How Interest Rates Change Discrete Insurance APVs

Why the Interest Rate Matters

A discrete life insurance benefit is paid at a specified time, commonly at the end of the policy year of death. Its actuarial present value (APV) depends on both the probability of that payment and how far it is discounted. To isolate the effect of interest, compare APVs using the same mortality model and the same benefit amounts, changing only the effective annual interest rate.

This distinction is useful in valuation and in exam calculations: if two APVs differ, the cause might be an interest-rate change, a mortality change, or a benefit change. In this tutorial, mortality and benefits are held fixed, so any difference comes from discounting. The treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: The effective annual interest rate $i$ is the rate at which a fund grows over one year. Its annual discount factor is $v=1/(1+i)$. A dollar payable at the end of year $j$ has present value $v^j$ dollars, provided $i>-1$. For the numerical examples below, $i\geq 0$.

Discounting a Discrete Death Benefit

Let the insured be age $x$ at issue. The probability of surviving $k$ years and dying in the following year is ${}_kp_xq_{x+k}$. A benefit paid at the end of that year is paid at time $k+1$, not time $k$. If the benefit for that death year is $B_k$, its contribution to the APV is the benefit multiplied by its death probability and its discount factor.

\text{Contribution from death in year }k+1 = B_k\,{}_kp_xq_{x+k}\,v^{k+1}.

For an $n$-year term insurance with benefits paid at the end of the year of death, the APV is the sum of these contributions over the covered years. The mortality probabilities and benefit schedule do not depend on $i$; only $v^{k+1}$ changes when the interest rate changes.

A^{1}_{x:\overline{n}|}(i) = \sum_{k=0}^{n-1} B_k\,{}_kp_xq_{x+k}(1+i)^{-(k+1)}.

For whole-life insurance with a constant benefit $B$, the analogous sum runs over all possible death years. The same timing rule applies: death in policy year $k+1$ produces a payment discounted for $k+1$ years.

A_x(i) = B\sum_{k=0}^{\infty}{}_kp_xq_{x+k}(1+i)^{-(k+1)}.

Worked Example: Comparing Two Rates for a Term Benefit

A hypothetical three-year term policy pays $10,000 at the end of the year of death. The probabilities of death in years 1, 2, and 3, respectively, are 0.10, 0.18, and 0.18. Find the APV at effective annual interest rates of 0% and 5%. Treat the stated death-year probabilities as fixed.

Method. Multiply each death-year probability by the $10,000 benefit, then discount the resulting payment by the number of years until it is paid. The year-end payments are at times 1, 2, and 3.

At $i=0$, $v=1$, so the APV is

10{,}000(0.10)+10{,}000(0.18)+10{,}000(0.18) =1{,}000+1{,}800+1{,}800 =\$4{,}600.00.

At $i=0.05$, $v=1/1.05$. The three discounted contributions are

\begin{aligned} \text{Year 1: }&\frac{1{,}000}{1.05}=\$952.38095\ldots,\\ \text{Year 2: }&\frac{1{,}800}{1.05^2} =\frac{1{,}800}{1.1025} =\$1{,}632.65306\ldots,\\ \text{Year 3: }&\frac{1{,}800}{1.05^3} =\frac{1{,}800}{1.157625} =\$1{,}554.90768\ldots. \end{aligned}

Adding without rounding the intermediate values gives an APV of $952.38095\ldots+1{,}632.65306\ldots+1{,}554.90768\ldots=\$4{,}139.94169\ldots$. Rounded to the nearest cent, the APV is $4,139.94. It is lower than the 0% APV because the same positive benefits are discounted at a positive rate.

Check and common mistake avoided. The three death probabilities sum to 0.46, so at 0% the expected payment is $10{,}000(0.46)=\$4{,}600$, agreeing with the direct calculation. Do not discount the year-1 payment for two years or the year-3 payment for only two years.

Why a Higher Rate Lowers the APV

For each covered death year, the probability and benefit are fixed and nonnegative. When $i$ increases above $-1$, the factor $(1+i)^{-(k+1)}$ decreases. Consequently, each term in the finite term-insurance sum decreases or remains zero. Adding the terms shows that the APV cannot increase. It decreases strictly if at least one covered death year has a positive benefit and positive probability.

