conditional on survival when it specifies ${}_{k+1}V_x$. Before dividing by the survival probability, the accumulated amount $(1+i)({}_kV_x+P_k)$ must cover the expected death-benefit amount ${}_1q_{x+k}b_{k+1}$ as well as the reserve needed for survivors. This is an annual recursion: it does not directly describe a model with premiums or benefits paid continuously within the year.

Comparing Rates from the Same Starting Reserve

At a fixed duration, abbreviate the starting reserve by $V$, the premium by $P$, the one-year survival probability by $p$, the death probability by $q$, and the benefit by $b$. Thus $p=1-q$, and the next reserve at rate $i$ is $$ V_{\text{next}}(i)=\frac{(1+i)(V+P)-qb}{p}. $$

If $V,P,p,q,$ and $b$ are identical in two calculations, the benefit and mortality terms cancel when the next reserves are subtracted. Consequently, the difference in next reserves is determined directly by the difference in accumulated interest.

Theorem: For two effective annual rates $i_1$ and $i_2$, with the same starting reserve, premium, mortality probabilities, and benefit, $$ V_{\text{next}}(i_2)-V_{\text{next}}(i_1) = \frac{(i_2-i_1)(V+P)}{p}. $$ The same difference applies to one-year reserve growth $V_{\text{next}}-V$, because the starting reserve is held fixed.

The factor $1/p$ appears because the recursion expresses the amount available for survivors as a reserve per survivor. If $V+P>0$, a higher rate increases the next reserve from that fixed starting value. If $V+P<0$, the sign reverses. The formula describes a change in the next reserve, not necessarily the percentage change in the reserve or the effect on a reserve path over many years.

Worked Example: Comparing Two Rates at a Fixed Duration

A hypothetical policy has a starting reserve of \$500, a beginning-of-year premium of \$100, a one-year death probability of $0.20$, and an end-of-year death benefit of \$700. Compare the one-year reserve growth at effective annual rates of $4\%$ and $9\%$. The mortality probabilities and all contract amounts are fixed.

Method. Apply the reserve recursion separately at each rate, using $p=1-0.20=0.80$. The expected death-benefit amount is $qb=0.20(700)=\$140$.

At $i=0.04$, the reserve and premium accumulate to $1.04(500+100)=1.04(600)=\$624$. After allowing for the death-benefit amount, the amount assigned to survivors is $624-140=\$484$. Therefore, $$ V_{\text{next}}(0.04)=\frac{484}{0.80}=\$605.00, \qquad \text{growth}=\$605.00-\$500.00=\$105.00. $$

At $i=0.09$, the accumulated amount is $1.09(600)=\$654$. The amount remaining after the death-benefit allowance is $654-140=\$514$, so $$ V_{\text{next}}(0.09)=\frac{514}{0.80}=\$642.50, \qquad \text{growth}=\$642.50-\$500.00=\$142.50. $$

The difference in growth is $\$142.50-\$105.00=\$37.50$. The sensitivity formula gives the same result: $(0.09-0.04)(600)/0.80=\$37.50$. The calculation is checked both by the two recursions and by the direct difference formula. Common mistake avoided: comparing only the interest earned on the starting reserve and omitting the premium. Interest applies to the full beginning-of-year amount $V+P$.

Why the Difference Formula Holds

The formula follows by writing the recursion at each rate and subtracting. For a fixed starting reserve and fixed contract quantities, both calculations contain the same $qb$, so that term disappears:

\begin{aligned} p\{V_{\text{next}}(i_2)-V_{\text{next}}(i_1)\} &=(1+i_2)(V+P)-qb-\bigl((1+i_1)(V+P)-qb\bigr)\\ &=(i_2-i_1)(V+P). \end{aligned}

Since $p>0$ in a model where a survivor reserve is defined, division by $p$ gives the result. The cancellation explains why mortality and the benefit do not appear in this particular local difference formula: they are held equal in both calculations. They still affect the level of each next reserve, and they can affect later comparisons by changing the reserves from which those later years begin.

