Life Annuities at Fractional Ages

Why Fractional Ages Matter

Life-table columns usually give survivor counts at integer ages, but an annuity valuation may begin when the annuitant is, for example, age 45.5. A payment date can also fall between integer ages. To value such payments, we need an assumption for mortality within each year of age. The uniform distribution of deaths assumption supplies a simple interpolation rule.

This rule is used to estimate survival at fractional ages from integer-age life-table values. Once the relevant values are found, the probability that a life survives from the fractional valuation age to a payment date is a ratio of survivor counts. The expected present value then combines that probability with the payment amount and its discount factor.

Definition: Under the uniform distribution of deaths (UDD) assumption, deaths are spread uniformly over each one-year age interval. If $0\leq r\leq 1$, the life-table survivor count at age $x+r$ is interpolated linearly: $$ \ell_{x+r}=(1-r)\ell_x+r\ell_{x+1}. $$ Here $\ell_x$ and $\ell_{x+1}$ are the survivor counts at integer ages $x$ and $x+1$. The interpolation applies within an age interval; for an age exactly at an integer, use the life-table value at that age.

From Interpolated Counts to Survival Probabilities

Let the annuitant be age $x+r$ at the valuation date, where $x$ is an integer and $0\leq r<1$. If a payment is due $t$ years later, the attained age at that date is $x+r+t$. Under the same life-table basis and fractional-age assumption, the probability of survival to the payment date is

{}_{t}p_{x+r}=\frac{\ell_{x+r+t}}{\ell_{x+r}}.

Interpolate the numerator if the attained age is fractional. If the attained age is an integer, use the life-table count at that integer age directly. This ratio uses the number alive at the payment date divided by the number alive at the valuation date; the initial age in the denominator must therefore be the actual fractional age, not the preceding integer age.

For payments $B_j$ at times $t_j$, each conditional on the annuitant being alive at that time, the expected present value at effective annual interest rate $i$ is

\operatorname{EPV}=\sum_j B_j v^{t_j}\,{}_{t_j}p_{x+r}, \qquad v=\frac{1}{1+i}.

This formula assumes the payment is made only if the annuitant is alive on the specified payment date. It does not apply unchanged to a payment guaranteed regardless of survival: such a payment is not multiplied by a survival probability. A payment at time zero, if due to a living annuitant, has survival probability one.

Why the Survival Ratio Works

The life-table count $\ell_y$ represents the number of lives from the table’s initial cohort expected to be alive at age $y$. Of the $\ell_{x+r}$ lives alive at the fractional valuation age, the fraction expected to remain alive at age $x+r+t$ is the count at that attained age divided by the starting count. Thus

P(\text{alive at }x+r+t\mid\text{alive at }x+r) =\frac{\ell_{x+r+t}}{\ell_{x+r}}.

For a single payment $B$ at time $t$, its present value is $Bv^t$ if paid and zero otherwise. The expected present value is therefore $Bv^t$ multiplied by the probability of payment. For several payments, expected values add, giving the sum above. This uses linearity of expectation and does not require payment-survival events at different dates to be independent.

Locate the valuation age.
Write the exact age as an integer age plus a fraction of a year, then interpolate its life-table count.

Find each payment age.
Add the payment time, measured in years, to the fractional valuation age. Interpolate each fractional attained age using the life-table interval containing it.

Calculate survival and discounting.
For each life-contingent payment, divide its attained-age count by the valuation-age count and multiply by the appropriate discount factor.

Add the expected present values.
Multiply each discounted survival probability by its payment amount, then sum the results.

Worked Applications

Worked Example: A Payment Halfway Through the First Year

A hypothetical annuity pays $1,000 at time 0.5 if the annuitant is alive. At valuation the annuitant is age 45.5. The table gives $\ell_{45}=8{,}000$ and $\ell_{46}=7{,}800$. The effective annual interest rate is $i=5\%$. Find the expected present value under UDD.

Method. Interpolate the life-table count at age 45.5, use the survival-count ratio to age 46, and discount for half a year.

