Whole-Life Annuity-Immediate Reserves

What a Whole-Life Annuity-Immediate Reserve Measures

A whole-life annuity-immediate pays an amount at the end of each year in which the annuitant is alive. The first payment is one year after the annuity begins. At a later duration, the prospective reserve is the expected present value of the payments that remain, conditional on the annuitant being alive at the valuation time. It is not the expected present value at issue of payments that might be made only if the annuitant survives to that duration.

This tutorial values the annuity’s future payments, with no future premiums payable. The same prospective principle applies more generally: value the future benefits and subtract the value of any future premiums. Here, the benefit stream is the annuity itself. The treatment of payment timing is essential: the standard annuity-immediate factor values payments at the ends of future years, not a payment due immediately.

Definition: For an annuitant aged $x$, a whole-life annuity-immediate of amount $B$ pays $B$ at the end of each future year in which the annuitant is alive. Its unit annuity factor is $a_x$. At duration $k$, immediately after the payment, if any, due at that time has been made, the prospective reserve conditional on survival to that time is $B a_{x+k}$.

Use an effective annual interest rate $i$, with discount factor $v=1/(1+i)$. The probability ${}_jp_x$ is the probability that a life aged $x$ survives $j$ years. Thus a payment $j$ years from the valuation date contributes only if the annuitant survives to receive it, and is discounted by $v^j$.

a_x=\sum_{j=1}^{\infty}v^j\,{}_jp_x

At duration $k$, use the attained age $x+k$ and the survival probabilities from that age. No factor ${}_kp_x$ is applied to $B a_{x+k}$: the reserve is conditional on being alive at duration $k$. The probability of reaching that duration is relevant when calculating an issue-time expected value, which is a different quantity.

A Recursion for the Annuity Factor

The annuity factor can be calculated either by summing discounted survival probabilities or by working forward one year at a time. The recursion is useful when future one-year survival probabilities are given by attained age, or when the survival probabilities become constant after a particular age.

Theorem: If $p_x={}_1p_x$ is the probability that a life aged $x$ survives one year, then $$ a_x=v p_x(1+a_{x+1}). $$ The factor $a_{x+1}$ is the value, at age $x+1$, of payments due at the ends of subsequent years.

To see why the recursion holds, condition on survival through the first year. If the annuitant does not survive, there are no future annuity payments. If the annuitant does survive, the first payment, of amount one, is made at the end of the year; the remaining payments have value $a_{x+1}$ at that time. The combined value at the end of the year is $1+a_{x+1}$, and its expected present value at age $x$ is therefore $v p_x(1+a_{x+1})$.

\begin{aligned} a_x &=\sum_{j=1}^{\infty}v^j\,{}_jp_x\\ &=v p_x\left(1+\sum_{r=1}^{\infty}v^r\,{}_rp_{x+1}\right)\\ &=v p_x(1+a_{x+1}). \end{aligned}

In the second line, the first payment is separated from all later payments. For a later payment $j=r+1$ years from age $x$, survival requires surviving the first year and then surviving $r$ more years from age $x+1$. This gives ${}_{r+1}p_x=p_x\,{}_rp_{x+1}$. The recursion assumes annual payments in arrears and valuation at a payment date, with the next payment one year away.

Worked Example: Summing Future Payments After a Duration

A hypothetical annuitant is aged 70 at a valuation date immediately after any payment due on that date. The annuity pays $1,500 at each future year-end if the annuitant is alive. The effective annual interest rate is $25\%$. The one-year survival probabilities are $p_{70}=0.90$, $p_{71}=0.80$, and $p_y=0.50$ for every integer age $y\geq 72$. Calculate the prospective reserve.

Method. Use the annuity-immediate sum, separating the first two payments from the infinite tail. This makes the payment timing and the survival condition for each payment explicit.