When the Starting Reserve Also Changes

In a multi-year comparison, the reserve at duration $k$ may differ across rates even though premiums, mortality, and benefits are unchanged. The fixed-starting-reserve formula alone then does not give the full difference in the next reserves. Let $i_2$ be the higher rate, $i_1$ the lower rate, and define $\Delta_k={}_kV_x(i_2)-{}_kV_x(i_1)$. Subtracting the two recursions gives

{}_1p_{x+k}\,\Delta_{k+1} = (1+i_2)\Delta_k +(i_2-i_1)\bigl({}_kV_x(i_1)+P_k\bigr).

There are two contributions. The term involving $\Delta_k$ carries forward the difference already present at duration $k$; the other term is the new interest effect on the lower-rate reserve plus premium. This equation assumes the same mortality probabilities, premium, and benefit in both scenarios. If the initial reserve is also the same, then $\Delta_0=0$, but a difference can arise in later durations.

Worked Example: Following the Rate Effect Through Two Years

A hypothetical two-year policy begins with a reserve of \$0. The premium is \$100 at the beginning of each year, and the death benefit is \$500 at the end of either year of death. The first-year death probability is $0.10$, and the second-year death probability is $0.20$. Compare the reserves at durations 1 and 2 under effective annual rates of $0\%$ and $10\%$, with all mortality and contract amounts held fixed.

Method. Apply the recursion year by year. In year 1, $p_1=0.90$ and the death-benefit amount is $0.10(500)=\$50$. At $0\%$, $$ {}_1V_x(0)=\frac{1.00(0+100)-50}{0.90} =\frac{50}{0.90} =\$55.5556. $$ At $10\%$, $$ {}_1V_x(0.10)=\frac{1.10(0+100)-50}{0.90} =\frac{60}{0.90} =\$66.6667. $$

For year 2, $p_2=0.80$ and the death-benefit amount is $0.20(500)=\$100$. At $0\%$, $$ {}_2V_x(0)=\frac{1.00(55.5556+100)-100}{0.80} =\frac{55.5556}{0.80} =\$69.4444. $$ At $10\%$, $$ {}_2V_x(0.10)=\frac{1.10(66.6667+100)-100}{0.80} =\frac{83.3334}{0.80} \mathrel{\approx}\$104.1667. $$

The duration-2 reserve difference is approximately $\$104.1667-\$69.4444=\$34.7223$, or $\$34.7222$ before rounding. The forward-difference equation checks it: $\Delta_1=100/9$, and $$ \Delta_2 = \frac{1.10(100/9)+0.10(55.5556+100)}{0.80} = \frac{250/9}{0.80} = \frac{625}{18} \mathrel{\approx}\$34.7222. $$ The rates, probabilities, benefits, and premiums have not been changed between scenarios; only the interest rate differs. Common mistake avoided: using the same duration-1 reserve in both year-2 calculations. The year-2 comparison must use each rate's own duration-1 reserve.

A useful differential form expresses the same path effect when the reserve is viewed as a function of the interest rate. If mortality, premiums, and benefits are fixed and the reserve is differentiable in $i$, let $W_k=d({}_kV_x)/di$. Differentiating the recursion gives

{}_1p_{x+k} W_{k+1} = (1+i)W_k+({}_kV_x+P_k).

This equation makes the two sources of sensitivity explicit: the current reserve's sensitivity is carried forward, and the current accumulated reserve-plus-premium creates a new interest effect. It is not a claim that reserves must always rise with interest. For example, a negative reserve-plus-premium can make the new direct effect negative.

Negative Growth and Sign Checks

Reserve growth need not be positive. A large expected death-benefit amount can make the reserve for survivors lower than the starting reserve, even after interest is credited. To find the growth directly from the recursion, subtract $V$ from the next reserve:

V_{\text{next}}-V = \frac{i(V+P)+P+q(V-b)}{p}.

This expression separates several effects, but the recursion remains the safest calculation route. In particular, the formula for the difference between rates concerns $V+P$, not the sign of the reserve growth itself.

