Deferred Term Insurance by Difference

Why a Difference of Deferred Values Works

A term insurance contract may cover a life only after a waiting period and only for a limited number of years. The valuation challenge is to include death benefits in the covered period while excluding both deaths during the waiting period and deaths after coverage ends. A difference of deferred whole-life values does exactly that: the first value includes deaths after the waiting period, and the second removes the deaths occurring after the term ends.

The identity is useful when deferred whole-life values are known or are straightforward to calculate. Its correct use depends on precisely identifying the first and last covered years and maintaining the same benefit amount and payment timing in both values. This treatment of discrete insurance is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Consider a life aged $x$ and a unit benefit payable at the end of the year of death. A deferred whole-life insurance with an $m$-year deferral pays only if death occurs after the life has survived $m$ years. Its value at age $x$ is denoted ${}_{m|}A_x$. For $m=0$, this is ordinary whole-life insurance, so ${}_{0|}A_x=A_x$. A deferred term insurance covering years $m+1$ through $n$, where $0\leq m<n$, pays at the end of the year of death if death occurs during one of those years, and pays nothing otherwise.

Here, “year $k$” means the interval from duration $k-1$ to duration $k$. Thus, death in year $m+1$ occurs after survival for $m$ complete years. Death in year $n$ is included; death after surviving $n$ years is not. This tutorial uses the discrete model in which the benefit is paid at the end of the death year, not immediately at the moment of death.

The Deferred Term Identity

Use the standard discrete whole-life insurance value $A_x$, with $v=1/(1+i)$, and survival probabilities ${}_kp_x$. The value of an $m$-year deferred whole-life insurance is

{}_{m|}A_x=v^m\,{}_mp_x\,A_{x+m}.

The factors $v^m$ and ${}_mp_x$ account for discounting to time 0 and for the requirement that the life survive the deferral period. Conditional on survival to age $x+m$, the remaining insurance has value $A_{x+m}$.

Theorem: For $0\leq m<n$, a unit-benefit discrete term insurance covering deaths in years $m+1,\ldots,n$, with the benefit paid at the end of the death year, has value

{}_{m|}A_x-{}_{n|}A_x.

For a benefit of amount $B$, its value is $B\bigl({}_{m|}A_x-{}_{n|}A_x\bigr)$.

The first deferred whole-life value includes every death after the life has survived $m$ years, including deaths in years $n+1,n+2,\ldots$. The second includes precisely those deaths after survival for $n$ years. Subtracting therefore leaves the benefits for years $m+1$ through $n$, inclusive.

Set the year boundaries.
For cover in years $m+1$ through $n$, the first deferred value uses $m$; the value to subtract uses $n$.

Use a common basis.
Both deferred whole-life values must use the same age, interest assumption, mortality basis, benefit amount, and end-of-death-year payment convention.

Subtract in the correct order.
Calculate ${}_{m|}A_x-{}_{n|}A_x$, then multiply by $B$ if the benefit is not one unit.

Derivation from the Death-Year Payments

Let ${}_{k|}A_x$ denote the present value of benefits for death in any year after year $k$. In the discrete model, the probability of death in year $j$ is ${}_{j-1}p_xq_{x+j-1}$, and the payment at the end of that year is discounted by $v^j$. Consequently,

{}_{m|}A_x=\sum_{j=m+1}^{\infty}v^j\,{}_{j-1}p_xq_{x+j-1}, \qquad {}_{n|}A_x=\sum_{j=n+1}^{\infty}v^j\,{}_{j-1}p_xq_{x+j-1}.

Subtract the second series from the first. Every term for $j>n$ appears in both and cancels. The remaining terms are exactly those with $m+1\leq j\leq n$:

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

This also verifies the identity directly from the expected present value of the covered death benefits. It requires the same whole-life payment basis in both deferred values. It does not describe a benefit payable at the moment of death, a survival benefit at the end of year $n$, or a contract whose benefit amount changes across the covered years.

Worked Examples

Worked Example: Constant Mortality and Coverage in Years 3 Through 5

A hypothetical policy pays one unit at the end of the year of death if a life aged $x$ dies in years 3, 4, or 5. Assume an effective annual interest rate $i=5\%$ and a constant annual mortality probability $q=0.10$ at every age. Find the value to four decimal places.

Method. Apply the deferred term difference with $m=2$ and $n=5$. Under constant mortality, $p=1-q=0.90$, $v=1/1.05=20/21$, and $vp=6/7$.

First calculate the whole-life insurance value. With constant mortality, the death benefit in year $j$ has present-value contribution $v^jp^{j-1}q$, so the geometric series gives

A_x=\sum_{j=1}^{\infty}v^jp^{j-1}q =\frac{vq}{1-vp} =\frac{(20/21)(1/10)}{1-6/7} =\frac{2}{3}.

The deferred whole-life values are

{}_{2|}A_x=(vp)^2A_x =\left(\frac67\right)^2\frac23, \qquad {}_{5|}A_x=(vp)^5A_x =\left(\frac67\right)^5\frac23.

Subtract:

{}_{2|}A_x-{}_{5|}A_x =\frac23\left[\left(\frac67\right)^2-\left(\frac67\right)^5\right] =\frac{3048}{16807} \approx 0.181353.

Rounded to four decimal places, the value is 0.1814. As a check, the directly covered death-year contributions are $v^3pq+v^4p^2q+v^5p^3q$, which are the same three terms retained by the subtraction.

Common mistake avoided: Using $m=3$ for the first deferred value would omit death in year 3. The deferral is two years because coverage begins in year 3, after survival through years 1 and 2.

