Deferred Annuity Reserves

Why Deferred Annuities Need Two Time Periods

A deferred life annuity has two distinct parts: a period when no annuity payments are due, followed by a period when payments may be made while the annuitant is alive. At an intermediate duration, the first part has shortened. A reserve calculation must account for the probability and interest discounting needed to reach the payment period, then value the annuity payments from that point onward.

The payment timing matters. In this tutorial, a deferred annuity-immediate has annual payments of amount $B$ at the ends of years $m+1,m+2,\ldots$, provided the annuitant is alive on each payment date. Thus the first payment follows $m$ full years without payments. An annuity-due has different timing; its value is discussed separately below.

Definition: The prospective reserve at duration $k$, conditional on the life aged $x+k$ being alive, is the expected present value at that duration of future benefits minus the expected present value of future premiums. A single-premium contract has no future premiums, so its reserve is just the value of its future benefits. The valuation assumes the stated interest and survival model and does not include expenses unless they are explicitly included.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. The probability that a life aged $y$ survives $t$ years is ${}_tp_y$. The annuity-immediate values used here have payments at the ends of future years: $a_y$ denotes a whole-life annuity-immediate, and $a_{y:\overline{n}|}$ denotes an $n$-year temporary life annuity-immediate.

Value Before the Payment Period Begins

Suppose the valuation is at duration $k<m$, and put $r=m-k$, the number of years remaining until the end of the deferral period. If the annuity then continues for life, the first payment is one year after duration $m$. The value at duration $k$ therefore includes survival and discounting through the $r$-year remaining deferral, followed by the whole-life annuity value at age $x+m$.

\text{EPV at duration }k = Bv^r\,{}_rp_{x+k}\,a_{x+m}, \qquad k<m.

For an annuity with only $n$ payments after the deferral period, replace the whole-life annuity factor with a temporary annuity factor:

\text{EPV at duration }k = Bv^r\,{}_rp_{x+k}\,a_{x+m:\overline{n}|}.

If the life has already reached duration $m$, there is no deferral left. For the annuity-immediate timing defined above, its value is $Ba_{x+k}$ for a whole-life annuity, or $Ba_{x+k:\overline{n}|}$ for a temporary annuity with the specified remaining payment term. These expressions value benefits only; a reserve for a premium-paying contract must also subtract the value of future premiums.

Derivation: For a temporary annuity with $n$ payments after a remaining deferral of $r$ years, payment $j$ in the subsequent annuity period occurs $r+j$ years from the valuation date, for $j=1,\ldots,n$. Its expected present value is therefore $$ B\sum_{j=1}^{n}v^{r+j}\,{}_{r+j}p_{x+k}. $$

Each survival probability factors as ${}_{r+j}p_{x+k}={}_rp_{x+k}\,{}_jp_{x+m}$, because surviving $r+j$ years from age $x+k$ means surviving the first $r$ years and then the next $j$ years. Taking $v^r{}_rp_{x+k}$ outside the sum gives $$ Bv^r\,{}_rp_{x+k}\sum_{j=1}^{n}v^j\,{}_jp_{x+m} = Bv^r\,{}_rp_{x+k}\,a_{x+m:\overline{n}|}. $$

The same reasoning applies to a whole-life annuity by extending the sum over all future payment dates, provided the annuity value is finite under the stated model. The factor $v^r{}_rp_{x+k}$ is not an extra payment: it accounts for reaching the age at which the subsequent annuity value is measured.

Worked Example: Whole-Life Deferred Annuity at an Intermediate Duration

A hypothetical single-premium contract pays $1,000 at the end of each year, starting at the end of year 3, for as long as the annuitant is alive. The annuitant is age $x$ at issue. Assume an effective annual interest rate $i=25\%$, and assume a conditional one-year survival probability of $0.75$ at every age. Find the reserve at duration 1, conditional on being alive then.

Method. This is a two-year deferral followed by a whole-life annuity-immediate. At duration 1, one year of deferral remains, so $r=2-1=1$. The single-premium contract has no future premiums.

