Deferred Insurance Net Level Premiums

Why Deferral Changes the Premium Calculation

A deferred insurance benefit is not payable for every death covered by an otherwise similar immediate policy. The insured must first survive the deferral period; only then does the stated coverage apply. Premiums, however, may be payable during the deferral period, throughout the benefit period, or for some other specified duration. A correct net level premium therefore requires two separate timelines: when benefits may be paid and when premiums are due.

The formulas below use an annual discrete model. A death benefit is paid at the end of the year of death, and a premium is paid at the beginning of a year if the insured is then alive. These timing assumptions matter. If a contract specifies a different benefit or premium timing, its expected present values must be adjusted accordingly.

Definition: An $m$-year deferred $n$-year term insurance pays a benefit at the end of the year of death if death occurs in policy years $m+1,\ldots,m+n$. It pays no benefit for death during the first $m$ years or after the $n$-year benefit period. An $m$-year deferred whole-life insurance pays at the end of the year of death if the insured survives the $m$-year deferral period and subsequently dies.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write ${}_kp_x$ for the probability that a life aged $x$ survives $k$ years, and ${}_kq_x=1-{}_kp_x$. For a life who has survived $k-1$ years, the probability of dying during the next year is $q_{x+k-1}$. Thus the unconditional probability of surviving $k-1$ years and then dying in the next year is ${}_{k-1}p_xq_{x+k-1}$.

Present Value of a Deferred Benefit

For an $m$-year deferred $n$-year term benefit of amount $B$, death in year $m+j$, where $j=1,\ldots,n$, has probability ${}_{m+j-1}p_xq_{x+m+j-1}$. The payment is discounted $m+j$ years. Therefore the expected present value of benefits is

B\sum_{j=1}^{n}v^{m+j}{}_{m+j-1}p_xq_{x+m+j-1}.

The corresponding deferred whole-life expression uses the same sum over all years after the deferral period. An equivalent way to express either value is to condition on survival to time $m$. Given survival, the policy is valued at age $x+m$, and its value at issue is further multiplied by the probability and discount factor for surviving to the deferral date:

{}_mE_x=v^m{}_mp_x,\qquad \text{EPV of deferred benefits}={}_mE_x\times \text{EPV of the applicable insurance at age }x+m.

This factorization is valid because no benefit is paid before the deferral period ends. It is a conditional-value calculation, not an assumption that the insured is certain to reach age $x+m$.

Premiums and the Equivalence Principle

Suppose level annual premiums are payable in advance at times $0,1,\ldots,r-1$, provided the insured is alive at each payment time. Their expected present value per dollar of premium is

\sum_{j=0}^{r-1}v^j{}_jp_x.

This is a temporary life annuity-due value. If $P$ is the net annual premium, the expected present value of premiums is $P$ times this factor. Under the equivalence principle, the net level premium is chosen to make the expected present values of premiums and benefits equal:

P= \frac{\text{EPV of benefits}} {\displaystyle\sum_{j=0}^{r-1}v^j{}_jp_x}.

The denominator depends on the premium-payment duration, not simply on the length of the benefit period. In particular, premiums payable during a deferral period must still be included if the insured is alive when they fall due.

Mark the benefit years.
For an $m$-year deferral followed by $n$ years of term coverage, benefits can arise only from death years $m+1$ through $m+n$.

Value each possible benefit.
For each covered year, multiply the death probability by the benefit amount and the discount factor for the payment date.

Mark the premium dates.
Include each premium only if it is due under the contract and the insured is alive at that date.

Apply equivalence.
Divide the benefit present value by the present value of premiums per dollar to obtain the net level premium.

Worked Examples

Worked Example: Two-Year Deferred Term With Two Premiums

A hypothetical policy pays $40,000 at the end of the year of death if death occurs in years 3 or 4; there is no benefit for death in years 1 or 2. Premiums are payable at times 0 and 1 if the insured is alive. Assume $i=10\%$, and assume an independent annual mortality probability $q=0.143$ at each age, so $p=0.857$. Find the net annual premium.

Method. Value the benefits in years 3 and 4, then divide by the expected present value of the two premiums per dollar. This keeps the benefit deferral separate from the premium duration.

For death in year 3, the insured must survive two years and then die: the probability is $p^2q=(0.857)^2(0.143)=0.105026207$. The discounted expected benefit per dollar of face amount is $v^3p^2q=(10/11)^3(0.105026207)=0.078907743$.

For death in year 4, the probability is $p^3q=(0.857)^3(0.143)$. Discounting three years of survival and the year-4 payment gives $v^4p^3q=(v^3p^2q)(vp)=0.078907743(0.857/1.1)=0.061476305$.

Thus the benefit present value is $40{,}000(0.078907743+0.061476305)=\$5{,}615.36$, to the nearest cent. The premium at time 0 has present-value factor 1. The premium at time 1 is paid only if the insured survives one year, so its factor is $vp=(10/11)(0.857)=0.779090909$. The premium factor is $1+0.779090909=1.779090909$.

Therefore $P=5{,}615.36/1.779090909=\$3{,}156.31$, rounded to the nearest cent. As a check, $3{,}156.31(1.779090909)$ is approximately $5,615.36, matching the benefit present value. The common mistake avoided here is treating the benefit as beginning before year 3 or omitting the survival condition on the second premium.

Worked Example: Premiums Paid Through the Benefit Period

A hypothetical policy defers coverage for 2 years and then provides a 3-year term benefit of $100,000, payable at the end of the year of death. Premiums are payable annually in advance at times 0 through 4 if the insured is alive. Assume $i=25\%$, $p=0.625$, and $q=0.375$ each year. Find the net annual premium.

