Deferred Annuities with Guaranteed Periods

Introduction

A deferred life annuity with a guaranteed period has two distinct features. The annuity may require the annuitant to survive a deferral period before any payments begin; once it has commenced, a specified number of payments are made even if the annuitant dies. Correct valuation depends on keeping those conditions separate.

This tutorial values annual, discrete deferred annuities with guaranteed payments. It distinguishes the survival requirement before commencement from the guarantee after commencement, and treats annuity-due and annuity-immediate payment timing separately. All products and data below are hypothetical.

Definition: Consider an annuity issued at age $x$, deferred for $m$ years, with annual payment amount $B$ and a guarantee of $n$ payments after commencement. The annuitant must survive to time $m$ for the annuity to commence. Once it commences, the first $n$ payments are made whether or not the annuitant remains alive. Payments after those $n$ guaranteed payments are made only while the annuitant is alive.

Payment Timing and Valuation

For an annuity-due commencing at time $m$, payments are scheduled at times $m,m+1,\ldots$. Its first $n$ payments are therefore at times $m$ through $m+n-1$. For an annuity-immediate commencing at time $m$, payments are scheduled at times $m+1,m+2,\ldots$, so its first $n$ payments are at times $m+1$ through $m+n$.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. The certain annuity factors for $n$ annual payments are

\ddot{a}_{\overline{n}|}=\sum_{j=0}^{n-1}v^j, \qquad a_{\overline{n}|}=\sum_{j=1}^{n}v^j.

These factors value payments from the start of a period: the annuity-due factor includes a payment at that time, while the annuity-immediate factor starts one year later. In the formulas below, ${}_tp_x$ is the probability that a life aged $x$ survives $t$ years, and $\ddot{a}_{x+t}$ and $a_{x+t}$ are whole-life annuity values at age $x+t$.

Theorem: For an annual deferred annuity-due with $n$ guaranteed payments, its expected present value at issue is

B\,{}_mp_xv^m\ddot{a}_{\overline{n}|} + B\,{}_{m+n}p_xv^{m+n}\ddot{a}_{x+m+n}.

For an annual deferred annuity-immediate with $n$ guaranteed payments, the expected present value at issue is

B\,{}_mp_xv^m a_{\overline{n}|} + B\,{}_{m+n}p_xv^{m+n}a_{x+m+n}.

The first term in each formula is the guaranteed segment, conditional on surviving the deferral. The second is the survivor-contingent tail, beginning after the guarantee.

These formulas assume level annual payments, a constant effective annual interest rate, and that the guarantee applies only after commencement. The annuity-due tail begins at time $m+n$; the annuity-immediate tail has its first payment at time $m+n+1$. The displayed formulas do not apply unchanged if payments are continuous, increase over time, or are guaranteed during the deferral period.

Why the Formula Separates into Two Parts

For the annuity-due, consider a guaranteed payment at time $m+j$, where $j=0,\ldots,n-1$. It is payable if the annuity commenced, which requires survival to time $m$, but it does not require survival to time $m+j$. Its expected present value is therefore $Bv^{m+j}{}_mp_x$. Summing these $n$ values gives the guaranteed-segment term:

\sum_{j=0}^{n-1}Bv^{m+j}{}_mp_x = B\,{}_mp_xv^m\sum_{j=0}^{n-1}v^j = B\,{}_mp_xv^m\ddot{a}_{\overline{n}|}.

After the guarantee, a due payment at time $m+n+j$, for $j\geq 0$, requires survival to that payment time. By conditioning on survival to $m+n$, its probability is ${}_{m+n}p_x\,{}_jp_{x+m+n}$. Summing those tail payments gives $B\,{}_{m+n}p_xv^{m+n}\ddot{a}_{x+m+n}$. For an annuity-immediate, the guaranteed payments are one year later than the due payments, and the post-guarantee tail is valued as an annuity-immediate. That timing difference produces $a_{\overline{n}|}$ and $a_{x+m+n}$, rather than their annuity-due counterparts.

Set the commencement time.
Identify the deferral $m$ and the survival probability ${}_mp_x$ needed for the annuity to start.

Count the guaranteed payment dates.
For a due annuity, they are $m$ through $m+n-1$; for an immediate annuity, they are $m+1$ through $m+n$.

Value the guaranteed segment.
Use the appropriate certain annuity factor and multiply by survival to commencement, not survival to each guaranteed payment date.

Value the tail separately.
Start it after the last guaranteed payment and use survival to the tail's valuation date together with the appropriate whole-life annuity factor.

Worked Valuations

Worked Example: Deferred Annuity-Due with Two Guaranteed Payments

A hypothetical annuity-due pays $B=\$1{,}000$ annually, beginning two years from issue. Two payments are guaranteed once the annuity commences. The effective annual interest rate is $i=0\%$. The mortality model is specified by ${}_2p_x=0.90$, ${}_4p_x=0.75$, and $\ddot{a}_{x+4}=6$; no parametric lifetime distribution is assumed. Find the expected present value at issue.

Method. The guaranteed due payments occur at times 2 and 3. Survival to time 2 is required for the annuity to commence, but survival during the guarantee is not required. The survivor-contingent tail starts at time 4.

At $i=0\%$, $v=1$, so $\ddot{a}_{\overline{2}|}=1+1=2$. The guaranteed segment is

\$1{,}000(0.90)(1)(2)=\$1{,}800.

