Deferred Term Recursion at Benefit Eligibility

Why the Final Waiting Year Matters

A deferred term insurance has a waiting period during which death does not trigger a benefit. If the insured survives the waiting period, a term insurance begins. At the start of the final waiting year, the contract’s value therefore depends on two events in sequence: survival through that year, and then the value of the term insurance at eligibility. Discounting is needed because the eligible-term value is one year in the future.

The key is to keep the contract’s time points distinct. The value at the start of the final waiting year is not simply the eligible term value, and it is not the value of a term that already covers the waiting year. It is the eligible term value multiplied by the probability of reaching eligibility and discounted one year.

Definition: Consider a unit-benefit \(n\)-year term insurance that becomes eligible after a waiting period. It pays 1 at the end of the year of death if death occurs during the eligible term, and pays nothing for death during the waiting period or survival through the term. Write \(A_{x:\overline{n}|}\) for the value at age \(x\) of a unit-benefit \(n\)-year term insurance. Write \({}_{1|}A_{x:\overline{n}|}\) for the value at age \(x\) of that term insurance deferred one year.

The Recursion at Eligibility

Assume a constant effective annual interest rate \(i\), with \(v=1/(1+i)\). Let \({}_1p_x\) be the probability that a life aged \(x\) survives one year, and let \({}_1q_x=1-{}_1p_x\). Assume the deferred contract pays no benefit during the waiting year and that the eligible term has a unit benefit payable at the end of the year of death. At the end of the waiting year, if the insured is alive, the age is \(x+1\), and the eligible term has value \(A_{x+1:\overline{n}|}\).

{}_{1|}A_{x:\overline{n}|} = v\,{}_1p_x\,A_{x+1:\overline{n}|}.

This is the deferred-term recursion at the final waiting year. The survival factor is the probability of reaching the date coverage becomes eligible. The discount factor moves the value at that future date back one year. Death during the waiting year contributes zero, so there is no death-benefit term for that year.

The recursion assumes annual discrete benefits and a full year remaining until eligibility. It does not directly describe a contract that pays during the waiting year, has a benefit payable immediately on eligibility, or uses a different payment timing. Those contracts require cash flows appropriate to their terms.

Why the Recursion Holds

The result follows by conditioning on what happens during the waiting year. If death occurs, the contract pays nothing. If the insured survives, the contract reaches eligibility and has the eligible term insurance value at that time. Thus the value one year before eligibility is the discounted expected value of the cash flow at the end of that year:

Separate the waiting-year outcomes.
Death during the year has probability \({}_1q_x\) and produces no payment. Survival has probability \({}_1p_x\) and leads to eligibility.

Assign the value at the end of the year.
The value is 0 on death during the waiting year and \(A_{x+1:\overline{n}|}\) on survival.

Discount the expected value.
The expected value at the end of the year is \({}_1q_x(0)+{}_1p_xA_{x+1:\overline{n}|}\). Multiplying by \(v\) gives the value at the start of the year.

v\left[{}_1q_x(0)+{}_1p_xA_{x+1:\overline{n}|}\right] = v\,{}_1p_xA_{x+1:\overline{n}|}.

For reference, under the same annual discrete assumptions, the eligible \(n\)-year term value can be calculated from its possible years of death:

A_{y:\overline{n}|} =\sum_{k=1}^{n}v^k\,{}_{k-1}p_y\,{}_1q_{y+k-1}.

The \(k\)th term represents a benefit paid at the end of year \(k\): the insured must survive the preceding \(k-1\) years and then die in year \(k\). The formula stops after \(n\) years because survival through the eligible term produces no benefit.

Worked Calculations

Worked Example: A Two-Year Term After One Waiting Year

A hypothetical contract has one waiting year remaining for a life aged 40. The probability of surviving that year is \(0.85\). If the insured reaches age 41, a two-year term begins: the conditional probability of death in the first eligible year is \(0.10\), and the conditional probability of death in the second eligible year, given survival through the first, is \(0.20\). Let \(i=0\%\). Find the deferred value at age 40.

Method. Calculate the eligible term value at age 41, then multiply by the waiting-year survival and discount factors. The death probabilities are conditional on reaching the start of each eligible year.

At \(i=0\%\), \(v=1\). The first-year term benefit has value \(1(0.10)=0.10\).

To receive the second-year benefit, the insured must survive the first eligible year, with probability \(1-0.10=0.90\), and then die in the second eligible year. Its value is \(1(0.90)(0.20)=0.18\).

Therefore the eligible two-year term value is \(0.10+0.18=0.28\).

For the final waiting year, \(v\,{}_1p_{40}=1(0.85)=0.85\). Hence \({}_{1|}A_{40:\overline{2}|}=0.85(0.28)=0.238\).

Check. With no interest, the result is the probability of surviving the waiting year and then receiving a benefit during the eligible term: \(0.85(0.28)=0.238\). A death during the waiting year is not insured. The calculation avoids treating the eligible term as if it had already started at age 40.

Worked Example: Discounting and Survival Both Apply

A hypothetical deferred contract has one waiting year remaining. The annual effective interest rate is \(i=25\%\), and the probability of surviving the waiting year is \(0.80\). At eligibility, a two-year term begins. Its conditional death probabilities are \(0.20\) in the first eligible year and \(0.30\) in the second eligible year, given survival through the first. Find the value at the start of the waiting year.