First calculate $v=1/(1+0.25)=0.8$. Under the constant survival assumption, the whole-life annuity-immediate at any age has value $$ a_y=\sum_{j=1}^{\infty}(vp)^j =\frac{vp}{1-vp} =\frac{0.8(0.75)}{1-0.8(0.75)} =\frac{0.6}{0.4} =1.5. $$

At duration 1 the remaining-deferral factor is $v\,{}_1p_{x+1}=0.8(0.75)=0.6$. Thus $$ {}_1V_x=1{,}000(0.6)(1.5)=\$900.00. $$

Check. The first payment is two years after duration 1. Its probability-weighted present value is $1{,}000(0.8^2)(0.75^2)=\$360$. The full reserve also includes later payments; the whole-life annuity factor accounts for all of them, giving a total of $900.00. The common error avoided is valuing $a_{x+2}$ at duration 1 without allowing for the remaining year of deferral.

Temporary Deferred Annuities

A deferred annuity may pay for a fixed number of years rather than for the annuitant’s entire future lifetime. The separation principle is unchanged: discount and condition on survival through the remaining deferral, then use the temporary life annuity value at the age when that payment period begins. Each survival probability in the temporary annuity is conditional on being alive at that age.

Worked Example: Temporary Payments After a Remaining Deferral

A hypothetical contract pays $250 at the ends of three consecutive years, with the first payment at the end of year 3, if the annuitant is alive on each payment date. Find its value at duration 1 for an annuitant then aged $x+1$. Assume $i=25\%$, so $v=0.8$, and the following one-year survival probabilities: ${}_1p_{x+1}=0.80$, ${}_1p_{x+2}=0.75$, ${}_1p_{x+3}=0.80$, and ${}_1p_{x+4}=0.50$.

Method. The deferral is two years from issue, leaving one year at duration 1. After that year, the three payments form a temporary annuity-immediate at age $x+2$.

Calculate that temporary annuity value, showing the survival and discount factor for each payment: $$ a_{x+2:\overline{3}|} =(0.8)(0.75) +(0.8^2)(0.75)(0.80) +(0.8^3)(0.75)(0.80)(0.50) =0.6000+0.3840+0.1536 =1.1376. $$

The probability and discount factor for surviving the remaining deferral are $v\,{}_1p_{x+1}=(0.8)(0.80)=0.64$. Therefore the value is $$ 250(0.64)(1.1376)=\$182.016=\$182.02. $$

Check. Directly from duration 1, the first payment is two years away and has survival probability $(0.80)(0.75)=0.60$; its value is $250(0.8^2)(0.60)=\$96.00$. The second is three years away, with survival probability $(0.80)(0.75)(0.80)=0.48$, giving $250(0.8^3)(0.48)=\$61.44$. The third is four years away, with survival probability $0.24$, giving $250(0.8^4)(0.24)=\$24.576$. Their total is $182.016, or $182.02 to the nearest cent. The common error avoided is using a survival probability from issue instead of conditioning on being alive at duration 1.

Reserves When Future Premiums Remain

If a deferred annuity is funded by premiums rather than a single premium, a prospective reserve subtracts the value of premiums still due. State whether the reserve is measured just before or just after a premium date: that convention determines whether a premium due at that date is included among future cash flows. For annual premiums payable at the ends of future years while the life is alive, a premium due $j$ years from the valuation date contributes a factor $v^j{}_jp_{x+k}$.

{}_kV_x = \text{EPV of future benefits at duration }k - \text{EPV of future premiums at duration }k.

Worked Example: Net Premium Reserve for a Deferred Whole-Life Annuity

A hypothetical contract pays $1,000 annually at the ends of years 4, 5, and later for as long as the annuitant is alive. It charges three equal annual net premiums at times 0, 1, and 2, each payable only if the annuitant is alive. Find the net premium and the reserve at duration 1, immediately after the premium at time 1. Assume $i=25\%$ and a conditional one-year survival probability of $0.75$ at every age.

Method. Use the equivalence principle at issue: the expected present value of the three premiums equals the expected present value of benefits. Here $v=0.8$, $vp=0.6$, and the whole-life annuity-immediate factor is $a_y=0.6/(1-0.6)=1.5$.