Worked Example: Age-Varying Mortality and a Two-Year Term

A hypothetical policy pays one unit at the end of the death year if a life aged $x$ dies in year 2 or year 3. The effective annual interest rate is $i=25\%$, so $v=0.8$. Assume $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.25$, and $q_{x+j}=0.20$ for all $j\geq3$. Find the value using deferred whole-life values.

Method. The required value is ${}_{1|}A_x-{}_{3|}A_x$. To evaluate these, calculate the whole-life values at the ages reached after the relevant deferral periods.

From age $x+3$ onward, mortality is constant at $q=0.20$, with $p=0.80$ and $vp=0.64$. Therefore,

A_{x+3}=\frac{vq}{1-vp} =\frac{0.8(0.20)}{1-0.64} =\frac49.

At age $x+2$, the first-year death probability is $0.25$, followed by the whole-life value at age $x+3$ if the life survives:

A_{x+2}=0.8(0.25)+(0.8)(0.75)A_{x+3} =0.20+0.60\left(\frac49\right) =\frac7{15}.

At age $x+1$, the first-year death probability is $0.20$, and survival for that year is $0.80$:

A_{x+1}=0.8(0.20)+(0.8)(0.80)A_{x+2} =0.16+0.64\left(\frac7{15}\right) =0.458666\ldots.

Now calculate both deferred values. Survival through year 1 is $0.90$, so

{}_{1|}A_x=(0.8)(0.90)A_{x+1} =0.72(0.458666\ldots) =0.33024.

Survival through three years is $(0.90)(0.80)(0.75)=0.54$, giving

{}_{3|}A_x=(0.8)^3(0.54)\left(\frac49\right) =0.12288.

Thus the term value is $0.33024-0.12288=\mathbf{0.20736}$. As a check, direct calculation gives $v^2(0.90)(0.20)+v^3(0.90)(0.80)(0.25)=0.1152+0.09216=0.20736$.

Common mistake avoided: Treating mortality as constant from age $x+1$ would ignore the specified $q_{x+2}=0.25$. The later mortality assumption applies only from age $x+3$ onward.

Worked Example: Converting a Unit Value to a Benefit Value

A hypothetical insurer considers a benefit of $50,000 payable at the end of the year of death if a life aged $x$ dies in years 3, 4, or 5. Assume $i=5\%$ and $q=0.10$ at every age. Find the expected present value of the benefit to the nearest cent.

Method. This has the same mortality and timing assumptions as the constant-mortality calculation above. The unit-benefit value is

{}_{2|}A_x-{}_{5|}A_x =\frac23\left[\left(\frac67\right)^2-\left(\frac67\right)^5\right] =\frac{3048}{16807} \approx 0.181353.

Multiply by the benefit amount:

50{,}000\left(\frac{3048}{16807}\right) =\frac{152{,}400{,}000}{16{,}807} \approx \$9{,}067.65038.

The expected present value is $9,067.65, rounded to the nearest cent. The scale check is that the unit value is about $18.14\%$, so a value near $9,068 is reasonable for a $50,000 benefit.

Common mistake avoided: The unit-benefit factor is not itself a dollar value. Multiply it by $50,000 once, and round only the final amount.

Boundary Checks and Common Traps

The difference identity is compact, but indexing errors can shift the coverage by a full year. Before evaluating, write the covered years explicitly and translate them into the two deferral periods. For coverage in years 1 through $n$, use $A_x-{}_{n|}A_x$. For coverage beginning in year $m+1$, the first deferred value is ${}_{m|}A_x$, not ${}_{m+1|}A_x$.

A useful boundary check is to consider what happens when $n=m+1$. The difference ${}_{m|}A_x-{}_{m+1|}A_x$ must contain only the death benefit for year $m+1$. Also, the result must be nonnegative: the first deferred whole-life insurance includes all payments in the second value, plus payments during the covered term.

Wrong upper boundary: Subtracting ${}_{n-1|}A_x$ would remove deaths only after year $n-1$, so deaths in year $n$ would be excluded from the result.
Wrong payment timing: The terms $v^j{}_{j-1}p_xq_{x+j-1}$ assume payment at the end of death year $j$. Do not substitute a continuous payment model.
Changed basis: Deferred values calculated using different interest or mortality assumptions cannot be subtracted to represent this contract.
Extra survival benefit: A payment at the end of year $n$ if the insured survives is not included in this death-only term value.

Key takeaway: For end-of-year-of-death benefits during years $m+1$ through $n$, value the contract as ${}_{m|}A_x-{}_{n|}A_x$. Identify the boundaries first, use a consistent basis, and scale the unit value by the benefit amount only after the difference is evaluated.

Check Your Understanding

For each problem, assume a discrete benefit payable at the end of the year of death. Give probability-scale values to four decimal places and dollar values to the nearest cent.

Assume constant annual mortality $q=0.20$ and effective annual interest $i=25\%$. Find the unit-benefit value for coverage in years 2 through 4. (Answer: 0.2099)
Assume constant annual mortality $q=0.10$ and $i=5\%$. A unit benefit is payable for death in years 1 or 2. Find its value. (Answer: 0.1769)
Assume constant annual mortality $q=0.20$ and $i=25\%$. A $15,000 benefit covers deaths in years 2 through 4. Find its expected present value. (Answer: $3,148.19)
Assume constant annual mortality $q=0.10$ and $i=5\%$. Find the unit-benefit value for coverage in years 3 through 6. (Answer: 0.2254)
For the same age, interest rate, mortality basis, and end-of-death-year timing, suppose ${}_{2|}A_x=0.2140$ and ${}_{6|}A_x=0.0715$. A $40,000 benefit covers deaths in years 3 through 6. Calculate its value. (Answer: $5,700.00)

What You'll Learn