The fraction of the age interval from 45 to 46 is $r=0.5$, so

\ell_{45.5}=(1-0.5)(8{,}000)+0.5(7{,}800) =4{,}000+3{,}900=7{,}900.

At time 0.5 the annuitant is age 46, so the survival probability is

{}_{0.5}p_{45.5}=\frac{\ell_{46}}{\ell_{45.5}} =\frac{7{,}800}{7{,}900}.

The discount factor is $v^{0.5}=(1.05)^{-0.5}$. Therefore

\operatorname{EPV} =1{,}000\left(\frac{7{,}800}{7{,}900}\right)(1.05)^{-0.5} \approx \$963.55.

Check. The survival probability is less than one and the half-year discount factor is also less than one, so the expected present value must be less than $1,000. The result is consistent with both reductions. The final monetary value is rounded to the nearest cent.

Common mistake avoided. Do not use $7{,}800/8{,}000$ as the conditional survival probability. That ratio starts at age 45, not at the annuitant’s valuation age 45.5.

Worked Example: Two Payments from a Fractional Valuation Age

A hypothetical annuity pays $400 at times 0.75 and 1.75, with each payment made only if the annuitant is alive. The annuitant is age 55.25 at valuation. The life table gives $\ell_{55}=12{,}000$, $\ell_{56}=11{,}700$, and $\ell_{57}=11{,}300$. The effective annual interest rate is $i=4\%$. Find the expected present value under UDD.

Method. First interpolate the count at the fractional starting age. Then calculate a separate conditional survival probability and discount factor for each payment.

At age 55.25, the fraction of the year from age 55 is $0.25$:

\ell_{55.25}=0.75(12{,}000)+0.25(11{,}700) =9{,}000+2{,}925=11{,}925.

The first payment is made at time 0.75, when the annuitant is age 56. The second is made at time 1.75, when the annuitant is age 57. Thus

{}_{0.75}p_{55.25}=\frac{11{,}700}{11{,}925}, \qquad {}_{1.75}p_{55.25}=\frac{11{,}300}{11{,}925}.

The present-value factors for the two dates are $(1.04)^{-0.75}$ and $(1.04)^{-1.75}$, respectively. Adding the two expected present values gives

\begin{aligned} \operatorname{EPV} &=400\left[ \left(\frac{11{,}700}{11{,}925}\right)(1.04)^{-0.75} +\left(\frac{11{,}300}{11{,}925}\right)(1.04)^{-1.75} \right]\\ &\approx \$734.97. \end{aligned}

Check. Both payments are conditional on survival, and both occur after valuation, so each term has a survival probability below one and a discount factor below one. The total is consequently below the certain, undiscounted total of $800. The final monetary value is rounded to the nearest cent.

Common mistake avoided. The denominator for both conditional probabilities is $11{,}925$, the count at age 55.25. Using $12{,}000$, the count at age 55, would incorrectly treat the annuitant as starting at age 55. That error gives approximately $730.38 instead of $734.97.

Worked Example: A Payment Date in a Later Fractional Year

A hypothetical annuity pays $2,500 once, at time 2, if the annuitant is alive. The annuitant is age 64.4 at valuation. The life table gives $\ell_{64}=30{,}000$, $\ell_{65}=29{,}400$, $\ell_{66}=28{,}700$, and $\ell_{67}=27{,}800$. The effective annual interest rate is $i=3\%$. Find the expected present value under UDD.

Method. Interpolate the starting count at age 64.4 and the attained-age count at age 66.4. Then use their ratio and discount for two years.

At the valuation age, the fraction of the interval from 64 to 65 is $0.4$:

\ell_{64.4}=0.6(30{,}000)+0.4(29{,}400) =18{,}000+11{,}760=29{,}760.

Two years after valuation, the annuitant is age 66.4. Interpolating within the interval from age 66 to age 67 gives

\ell_{66.4}=0.6(28{,}700)+0.4(27{,}800) =17{,}220+11{,}120=28{,}340.

The probability of surviving from age 64.4 to age 66.4 is

{}_{2}p_{64.4}=\frac{28{,}340}{29{,}760}.

