Deriving the Net Level Premium Equation

Why a Net Level Premium Is a Ratio

A life insurance contract may promise benefits at different times and collect premiums only while the policy remains in force. To find a net level premium, the actuary must account for both sides of that arrangement: the expected present value of the benefits and the expected present value of the premium payments. The key is to value each stream using the same interest and survival assumptions, then apply the equivalence principle.

A net premium covers the expected cost of benefits under the stated assumptions; it does not include an allowance for expenses, profit, or a risk margin. A level premium is a fixed amount at each scheduled premium date, although the number of payments actually made may depend on survival or whether the policy remains in force.

Definition: Let $Z$ be the present value of all benefits under a contract, and let $W$ be the present value of all scheduled payments of one unit of level premium. The net level premium $P$, under the equivalence principle, satisfies $P E[W]=E[Z]$. If $E[W]>0$, then $$ P=\frac{E[Z]}{E[W]}. $$ The numerator is the expected present value of benefits. The denominator is the expected present value of one unit of premium payments.

The ratio follows because a premium of $P$ dollars at each payment date has present value $P W$: multiplying every unit premium by $P$ multiplies the present value of the unit-premium stream by $P$. Taking expectations gives $E[PW]=P E[W]$. Equating this expected value to $E[Z]$ and dividing by $E[W]$ gives the formula. No assumption that benefits and premiums are independent is needed.

The denominator is not automatically the present value of a certain annuity. If a premium is due only while the insured is alive and the policy is in force, its expected present value includes the probability that each payment is due. Timing matters too: a payment at time 0 is not discounted, while a payment at time $k$ is discounted by $v^k$, where $v=1/(1+i)$.

Building the Premium Factor

For annual premiums in advance, suppose payments are scheduled at times $0,1,\ldots,n-1$, and each is made only if the insured is alive and the policy is in force at that time. Let ${}_kp_x$ be the probability that a life aged $x$ survives $k$ years. The time-0 premium is certain under this setup, so its payment probability is 1. For $k\geq 1$, the probability that the payment at time $k$ is due is ${}_kp_x$, provided the contract has not otherwise ended.

E[W]=\sum_{k=0}^{n-1}v^k\,{}_kp_x, \qquad {}_0p_x=1.

This expression is the expected present value of a stream of one-dollar premiums. Each term combines the discount factor for its payment date with the probability that the payment is made. If a payment is due under a different condition, such as only after surviving to the end of a year, the probability and timing in that term must reflect the contract wording instead.

Identify the benefit cash flows.
List each possible benefit amount, the time it is paid, and the event that triggers payment.

Calculate the benefit EPV.
For each possible payment, multiply its amount by its probability and discount to the valuation date. Add the results to obtain $E[Z]$.

Identify the premium cash flows.
List scheduled payment dates and the conditions under which each payment is due.

Calculate and use the premium factor.
Add the discounted probabilities of the unit premiums to obtain $E[W]$, then divide the benefit EPV by that factor.

Worked Applications

Worked Example: Two Possible Benefit Dates

A hypothetical contract pays $10,000 at the end of year 1 if the insured dies in that year, or $20,000 at the end of year 2 if the insured dies in year 2. The annual effective interest rate is $i=0.05$. The probability of death in year 1 is 0.10; given survival through year 1, the probability of death in year 2 is 0.20. Level annual premiums are payable in advance at times 0 and 1, if the insured is alive at the payment date. Find the net level premium.

Method. Calculate the expected present value of the death benefits, then divide by the expected present value of one dollar of premiums. The common mistake to avoid is treating the time-1 premium as certain even though it is payable only if the insured survives year 1.

First, $v=1/1.05=0.95238095$, and $v^2=1/1.1025=0.90702948$. The probability of death in year 2, measured from the start of the contract, is $0.90(0.20)=0.18$.

The benefit EPV is

E[Z]=10{,}000(0.10)v+20{,}000(0.18)v^2 =1{,}000(0.95238095)+3{,}600(0.90702948) =952.38+3{,}265.31 =4{,}217.69.

The unit premium at time 0 is certain. The unit premium at time 1 is due only if the insured survives year 1, with probability 0.90. Therefore,

$$ E[W]=1+0.90v =1+0.90(0.95238095) =1.85714286. $$

Thus,

P=\frac{4{,}217.69}{1.85714286} =2{,}271.06.

The net level premium is $2,271.06 per scheduled payment, rounded to the nearest cent. As a check, $2{,}271.06(1.85714286)$ is approximately $4,217.69, the benefit EPV, with the small difference due to rounding.

The incorrect factor $1+v=1.95238095$ would count the time-1 premium as payable with certainty. That gives $4{,}217.69/1.95238095\approx \$2{,}160.28$, which is too low. The correction is to multiply the time-1 unit premium by the 0.90 probability that it is due.

Worked Example: Benefits Triggered by Different Outcomes

A hypothetical two-year contract pays $5,000 at the end of year 1 if death occurs in year 1, $9,000 at the end of year 2 if death occurs in year 2, and $2,000 at the end of year 2 if the insured survives the full two years. The annual effective interest rate is $i=0.05$. The probability of death in year 1 is 0.10, and the probability of death in year 2 conditional on surviving year 1 is 0.20. Annual level premiums are payable in advance at times 0, 1, and 2 if the insured is alive at each payment date. Find the net level premium.

Method. Separate the mutually exclusive benefit outcomes, and then build the unit-premium factor using survival to each premium date. In particular, the time-2 premium is payable only if the insured survives two years.

