How Interest and Mortality Change Net Premiums

Why Interest and Mortality Must Be Considered Together

A net premium balances the expected present value of benefits against the expected present value of premiums. Interest changes the present value of payments at different times. Mortality changes both the probability and timing of insurance claims, and it can also change how many premiums are expected to be paid. These effects are related, but they are not interchangeable.

The distinction matters especially for a net level premium. Its numerator is the expected present value of benefits, while its denominator is the expected present value of the premium-payment stream. A change in interest or mortality can affect both. It is therefore unsafe to infer the direction of a level-premium change by looking only at the benefit side.

Definition: A net single premium is the expected present value, at issue, of the future benefits under the stated assumptions. A net level premium is a constant premium satisfying the equivalence principle: the expected present value of premiums equals the expected present value of benefits. These premiums exclude expenses.

For a discrete insurance benefit paid at the end of the year of death, let $B_k$ be the benefit if death occurs in policy year $k+1$, and let ${}_{k|}q_x$ be the probability that a life aged $x$ survives $k$ years and dies in the following year. At effective annual interest rate $i$, with $v=1/(1+i)$, the net single premium for a benefit payable only within the first $n$ years is

\operatorname{NSP} = \sum_{k=0}^{n-1} v^{k+1}\,{}_{k|}q_x B_k.

If premiums are paid annually in advance while the insured is alive, for at most $n$ payments, their expected present value per dollar of annual premium is

\ddot{a}_{x:\overline{n}|} = \sum_{k=0}^{n-1} v^k\,{}_kp_x.

Here ${}_kp_x$ is the probability of surviving $k$ years. Equating the two present values gives the net level premium $P$:

P = \frac{\operatorname{NSP}}{\ddot{a}_{x:\overline{n}|}}.

How Interest Changes Present Values

For a fixed payment $C$ due at time $t$, its present value is $Cv^t=C(1+i)^{-t}$. If $C>0$ and $t>0$, increasing $i$ decreases this present value. Thus, with the mortality probabilities and benefits held fixed, a higher effective interest rate decreases the net single premium for a positive death benefit paid in the future.

A level premium has a second interest-sensitive quantity: the premium annuity present value. As the interest rate rises, that value also falls when premiums are due after issue. Since both the numerator and denominator of the level-premium formula decrease, the direction of the ratio is not established by either decrease alone. It depends on the benefit and premium timing.

Important qualification: A higher interest rate reduces the present value of a fixed future payment. It does not, by itself, guarantee that every net level premium decreases: the expected present value of the premium stream also changes. Calculate or compare the full ratio for the contract being considered.

Worked Example: Interest and a One-Year Net Single Premium

A hypothetical one-year term policy pays $12,000 at the end of the year if the insured dies during that year. Assume $q_x=0.03$, with no benefit on survival, and compare effective annual interest rates of 2% and 8%. Find the net single premium at each rate and explain the change.

Method. For one year, the probability of a claim is $q_x=0.03$, and a claim is discounted for one year. The single premium is $Bv q_x$.

At $i=0.02$, $v=1/1.02=0.9803922$. The expected claim amount before discounting is $12{,}000(0.03)=\$360$, so

\operatorname{NSP}_{0.02} = 360\left(\frac{1}{1.02}\right) = \$352.9412 \approx \$352.94.

At $i=0.08$, $v=1/1.08=0.9259259$, and

\operatorname{NSP}_{0.08} = 360\left(\frac{1}{1.08}\right) = \$333.3333 \approx \$333.33.

The premium falls by $\$352.94-\$333.33=\$19.61$, to the cent. This agrees with the present-value argument: the claim amount and probability are unchanged, but the higher rate discounts the year-end benefit more heavily. The common mistake to avoid is multiplying by the discount factor without first taking the probability-weighted claim amount, or using $1+i$ instead of $v=1/(1+i)$.

Why Mortality Can Change Both Parts of a Level Premium

For life insurance, increasing mortality can raise the expected present value of benefits because a claim becomes more likely or occurs earlier. But the same increase in mortality can reduce the expected present value of future premiums: fewer policies remain in force to make premiums due in later years.

For a level-premium term policy with a positive benefit payable on death during the term, assume $i\geq 0$ and increase one conditional death probability while keeping the other conditional probabilities fixed. The probability shifted out of later years may instead produce a claim in the year whose death probability increased, or may produce no covered claim if the policy term has ended. A claim made earlier has a present value at least as large as a later equal claim when $i\geq 0$; an uncovered outcome has value zero. Therefore, the expected present value of benefits cannot decrease. At $i=0$, it can remain unchanged if the shifted probability would otherwise have produced a later covered claim.