First, $v=1/(1+0.25)=0.80$. The survival probabilities from age 70 are ${}_1p_{70}=0.90$ and ${}_2p_{70}=0.90(0.80)=0.72$. For $j\geq 3$, survival to year $j$ is $0.72(0.50)^{j-2}$.

The first payment contributes $0.80(0.90)=0.72$ per dollar of annuity. The second contributes $0.80^2(0.72)=0.4608$. The tail, starting with the third year-end, is a geometric series. Its first term is $0.80^3(0.72)(0.50)=0.18432$, and its common ratio is $0.80(0.50)=0.40$. Therefore, its value is $0.18432/(1-0.40)=0.3072$.

a_{70}=0.72+0.4608+0.3072=1.488

Multiplying by the annual payment gives the reserve: $1{,}500(1.488)=\$2{,}232.00$. The result is conditional on the annuitant being alive at age 70. Check: all terms are nonnegative, and the calculated factor includes the year-one and year-two payment values plus the later tail.

Calculating a Reserve from an Attained Age

At duration $k$, the calculation begins from the attained age $x+k$, not the original issue age. If the future one-year survival probability is constant at $p$ from that attained age onward, the annuity factor has a geometric-series form. This special case requires both a constant effective annual interest rate and the stated constant survival probability for every future year.

a_y=\sum_{j=1}^{\infty}(vp)^j=\frac{vp}{1-vp}, \qquad vp<1

When the one-year survival probabilities differ by age, the recursion can combine a single age-specific probability with a constant-probability tail. The following example also shows how to convert a factor into a dollar reserve.

Worked Example: Using a Constant-Survival Tail

A hypothetical life annuity was issued at age 67. At duration 7, the annuitant is alive and age 74, immediately after any payment due at that time. The annual payment is $12,000. The effective annual interest rate is $4\%$. At age 74, the one-year survival probability is $0.80$; from age 75 onward, it is $0.60$ each year. Calculate the reserve at duration 7.

Method. First calculate the annuity factor from age 75 using the constant-survival tail, then use the recursion to move back to age 74. Do not multiply by the probability of surviving the first seven policy years: survival to duration 7 is already part of the valuation condition.

The discount factor is $v=1/1.04=25/26$. From age 75 onward, $vp=(25/26)(0.60)=15/26$, so $a_{75}=(15/26)/(1-15/26)=15/11$. Then the age-74 factor is $a_{74}=v p_{74}(1+a_{75})$.

a_{74}=\frac{25}{26}(0.80)\left(1+\frac{15}{11}\right) =\frac{10}{13}\cdot\frac{26}{11} =\frac{20}{11}

The reserve is $12{,}000(20/11)=\$21{,}818.18$, rounded to the nearest cent. Check: the annuity factor exceeds one because there is a possible payment at the end of the first future year, plus possible payments in later years; the dollar reserve is the factor multiplied by the stated annual amount.

Payment-Date Timing: Just Before or Just After

The usual duration reserve for an annuity-immediate is stated immediately after any payment due at that time has been made. At that point, the next possible payment is one full year away, so the factor is $a_{x+k}$. Immediately before a scheduled payment date, conditional on the annuitant being alive then, the payment due at that date is still outstanding. Its value is added to the reserve for the later payments.

\text{Reserve just before the payment date} = B+B a_{x+k} = B(1+a_{x+k})

This distinction is about timing, not a change in mortality or interest assumptions. The payment due at the current date is certain conditional on being alive at that date; it is not discounted another year and is not multiplied by a one-year survival probability. The later payments remain contingent on future survival.

Worked Example: Comparing Values Around a Payment

A hypothetical annuity pays $1,000 annually. Immediately after the payment at time 4, the annuitant is age 74. The effective annual interest rate is $4\%$; the one-year survival probability at age 74 is $0.80$, and it is $0.60$ at age 75 and every later integer age. Find the value immediately after the time-4 payment and immediately before it, conditional on being alive at time 4.

