Whole-Life Net Level Premiums

A whole-life contract provides a benefit upon the insured’s death, whenever that death occurs. To price its annual net level premium, the actuary balances the present value of the benefit against the present value of the premiums. The timing of both cash flows matters: in this tutorial, the benefit is paid at the end of the year of death, and premiums are paid annually in advance, beginning at issue and continuing only while the insured is alive.

The net premium covers the expected present value of the insurance benefit. It does not include expenses, commissions, profit margins, or other loadings. The calculation uses the same interest and mortality assumptions for both sides of the equivalence equation. This is the discrete whole-life setting aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: For an insured aged $x$, a unit whole-life insurance benefit is paid at the end of the year of death. Its actuarial present value is $A_x$. The present value of an annuity-due paying 1 at time 0 and at each subsequent integer time while the insured is alive is $\ddot{a}_x$. The net annual level premium per unit of benefit is $P_x$. For a benefit amount $B$, the annual net premium is $B P_x$.

The Equivalence Principle for Whole Life

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Under the mortality model, ${}_kp_x$ is the probability that a life aged $x$ survives $k$ years, and $q_{x+k}$ is the probability that a life aged $x+k$ dies within the next year. Thus the probability of dying during year $k+1$, for a life aged $x$ at issue, is ${}_kp_x q_{x+k}$.

The unit benefit is discounted for $k+1$ years if death occurs in year $k+1$. The premium due at time $k$ is paid only if the life survives to that time. Therefore the respective present values are:

A_x=\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}, \qquad \ddot{a}_x=\sum_{k=0}^{\infty}v^k{}_kp_x.

The sum for $A_x$ starts with the first possible death benefit at time 1. The annuity-due sum starts with the first premium at time 0, for which ${}_0p_x=1$. These distinctions are essential: a whole-life contract does not mean that premiums continue after death.

Theorem: Under the equivalence principle, with annual premiums payable in advance while the insured is alive and a unit death benefit payable at the end of the year of death,

P_x\ddot{a}_x=A_x, \qquad P_x=\frac{A_x}{\ddot{a}_x}.

For benefit $B$, the benefit present value is $B A_x$, so the annual net premium is $B A_x/\ddot{a}_x$. The denominator is the expected present value of a stream of unit premiums; it is not the number of years the policy is expected to remain in force.

Why the Insurance-Annuity Identity Holds

A useful check is the discrete whole-life identity $A_x+d\ddot{a}_x=1$, where $d=1-v$. It follows directly by telescoping the survival probabilities. Write ${}_{k+1}p_x={}_kp_x(1-q_{x+k})$, so that ${}_kp_xq_{x+k}={}_kp_x-{}_{k+1}p_x$. Then:

\begin{aligned} A_x &=\sum_{k=0}^{\infty}v^{k+1}\left({}_kp_x-{}_{k+1}p_x\right),\\ d\ddot{a}_x &=(1-v)\sum_{k=0}^{\infty}v^k{}_kp_x =\sum_{k=0}^{\infty}v^k{}_kp_x-\sum_{k=0}^{\infty}v^{k+1}{}_kp_x. \end{aligned}

Adding the two expressions cancels the terms involving $v^{k+1}{}_kp_x$, leaving the telescoping sum $1-\lim_{n\to\infty}v^{n+1}{}_{n+1}p_x$. For a positive interest rate, $0<v<1$, and survival probabilities are at most 1, so this limit is 0. Hence $A_x+d\ddot{a}_x=1$. The identity applies to a discrete end-of-year death benefit and an annuity-due premium stream. It does not justify substituting continuous-insurance values or premiums payable in arrears.

Calculating Present Values from Mortality Rates

When mortality is supplied as one-year probabilities, calculate the probability of each possible year of death by multiplying conditional survival probabilities. The annuity present value uses the probability of being alive at each premium date. A finite model can include a terminal age at which death is certain; after that age, survival and premium probabilities are zero.

Set the contract timing.
Record when the benefit is paid and when each premium is due.

Build survival and death probabilities.
For each year, find ${}_kp_x$ and ${}_kp_xq_{x+k}$ from the mortality assumptions.

Calculate both present values.
Discount each death benefit by its time of payment and each premium by its payment time.

Apply equivalence.
Divide the benefit present value by the annuity-due present value, then scale for the benefit amount if needed.

Worked Example: A Whole-Life Policy with Three Possible Years of Death

A hypothetical insurer considers a whole-life policy issued to a life aged 40. The benefit is $100,000, payable at the end of the year of death. Annual premiums are due in advance while the insured is alive. The effective annual interest rate is 5%. The assumed mortality probabilities are $q_{40}=0.10$, $q_{41}=0.20$, and $q_{42}=1$. Calculate the annual net level premium.

Method. Find the death probabilities and survival probabilities, calculate the two present values, and apply equivalence. This avoids treating the time-0 premium as if it were due at the end of the first year.

Here $v=1/1.05=20/21$. The probabilities of death in years 1, 2, and 3 are respectively $0.10$, $0.90(0.20)=0.18$, and $0.90(0.80)(1)=0.72$. The survival probabilities at times 0, 1, and 2 are $1$, $0.90$, and $0.72$.

The unit-benefit present value is:

A_{40}=0.10v+0.18v^2+0.72v^3 =0.095238+0.163265+0.621963 =0.880466.

The annuity-due present value is:

\ddot{a}_{40}=1+0.90v+0.72v^2 =1+0.857143+0.653061 =2.510204.