The premium-payment annuity value cannot increase under that same mortality change: survival to later premium dates becomes no more likely. Consequently, the level premium, which is the benefit value divided by the premium annuity value, cannot decrease when its numerator is positive. It can remain unchanged in boundary cases where neither the benefit value nor the premium annuity value changes.

The condition on interest is essential. If $i<0$, a later payment has a greater discount factor than an earlier payment. Moving probability from a later claim to an earlier claim can then reduce the expected present value of benefits. Do not state the nondecrease result for mortality changes without its nonnegative-interest and contract assumptions.

P=\frac{\operatorname{EPV}(\text{benefits})} {\operatorname{EPV}(\text{premiums per dollar of level premium})}.

This ratio also explains the contrast with a life annuity. A life annuity pays while the annuitant is alive, so increased mortality generally reduces the expected present value of its payments. For life insurance, death triggers the benefit; under the stated assumptions, increased mortality does not reduce the expected present value of a positive term death benefit. The direction depends on which event gives rise to the payment.

Worked Example: Comparing Net Level Premiums Under Two Interest Rates

A hypothetical two-year term policy pays $10,000 at the end of the year of death. Annual premiums are paid in advance while the insured is alive, for at most two payments. Assume $q_x=0.10$, $q_{x+1}=0.20$, and the same mortality assumptions at both interest rates. Compare the net level premiums at $i=0$ and $i=0.25$.

Method. The probability of death in year 1 is $q_x=0.10$. The probability of death in year 2 is ${}_1p_xq_{x+1}=0.90(0.20)=0.18$. The benefit present value is the sum of these probabilities multiplied by their discounted benefits. The premium annuity value is $1+v{}_1p_x$, since the first premium is due at issue and the second is due only if the insured survives one year.

At $i=0$, $v=1$. The benefit present value is

10{,}000\left[0.10+0.18\right] = \$2{,}800.

The premium annuity value is $1+1(0.90)=1.90$. Thus the annual net level premium is

P_0 = \frac{\$2{,}800}{1.90} = \$1{,}473.6842 \approx \$1{,}473.68.

At $i=0.25$, $v=1/1.25=0.80$ and $v^2=0.64$. The benefit present value is

10{,}000\left[0.10(0.80)+0.18(0.64)\right] = 10{,}000(0.080+0.1152) = \$1{,}952.

The premium annuity value is $1+0.80(0.90)=1.72$. Therefore,

P_{0.25} = \frac{\$1{,}952}{1.72} = \$1{,}134.8837 \approx \$1{,}134.88.

The level premium is lower at the higher rate in this example. The result is verified by the full ratio, not inferred merely from the lower benefit present value: the premium annuity value also fell, from 1.90 to 1.72. The common mistake is to discount the second benefit for one year rather than two; it is payable at the end of year 2 and therefore uses $v^2$.

Worked Mortality Comparison

A conditional mortality change affects the distribution of death times and the expected number of premium payments. The next example keeps interest and the second-year conditional death probability fixed, and increases the first-year conditional death probability. This makes the two parts of the net level premium visible.

Worked Example: Increasing First-Year Mortality

A hypothetical two-year term policy pays $10,000 at the end of the year of death and charges annual premiums in advance while the insured is alive, for at most two years. Assume $i=0.05$ and $q_{x+1}=0.20$. Compare $q_x=0.10$ with $q_x=0.20$. Calculate the net single premium and annual net level premium in both cases.

Method. Use $v=1/1.05=20/21$. For a first-year death probability $q_x$, the year-2 death probability is $(1-q_x)q_{x+1}$. The net single premium is the discounted value of both possible claims. Divide it by $1+v(1-q_x)$ to obtain the annual net level premium.

Case 1: $q_x=0.10$. The probability of death in year 1 is $0.10$; the probability of death in year 2 is $0.90(0.20)=0.18$. The benefit present value, as a proportion of the $10,000 benefit, is

0.10v+0.18v^2 = \frac{2}{21}+\frac{8}{49} = \frac{38}{147}.

Thus the net single premium is $10{,}000(38/147)=\$2{,}585.0340$, or $\$2{,}585.03$ to the cent. The premium annuity value is

1+0.90v = 1+\frac{6}{7} = \frac{13}{7}.

The annual net level premium is

P = \frac{10{,}000(38/147)}{13/7} = 10{,}000\left(\frac{38}{273}\right) = \$1{,}391.9414 \approx \$1{,}391.94.

