Endowment Insurance Net Level Premiums

Why Endowment Premiums Have Two Benefit Components

An endowment insurance contract combines two possible benefit payments: a death benefit if the insured dies during a specified term, and a maturity benefit if the insured survives to the end of that term. To calculate a net level premium, value both benefits at issue, then divide their expected present value by the expected present value of the premium stream.

The key distinction from term insurance is the payment on survival to the end of the contract. That maturity benefit is not an additional payment on top of a death benefit: the contract pays the specified benefit either on death during the term or on survival to maturity. The premium calculation must account for both possibilities.

Definition: An $n$-year discrete endowment insurance with benefit amount $B$ pays $B$ at the end of the year of death if the insured dies within the $n$-year term, or $B$ at time $n$ if the insured survives to the end of the term. A net level premium is a fixed premium amount payable at the beginning of each year while the insured is alive and the contract remains in force, determined by the equivalence principle without expenses.

Assume an effective annual interest rate $i$, with $v=1/(1+i)$. Let ${}_kp_x$ be the probability that a life aged $x$ survives $k$ years, and let $q_{x+k}=1-p_{x+k}$ be the probability of death during the following year, conditional on being alive at age $x+k$. Death benefits are paid at the end of the year of death. Premiums are payable at times $0,1,\ldots,n-1$, provided the insured is alive at each payment time. No premium is paid at time $n$.

Expected Present Values

For death during year $k+1$, where $k=0,\ldots,n-1$, the probability is ${}_kp_xq_{x+k}$, and the benefit is paid at time $k+1$. If the insured survives the full term, the maturity benefit is paid at time $n$, with probability ${}_np_x$. Therefore, the expected present value of benefits is

\operatorname{EPV}(\text{benefits}) = B\left( \sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} + v^n{}_np_x \right).

The expected present value of one dollar of annual premium is the sum of the discounted probabilities that each premium is payable. The premium at time $k$ is paid only if the insured has survived to that time. Thus,

\operatorname{EPV}(\text{premiums per dollar}) = \sum_{k=0}^{n-1}v^k{}_kp_x.

The time-zero premium is included with probability one and discount factor one. The last premium is at time $n-1$, not time $n$. By the equivalence principle, the expected present values of benefits and premiums are equal. If $P$ is the annual net premium, then

P = B\, \frac{ \displaystyle\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} +v^n{}_np_x }{ \displaystyle\sum_{k=0}^{n-1}v^k{}_kp_x }.

This formula assumes the same benefit amount is payable on death and on survival to maturity, and that premiums stop at death or at the end of the term. If the death and maturity benefits differ, value each benefit using its own amount. If premiums are payable at a different frequency or time, the premium present-value denominator must be adjusted to match the contract.

Why the Formula Holds

The numerator is the expected present value of the contract’s benefit, because it sums the discounted benefit in each mutually exclusive outcome: death in each term year, or survival through the entire term. The denominator is the expected present value of a unit premium at each scheduled payment time, including only those payments that are made while the insured is alive.

Partition the outcomes.
Death in one of the $n$ years and survival to time $n$ are mutually exclusive and cover all possible outcomes during the term.

Discount each benefit payment.
For death in year $k+1$, multiply $Bv^{k+1}$ by the probability ${}_kp_xq_{x+k}$. For survival to maturity, multiply $Bv^n$ by ${}_np_x$.

Discount each possible premium.
At time $k$, a unit premium has present value $v^k$ and is paid with probability ${}_kp_x$. Sum these expected present values.

Apply equivalence.
Set $P$ times the expected present value of unit premiums equal to the expected present value of benefits, then solve for $P$.

There is also a useful benefit-value check. The endowment benefit can be separated into term insurance, which pays on death during the term, and a pure endowment, which pays only if the insured survives to time $n$. The endowment benefit value is the sum of those two present values. This decomposition is a check on the benefit calculation, not a change to the contract.