Worked Example: A Higher Rate Makes Negative Growth Less Negative

A hypothetical policy has a starting reserve of \$300, a beginning-of-year premium of \$50, a one-year death probability of $0.20$, and a year-end death benefit of \$1,000. Compare reserve growth at $3\%$ and $8\%$ effective annual interest.

Method. Here $p=0.80$ and the death-benefit amount is $0.20(1{,}000)=\$200$. At $3\%$, the accumulated reserve and premium are $1.03(300+50)=\$360.50$. Thus, $$ V_{\text{next}}(0.03)=\frac{360.50-200}{0.80}=\$200.625, \qquad \text{growth}=\$200.625-\$300=-\$99.375. $$ At $8\%$, the accumulated amount is $1.08(350)=\$378$, giving $$ V_{\text{next}}(0.08)=\frac{378-200}{0.80}=\$222.50, \qquad \text{growth}=\$222.50-\$300=-\$77.50. $$

Growth is negative at both rates, but it is less negative at the higher rate. The reserve difference is $\$222.50-\$200.625=\$21.875$, which agrees with $(0.08-0.03)(350)/0.80=\$21.875$. All amounts are calculated without intermediate rounding; reserve values are shown to the nearest cent where applicable. Common mistake avoided: concluding that negative reserve growth means a higher interest rate must lower the next reserve. Here $V+P=\$350>0$, so the higher rate raises the next reserve even though both one-year growth amounts are negative.

Common Exam Traps

Changing more than the interest rate. If the premium is recalculated when the rate changes, this is not a comparison with premiums held fixed. The direct formula then needs to account for the changed premium as well.
Omitting the premium from the amount earning interest. Under the stated timing, $V+P$, not just $V$, accumulates during the year.
Using the wrong survival probability. The next reserve is conditional on survival, so the difference in accumulated amounts is divided by $p$, not multiplied by it.
Confusing a local comparison with a path comparison. The fixed-starting-reserve formula applies only when the reserve at the start of the compared year is the same. If prior reserves differ, carry that difference forward through the recursion.
Assuming the direction without checking the sign. The local interest effect has the sign of $(i_2-i_1)(V+P)$. A negative $V+P$ reverses the usual direction.

Key takeaway: With a fixed starting reserve and fixed contract quantities, changing the annual rate changes the next reserve by $(i_2-i_1)(V+P)/p$. Across multiple years, earlier rate effects alter later starting reserves, so use the reserve-difference recursion rather than treating every year as an independent fixed-start comparison.

Check Your Understanding

For each problem, keep the stated premiums, benefits, and mortality probabilities fixed across any interest-rate comparison.

A hypothetical policy has $V=\$1{,}200$, $P=\$80$, $q=0.05$, and $b=\$1{,}500$. Find the next reserve at $2\%$ and $5\%$, and compare the one-year growth amounts. (Answer: $p=0.95$; next reserves are \$1,295.37 and \$1,335.79, respectively; growth increases by \$40.42.)
A policy has a fixed starting reserve of \$700, a premium of \$50, $q=0.10$, and a death benefit of \$600. Compare next reserves at $3\%$ and $7\%$. (Answer: \$791.67 at $3\%$ and \$825.00 at $7\%$; the difference is \$33.33.)
A two-year policy starts with $V_0=\$20$. Premiums are \$80 each year and death benefits are \$300 in year 1 and \$400 in year 2. The death probabilities are $0.10$ and $0.25$, respectively. Calculate the duration-2 reserve at $0\%$ and $5\%$. (Answer: \$77.0370 at $0\%$ and \$95.3333 at $5\%$; the difference is \$18.2963.)
A one-year policy has $V=\$400$, $P=\$100$, $p=0.80$, $q=0.20$, and $b=\$900$. What effective annual interest rate gives a next reserve of \$600? (Answer: $i=0.32$, or $32\%$.)
A policy has $V=-\$200$, $P=\$50$, $p=0.90$, $q=0.10$, and $b=\$500$. Compare next reserves at $2\%$ and $6\%$, and explain the direction of the change. (Answer: $-\$225.56$ at $2\%$ and $-\$232.22$ at $6\%$; the higher rate lowers the next reserve by \$6.67 because $V+P=-\$150$.)

What You'll Learn