Case 2: $q_x=0.20$. The probability of death in year 1 is $0.20$; the probability of death in year 2 is $0.80(0.20)=0.16$. The benefit present value, as a proportion of the benefit, is

0.20v+0.16v^2 = \frac{4}{21}+\frac{64}{441} = \frac{148}{441}.

The net single premium is $10{,}000(148/441)=\$3{,}356.0091$, or $\$3{,}356.01$ to the cent. The premium annuity value is $1+0.80v=1+16/21=37/21$. Therefore,

P = \frac{10{,}000(148/441)}{37/21} = 10{,}000\left(\frac{148}{777}\right) = \$1{,}905.4054 \approx \$1{,}905.41.

Both the net single premium and the annual net level premium increase. The benefit present value rises, while the premium annuity value falls from $13/7$ to $37/21$. As a check, $37/21<13/7$, and the level-premium ratio in the second case is larger. The common mistake is to treat the year-2 death probability as $q_{x+1}$ alone; it must be multiplied by the probability $1-q_x$ of surviving year 1.

A Short Justification for the Mortality Direction

The claim-value argument can be made explicitly. Consider a cohort of policies alive at the start of a particular year, and increase the conditional probability of death in that year. The extra probability mass assigned to death in that year must be removed from outcomes that previously involved survival through that year. Under a term contract, those later outcomes either produce a covered death benefit in a later year or no benefit after the term expires.

For $i\geq0$, $v\leq1$, so for a fixed positive benefit $B$ and a later claim year $k>j$,

Bv^{j+1}\geq Bv^{k+1}.

A claim moved to the earlier year therefore has present value at least as large as a claim in any later covered year. If the later outcome is no covered claim, its value is zero, also no greater than the earlier positive benefit. Taking expectations preserves this ordering, so the expected present value of benefits cannot decrease. At $i=0$, equal benefits in covered years have equal present values, which is why the benefit value can stay unchanged.

The same mortality change cannot increase the value of the premium annuity: a premium due in a later year is paid only if the insured survives to that date. The first premium is unaffected if it is due at issue, but later survival probabilities do not rise. With a positive benefit present value, a numerator that cannot decrease and a positive denominator that cannot increase imply that their ratio cannot decrease.

Common Mistakes and Exam Traps

Claiming mortality always raises a benefit present value. The stated nondecrease result for a term death benefit requires $i\geq0$. With negative interest, an earlier claim can have a lower present value than a later claim.
Ignoring the premium annuity. A level premium is a ratio. Mortality can increase the benefit value and decrease the expected number of premium payments; both effects matter.
Assuming the interest effect on a level premium from the benefit side alone. A higher rate discounts benefits more heavily, but it also changes the present value of future premiums. Use the contract's full premium equation.
Using the wrong payment time. A death benefit payable at the end of year $k+1$ is discounted by $v^{k+1}$. A premium due at the start of year $k+1$ is discounted by $v^k$, conditional on survival to that payment date.
Changing several assumptions at once. To isolate an interest effect, hold mortality fixed; to isolate a mortality effect, hold interest fixed. State explicitly what is being compared.

Key takeaway: Interest discounts payment timing; mortality changes claim probabilities and the probability of future premium payments. For a level net premium, compare both expected present values. Under nonnegative interest, increasing a conditional death probability cannot reduce the benefit value or the level premium for a positive-benefit term policy, but that direction need not hold with negative interest.

Check Your Understanding

For each problem, identify which present values are affected before drawing a conclusion about the premium.

A hypothetical one-year term policy pays $5,000 at year-end. Assume $q_x=0.04$ and $i=0.25$. Find the net single premium. (Answer: $160.00)
A hypothetical one-year term policy pays $12,000 at year-end with $q_x=0.03$. Find and compare the net single premiums at $i=0.02$ and $i=0.08$, to the cent. (Answer: $352.94 at 2%; $333.33 at 8%)
A hypothetical two-year term policy pays $1,000 at the end of the year of death. Let $i=0$, $q_x=0.10$, and $q_{x+1}=0.30$. Find the net single premium. (Answer: $370.00)
A hypothetical two-year term policy pays $5,000 at the end of the year of death. Premiums of $P$ are due at issue and one year later if the insured survives. If $i=0$, $q_x=0.20$, and $q_{x+1}=0.10$, calculate $P$. (Answer: $777.78)
A hypothetical life annuity-due pays $1,000 at time 0, and at times 1 and 2 only if the annuitant survives to each payment date. Assume $i=0$, with one-year survival probabilities $p_x=0.90$ and $p_{x+1}=0.80$. Find its expected present value. Then repeat if $p_x=0.80$ and $p_{x+1}=0.80$. (Answer: $2,620.00 under the first mortality assumption; $2,440.00 under the second)

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