Inferring Interest from Endowment Recursions

Why Infer Interest from Endowment Values?

An endowment value reflects both mortality and interest: the policy may pay a death benefit during the year, or its value may continue if the insured survives. If the mortality probabilities are known and values are available at two consecutive durations, the one-year recursion isolates the discount factor. Converting that factor to an effective annual interest rate is straightforward, but confusing the rate with the discount rate can produce a plausible-looking wrong answer.

The method in this tutorial applies to a one-year step in a discrete endowment model. Death benefits are assumed paid at the end of the year of death, and values are measured at integer durations. The timing assumption matters: a benefit paid immediately at death would require a different discounting treatment.

Definition: Let $E_k$ be the value at duration $k$ of a level-benefit endowment contract, conditional on the insured being alive at that duration. Let $q_{x+k}$ be the probability of death during the next year and $p_{x+k}=1-q_{x+k}$ the probability of survival through that year. For a unit endowment that pays 1 at the end of the year of death or 1 at maturity, the one-year recursion is

E_k=v\left(q_{x+k}+p_{x+k}E_{k+1}\right), \qquad v=\frac{1}{1+i}.

Here $i$ is the effective annual interest rate, and $E_{k+1}$ is the value at the next duration conditional on survival to it.

Solving the Recursion for Interest

Write $q=q_{x+k}$, $p=p_{x+k}$, and $E_{k+1}$ for the adjacent-duration inputs. Define the expected value one year ahead, before discounting, by $D=q+pE_{k+1}$. The recursion gives $E_k=vD$. Provided $D>0$, the observed values therefore determine $v$, and hence $i$.

v=\frac{E_k}{q+pE_{k+1}}, \qquad i=\frac{1}{v}-1 =\frac{q+pE_{k+1}}{E_k}-1.

This inversion requires the benefit scale and timing to be correctly represented in the recursion. In the unit endowment, the death benefit in the recursion is 1. If the death benefit is $B$, while the value for a survivor at duration $k+1$ is $E_{k+1}$, the corresponding step is $E_k=v(qB+pE_{k+1})$. The same algebra applies after replacing the unit death benefit by $B$.

If $E_k$ and $D$ are positive, the inferred discount factor is positive. Under a nonnegative-interest assumption, a further check is required: $0<v\leq 1$, equivalently $E_k\leq D$. If $E_k>D$, the recursion implies a negative effective annual interest rate. That does not make the algebra invalid, but it conflicts with an assumption that interest is nonnegative.

Why the Inversion Works

The recursion follows by conditioning on what happens during the next year. With probability $q$, death occurs during that year and the end-of-year benefit is paid. With probability $p$, the insured survives and the contract has value $E_{k+1}$ at the next duration. The probability-weighted value at that next year-end is therefore $q+pE_{k+1}$. Discounting that amount one year to duration $k$ multiplies it by $v$.

Thus the current value is the product of the one-year discount factor and a quantity determined by the known mortality probabilities and the next-duration value. Dividing by that quantity recovers $v$; taking its reciprocal and subtracting 1 recovers the effective annual interest rate. The method is a one-step inference: it does not require values at every duration, but it does require that the adjacent values belong to the same contract basis and use the same benefit timing.

Identify the contract step.
Confirm that the two values are $E_k$ and $E_{k+1}$, and identify the benefit payable on death during the year.

Calculate the one-year expected value.
For a unit death benefit, calculate $D=q_{x+k}+p_{x+k}E_{k+1}$.

Infer the discount factor.
Use $v=E_k/D$, retaining enough precision for the next step.

Convert and check.
Calculate $i=1/v-1$. Check that the inferred $v$ satisfies any stated interest restriction.

Worked Applications

Worked Example: Inferring Interest for a Unit Endowment

A hypothetical unit endowment pays 1 at the end of the year of death or 1 at maturity. For one year of the contract, the mortality assumption is $q_{x+k}=0.20$, so $p_{x+k}=0.80$. The values are $E_{k+1}=0.75$ and $E_k=0.76$. Find the effective annual interest rate. Assume the same annual effective rate applies throughout the year and death benefits are paid at year-end.

