Net Premiums for Continuous Insurance

Why Continuous Payment Timing Matters

A life insurance benefit may be paid at the moment of death rather than at the end of the year of death. Premiums may also be collected continuously while the insured is alive. These contracts require present values that account for the actual payment times: a benefit paid sooner is discounted for less time, and a continuous premium stream is valued over the time the insured survives.

The key distinction is between a benefit amount and a premium rate. A continuous premium rate of $P$ dollars per year does not mean one annual payment of $P$. It means premiums accumulate at rate $P$ while coverage conditions for payment are met. The net premium is determined by equating the expected present value of benefits and the expected present value of premiums.

Definition: Let $T_x$ be the future lifetime of a life aged $x$, and let $\delta$ be a constant force of interest. A continuous whole-life insurance with benefit $b$, payable immediately at death, has present value $b e^{-\delta T_x}$. Its net single premium is $b\bar A_x$, where $\bar A_x=E[e^{-\delta T_x}]$. A continuous premium stream at rate $P$, payable while the insured is alive, has present value $P\int_0^{T_x}e^{-\delta t}\,dt$. The corresponding whole-life continuous life annuity value is $\bar a_x=E[\int_0^{T_x}e^{-\delta t}\,dt]$.

Present Values for Continuous Benefits and Premiums

For a continuous future lifetime distribution, write ${}_tp_x=P(T_x>t)$ for the probability the life survives at least $t$ years and $\mu_{x+t}$ for the force of mortality at duration $t$. The density of death at time $t$ is ${}_tp_x\mu_{x+t}$. Thus the expected present value of a benefit payable at death is found by integrating its discounted amount against this density.

\bar A_x =E[e^{-\delta T_x}] =\int_0^\infty e^{-\delta t}{}_tp_x\mu_{x+t}\,dt

If the insurance pays only when death occurs within $n$ years, the integral ends at $n$. Denote its value for a unit benefit by $\bar A^1_{x:\overline{n}|}$. A continuous premium stream payable only while the insured is alive and for at most $n$ years has value $\bar a_{x:\overline{n}|}$. Its expected present value follows by integrating the probability that a premium is still being paid at each time.

\bar A^1_{x:\overline{n}|} =\int_0^n e^{-\delta t}{}_tp_x\mu_{x+t}\,dt, \qquad \bar a_{x:\overline{n}|} =\int_0^n e^{-\delta t}{}_tp_x\,dt

For a benefit amount $b$, multiply the unit-benefit insurance value by $b$. Under the equivalence principle, the net level continuous premium rate is the benefit present value divided by the premium annuity value. This calculation assumes the same contract conditions and mortality and interest basis are used for both present values.

P=\frac{\text{expected present value of benefits}} {\text{expected present value of premiums}}

Why the Continuous Net Premium Equation Holds

The benefit integral and premium integral arise from two different random payment patterns. For a death benefit, payment occurs at the random time $T_x$, so its discounted amount is evaluated at death. For premiums, a small payment over an interval of length $dt$ at time $t$ is made only if the insured is alive at $t$. Its expected discounted amount is therefore $P e^{-\delta t}{}_tp_x\,dt$.

Value the benefit.
For a unit whole-life benefit, discount to time 0 at the random death time and take the expectation: $E[e^{-\delta T_x}]=\bar A_x$.

Value the premium stream.
At time $t$, a premium rate $P$ contributes $P\,dt$ if the insured is alive. Its expected discounted contribution is $P e^{-\delta t}{}_tp_x\,dt$.

Equate expected present values.
Integrating the premium contributions gives $P\bar a_x$. Equivalence requires $b\bar A_x=P\bar a_x$, so $P=b\bar A_x/\bar a_x$.

The same reasoning applies to term coverage or a limited premium period by changing the integration limits. For example, if a whole-life benefit remains payable at death but premiums stop after $m$ years, the benefit integral remains over the whole lifetime while the premium integral ends at $m$. This distinction is essential: limiting premiums does not, by itself, limit insurance coverage.

Constant Force of Mortality

A useful model assumes a constant force of mortality $\mu$ and constant force of interest $\delta$. Under this assumption, ${}_tp_x=e^{-\mu t}$, and the death-time density is $\mu e^{-\mu t}$. For whole-life insurance and premiums, the resulting present values have simple forms.

\bar A_x=\int_0^\infty \mu e^{-(\delta+\mu)t}\,dt =\frac{\mu}{\delta+\mu}, \qquad \bar a_x=\int_0^\infty e^{-(\delta+\mu)t}\,dt =\frac{1}{\delta+\mu}

For $n$-year term insurance, both the benefit and a premium stream limited to the same $n$-year period use the same factor $1-e^{-(\delta+\mu)n}$. Consequently, if premiums are payable continuously while alive and only during the term, the net rate per year is $b\mu$ in this particular model. This simplification depends on the constant-force assumption and matching coverage and premium periods; it is not a general identity for other mortality models or payment patterns.

\bar A^1_{x:\overline{n}|} =\frac{\mu}{\delta+\mu}\left(1-e^{-(\delta+\mu)n}\right), \qquad \bar a_{x:\overline{n}|} =\frac{1-e^{-(\delta+\mu)n}}{\delta+\mu}

Worked Example: Whole-Life Benefit and Continuous Net Premium

A hypothetical policy pays $ \$120{,}000$ immediately at death. Assume constant force of mortality $\mu=0.02$ per year and constant force of interest $\delta=0.04$ per year. Find the net single premium and the level premium rate payable continuously while the insured is alive.

Method. Calculate the whole-life benefit value and whole-life continuous life annuity value under the constant-force model, then apply equivalence.

First, $\delta+\mu=0.04+0.02=0.06$. The unit-benefit insurance value is

\bar A_x=\frac{0.02}{0.06}=\frac{1}{3}.