Method. Calculate the value of payments after time 4 using the factor from the prior example’s age-74 assumptions, then add the payment due at time 4 for the before-payment value.

For the after-payment value, $a_{74}=20/11$, so the reserve is $1{,}000(20/11)=\$1{,}818.18$, rounded to the nearest cent. Immediately before the payment, the time-4 payment of $1,000 is also due. Thus the value is $\$1{,}000+\$1{,}818.18=\$2{,}818.18$.

Check: the before-payment value exceeds the after-payment value by exactly $1,000, the payment just about to be made. Common mistake avoided: using $1+a_{74}$ for the standard after-payment reserve would incorrectly include a payment that has already been made.

Common Exam Traps

Several errors arise from mixing the reserve’s valuation condition with the probability that the annuitant reaches the valuation date, or from shifting the payment dates by one year. In the age-74 example, the standard after-payment reserve is $12{,}000(20/11)=\$21{,}818.18$. The following are different calculations, not alternative ways to obtain that reserve.

Multiplying by survival to the duration. If ${}_7p_{67}$ were, for example, $0.70$, then $0.70(\$21{,}818.18)=\$15{,}272.73$ would weight the reserve by the probability of reaching duration 7. It would not be the reserve conditional on being alive at that duration.
Using an annuity-due factor after the payment date. The incorrect factor $1+a_{74}=31/11$ gives $12{,}000(31/11)=\$33{,}818.18$. It includes an immediate payment. For the standard after-payment reserve, use $a_{74}$, not $1+a_{74}$.
Applying survival to a payment due now. Just before a payment date, conditional on being alive at that date, the scheduled payment is due. Do not multiply it by another one-year survival probability.

To keep the calculation organized, identify the valuation time, establish whether the payment at that time has already been made, state that the value is conditional on survival to the valuation time, and then use survival probabilities from the attained age for future payments.

Key takeaway: Immediately after the payment date at duration $k$, the prospective reserve for a level whole-life annuity-immediate is $B a_{x+k}$, where $a_{x+k}=\sum_{j=1}^{\infty}v^j\,{}_jp_{x+k}$. A payment due immediately before that date is added only if the valuation is explicitly made before the payment.

Check Your Understanding

For each problem, state whether the requested value is conditional on survival to the valuation time and use the specified payment timing.

A hypothetical annuity pays $600 per year. Immediately after a payment date, the annuitant is age $x+4$. The effective annual interest rate is $10\%$; the one-year survival probability at age $x+4$ is $0.80$, and from age $x+5$ onward it is $0.50$ each year. Find the reserve. (Answer: $800.00)
A $2,200 whole-life annuity-immediate is valued immediately after a payment date, conditional on the annuitant being alive at age $y$. The effective annual interest rate is $5\%$; $p_y=0.75$, and the one-year survival probability at every age $y+1$ and later is $0.50$. Find the reserve. (Answer: $3,000.00)
A $900 annuity is valued at a payment date when the annuitant is alive. The effective annual interest rate is $25\%$, and the one-year survival probability is $0.60$ at this age and every future age. Find the value immediately after the payment and immediately before it. (Answer: After: $600.00; before: $1,500.00)
A $2,500 annuity-immediate is valued immediately after a payment date at age $z$. The effective annual interest rate is $10\%$, $p_z=0.90$, and the one-year survival probability at age $z+1$ and every later age is $0.60$. Calculate the reserve using the recursion. (Answer: $4,500.00)
A $1,800 annuity-immediate is valued immediately after a payment date at duration 3, conditional on the annuitant being alive then. The effective annual interest rate is $25\%$; at the attained age, the one-year survival probability is $0.70$, and from the following age onward it is $0.40$ each year. Also, ${}_3p_x=0.45$ at issue. Find both the duration-3 reserve and its issue-time value weighted by survival to duration 3 and discounted for three years. (Answer: Duration-3 reserve: $1,482.35; issue-time weighted value: $341.53)

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