Method. Enumerate the three benefit years and all five premium dates. Here $v=0.8$ and $vp=0.8(0.625)=0.5$.

For death in year 3, the discounted probability is $v^3p^2q=(0.8)^3(0.625)^2(0.375)=0.075$. For year 4 it is the year-3 factor multiplied by $vp$, giving $0.075(0.5)=0.0375$. For year 5 it is $0.0375(0.5)=0.01875$. The total factor is $0.075+0.0375+0.01875=0.13125$, so the benefit present value is $100{,}000(0.13125)=\$13{,}125$.

The five premium dates have expected present-value factors $1,(vp),(vp)^2,(vp)^3,(vp)^4$, or $1,0.5,0.25,0.125,0.0625$. Their sum is $1.9375$. Hence $P=13{,}125/1.9375=210{,}000/31=\$6{,}774.19$, rounded to the nearest cent.

The check is $6{,}774.1935(1.9375)=\$13{,}125$, before rounding. The common mistake avoided is using only the three-year benefit duration to count premiums: the contract specifies five possible premium dates, including dates in the deferral period.

Worked Example: Deferred Whole-Life Cover With Lifetime Premiums

A hypothetical policy pays $100,000 at the end of the year of death, provided death occurs after a 3-year deferral. Premiums are payable annually in advance from time 0 for as long as the insured is alive. Assume $i=25\%$, $p=0.8$, and $q=0.2$ each year. Find the net annual premium.

Method. First find the immediate whole-life insurance value under the stated constant mortality model, then defer it for 3 years. Separately value the lifetime premium annuity-due.

For immediate whole-life insurance, the discounted probability of death in year $k$ is $v^kp^{k-1}q=vq(vp)^{k-1}$. Summing this geometric series gives $A=vq/[1-vp]$. Here $v=0.8$, $vp=0.64$, and $vq=0.16$, so $A=0.16/(1-0.64)=0.16/0.36=4/9$.

Survival and discounting through the 3-year deferral contribute $(vp)^3=(0.64)^3=0.262144$. Therefore the deferred benefit value per dollar is $0.262144(4/9)=0.116508444$, and the benefit present value is $100{,}000(0.116508444)=\$11{,}650.84$, rounded to the nearest cent.

The lifetime premium factor is the sum $1+vp+(vp)^2+\cdots=1/(1-vp)=1/0.36=25/9$. Thus $P=11{,}650.8444/(25/9)=\$4{,}194.30$, rounded to the nearest cent. Multiplying the unrounded premium by $25/9$ recovers the unrounded benefit value. The common mistake avoided is charging premiums only after the deferral; this contract requires premiums from issue.

Why the Formula Works—and When to Adjust It

The equivalence equation follows directly from the definition of the net level premium. Let $Z$ be the present value of the benefit and let $W$ be the present value of the premiums per dollar of level premium. The present value of premiums is $PW$. Equivalence requires equal expected present values, so $E[Z]=P E[W]$. Since $E[W]$ is the premium factor, solving gives $P=E[Z]/E[W]$. The formula does not require benefits and premiums to have the same duration; it requires that each cash flow be included at its actual time and under its applicable survival condition.

The conditional factorization for deferral has the same logic. A payment after time $m$ is possible only if the insured is alive then. Its present value from issue equals the probability of surviving to time $m$, multiplied by the discount to that time and by the conditional value of the remaining contract. Hence the factor ${}_mE_x=v^m{}_mp_x$. If mortality varies by age, use the appropriate survival probabilities and conditional insurance value at age $x+m$; do not substitute a constant-$q$ shortcut.

These results assume annual premiums in advance and benefits at the end of the year of death. If the contract instead pays continuously at death or specifies premiums in arrears, the cash-flow timing changes the relevant present-value factors. A net premium also excludes expenses and profit loads; those are not part of the equivalence calculation presented here.

Common Timing Traps

Discounting from the wrong date. A benefit for death in year $m+j$ is discounted $m+j$ years when paid at the end of that year.
Forgetting survival to the covered year. A year-$k$ death probability is ${}_{k-1}p_xq_{x+k-1}$, not merely $q_{x+k-1}$.
Stopping premiums when the deferral ends by default. The contract, not the benefit period, determines how long premiums are payable.
Including a premium after death. An advance premium at time $j$ is included only if the insured survives to time $j$.
Confusing conditional and unconditional values. A value at age $x+m$ must be multiplied by ${}_mE_x$ to express its value at issue.

The central exam habit is to write the benefit dates and premium dates before calculating. Once those timelines are explicit, the survival probabilities, discount powers, and payment counts follow from the contract rather than from a memorized label.

Check Your Understanding

For each hypothetical policy, use annual discrete benefits paid at the end of the year of death and premiums paid in advance when due. Assume the stated mortality probabilities apply independently each year.

A $10,000 policy defers coverage for 1 year and then insures the next 2 years. Premiums are due at times 0 and 1. If $i=0$ and $q=0.20$, find the net annual premium. (Answer: $1,600.00)
A $20,000 policy has a 2-year deferral followed by 1 year of term coverage. Three premiums are due at times 0, 1, and 2. If $i=0$, $p=0.75$, and $q=0.25$, find the net annual premium. (Answer: $1,216.22)
A $40,000 policy defers coverage for 1 year and then provides a 2-year term benefit. Three premiums are due at times 0, 1, and 2. If $i=25\%$, $p=0.625$, and $q=0.375$, find the net annual premium. (Answer: $5,142.86)
A $50,000 deferred whole-life policy has a 2-year deferral. Four premiums are due at times 0, 1, 2, and 3. If $i=25\%$, $p=0.8$, and $q=0.2$ each year, find the net annual premium. (Answer: $3,937.38)

What You'll Learn