The net single premium is therefore $ \$120{,}000(1/3)=\$40{,}000.00$. The premium annuity value is

\bar a_x=\frac{1}{0.06}=16.6666667.

The level continuous premium rate is $P=40{,}000/16.6666667=\$2{,}400.00$ per year. Equivalently, $P=120{,}000(0.02)=\$2{,}400$ per year. Check: $P\bar a_x=2{,}400(16.6666667)=\$40{,}000.00$, equal to the net single premium. Money is rounded to the nearest cent; the displayed annuity value is carried at additional precision for the check.

Common mistake avoided: Treating the continuous premium rate as an annual premium paid in advance would use a different payment pattern and present value. Here the premium is paid continuously while the insured is alive.

Worked Example: Five-Year Continuous Term Insurance

A hypothetical policy pays $ \$80{,}000$ immediately at death if death occurs in the next five years, and pays nothing otherwise. Assume constant forces $\mu=0.04$ and $\delta=0.06$ per year. Find the net single premium and the level continuous premium rate, payable while alive during the five-year term.

Method. Use the term insurance and temporary life annuity integrals, which have the same duration and survival factor under the stated constant-force model.

Here $\delta+\mu=0.10$, and $e^{-0.10(5)}=e^{-0.5}=0.60653066$. The common factor is

\frac{1-e^{-0.5}}{0.10} =\frac{1-0.60653066}{0.10} =3.9346934.

The unit-benefit term insurance value is $0.04(3.9346934)=0.15738774$. Thus the net single premium is $80{,}000(0.15738774)=\$12{,}591.02$.

The temporary premium annuity value is $3.9346934$. Therefore,

P=\frac{12{,}591.0189}{3.9346934} =\$3{,}200.00\text{ per year}.

Check: The benefit value is $80{,}000(0.04)(3.9346934)=\$12{,}591.02$, and the premium value is $3{,}200(3.9346934)=\$12{,}591.02$, after rounding. Money is rounded to the nearest cent. Common mistake avoided: The death benefit is paid immediately at death, not at the end of the year of death. The continuous integral discounts it to the actual death time.

Worked Example: Whole-Life Coverage with Premiums Limited to Ten Years

A hypothetical policy pays $ \$100{,}000$ immediately at death whenever death occurs. Premiums are payable continuously while the insured is alive, but only during the first ten years. Assume constant forces $\mu=0.03$ and $\delta=0.02$ per year. Find the net single premium and the continuous premium rate.

Method. Value the whole-life benefit over the entire future lifetime, but value premiums only over the ten-year payment period.

Since $\delta+\mu=0.05$, the unit whole-life insurance value is $0.03/0.05=0.6$. Thus the net single premium is $100{,}000(0.6)=\$60{,}000.00$.

The premium annuity value for at most ten years is

\bar a_{x:\overline{10}|} =\frac{1-e^{-0.05(10)}}{0.05} =\frac{1-e^{-0.5}}{0.05} =\frac{0.39346934}{0.05} =7.8693868.

Therefore the continuous premium rate is $P=60{,}000/7.8693868=\$7{,}624.48$ per year. Check: $7{,}624.4823(7.8693868)=\$60{,}000.00$, to the nearest cent. Common mistake avoided: Stopping the benefit calculation after ten years would incorrectly price ten-year coverage. The contract described has whole-life coverage; only premiums are limited to ten years.

When the Simple Formulas Do Not Apply

The integral definitions remain valid when mortality varies with age or duration, but the constant-force closed forms do not. In that case, use the appropriate survival probabilities and forces of mortality in the integrals, or use other supplied actuarial values that represent the same continuous timing.

Also check the payment conditions before selecting an annuity value. Premiums might be payable for the whole term, for a shorter fixed period, or only while the insured is alive. Benefits might be payable at death only within a term or at any future death. The numerator and denominator of the net premium equation must reflect these conditions separately.

Key takeaway: For continuous insurance, value a death benefit at the time of death and value premiums as a continuous stream while the insured is eligible to pay. Equate those expected present values, keeping coverage duration and premium duration distinct.

Check Your Understanding

For each hypothetical contract, use the stated constant forces and round money to the nearest cent. Continuous premium rates are in dollars per year.

A whole-life policy pays $ \$200{,}000$ immediately at death. Let $\mu=0.025$ and $\delta=0.025$. Find the net single premium. (Answer: $ \$100{,}000.00$)
A four-year term policy pays $ \$50{,}000$ immediately at death within the term. Let $\mu=0.05$ and $\delta=0.05$. Find the net single premium and the continuous level premium rate payable while alive during the term. (Answer: Net single premium $ \$8{,}242.00$; rate $ \$2{,}500.00$ per year)
A hypothetical policy pays $ \$100{,}000$ immediately at death if death occurs from year 3 through year 7 after issue. Premiums are payable continuously while alive from issue through year 7. Let $\mu=0.02$ and $\delta=0.03$. Find the net single premium and the level continuous premium rate. (Answer: Net single premium $ \$6{,}240.80$; rate approximately $ \$1{,}056.62$ per year)
A whole-life policy pays $ \$90{,}000$ immediately at death, while premiums are payable continuously only during the first eight years. Let $\mu=0.04$ and $\delta=0.01$. Find the continuous net premium rate. (Answer: approximately $ \$10{,}919.63$ per year)
A six-year term policy pays $ \$100{,}000$ immediately at death within six years. Premiums are payable continuously while alive for only the first three years. Let $\mu=0.03$ and $\delta=0.02$. Find the net single premium and the continuous premium rate. (Answer: Net single premium approximately $ \$15{,}550.91$; rate approximately $ \$5{,}582.28$ per year)

What You'll Learn