The benefit is deferred three years and then paid as a whole-life annuity-immediate. Its issue value is $$ 1{,}000(0.6)^3(1.5)=\$324.00. $$

The premium dates are 0, 1, and 2. Their issue-date factors are $1$, $v\,{}_1p_x=0.6$, and $v^2\,{}_2p_x=(0.6)^2=0.36$. Thus the total premium factor is $1+0.6+0.36=1.96$, and the level premium $P$ satisfies $$ 1.96P=324, \qquad P=\frac{324}{1.96}=\$165.3061\ldots=\$165.31. $$

At duration 1, two years of deferral remain, so the future benefit value is $$ 1{,}000(0.6)^2(1.5)=\$540.00. $$

The premium at time 1 has already been paid. Only the premium at time 2 remains, one year from the valuation date. Its expected present value is $$ P(vp)=\$165.3061\ldots(0.6)=\$99.1837\ldots. $$

Therefore, immediately after the premium at time 1, $$ {}_1V_x=540.00-99.1837\ldots=\$440.82. $$

Check. At duration 2, immediately after the final premium, no premiums remain and one year of deferral remains. The benefit value is $1{,}000(0.6)(1.5)=\$900.00$. This later reserve is larger, as expected here because the remaining deferral is shorter and the future premium obligation has ended. The common error avoided is subtracting a premium already paid at duration 1, or counting a premium at time 3 when the contract specifies only three payments at times 0, 1, and 2.

Payment Timing and Common Traps

An annuity-immediate and an annuity-due differ by one payment date. Under the convention used here, the deferred annuity-immediate’s first payment is at time $m+1$. If instead a deferred annuity-due pays its first amount immediately at time $m$, conditional on survival to that date, the value before the payment period is

Bv^{m-k}\,{}_{m-k}p_{x+k}\,\ddot{a}_{x+m}, \qquad k<m,

for a whole-life annuity-due beginning at duration $m$. The annuity-due factor includes a payment at the start of its payment period; do not use it for the annuity-immediate timing specified earlier.

A frequent incorrect calculation at duration $k<m$ is to write $Ba_{x+m}$ and stop. That values the payment period as if it had already been reached. The correction is to multiply by both $v^{m-k}$ and ${}_{m-k}p_{x+k}$. Omitting one factor ignores, respectively, interest discounting or the chance the life does not survive to the payment period.

The separation formula also relies on the annuity value being appropriate for the payments that actually remain. A temporary annuity must use the correct remaining number of payments; a whole-life annuity must reflect payments for life. At or after the start of payments, do not keep applying a deferral factor for a period that has already elapsed.

Key takeaway: At a duration before payments begin, first identify the years remaining until the payment period. Multiply the annuity value at the start of that period by the discount and survival factor for reaching it. For a premium-paying contract, subtract only premiums still due after the valuation date.

Check Your Understanding

For each problem, state the payment timing and use the appropriate remaining deferral and annuity value.

A hypothetical single-premium contract pays $1,000 annually as a whole-life annuity-immediate after a two-year deferral. Find its value at duration 1, conditional on survival, assuming $i=25\%$ and a constant one-year survival probability of $0.75$. (Answer: $900.00)
A temporary deferred annuity-immediate has two payments of $250, the first at the end of year 3. Find its value at duration 1 if $i=11.1111\ldots\%$, ${}_1p_{x+1}=0.80$, ${}_1p_{x+2}=0.50$, and ${}_1p_{x+3}=0.40$. (Answer: $110.16)
A whole-life annuity-immediate of $600 per year is deferred for two years. Find its issue value under $i=25\%$ and constant one-year survival probability $0.75$. (Answer: $324.00)
A $1,000 whole-life annuity-immediate is deferred for three years. Three equal premiums are payable at times 0, 1, and 2 while alive. Under $i=25\%$ and constant one-year survival probability $0.75$, find the net premium and the reserve at duration 1 immediately after the premium then due. (Answer: net premium $165.31; reserve $440.82)
A hypothetical whole-life annuity-due of $400 begins at the end of a two-year deferral, with its first payment made immediately then if the annuitant is alive. Find its value at duration 1 under $i=25\%$ and constant one-year survival probability $0.75$. (Answer: $600.00)

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