Result: With mortality and nonnegative benefits held fixed, a discrete term-insurance APV is a nonincreasing function of the effective annual interest rate $i>-1$. It is strictly decreasing when at least one possible covered payment has a positive amount.

The same comparison works for whole-life insurance: each possible death-year payment is discounted less at a lower rate, so its present value is no smaller. For nonnegative $i$, the whole-life sum is finite when benefits are bounded, since the sum of the death probabilities cannot exceed 1 and the discount factors are at most 1. In particular, at $i=0$, the APV is the benefit multiplied by the probability that a covered death eventually occurs.

For a finite term with general fixed amounts, the rate sensitivity can also be seen by differentiating its APV with respect to $i$. Write $w_k=B_k\,{}_kp_xq_{x+k}$, which is fixed and nonnegative. Differentiating each of the finitely many terms gives the following expression.

\frac{d}{di}A^{1}_{x:\overline{n}|}(i) = -\sum_{k=0}^{n-1} (k+1)w_k(1+i)^{-(k+2)} \leq 0,\qquad i>-1.

Hold the death-year weights fixed.
Each $w_k$ combines the benefit and the probability of death in year $k+1$; neither changes in this comparison.

Differentiate the discount factor.
The derivative of $(1+i)^{-(k+1)}$ is $-(k+1)(1+i)^{-(k+2)}$.

Use nonnegative weights.
Every term in the derivative is nonpositive, so the sum is nonpositive. If any $w_k>0$, its derivative term is negative and the APV decreases strictly.

This argument also explains why later payments are more affected in absolute terms: the derivative term includes the factor $k+1$, the payment time. The conclusion concerns the APV, not the mortality probabilities. A change in mortality is a different comparison and must not be silently combined with an interest-rate change.

Worked Example: Whole-Life Insurance Under Constant Mortality

A hypothetical whole-life policy pays $10,000 at the end of the year of death. Assume a constant annual mortality model: in every year, the conditional probability of death is $q=0.10$, so $p=1-q=0.90$. Calculate the APVs at effective annual rates of 5% and 10%.

Method and condition. Under this model, the probability of surviving $k$ years and dying in the next year is $p^kq$. The whole-life APV is therefore a geometric series with first term $Bqv$ and common ratio $pv$. For $q>0$ and $i\geq0$, $pv=0.90/(1+i)<1$, so the series converges.

Summing it gives

A_x(i) =B\sum_{k=0}^{\infty}p^kqv^{k+1} =\frac{Bqv}{1-pv} =\frac{Bq}{q+i}.

For the last equality, substitute $v=1/(1+i)$: $qv/(1-pv)=q/[1+i-p]=q/(q+i)$, because $p=1-q$. At 5%, the APV is

A_x(0.05) =\frac{10{,}000(0.10)}{0.10+0.05} =\frac{1{,}000}{0.15} =\$6{,}666.666\ldots \approx \boxed{\$6{,}666.67}.

At 10%, it is

A_x(0.10) =\frac{10{,}000(0.10)}{0.10+0.10} =\frac{1{,}000}{0.20} =\boxed{\$5{,}000.00}.

The 10% value is lower, even though the mortality model and benefit are unchanged. As a check, at $i=0$ this model gives $A_x(0)=Bq/q=B$: with $q>0$, death eventually occurs with probability 1 and the undiscounted benefit is $10,000.

Common mistake avoided. The geometric-series formula requires $pv<1$, and the simplified expression $Bq/(q+i)$ requires a nonzero denominator. Here $q>0$ and $i\geq0$, so both conditions hold. If $q=0$, the insured never dies under this model and the APV is 0 for $i\geq0$; at $q=i=0$, the simplified fraction is undefined and must not be used.

Benefits Paid at Different Times

The effect of a rate change is not determined by the total expected benefit alone. The timing of the possible payments matters because a payment in a later year has a higher power of $v$. If a benefit schedule gives larger amounts in later years, those amounts may contribute substantially at a low rate but are more heavily discounted at a higher rate.