Worked Premium Calculations

Worked Example: A Three-Year Endowment with Changing Survival Probabilities

A hypothetical policy pays $1,000 on death during a three-year term or on survival to the end of the term. The annual effective interest rate is $5\%$. Conditional one-year survival probabilities are $p_x=0.90$, $p_{x+1}=0.80$, and $p_{x+2}=0.75$. Find the annual net premium payable in advance while the insured is alive.

Method. Calculate the probability of each benefit outcome and each premium payment, then apply equivalence. Here $v=1/1.05=20/21$, $v^2=400/441$, and $v^3=8000/9261$.

The probability of death in year 1 is $q_x=1-0.90=0.10$. The probability of death in year 2 is ${}_1p_xq_{x+1}=0.90(1-0.80)=0.18$. The probability of death in year 3 is ${}_2p_xq_{x+2}=0.90(0.80)(1-0.75)=0.18$. The probability of survival to maturity is ${}_3p_x=0.90(0.80)(0.75)=0.54$. These probabilities sum to $0.10+0.18+0.18+0.54=1.00$.

For a $1,000 benefit, the expected present value of benefits is

1000\left( 0.10v+0.18v^2+0.18v^3+0.54v^3 \right) = 1000\left( \frac{2}{21}+\frac{8}{49}+\frac{640}{1029} \right) = 1000\left(\frac{906}{1029}\right) \approx \$880.47.

The premium is paid at time 0 with probability 1, at time 1 with probability 0.90, and at time 2 with probability $0.90(0.80)=0.72$. The expected present value of premiums per dollar is

1+0.90v+0.72v^2 = 1+\frac{6}{7}+\frac{32}{49} = \frac{123}{49} \approx 2.510204.

Equating the two present values gives

P=\frac{1000(906/1029)}{123/49} \approx \$350.75.

Check. The $880.47 benefit value is divided by an expected premium value of about 2.510204, giving approximately $350.75 per year. The maturity contribution is included as $1000(0.54)v^3$. Omitting it would price a term policy instead of the stated endowment.

Worked Example: A Two-Year Endowment with a Maturity-Payment Trap

A hypothetical policy pays $10,000 either at the end of the year of death or at the end of a two-year term if the insured survives. The annual effective interest rate is $4\%$, with $p_x=0.80$ and $p_{x+1}=0.75$. Find the annual net premium payable in advance while alive.

Method. Value death in each year and survival to maturity separately. Here $v=25/26$ and $v^2=625/676$. The outcome probabilities are $q_x=0.20$, ${}_1p_xq_{x+1}=0.80(0.25)=0.20$, and ${}_2p_x=0.80(0.75)=0.60$. They sum to 1.

The expected present value of benefits is

10000\left(0.20v+(0.20+0.60)v^2\right) = 10000\left(\frac{5}{26}+\frac{125}{169}\right) = 10000\left(\frac{315}{338}\right) \approx \$9{,}319.53.

The first premium is paid at time 0. The second is paid at time 1 only if the insured survives the first year. Therefore, the expected present value per dollar of premium is

1+0.80v = 1+\frac{20}{26} = \frac{23}{13}.

The annual premium is

P = \frac{10000(315/338)}{23/13} = 10000\left(\frac{315}{598}\right) \approx \$5{,}267.56.

Common mistake and correction. A term-only calculation would use $10000(0.20v+0.20v^2)$, approximately $3,236.76, as the benefit value. That calculation omits the maturity payment. The stated contract pays $10,000 on survival as well, so the correct benefit value includes $10000(0.60)v^2$, giving $9,319.53. Dividing that full value by $23/13$ gives the correct premium of $5,267.56.

Check. Multiplying $5,267.56 by $23/13$ gives approximately $9,319.53, subject to rounding. The premium stream has two possible payments, not three: no premium is due at maturity.

