Discounting Benefits Under Yearly Changing Interest Rates

Why the Timing of Interest Rates Matters

A life insurance benefit paid at the end of policy year 3 is discounted through three years, not just through the interest rate in year 3. If the effective interest rate changes from one year to the next, each year's accumulation factor contributes to the value at issue. The discount factor for a payment therefore depends on the whole sequence of rates up to its payment date.

This tutorial develops that sequence for a discrete death benefit and combines it with the probability of death in each policy year. The method applies when the yearly interest rates are known and deterministic. If future rates are uncertain, a single fixed discount sequence is not enough; the valuation must account for the rate uncertainty as well.

Definition: Let $i_j$ be the effective annual interest rate applying during policy year $j$, for $j=1,2,\ldots$, with $i_j>-1$. The accumulated value at time $k$ of 1 invested at time 0 is $\prod_{j=1}^{k}(1+i_j)$. Thus the discount factor from time $k$ to time 0 is $$ D_k=\prod_{j=1}^{k}\frac{1}{1+i_j} =\frac{1}{\prod_{j=1}^{k}(1+i_j)},\qquad D_0=1. $$

The subscript $j$ identifies the policy year in which the rate applies. In particular, $i_k$ is the rate for year $k$, but a payment at the end of year $k$ is discounted using $i_1,\ldots,i_k$. A rate sequence is not the same thing as one interest rate applied repeatedly.

Deriving the Policy-Year Discount Factor

Suppose 1 is invested at time 0. At the end of year 1 it has accumulated to $1+i_1$. During year 2, that amount earns interest at rate $i_2$, so the value becomes $(1+i_1)(1+i_2)$. Repeating this year by year gives the accumulation through year $k$. Discounting reverses that accumulation, which gives the product of the reciprocal factors.

Accumulate through year 1.
One unit at time 0 becomes $1+i_1$ at time 1.

Continue one year at a time.
Each year multiplies the existing accumulated value by that year's factor $1+i_j$.

Reverse the accumulation.
To express a time-$k$ payment at time 0, divide by the full accumulated factor.

For a check, the discount factors obey the recursion $D_k=D_{k-1}/(1+i_k)$. This follows directly by separating the final factor from the product:

D_k =\frac{1}{\prod_{j=1}^{k-1}(1+i_j)}\frac{1}{1+i_k} =D_{k-1}\frac{1}{1+i_k}.

This recursion is also a practical way to calculate the factors without repeatedly multiplying the full sequence.

Using the Factors in a Discrete Insurance APV

Assume a life aged $x$ is covered by an $n$-year term insurance. A benefit $B_k$ is paid at the end of policy year $k$ if death occurs during that year. Mortality follows the usual discrete-year convention: death in year $k$ means survival through $k-1$ years followed by death during the next year. Interest rates $i_1,\ldots,i_n$ are known and do not depend on the insured's survival.

Theorem: Under these assumptions, the expected present value of the death benefits is $$ \operatorname{APV} =\sum_{k=1}^{n} B_k D_k\,{}_{k-1}p_xq_{x+k-1} =\sum_{k=1}^{n} B_k \left(\prod_{j=1}^{k}\frac{1}{1+i_j}\right) {}_{k-1}p_xq_{x+k-1}. $$ Here $B_k$ is the benefit payable for death in year $k$, and ${}_{k-1}p_xq_{x+k-1}$ is the probability of surviving $k-1$ years and then dying in year $k$.

To justify the APV formula, define the present value of the benefit as a random variable. It equals $B_kD_k$ if death occurs in year $k$, and zero if death occurs after the term. The events of death in the different policy years are disjoint. The expected value is therefore the sum, over years, of each possible present value multiplied by the probability of its corresponding event. This gives the stated expression.

When the benefit is level, set $B_k=B$ for every covered year. When it varies by year, retain the appropriate $B_k$ in each term. The rate sequence changes the discount factors; it does not change the mortality probabilities.

Worked Example: Three-Year Term Insurance With Changing Rates

A hypothetical three-year term policy on a life aged $x$ pays $10,000 at the end of the year of death. The annual mortality probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. The effective interest rates are 4% in year 1, 5% in year 2, and 6% in year 3. Find the APV of the death benefit.

Method. Calculate each year's unconditional death probability, calculate the discount factor through that year, and multiply the benefit by both quantities.

Year 1. The probability of death in year 1 is $q_x=0.10$. The discount factor is $D_1=1/1.04=0.96153846$. The expected present value contribution is $10{,}000(0.10)(0.96153846)=$961.53846$.

Year 2. Survival through year 1 is $1-0.10=0.90$, so the probability of death in year 2 is $0.90(0.20)=0.18$. The discount factor is $D_2=1/(1.04\times1.05)=1/1.092=0.91575092$. The contribution is $10{,}000(0.18)(0.91575092)=$1{,}648.35166$.

Year 3. Survival through two years is $0.90(1-0.20)=0.90(0.80)=0.72$. The probability of death in year 3 is $0.72(0.25)=0.18$. The discount factor is $D_3=1/(1.04\times1.05\times1.06)=1/1.15752=0.86391500$. The contribution is $10{,}000(0.18)(0.86391500)=$1{,}555.04700$.

Adding the three contributions gives $$961.53846+$1{,}648.35166+$1{,}555.04700=$4{,}164.93712$. Rounded to the nearest cent, the APV is $4,164.94.

Check. The three death probabilities sum to $0.10+0.18+0.18=0.46$, which is the probability of death within three years. Each contribution is less than the corresponding undiscounted expected benefit. The calculation avoids the mistake of discounting every payment using only the rate in its payment year.

