Net Single Premiums for Life Insurance

What a Net Single Premium Measures

A single premium is paid once, at the start of an insurance contract, to provide its stated benefit. The net single premium is the expected present value of that benefit under the specified interest and mortality assumptions. It prices the benefit alone: it does not include expenses, commissions, profit margins, or other loadings.

The mortality assumption determines the probability and timing of a claim; the interest assumption determines how much a future payment is worth at issue. Both are essential. A benefit payable at the end of the policy year of death has a different present value from the same amount payable at the moment of death.

Definition: Let $Z$ be the present value, at issue, of the insurance benefit. The net single premium is $E[Z]$. For a fixed benefit amount, $Z$ depends on the random time of death and on the contract’s benefit-payment rule. A calculation must specify the interest basis, mortality model, benefit amount, and payment timing.

Discrete Term Insurance

Suppose an $n$-year term policy issued to a life aged $x$ pays a benefit $B$ at the end of the policy year of death, provided death occurs within the $n$-year term. If the insured survives to the end of the term, no death benefit is paid. Assume a constant effective annual interest rate $i$, with $v=1/(1+i)$.

For $k=0,\ldots,n-1$, the probability of surviving $k$ years and then dying during the following year is ${}_kp_xq_{x+k}$. The payment is then made $k+1$ years after issue, so its present value is $Bv^{k+1}$. Adding the expected present values over the mutually exclusive possible death years gives the net single premium:

E[Z] = B\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k}.

Here ${}_0p_x=1$, and $q_{x+k}=1-p_{x+k}$. This expression uses the mortality rates applicable at each attained age; it does not assume that the annual mortality rate is constant. The probability of receiving no death benefit during the term is ${}_np_x$, which can be used to check the death probabilities: $\sum_{k=0}^{n-1}{}_kp_xq_{x+k}+{}_np_x=1$.

Worked Example: A Three-Year Term Benefit Paid at Year-End

A hypothetical three-year term policy provides a benefit of $B=\$100{,}000$, payable at the end of the policy year of death. The effective annual interest rate is $i=4\%$. Assume $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Calculate the net single premium. The method is to find each year’s unconditional death probability, discount its benefit to issue, and add the expected values.

The discount factor is $v=1/1.04=0.96153846$. The probability of death in year 1 is $q_x=0.10$. The probability of death in year 2 is survival through year 1 followed by death: ${}_1p_xq_{x+1}=0.90(0.20)=0.18$. The probability of death in year 3 is ${}_2p_xq_{x+2}=0.90(0.80)(0.25)=0.18$. Survival through all three years is $0.90(0.80)(0.75)=0.54$; the probabilities sum to $0.10+0.18+0.18+0.54=1.00$.

The present values per dollar of benefit, conditional on each death year, are $v=0.96153846$, $v^2=0.92455621$, and $v^3=0.88899636$. Thus the expected present value per dollar is $0.10(0.96153846)+0.18(0.92455621)+0.18(0.88899636)=0.42259331$.

Multiplying by the benefit gives $100{,}000(0.42259331)=\$42{,}259.33$, rounded to the nearest cent. This is the net single premium. The calculation avoids the common mistake of discounting a year-2 death benefit by $v$: a death in policy year 2 produces a payment at time 2, so its discount factor is $v^2$.

Whole-Life Insurance

For whole-life insurance with benefit $B$ paid at the end of the year of death, there is no fixed term boundary. If $K_x$ is the curtate future lifetime, then death in policy year $k+1$ has probability ${}_kp_xq_{x+k}$ and produces a payment at time $k+1$. The unit-benefit whole-life insurance value is $A_x$, so the net single premium for benefit $B$ is $BA_x$. In terms of the annual mortality probabilities,

A_x=\sum_{k=0}^{\infty}v^{k+1}\,{}_kp_xq_{x+k}, \qquad \text{net single premium}=BA_x.

The infinite sum accounts for every possible policy year of death. Its terms are expected present values, not merely death probabilities: each death probability is multiplied by the discount factor for the actual payment time. In a valid model, the series converges; for example, it does so when $i>0$, since each term is bounded above by the corresponding discounted probability and the discount factors decrease geometrically.

Worked Example: Whole-Life Insurance with Constant Annual Mortality

A hypothetical whole-life policy pays $\$50{,}000$ at the end of the year of death. Assume the effective annual interest rate is $i=5\%$, and assume a constant annual probability of death $q=0.20$ at every policy duration. Thus the annual survival probability is $p=0.80$. Find the net single premium.

The discount factor is $v=1/1.05=20/21$. The probability of death in year $k+1$, for $k=0,1,2,\ldots$, is $p^kq=(0.80)^k(0.20)$. Therefore the unit-benefit value is the geometric series $A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq=qv\sum_{k=0}^{\infty}(pv)^k$.

The ratio is $pv=0.80(20/21)=16/21$, which is less than 1, so the geometric-series formula applies. Also $qv=0.20(20/21)=4/21$. Hence $A_x=(4/21)/(1-16/21)=(4/21)/(5/21)=4/5=0.8$.

The net single premium is $50{,}000(0.8)=\$40{,}000.00$, rounded to the nearest cent. As a check, the first-year death probability is 0.20 and the first-year expected present value per dollar is $0.20(20/21)=4/21$. The remaining years contribute the rest of the convergent series. The method correctly discounts the year-$(k+1)$ benefit by $v^{k+1}$, rather than treating the whole-life benefit as payable immediately.