Method. First value the eligible term at the start of eligibility. Then apply both the waiting-year survival probability and the one-year discount factor.

The discount factor is \(v=1/(1.25)=0.80\). For the first eligible year, the value of a death benefit is \(v(0.20)=0.80(0.20)=0.16\).

The probability of surviving that first eligible year is \(1-0.20=0.80\). The probability of surviving the first eligible year and dying in the second is \(0.80(0.30)=0.24\). The second-year benefit has value \(v^2=0.80^2=0.64\) per dollar payable at that time, so its contribution is \(0.64(0.24)=0.1536\).

The eligible term value is therefore \(0.16+0.1536=0.3136\). For the final waiting year, the combined survival and discount factor is \(v\,{}_1p=0.80(0.80)=0.64\).

Thus the value at the start of the waiting year is \(0.64(0.3136)=0.200704\).

Check. The value is below the eligible term value because it is discounted and weighted by the probability of reaching eligibility. The two factors are both \(0.80\); using just one would give \(0.80(0.3136)=0.25088\), which is too large.

Worked Example: Three Eligible Years With Zero Interest

A hypothetical benefit becomes eligible after one final waiting year. The probability of surviving the waiting year is \(0.90\). At eligibility, the term lasts three years; the conditional death probabilities in years 1, 2, and 3 are \(0.10\), \(0.20\), and \(0.25\), respectively, conditional on reaching each year. The effective annual interest rate is \(0\%\). Find the value at the start of the waiting year.

Method. Sum the probabilities of death in each eligible year to obtain the unit-benefit term value, then multiply by the probability of surviving the waiting year. With zero interest, no discount adjustment changes the values.

The probability of death in eligible year 1 is \(0.10\). The probability of death in year 2 is \(0.90(0.20)=0.18\), because the insured must survive year 1. The probability of death in year 3 is \(0.90(0.80)(0.25)=0.18\), because survival is required in both earlier years.

The eligible term value is \(0.10+0.18+0.18=0.46\). Since \(i=0\%\), \(v=1\), and the final waiting-year factor is \(1(0.90)=0.90\).

The deferred value is \(0.90(0.46)=0.414\).

Check. The eligible death probabilities sum to \(0.46\), which is less than 1 because the insured may survive the three-year term. The deferred value is smaller still because the insured must also survive the waiting year.

Common Exam Trap: Omitting One of the Two Factors

A frequent error is to multiply the eligible term value by survival alone, as though the future value were already measured at the start of the waiting year. The opposite error is to discount by \(v\) but ignore the chance that the insured dies before eligibility. Both factors are required: \(v\) accounts for time value of money, and \({}_1p_x\) accounts for reaching the eligible term.

For the second worked example, the eligible term value is \(0.3136\). Using survival alone gives the incorrect result \(0.80(0.3136)=0.25088\). Using discount alone also gives \(0.80(0.3136)=0.25088\). The correct calculation is \(v\,{}_1p\,A=0.80(0.80)(0.3136)=0.200704\).

Another trap is to treat the waiting year’s death probability as an insured death probability. For this contract, death during the waiting year produces no benefit, so its contribution to the value is zero. In contrast, a death during any of the eligible term years can produce a benefit, subject to survival to that year.

Key takeaway: At the start of the final waiting year, a deferred unit-benefit term insurance is worth the eligible term value multiplied by the waiting-year survival probability and the one-year discount factor: \({}_{1|}A_{x:\overline{n}|}=v\,{}_1p_xA_{x+1:\overline{n}|}\). Check the benefit trigger, the valuation date, and both factors before calculating.

Check Your Understanding

For each problem, assume annual discrete benefits payable at the end of the year of death, no benefit during the waiting year, and the mortality assumptions stated in the problem. Compute the deferred value at the start of the final waiting year.

A hypothetical one-year waiting period has survival probability \(0.90\). At eligibility, a two-year term has conditional death probabilities \(0.20\) in year 1 and \(0.25\) in year 2. If \(i=0\%\), what is the deferred value? (Answer: \(0.3600\))
A hypothetical contract has one waiting year with survival probability \(0.90\). At eligibility, a two-year term has conditional death probabilities \(0.10\) and \(0.20\) in its successive years. If \(i=10\%\), what is the value at the start of the waiting year? (Answer: \(0.1961\))
A hypothetical waiting year has survival probability \(0.75\). The eligible two-year term has conditional death probabilities \(0.10\) and \(0.20\), and \(i=25\%\). Find the deferred value. (Answer: \(0.11712\))
A hypothetical contract has one waiting year with survival probability \(0.80\), followed by a three-year term with conditional death probabilities \(0.15\), \(0.10\), and \(0.20\) in its successive years. If \(i=0\%\), find the deferred value. (Answer: \(0.3104\))
A hypothetical waiting year has survival probability \(0.95\). At eligibility, a two-year term has conditional death probabilities \(0.08\) and \(0.12\). If \(i=10\%\), find the deferred value, rounded to four decimal places. (Answer: \(0.1416\))

What You'll Learn