Worked Example: A Changing Benefit Schedule

A hypothetical three-year term policy pays $1,000 if death occurs in year 1, $2,000 if it occurs in year 2, and $3,000 if it occurs in year 3. The death-year probabilities are 0.20, 0.30, and 0.50. Calculate the APV at 0% and 10%.

Method. For each year, first multiply the benefit by its death-year probability. This gives expected payments of $1{,}000(0.20)=\$200$, $2{,}000(0.30)=\$600$, and $3{,}000(0.50)=\$1{,}500$. Then discount each at its payment time.

At 0%, there is no discounting:

200+600+1{,}500=\boxed{\$2{,}300.00}.

At 10%, $v=1/1.10$, and the APV is

\begin{aligned} \frac{200}{1.10}&=\$181.81818\ldots,\\ \frac{600}{1.10^2}&=\frac{600}{1.21}=\$495.86777\ldots,\\ \frac{1{,}500}{1.10^3}&=\frac{1{,}500}{1.331}=\$1{,}126.97220\ldots,\\ \text{Total}&=\$1{,}804.65815\ldots\approx\boxed{\$1{,}804.66}. \end{aligned}

The result is below $2,300.00, as expected for nonnegative payments discounted at a positive rate. The year-3 payment has the largest expected amount, but it is also discounted for the longest period. The sum of the stated death probabilities is 1, providing a check that this example assigns a death year within the three-year coverage period.

Common mistake avoided. Do not apply the year-1 benefit to every death year, or discount all three expected payments by the same factor. The schedule specifies both the amount and the time of each possible payment.

Exam Traps and Practical Checks

A reliable comparison begins by writing the death-year probabilities and benefit amounts before substituting an interest rate. This keeps the mortality-and-benefit part of the calculation separate from the discounting part. Recalculate the discount factors when changing $i$, but do not alter the death probabilities or benefit schedule unless the problem explicitly changes those assumptions.

Discounting from the wrong time: a benefit paid at the end of policy year $k+1$ uses $v^{k+1}$. Using $v^k$ discounts it for one year too few.
Confusing effective annual rates: if the stated rate is effective annually, use $v=1/(1+i)$. Do not treat it as a nominal rate convertible more than once per year.
Changing assumptions unintentionally: when isolating an interest effect, keep all mortality probabilities and benefits fixed.
Applying the geometric sum without checking it: for the constant-mortality whole-life formula, verify $pv<1$. Handle a zero death probability separately rather than substituting into an undefined closed form.
Rounding too early: retain full precision in discount factors and contributions, then round the final dollar amount to the nearest cent.

Key takeaway: For fixed mortality and nonnegative benefits, increasing the effective annual interest rate reduces or leaves unchanged the APV of discrete term or whole-life insurance. The payment date determines the power of the annual discount factor; death in year $k+1$ means discounting by $v^{k+1}$.

Check Your Understanding

For each problem, keep the stated mortality and benefit assumptions fixed when comparing interest rates. Round dollar answers to the nearest cent.

A one-year policy pays $20,000 at the end of the year if death occurs. The probability of death during the year is 0.04. Find its APV at an effective annual interest rate of 4%. (Answer: $769.23)
A two-year policy pays $5,000 at the end of the year of death. The death-year probabilities are 0.10 and 0.18. Find the APV at 0% and at 10%. (Answer: $1,400.00 at 0%; $1,198.35 at 10%)
A whole-life policy pays $8,000 at the end of the year of death. Under a constant annual mortality model, $q=0.20$. Find the APV at an effective annual interest rate of 5%. (Answer: $6,400.00)
A two-year term policy pays $1,000 for death in year 1 and $3,000 for death in year 2. The death-year probabilities are 0.10 and 0.20. Find the APV at 5%. (Answer: $639.46)
A three-year policy pays $4,000 at the end of the year of death. The death-year probabilities are 0.05, 0.09, and 0.12. Find the APV at 0%, then state whether the APV at 6% must be lower, equal, or higher. (Answer: $1,040.00 at 0%; lower at 6%)

What You'll Learn