The premium per dollar of benefit is $0.880466/2.510204=0.350755$. Thus the annual premium is $100{,}000(0.350755)=\$35{,}075.49$, rounded to the nearest cent. As a check, multiplying the unrounded premium factor by $2.510204$ recovers the unit-benefit present value $0.880466$.

Worked Example: A Whole-Life Policy with a Terminal Age

A hypothetical policy provides a $50,000 benefit at the end of the year of death. Premiums are paid annually in advance while the insured is alive. For a life aged 60, assume an effective annual interest rate of 4% and mortality probabilities $q_{60}=0.20$, $q_{61}=0.25$, $q_{62}=0.40$, and $q_{63}=1$. Find the annual net level premium.

Method. Enumerate all death years and premium dates. The terminal mortality probability of 1 means that no one survives age 63 to make a premium payment at time 4.

Here $v=1/1.04=25/26$. The death probabilities in years 1 through 4 are $0.20$, $0.80(0.25)=0.20$, $0.80(0.75)(0.40)=0.24$, and $0.80(0.75)(0.60)(1)=0.36$. The probabilities of being alive at premium times 0 through 3 are $1$, $0.80$, $0.60$, and $0.36$.

\begin{aligned} A_{60} &=0.20v+0.20v^2+0.24v^3+0.36v^4\\ &=0.192308+0.184911+0.213359+0.307730\\ &=0.898308,\\ \ddot{a}_{60} &=1+0.80v+0.60v^2+0.36v^3\\ &=1+0.769231+0.554734+0.320039\\ &=2.644003. \end{aligned}

Thus the premium per dollar of benefit is $0.898308/2.644003=0.339753$. The annual premium is $50{,}000(0.339753)=\$16{,}987.64$, rounded to the nearest cent. As a check on the timing, the annuity includes four possible premiums, at times 0, 1, 2, and 3; it does not include one at time 4.

Worked Example: Constant Mortality and a Geometric Series

A hypothetical whole-life policy pays $80,000 at the end of the year of death. Premiums are due annually in advance while the insured is alive. Assume a 5% effective annual interest rate and a constant one-year death probability $q_{x+k}=0.10$ at every future age. Find the annual net level premium.

Method. Use the geometric series for the survival-weighted payments, then check the resulting premium using the equivalence equation. The constant mortality rate is a mathematical assumption for this example, not a claim about actual mortality.

The probability of surviving $k$ years is ${}_kp_x=0.90^k$, and $v=20/21$. Therefore:

A_x=\sum_{k=0}^{\infty}0.10v^{k+1}(0.90)^k =\frac{0.10v}{1-0.90v} =\frac{2}{3}.

The annuity-due value is:

\ddot{a}_x=\sum_{k=0}^{\infty}(0.90v)^k =\frac{1}{1-0.90v} =7.

Consequently $P_x=(2/3)/7=2/21$ per dollar of benefit. The annual premium is $80{,}000(2/21)=\$7{,}619.05$, rounded to the nearest cent. The check $7(2/21)=2/3$ confirms that the premium and benefit present values agree. Here the constant mortality assumption makes the premium factor $qv$, but that shortcut depends on this particular model and should not be applied to varying mortality rates.

Common Timing Traps

The most frequent error is using an annuity-immediate value for premiums that are actually due in advance. An annuity-immediate starts its payments at time 1; the stated contract starts at time 0. Since the time-0 premium is certain, omitting it understates the premium present value and overstates the premium required by equivalence.

For the first example, the annuity-immediate value would be $\ddot{a}_{40}-1=1.510204$. The incorrect calculation $0.880466/1.510204$ gives a premium factor of about $0.5830$, rather than the correct factor $0.3508$. The correction is to use the annuity-due value $2.510204$, because a premium is paid at issue.

A second trap is discounting the benefit for $k$ years when death occurs in year $k+1$. The benefit is paid at the end of that year, so its discount factor is $v^{k+1}$. A third is including a premium at a time when the insured is not alive. Premiums at time $k$ are weighted by ${}_kp_x$, not by the probability of dying during year $k+1$.

Key takeaway: For a discrete whole-life benefit payable at the end of the year of death and annual premiums payable in advance while alive, the net level premium is benefit present value divided by annuity-due present value: $B A_x/\ddot{a}_x$. Confirm the cash-flow dates before calculating either value.

Check Your Understanding

For each hypothetical contract, use the stated annual interest and mortality assumptions. Benefits are payable at the end of the year of death, and premiums are payable in advance while the insured is alive unless the question states otherwise.

A whole-life policy has a $125,000 benefit, a 4% effective annual interest rate, and a constant one-year death probability of 0.08 at every future age. Find the annual net level premium. (Answer: $9,615.38)
For a life aged $x$, assume $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=1$, with a 5% effective annual interest rate. A $20,000 whole-life benefit is payable at death. Find the annual net level premium. (Answer: $7,015.10)
A life aged $x$ has $q_x=0.15$, $q_{x+1}=0.25$, and $q_{x+2}=1$. The effective annual interest rate is 4%, and the whole-life benefit is $80,000. Calculate the annual net level premium. (Answer: $30,163.44)
A hypothetical whole-life contract has unit-benefit value $A_x=0.72$ and annuity-due value $\ddot{a}_x=4.5$, under the same interest and mortality assumptions. The benefit is $60,000. Find the annual net level premium and state when the first payment is due. (Answer: $9,600.00 annually; first payment at issue)
A hypothetical whole-life policy has $A_x=0.84$ and $\ddot{a}_x=3.2$. Its annual net level premium is $7,875. Find the benefit amount. (Answer: $30,000)

Whole-Life Net Level Premiums

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