Combining Temporary and Deferred Annuity Segments

Why Split an Annuity into Phases?

A life annuity need not pay the same amount throughout its term. A hypothetical contract might pay one amount during an initial period and a different amount afterward, or might begin paying only after a deferment. The direct way to value such a contract is to list its payment dates and add the expected present value of every possible payment. When the cash flows fall into segments, it is often clearer to value each segment separately and then add the segment values.

The key is to keep three features aligned for every segment: its payment amount, its payment dates, and the probability the annuitant is alive on each payment date. A payment scheduled at the end of year $k$ in an annuity-immediate is made only if the annuitant survives to that date.

Definition: For a life aged $x$, an annual life annuity-immediate pays at the end of each year while the life is alive. Under a constant effective annual interest rate $i$, let $v=1/(1+i)$. The expected present value of a payment of 1 at the end of year $k$, conditional on the life being alive at issue, is $v^k{}_kp_x$. A phased annuity has different payment amounts or different payment periods in successive segments; its value is the sum of the expected present values of those segments.

Valuing Each Segment

Suppose a level annual payment $C$ is made at the ends of years $m+1,\ldots,m+n$, provided the annuitant is alive at each payment date. Here $m$ is the number of complete years before this segment begins, and $n$ is the number of payments in the segment. Its value at issue is

C\sum_{k=m+1}^{m+n}v^k{}_kp_x.

The same value can be written by first conditioning on survival to time $m$. If the life survives to time $m$, the remaining $n$ payments are an $n$-year temporary annuity-immediate at age $x+m$. Thus the segment value is

v^m{}_mp_x\;C\sum_{j=1}^{n}v^j{}_jp_{x+m}.

These expressions agree because, for $k=m+j$, surviving to time $m$ and then surviving a further $j$ years has probability ${}_mp_x{}_jp_{x+m}={}_{m+j}p_x$. This conditioning approach is especially useful when a segment is deferred: survival to its start is part of its value, not an optional adjustment.

Result: If the annuity pays $C_r$ at the end of year $r$ when the annuitant is alive, then its expected present value is $$ \sum_r C_rv^r{}_rp_x. $$ For a contract divided into disjoint segments, calculate this sum over the payment dates in each segment and add the segment values. This additivity follows from linearity of expectation; the payment events do not have to be independent.

Worked Example: Two Finite Payment Phases

A hypothetical annuity-immediate pays $100 at the end of each of years 1 and 2, then $200 at the end of each of years 3 and 4, provided the annuitant is alive. Assume an effective annual interest rate of $i=10\%$ and a constant annual survival probability of $0.9$ at every age. Find the expected present value, to the nearest cent.

Method. Value each payment at its own date. Under the stated assumptions, $v=1/1.1=10/11$, and ${}_kp_x=(0.9)^k=(9/10)^k$. Therefore $v^k{}_kp_x=(9/11)^k$.

Years 1 and 2. Their expected present value is

100\left(\frac{9}{11}+\left(\frac{9}{11}\right)^2\right) =100\left(\frac{99+81}{121}\right) =\frac{18{,}000}{121} =\$148.7603\ldots.

Years 3 and 4. Their expected present value is

200\left(\left(\frac{9}{11}\right)^3+\left(\frac{9}{11}\right)^4\right) =200\left(\frac{7{,}290+6{,}561}{14{,}641}\right) =\frac{2{,}770{,}200}{14{,}641} =\$189.21\text{ (approximately)}.

The numerator in the last fraction is $200(7{,}290+6{,}561)=2{,}770{,}200$. Adding the unrounded phase values gives $148.7603306\ldots+189.1925415\ldots=\$337.9528721\ldots$, so the expected present value is $337.95 to the nearest cent.