The unconditional probabilities are $0.10$ for death in year 1, $0.90(0.20)=0.18$ for death in year 2, and $0.90(0.80)=0.72$ for survival through year 2. Using $v=0.95238095$ and $v^2=0.90702948$, the benefit EPV is

\begin{aligned} E[Z] &=5{,}000(0.10)v+9{,}000(0.18)v^2+2{,}000(0.72)v^2\\ &=500(0.95238095)+1{,}620(0.90702948)+1{,}440(0.90702948)\\ &=476.19+1{,}469.39+1{,}306.12\\ &=3{,}251.70. \end{aligned}

The premium at time 0 is certain. The time-1 premium has probability 0.90 of being due, and the time-2 premium has probability 0.72 of being due. Therefore,

$$ E[W]=1+0.90v+0.72v^2 =1+0.85714286+0.65306122 =2.51020408. $$

The net level premium is

P=\frac{3{,}251.70}{2.51020408} =1{,}295.39.

The answer is $1,295.39 per scheduled payment, rounded to the nearest cent. Multiplying this premium by the premium factor gives approximately $3,251.70, confirming the calculation. Counting all three payments as certain would use the wrong denominator: the time-1 and time-2 payments are contingent on survival.

Worked Example: Solving for an Unknown Benefit

A hypothetical contract has an annual effective interest rate of $i=0$. The probability of death in year 1 is 0.20; conditional on surviving year 1, the probability of death in year 2 is 0.125. If death occurs in year 1, the benefit is $1,000. If death occurs in year 2, the benefit is $K$. If the insured survives two years, the benefit is $1,000 at the end of year 2. Level premiums of $1,000 are payable in advance at times 0, 1, and 2 while the insured is alive. Find $K$ under the equivalence principle.

Method. First calculate the expected present value of the premiums. Since $i=0$, there is no discounting. The survival probabilities to times 1 and 2 are $0.80$ and $0.80(0.875)=0.70$, so the unit-premium factor is

$$ E[W]=1+0.80+0.70=2.50. $$

The expected present value of the $1,000 premiums is $1{,}000(2.50)=\$2{,}500$. The unconditional probability of death in year 2 is $0.80(0.125)=0.10$, and the probability of surviving two years is 0.70. Thus, the benefit EPV is

$$ E[Z]=1{,}000(0.20)+K(0.10)+1{,}000(0.70) =200+0.10K+700 =900+0.10K. $$

Equating benefit and premium EPVs gives $900+0.10K=2{,}500$. Therefore, $0.10K=1{,}600$, so $K=\$16{,}000$. Checking, the benefit EPV is $200+1{,}600+700=\$2{,}500$, equal to the premium EPV.

Conditions and Common Pitfalls

The ratio is useful only when the benefit EPV and unit-premium EPV are built from the actual contract cash flows. The model must specify the interest rate, benefit payment times, premium payment times, and the conditions under which each payment is made. In a life-contingent model, the survival probabilities used for benefits and premiums must also be consistent with the stated mortality assumptions.

Using the wrong premium condition. A scheduled payment is not necessarily made. If it is payable only while the insured is alive, include the probability of survival to its due date.
Discounting from the wrong time. A payment at time 0 has discount factor 1. A payment at the end of year $k$ has discount factor $v^k$.
Confusing conditional and unconditional probabilities. A death probability conditional on surviving year 1 must be multiplied by the probability of surviving year 1 to obtain the probability of death in year 2 from the contract’s start.
Including premiums that are not part of the contract. A payment schedule that ends after a fixed number of years should not be extended beyond that period.
Calling the net premium a full commercial premium. The equivalence calculation covers expected benefits under its assumptions. It does not add expenses or other loadings.

If the unit-premium EPV is zero, the ratio is undefined: no scheduled premium payments have positive expected present value. Otherwise, the equivalence principle determines a unique net level premium for the specified benefit and premium cash flows. The resulting expected-value equality does not imply that benefits and premiums match on every individual outcome.

Check Your Understanding

For each problem, identify the benefit EPV and unit-premium factor before calculating or interpreting the premium.

A hypothetical contract pays $6,000 at the end of year 1 if death occurs in year 1 and $8,000 at the end of year 2 if death occurs in year 2. The annual effective interest rate is 0. The probabilities of death in years 1 and 2, from the contract’s start, are 0.10 and 0.18. Premiums are due in advance at times 0 and 1 if alive. Find the net level premium. (Answer: $1,073.68)
A contract pays $5,000 at the end of year 1 on death in that year, or $7,000 at the end of year 2 on death in that year. The annual effective interest rate is 0.04, the first-year death probability is 0.12, and the second-year death probability conditional on surviving year 1 is 0.10. Premiums are payable in advance at times 0, 1, and 2 if alive. Find the net level premium. (Answer: $444.64)
A benefit has an expected present value of $5,000. The annual effective interest rate is 0.05. One-dollar premiums are payable at the ends of years 1 and 2 only if alive then; the probabilities of survival to those dates are 0.90 and 0.72. Find the net level premium. (Answer: $3,310.81)
A hypothetical contract has no interest. The insured survives to time 1 with probability 0.80 and to time 2 with probability 0.70. Benefits are $1,000 on death in year 1, $K$ on death in year 2, and $1,000 on survival through year 2. Premiums of $1,000 are payable in advance at times 0, 1, and 2 while alive. Find $K$ using the equivalence principle. (Answer: $16,000)

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