The expected present value is therefore

\operatorname{EPV} =2{,}500\left(\frac{28{,}340}{29{,}760}\right)(1.03)^{-2} \approx \$2{,}244.05.

Check. The payment is contingent on survival and is made two years in the future, so its expected present value must be less than $2,500. The final monetary value is rounded to the nearest cent.

Common mistake avoided. The attained age is 66.4, not 66. The numerator must be interpolated in the age 66 to 67 interval; using $\ell_{66}$ would omit the mortality between ages 66 and 66.4.

Assumptions, Scope, and Exam Traps

UDD is an assumption about the distribution of deaths within each integer-age interval. It makes the survivor count linear between the table’s integer-age values. It does not mean that survival probabilities over equal-length portions of different years are identical: the slope of the interpolation depends on the life-table values in each interval. Nor should the linear interpolation be extended beyond the interval to which its endpoint values apply.

When a payment date is fractional, identify the attained age before selecting life-table entries. For instance, if valuation is at age 60.3 and a payment is due 1.4 years later, the attained age is 61.7. The count at that age is interpolated between ages 61 and 62, while the starting count is interpolated between ages 60 and 61. Mixing those intervals or using the elapsed time as though it were an age fraction can produce an incorrect probability.

Use the correct denominator. For survival conditional on being alive at the fractional valuation age, divide by $\ell$ at that fractional age.
Keep age and time distinct. A payment at time $t$ is made at attained age equal to valuation age plus $t$; $t$ alone is not the attained age.
Discount for the payment time. A payment at time $t$ uses $v^t$, not a discount factor based on the attained age.
Separate guaranteed and life-contingent payments. Apply a survival probability only to the part payable if the annuitant is alive.
Use the assumed mortality model. Fractional-age interpolation requires an assumption such as UDD; integer-age table values alone do not uniquely determine fractional-age survival.

The central procedure is to interpolate the valuation-age and payment-age counts separately, form the conditional survival ratio for each payment, and then discount and add. This method keeps the mortality assumption, payment timing, and interest calculation distinct, which is especially useful when an annuity has several payment dates at fractional ages.

Check Your Understanding

For each problem, assume UDD within every one-year age interval and an effective annual interest rate as stated. Round monetary answers to the nearest cent.

A hypothetical annuity pays $1,000 at time 0.7 if the annuitant is alive. The valuation age is 40.3, with $\ell_{40}=5{,}000$ and $\ell_{41}=4{,}900$; $i=4\%$. Find the expected present value. (Answer: $959.22)
A payment of $500 is due at time 0.4 if an annuitant aged 52.6 at valuation is alive. Given $\ell_{52}=9{,}000$, $\ell_{53}=8{,}700$, and $i=3\%$, find the expected present value. (Answer: $487.40)
A hypothetical contract pays $1,200 at time 1.25 if a life aged 60.25 at valuation is alive. Given $\ell_{60}=20{,}000$, $\ell_{61}=19{,}600$, $\ell_{62}=19{,}000$, and $i=4\%$, find the expected present value. (Answer: approximately $1,108.14)
An annuity pays $300 at times 0.6 and 1.6 if a life aged 70.4 at valuation is alive at each date. Given $\ell_{70}=10{,}000$, $\ell_{71}=9{,}600$, $\ell_{72}=9{,}100$, and $i=5\%$, write the expected present value as a numerical expression using the interpolated counts. (Answer: $300[(9{,}600/9{,}840)(1.05)^{-0.6}+(9{,}100/9{,}840)(1.05)^{-1.6}]$, approximately $540.84)
At valuation age 23.75, a life table gives $\ell_{23}=12{,}000$, $\ell_{24}=11{,}800$, and $\ell_{25}=11{,}500$. Under UDD, find the conditional probability that the life survives to age 24.5. (Answer: $\ell_{23.75}=11{,}850$, $\ell_{24.5}=11{,}650$, so the probability is $11{,}650/11{,}850\approx0.9831$)

What You'll Learn