Continuous Payment at the Moment of Death

Some contracts pay the benefit immediately when death occurs rather than at the end of the policy year. For a continuous model, let $T_x$ be the future lifetime and let $\delta$ be the constant force of interest. A benefit $B$ payable at death has present value $Be^{-\delta T_x}$. If the policy is an $n$-year term, the benefit is paid only when $T_x\le n$, so its net single premium is $B E[e^{-\delta T_x}\mathbf{1}_{\{T_x\le n\}}]$.

For example, suppose future lifetime has an exponential distribution with constant force of mortality $\mu$, so $S(t)=P(T_x>t)=e^{-\mu t}$ and density $f(t)=\mu e^{-\mu t}$ for $t\ge0$. Under these assumptions, continuous term insurance has value $B\int_0^n e^{-\delta t}\mu e^{-\mu t}\,dt$. This formula depends on the exponential mortality model and continuous payment at death; it is not the end-of-year discrete term formula.

Derivation: Under the exponential model, the probability density of death at time $t$ is $f(t)=\mu e^{-\mu t}$. A death at that time produces present value $Be^{-\delta t}$. The expected present value over deaths during the term is therefore the integral of present value times density: $$ B\int_0^n e^{-\delta t}\mu e^{-\mu t}\,dt = B\mu\int_0^n e^{-(\mu+\delta)t}\,dt = B\frac{\mu}{\mu+\delta}\left(1-e^{-(\mu+\delta)n}\right). $$ The final equality follows by integrating the exponential function. For whole-life insurance under the same model, take the limit as $n$ tends to infinity; if $\mu+\delta>0$, the exponential term tends to zero and the value is $B\mu/(\mu+\delta)$.

Worked Example: One-Year Term Insurance Paid Immediately at Death

A hypothetical one-year term policy pays $\$10{,}000$ immediately on death if death occurs within the year. Assume an exponential future lifetime with constant force of mortality $\mu=0.5$ per year, and a constant force of interest $\delta=0.5$ per year. Calculate the net single premium. The method is to integrate the discounted benefit against the death-time density.

The density is $f(t)=0.5e^{-0.5t}$ for $t\ge0$. For a death at time $t$, the discount factor is $e^{-0.5t}$. Therefore the expected present value per dollar of benefit is $\int_0^1 e^{-0.5t}(0.5e^{-0.5t})\,dt=0.5\int_0^1e^{-t}\,dt$.

Evaluating the integral gives $0.5(1-e^{-1})=0.3160602794\ldots$. The unrounded premium is $10{,}000[0.5(1-e^{-1})]=\$3{,}160.602794\ldots$, so the net single premium is $\$3{,}160.60$, rounded to the nearest cent.

As a check, the probability of death within the year is $1-e^{-0.5}=0.39346934$, which exceeds the insurance value per dollar, $0.31606028$, because a death benefit is discounted when death occurs after issue. The calculation keeps the full-precision value through multiplication; rounding the unit value too early can produce a misleading displayed multiplication.

Checks That Prevent Common Errors

Before accepting a net single premium, verify that the mortality probabilities correspond to the contract’s covered period, that the discount exponent matches the actual payment time, and that the benefit is included only in states where it is payable. For a discrete $n$-year term, the death probabilities plus the probability of survival to time $n$ must sum to one. For continuous insurance, the integration limits must match the contract term.

A frequent exam trap is to use the right death probabilities with the wrong payment time. If a benefit is paid at the end of the year of death, death during year $k+1$ means discount by $v^{k+1}$. Using $v^k$ discounts one year too little. Conversely, a benefit paid immediately at death is discounted by the exact death time, not by an integer policy-year exponent. Another error is to apply a constant mortality rate when the problem gives age-specific $q_x$ values; in that case, multiply the appropriate successive survival probabilities and use the stated rate for each year.

Identify the benefit event.
Decide whether the policy pays on death during a term, on death at any future duration, or only under another stated condition.

Match the timing convention.
Use the end-of-year discount factor for a discrete end-of-year benefit, or discount at the exact death time for continuous payment.

Calculate expected present value.
Multiply each possible present value by its probability, or integrate present value against the death-time density, then apply the benefit amount.

Check and round.
Check probability totals or the model’s limiting behavior, retain full precision during calculations, and round the final dollar amount to cents.

Net single premiums are a direct application of expected present value: the contract’s benefit pattern determines the possible payments, while interest and mortality determine their probabilities and values at issue. This distinction is important when comparing insurance forms with different terms or payment timing. These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the benefit timing before calculating. Answers are rounded only at the final step.

A hypothetical two-year term policy pays $\$20{,}000$ at the end of the year of death. Assume $i=5\%$, $q_x=0.10$, and $q_{x+1}=0.20$. Find the net single premium. (Answer: $\$5{,}170.07$)
A hypothetical three-year term policy pays $\$1{,}000$ at the end of the year of death. Assume $q=0.05$ in each of the three years and $i=0\%$. Find the net single premium. (Answer: $\$142.63$)
A hypothetical two-year term policy pays $\$8{,}000$ immediately at death. Assume exponential future lifetime with $\mu=0.25$ and force of interest $\delta=0.25$, both per year. Find the net single premium. (Answer: $\$2{,}528.48$)
A hypothetical whole-life policy pays $\$10{,}000$ at the end of the year of death. Assume a constant annual $q=0.10$ and $i=10\%$. Find the net single premium. (Answer: $\$5{,}000.00$)
A hypothetical two-year term policy pays $\$30{,}000$ at the end of the year of death. Assume $i=4\%$, $q_x=0.20$, and $q_{x+1}=0.25$. Find the net single premium and verify the probability of survival through the term. (Answer: $\$11{,}316.57$; survival probability $0.60$)

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