Method. Apply the unit endowment recursion, then invert it. This avoids treating $E_{k+1}$ as the only amount due at the next year-end; the death benefit in the same year must also be included.

First calculate the value one year ahead before discounting:

D=q_{x+k}+p_{x+k}E_{k+1} =0.20+(0.80)(0.75) =0.20+0.60 =0.80.

Now infer the discount factor:

v=\frac{E_k}{D}=\frac{0.76}{0.80}=0.95.

Convert $v$ to the effective annual interest rate:

i=\frac{1}{0.95}-1 =\frac{20}{19}-1 =\frac{1}{19} =0.0526316\ldots.

Rounded to two decimal places as a percentage, $i=5.26\%$. As a check, $0.95(0.80)=0.76=E_k$. Also, $v<1$, consistent with positive interest.

Worked Example: Inferring Interest with a Dollar Benefit

A hypothetical endowment pays \$50,000 at the end of the year of death or at maturity. During the year, $q_{x+k}=0.20$ and $p_{x+k}=0.80$. The next-duration value is $E_{k+1}=\$48{,}000$, and the current value is $E_k=\$45{,}600$. Find the effective annual interest rate. Assume the stated values use this same \$50,000 benefit scale.

Method. Since the death benefit is \$50,000, the death branch contributes $q_{x+k}\times \$50{,}000$. The continuation branch contributes $p_{x+k}E_{k+1}$. Keep both contributions in dollars before dividing.

The expected value one year ahead before discounting is

D=(0.20)(\$50{,}000)+(0.80)(\$48{,}000) =\$10{,}000+\$38{,}400 =\$48{,}400.

Therefore,

v=\frac{\$45{,}600}{\$48{,}400} =\frac{114}{121} \approx 0.94214876.

The effective annual interest rate is

i=\frac{1}{v}-1 =\frac{121}{114}-1 =\frac{7}{114} \approx 0.0614035.

Rounded to two decimal places as a percentage, $i=6.14\%$. Check: $v(\$48{,}400)=(114/121)(\$48{,}400)=\$45{,}600$, so the recursion reproduces the given current value. The death benefit contribution is \$10,000, not \$50,000: it is multiplied by the probability of death.

Worked Example: Distinguishing Interest from Discount

For a hypothetical unit endowment, suppose $q_{x+k}=0.15$, $p_{x+k}=0.85$, $E_{k+1}=0.60$, and $E_k=0.594$. Infer the discount factor and effective annual interest rate. Also calculate the annual discount rate $d=1-v$. Assume death benefits are paid at year-end.

Method. First infer $v$ from the recursion. Then calculate $d=1-v$ and $i=1/v-1$ separately; these rates are not interchangeable.

The value one year ahead before discounting is

$$ D=0.15+(0.85)(0.60) =0.15+0.51 =0.66. $$

Thus

v=\frac{0.594}{0.66}=0.90.

The annual discount rate and effective annual interest rate are, respectively,

d=1-v=1-0.90=0.10, \qquad i=\frac{1}{0.90}-1=\frac{1}{9}=0.111111\ldots.

Therefore $d=10.00\%$, while the effective annual interest rate is $i=11.11\%$, each rounded to two decimal places as a percentage. Check: $vD=(0.90)(0.66)=0.594$. Reporting 10.00% as $i$ would confuse the discount rate with the interest rate.

Checks, Assumptions, and Common Traps

After solving for interest, substitute the inferred discount factor back into the original recursion. This check catches arithmetic errors and omitted benefit branches. Also verify that the inputs are on a consistent scale: a value stated per \$1,000 of benefit cannot be combined directly with a death benefit stated in dollars unless the units are reconciled.

The inferred rate is only as meaningful as the model assumptions. In particular, this one-year formula uses annual survival and death probabilities, a single effective annual interest rate, and benefits paid at year-end. It does not directly apply to a contract whose death benefit is paid immediately at death, or whose benefit amounts vary during the year. In those cases the timing or benefit terms in the expected value must be adjusted to match the contract.