Check. The value of the second phase can also be computed by deferring a two-payment temporary annuity for two years: $200(9/11)^2[9/11+(9/11)^2]$. This equals $200(81/121)(180/121)=2{,}916{,}000/14{,}641=\$199.17$, not the value calculated above. This check reveals that the displayed phase calculation must use the correct powers and numerator: $200[(9/11)^3+(9/11)^4]=200(729/1331+6561/14641)=200(8019+6561)/14641=2{,}916{,}000/14{,}641=\$199.17$. Hence the correct total is $\$148.76+\$199.17=\mathbf{\$347.93}$. The common mistake avoided is misaligning the payment dates with their powers of $v$ and survival.

Different Survival Probabilities by Year

A constant survival probability is not required. If annual survival probabilities vary, obtain ${}_kp_x$ by multiplying the one-year survival probabilities up to year $k$. For instance, ${}_3p_x=p_xp_{x+1}p_{x+2}$. Each segment still uses the survival probability to each actual payment date.

Worked Example: A Deferred Phase with Changing Survival

A hypothetical annuity-immediate pays $300 at the ends of years 1 and 2, followed by $500 at the ends of years 3 and 4, while the annuitant is alive. Assume $i=10\%$, $p_x=0.90$, $p_{x+1}=0.80$, $p_{x+2}=0.75$, and $p_{x+3}=0.50$. Find the expected present value to the nearest cent.

Method. Set $v=1/1.10=10/11$. The survival probabilities to the four payment dates are ${}_1p_x=0.90$, ${}_2p_x=0.90(0.80)=0.72$, ${}_3p_x=0.90(0.80)(0.75)=0.54$, and ${}_4p_x=0.90(0.80)(0.75)(0.50)=0.27$.

For years 1 and 2, the discounted survival factors are $v(0.90)=0.8181818\ldots$ and $v^2(0.72)=0.5950413\ldots$. The first phase is therefore

300(0.8181818\ldots+0.5950413\ldots) =\$423.9669\ldots.

For years 3 and 4, the factors are $v^3(0.54)=0.4057100\ldots$ and $v^4(0.27)=0.1844134\ldots$. The deferred phase is

500(0.4057100\ldots+0.1844134\ldots) =\$295.0617\ldots.

Adding the unrounded phase values gives $\$719.0286\ldots$, so the expected present value is $719.03. A useful check is that the year-4 payment uses survival through all four years, $0.27$, not merely the last one-year survival probability, $0.50$.

A Deferred Whole-Life Phase

A later phase can continue for the rest of the annuitant’s life. If a level annuity-immediate begins with its first payment at the end of year $m+1$, its value at issue is $v^m{}_mp_x$ times the whole-life annuity-immediate value at age $x+m$. Under a constant annual survival probability $p$ and constant effective annual interest rate $i$, define $r=vp$. The whole-life annuity-immediate factor is the geometric series $r+r^2+r^3+\cdots=r/(1-r)$, provided $r<1$.

Justification: A payment $C$ at the end of year $k$ has expected present value $C v^k{}_kp_x$. For a whole-life annuity-immediate with constant $p$, this becomes $C(vp)^k=Cr^k$. Summing from $k=1$ to infinity gives $C(r+r^2+\cdots)=Cr/(1-r)$. If the first payment is deferred until the end of year $m+1$, factor out the first $m$ years of discounting and survival, giving $v^mp^m$ times the same whole-life factor.

Worked Example: Temporary Payments Followed by a Whole-Life Phase

A hypothetical annuity-immediate pays $200 at the ends of years 1 and 2, then $300 at the end of every later year while the annuitant is alive. Assume $i=10\%$ and a constant annual survival probability $p=0.90$. Find the expected present value to the nearest cent.

Method. For the first phase, $vp=(10/11)(9/10)=9/11$. The value of the two temporary payments is

200\left(\frac{9}{11}+\left(\frac{9}{11}\right)^2\right) =200\left(\frac{180}{121}\right) =\$297.5207\ldots.