Varying Benefits and Rate Sequences

The same calculation works when benefits differ by policy year. The probability of death in year $k$ remains ${}_{k-1}p_xq_{x+k-1}$, while the amount discounted is the benefit actually payable in that year. Care is needed to keep three sequences aligned: the death probability, the benefit, and the discount factor must all refer to the same policy year.

Worked Example: A Benefit That Changes by Policy Year

A hypothetical two-year term policy pays $5,000 for death in year 1 and $8,000 for death in year 2. The mortality assumptions are $q_x=0.20$ and $q_{x+1}=0.25$. The effective interest rate is 0% in year 1 and 25% in year 2. Find the APV.

Method. Match each year's benefit to its unconditional death probability and cumulative discount factor.

For year 1, the death probability is $0.20$, and $D_1=1/(1+0)=1$. The contribution is $5{,}000(0.20)(1)=$1{,}000$.

For year 2, survival through year 1 is $1-0.20=0.80$, so the death probability is $0.80(0.25)=0.20$. The discount factor includes both annual rates: $D_2=1/[(1+0)(1+0.25)]=1/1.25=0.80$. The contribution is $8{,}000(0.20)(0.80)=$1{,}280$.

The APV is $$1{,}000+$1{,}280=$$2,280.00. The death probabilities sum to $0.40$, consistent with survival through two years of $0.80(0.75)=0.60$. In particular, the year-2 benefit is discounted through year 1 even though year 1's rate is zero.

The order of rates can matter even when two sequences have the same product over the full term. A year-1 payment uses only the first rate, while a year-2 payment uses both. Therefore, equal total accumulation to the end of year 2 does not by itself imply equal APVs.

Worked Example: Why the Order of Rates Can Change the APV

Use the two-year policy in the preceding example: benefits are $5,000 in year 1 and $8,000 in year 2; $q_x=0.20$ and $q_{x+1}=0.25$. Compare the rate sequences (0%, 25%) and (25%, 0%).

The unconditional death probabilities are $0.20$ in year 1 and $(1-0.20)(0.25)=0.20$ in year 2.

For rates (0%, 25%), the discount factors are $D_1=1$ and $D_2=1/(1\times1.25)=0.80$. The APV is $5{,}000(0.20)(1)+8{,}000(0.20)(0.80)=$1{,}000+$1{,}280=$2{,}280$.

For rates (25%, 0%), the discount factors are $D_1=1/1.25=0.80$ and $D_2=1/(1.25\times1)=0.80$. The APV is $5{,}000(0.20)(0.80)+8{,}000(0.20)(0.80)=$800+$1{,}280=$2{,}080$.

Both sequences have the same two-year accumulation factor, $1.25$, and the same year-2 discount factor. However, the first-year payment is discounted differently, so the APVs differ by $200. The comparison avoids the error of treating a matching end-of-term accumulation factor as proof that all earlier cash flows have matching values.

Common Mistakes and Useful Checks

A frequent error is to use $1/(1+i_k)^k$ for a year-$k$ payment. That expression assumes the same rate applies in every year. With rates that vary, the correct denominator is $\prod_{j=1}^{k}(1+i_j)$. Another tempting shortcut is to average the annual rates and use that average in every year. In general, this does not preserve the accumulated factors for each payment date.

For example, in the two-year policy above, the incorrect approach of applying 25% in both years would use $D_1=0.80$ and $D_2=1/(1.25^2)=0.64$. It would produce $5{,}000(0.20)(0.80)+8{,}000(0.20)(0.64)=$800+$1{,}024=$1{,}824$, rather than the correct $2,280 under rates (0%, 25%). The error comes from replacing the actual first-year rate of 0% with 25%.

Check the payment date. A benefit payable at the end of year $k$ uses exactly $k$ annual rates.
Check the death probability. The probability for year $k$ includes survival through years $1$ to $k-1$, followed by death during year $k$.
Check the benefit schedule. Use the benefit for the year of death; do not substitute a different year's amount.
Check the discount factors. For positive rates, $D_k$ should decrease as $k$ increases.

These checks help separate the mortality calculation from the interest calculation. The mortality model determines when a benefit is payable; the accumulated rate sequence determines how much that payment is worth at time 0. Keeping those jobs distinct is central to discrete insurance valuation and is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical policy, use end-of-year death benefits and known effective annual rates. Give dollar answers to the nearest cent.

A one-year policy pays $10,000 if death occurs during the year. If $q_x=0.20$ and the annual effective rate is 5%, find the APV. (Answer: $1,904.76)
A two-year policy pays $6,000 in either year. Let $q_x=0.10$, $q_{x+1}=0.20$, and let the rates be 0% in year 1 and 25% in year 2. Find the APV. (Answer: $1,464.00)
A three-year policy pays $1,000 in each year. Let $q_x=0.10$, $q_{x+1}=0.10$, $q_{x+2}=0.20$, and let the rates be 10%, 0%, and 25%, respectively. Find the APV. (Answer: $290.55)
A two-year policy pays $4,000 in year 1 and $9,000 in year 2. Let $q_x=0.30$, $q_{x+1}=0.20$, and let the rates be 25% in year 1 and 0% in year 2. Find the APV. (Answer: $1,968.00)
A two-year policy pays $10,000 in either year, with $q_x=0.10$ and $q_{x+1}=0.20$. Compare rates (0%, 25%) with rates (25%, 0%). Find the APV under each sequence. (Answer: $2,440.00 for (0%, 25%); $2,240.00 for (25%, 0%))

What You'll Learn