Worked Example: Checking an Endowment by Separating Its Components

A hypothetical two-year endowment pays $1,000 on death during the term or survival to maturity. The annual effective interest rate is $10\%$, and the conditional survival probability in each year is $0.90$. Find the annual net premium payable in advance while alive, and verify the benefit value by separating the death and maturity components.

Method. Use $v=10/11$ and $v^2=100/121$. The probabilities of death in years 1 and 2 are $0.10$ and $0.90(0.10)=0.09$, respectively; the probability of survival to maturity is $0.90^2=0.81$. The probabilities sum to 1.

The death-benefit present value per dollar of benefit is

0.10v+0.09v^2 = \frac{11}{121}+\frac{9}{121} = \frac{20}{121}.

The maturity-benefit present value per dollar is $0.81v^2=81/121$. Adding the components gives the endowment benefit value per dollar:

\frac{20}{121}+\frac{81}{121} = \frac{101}{121}.

The expected present value of the $1,000 benefit is therefore $1000(101/121)$, approximately $834.71. The expected present value of one dollar of premiums is

1+0.90v=1+\frac{9}{11}=\frac{20}{11}.

Thus,

P = \frac{1000(101/121)}{20/11} = 1000\left(\frac{101}{220}\right) \approx \$459.09.

Check. The death and maturity values add to $101/121$, the same as the direct calculation $0.10v+(0.09+0.81)v^2$. Dividing the benefit value by $20/11$ gives the stated annual premium. The component check helps catch a maturity benefit accidentally counted at the wrong time.

Common Exam Traps and Checks

An endowment premium calculation combines benefit timing, survival probabilities, interest, and premium timing. Before finalizing a result, check the contract against the calculation.

Leaving out the maturity benefit. The survival term $v^n{}_np_x$ belongs in the benefit value whenever the policy pays on survival to the end of the term.
Using the wrong payment time. A death benefit for death during year $k+1$ is paid at time $k+1$, so its discount factor is $v^{k+1}$. The maturity benefit is paid at time $n$, so its discount factor is $v^n$.
Adding an extra premium at maturity. With premiums payable at the beginning of each year while the policy is in force, the times are $0$ through $n-1$. There is no premium at time $n$.
Confusing conditional and unconditional probabilities. The probability of death in year $k+1$ is ${}_kp_xq_{x+k}$, not just $q_{x+k}$. The latter is conditional on survival to that year.
Forgetting the benefit amount. The fraction in the premium formula gives the premium per dollar of benefit. Multiply by $B$ to obtain the premium for the stated benefit.

A final reasonableness check is to confirm that the benefit-outcome probabilities sum to one and that the benefit value includes exactly one payment in each outcome. Also confirm that the premium denominator reflects only payments the contract actually requires.

Check Your Understanding

For each hypothetical contract, calculate the annual net premium payable in advance while the insured is alive. Give the premium per unit of benefit where the benefit amount is not specified.

A two-year endowment has benefit 1, annual effective interest $i=5\%$, and conditional survival probabilities $p_x=p_{x+1}=0.80$. (Answer: $P=404/777\approx 0.5201$.)
A three-year endowment pays $5,000 on death during the term or survival to maturity. The annual effective interest rate is $4\%$, and conditional survival probabilities are $0.90$ in each year. (Answer: approximately $1,720.03 per year.)
A two-year endowment pays $20,000 on death during the term or survival to maturity. The annual effective interest rate is $5\%$, and $p_x=0.70$, $p_{x+1}=0.60$. (Answer: approximately $11,047.62 per year.)
A two-year endowment pays $1,000 on death during the term or survival to maturity. The annual effective interest rate is $10\%$, and conditional survival probabilities are $0.95$ in each year. (Answer: approximately $445.68 per year.)
A two-year endowment pays $25,000 on death during the term or survival to maturity. The annual effective interest rate is $4\%$, and $p_x=0.80$, $p_{x+1}=0.75$. (Answer: approximately $13,168.90 per year.)

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