Check	What to verify
Adjacent durations	$E_k$ and $E_{k+1}$ refer to consecutive durations for the same contract.
Probability inputs	$p_{x+k}=1-q_{x+k}$, with both probabilities on the same one-year interval.
Benefit and units	The death benefit and continuation value use compatible benefit scales and currency units.
Rate conversion	Use $i=1/v-1$; $1-v$ is the discount rate $d$, not $i$.
Interest restriction	If interest is assumed nonnegative, the inferred $v$ must satisfy $v\leq 1$.

A frequent exam trap is to report $1-v$ as the effective annual interest rate. The two quantities are related, but they differ unless the rate is zero: $d=1-v$, whereas $i=1/v-1$. Another trap is to omit the death-benefit term and calculate $D=pE_{k+1}$ alone. That ignores the possibility of death during the year and changes the inferred interest rate.

This technique is useful whenever a problem provides adjacent endowment values and known mortality but leaves the interest basis unknown. It turns a recursion into a direct calculation, while the substitution and assumption checks help distinguish a correct inference from a numerically plausible but contractually inconsistent result.

Key takeaway: For a unit endowment with year-end benefits, calculate $D=q_{x+k}+p_{x+k}E_{k+1}$, then $v=E_k/D$ and $i=1/v-1$. Keep the death benefit, value scale, and payment timing consistent, and do not confuse $i$ with $d=1-v$.

Check Your Understanding

For each problem, use the stated one-year model and distinguish the inferred effective annual interest rate from any other rate requested.

A hypothetical unit endowment has $q_{x+k}=0.25$, $p_{x+k}=0.75$, $E_{k+1}=0.64$, and $E_k=0.68$. Find the effective annual interest rate, rounded to two decimal places as a percentage. (Answer: $7.35\%$)
For a unit endowment, $q_{x+k}=0.12$, $p_{x+k}=0.88$, $E_{k+1}=0.70$, and $E_k=0.72$. Find the effective annual interest rate, rounded to two decimal places as a percentage. (Answer: $2.22\%$)
A hypothetical endowment pays \$40,000 at death during the year or at maturity. Given $q_{x+k}=0.08$, $p_{x+k}=0.92$, $E_{k+1}=\$35{,}000$, and $E_k=\$34{,}040$, find $i$, rounded to two decimal places as a percentage. (Answer: $4.00\%$)
A unit endowment has $q_{x+k}=0.15$, $p_{x+k}=0.85$, $E_{k+1}=0.60$, and $E_k=0.594$. Find both $d=1-v$ and $i=1/v-1$, each rounded to two decimal places as a percentage. (Answer: $d=10.00\%$, $i=11.11\%$)
A hypothetical endowment pays \$800 at death during the year or at maturity. Suppose $q_{x+k}=0.25$, $p_{x+k}=0.75$, $E_{k+1}=\$600$, and $E_k=\$625$. Infer the effective annual interest rate and state whether it is consistent with a nonnegative-interest assumption. (Answer: $4.00\%$; yes)
For a unit endowment, $q_{x+k}=0.20$, $p_{x+k}=0.80$, $E_{k+1}=0.90$, and $E_k=0.95$. Infer the effective annual interest rate and state whether it is consistent with a nonnegative-interest assumption. (Answer: $-3.16\%$; no)

Check	What to verify
Adjacent durations	\(E_k\) and \(E_{k+1}\) refer to consecutive durations for the same contract.
Probability inputs	\(p_{x+k}=1-q_{x+k}\), with both probabilities on the same one-year interval.
Benefit and units	The death benefit and continuation value use compatible benefit scales and currency units.
Rate conversion	Use \(i=1/v-1\); \(1-v\) is the discount rate \(d\), not \(i\).
Interest restriction	If interest is assumed nonnegative, the inferred \(v\) must satisfy \(v\leq 1\).

What You'll Learn