The second phase is deferred for two years. At age $x+2$, its whole-life annuity-immediate factor is $(9/11)/(1-9/11)=(9/11)/(2/11)=9/2$. The two-year discount-and-survival factor is $v^2{}_2p_x=(9/11)^2=81/121$. Thus the second phase is

300\left(\frac{81}{121}\right)\left(\frac{9}{2}\right) =300\left(\frac{729}{242}\right) =\$903.7190\ldots.

The total is $\$297.5207\ldots+\$903.7190\ldots=\$1{,}201.2397\ldots$, or $1,201.24 to the nearest cent.

Common mistake and correction. The post-deferment whole-life factor is $r/(1-r)$, not $1/(1-r)$. Omitting its numerator would give the incorrect phase calculation $300(81/121)/(2/11)=300(891/242)=\$1{,}104.55$, overstating the phase value. The correct factor includes $9/11$, so the phase value is $300(729/242)=\$903.72$. This check also confirms that the deferred phase starts with a year-3 payment, not a year-2 payment.

Timing and Boundary Checks

The segment method works for annuity-due payments as well, but their payment dates differ. An annuity-due pays at the beginning of each payment period: a payment at time $k$ is discounted by $v^k$ and, for a life-contingent payment, weighted by survival to time $k$. In particular, a payment at time 0 is certain at issue and has neither a discount factor nor a survival factor. Do not use an annuity-due factor for end-of-year payments, or an annuity-immediate factor for a payment at the beginning of a period.

At a phase boundary, record the dates explicitly before using a standard annuity value. If the first phase pays at the ends of years 1 through $m$, a following immediate phase begins at the end of year $m+1$. The first phase has $m$ payments; the second begins after survival through $m$ years. The payment at the boundary must be counted once, not omitted or counted twice.

Mark payment dates.
Write the actual time of every payment in each phase, distinguishing beginning-of-year from end-of-year payments.

Attach survival and discounting.
For a payment at time $k$, use the probability of being alive at that payment date and discount from time $k$ to issue.

Value the segments.
Sum payment-level expected present values within each phase, or use an appropriate temporary or whole-life factor after matching its timing.

Add and check.
Add the unrounded segment values. Check the boundary date, the survival period, and whether each factor includes the first payment at the correct time.

Splitting a phased annuity is valuable because it separates the cash-flow pattern from the survival calculation. It also makes the result easier to audit: each segment has a visible start date, payment amount, and set of survival probabilities. For exam problems aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus, a timeline and a payment-date sum are reliable checks when a compact annuity-factor expression is easy to misapply.

Key takeaway: A phased life annuity is valued by adding the expected present values of its payment segments. For every payment, match the amount, discount period, and probability of survival to its actual payment date.

Check Your Understanding

For each problem, state the payment timing and survival assumptions before calculating. Round money to the nearest cent.

A hypothetical annuity-immediate pays $120 at the ends of years 1 and 2 and $240 at the end of year 3 if alive. Assume $i=0$ and a constant annual survival probability of $0.8$. Find its expected present value. (Answer: $326.40)
A hypothetical annuity-immediate pays $400 at the ends of years 2 and 3. Assume $i=25\%$, $p_x=0.80$, and $p_{x+1}=0.75$. Find its expected present value. (Answer: $384.00)
A hypothetical annuity-immediate pays $100 at the end of year 1 and $250 at the ends of years 2 and 3. Assume $i=10\%$ and constant annual survival probability $p=0.90$. Find its expected present value. (Answer: $439.52)
A hypothetical annuity-immediate pays $150 at the ends of years 1 and 2, then $250 at the end of every later year while the annuitant is alive. Assume $i=25\%$ and constant annual survival probability $p=0.80$. Find its expected present value. (Answer: $536.00)
A hypothetical annuity-due pays $500 at time 0 and $200 at time 1 if the annuitant is alive then. Assume $i=10\%$ and $p_x=0.90$. Find its expected present value. (Answer: $663.64)

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