The tail value is

\$1{,}000(0.75)(1)\ddot{a}_{x+4} = \$1{,}000(0.75)(6) = \$4{,}500.

Therefore the expected present value is $\$1{,}800+\$4{,}500=\$6{,}300$. Check: the guaranteed payments are two payments of $\$1{,}000$, each weighted by the $0.90$ probability of surviving to commencement; the tail is weighted by the $0.75$ probability of surviving to time 4. The total is $\boxed{\$6{,}300.00}$. The common mistake to avoid is counting guaranteed payments without applying the survival requirement during deferral.

Worked Example: Discounting Both Segments of a Deferred Annuity-Due

A hypothetical annuity-due pays $\$2{,}000$ annually, beginning one year from now, with two payments guaranteed after commencement. The effective annual interest rate is $i=5\%$. The mortality model is specified by ${}_1p_x=0.98$, ${}_3p_x=0.90$, and $\ddot{a}_{x+3}=7.0$; no parametric lifetime distribution is assumed. Find the expected present value at issue, rounded to the nearest cent.

Here $m=1$, $n=2$, and $v=1/1.05=20/21$. The guaranteed payments occur at times 1 and 2. The annuity must survive to time 1 to commence, while the tail begins at time 3.

The guaranteed-segment value is

\$2{,}000(0.98)\left(\frac{20}{21}\right) \left(1+\frac{20}{21}\right) = \$3{,}644.4444\ldots

The tail value is

\$2{,}000(0.90)\left(\frac{20}{21}\right)^3(7.0) = \$10{,}884.3537\ldots

Adding the unrounded values gives $\$14{,}528.7981\ldots$, so the expected present value is $\boxed{\$14{,}528.80}$. Check: the first payment is discounted for one year, the two-payment due factor includes dates 1 and 2, and the tail uses survival to time 3. The common mistake is to discount the guaranteed segment from time 0 as though its first due payment were immediate.

Worked Example: Deferred Annuity-Immediate with a Guarantee

A hypothetical annuity-immediate pays $\$1{,}500$ annually. It is deferred for two years and has three guaranteed payments after commencement. The effective annual interest rate is $i=0\%$. The mortality model is specified by ${}_2p_x=0.94$, ${}_5p_x=0.82$, and $a_{x+5}=6.4$; no parametric lifetime distribution is assumed. Find its expected present value at issue.

An annuity-immediate commencing at time 2 first pays at time 3. Its three guaranteed payments are at times 3, 4, and 5. Survival to time 2 is required for commencement, but death after that time does not remove any of those three payments. The tail starts with a payment at time 6 and requires survival to time 5.

At $i=0\%$, $v=1$ and $a_{\overline{3}|}=1+1+1=3$. Thus the guaranteed segment is

\$1{,}500(0.94)(1)(3)=\$4{,}230.

The tail value is

\$1{,}500(0.82)(1)a_{x+5} = \$1{,}500(0.82)(6.4) = \$7{,}872.

The total is $\$4{,}230+\$7{,}872=\boxed{\$12{,}102.00}$. Check: the immediate certain factor has three payments one, two, and three years after commencement; the tail is an annuity-immediate at age $x+5$, not an annuity-due. The common mistake is shifting the guaranteed immediate payments one year too early.

Common Exam Trap

The word “guaranteed” does not mean the annuity necessarily pays if the annuitant dies during deferral. In the setup here, survival to commencement is still required. Once the annuity commences, however, the guarantee means the scheduled payments continue through the guaranteed period even if the annuitant dies. These are separate conditions, and they use different survival probabilities.

For example, for a due annuity deferred $m$ years, it is wrong to value guaranteed payment $j$ using ${}_{m+j}p_x$. That would require survival to each payment date and would remove guaranteed payments after death during the guarantee period. The correct expected present value for that payment is $Bv^{m+j}{}_mp_x$. Conversely, omitting ${}_mp_x$ would incorrectly pay the annuity even when the annuitant dies before commencement. The tail is different: each tail payment is contingent on survival to its own payment date, which is why the tail begins with ${}_{m+n}p_x$ and a life-annuity factor.

A final timing check helps prevent off-by-one errors. For a due annuity with $n$ guarantees, the last guaranteed payment is at $m+n-1$, and the tail starts at $m+n$. For an immediate annuity, the last guaranteed payment is at $m+n$, and the tail starts at $m+n+1$. The annuity factor at the tail valuation date accounts for this due-versus-immediate distinction.

Key takeaway: Multiply the guaranteed segment by the probability of surviving the deferral, then treat its payments as certain after commencement. Value the later tail separately, requiring survival to the tail and using the annuity factor that matches the payment timing.

Check Your Understanding

For each problem, identify the commencement and payment dates before applying the valuation formula. Unless stated otherwise, no parametric lifetime distribution is assumed; use the stated survival probabilities and annuity factors.

A hypothetical annuity-due pays $\$500$ annually after a one-year deferral, with two guaranteed payments. At $i=0\%$, ${}_1p_x=0.96$, ${}_3p_x=0.80$, and $\ddot{a}_{x+3}=5$. Find its expected present value at issue. (Answer: $\$2{,}960.00$)
A hypothetical annuity-immediate pays $\$1{,}000$ annually after a two-year deferral, with two guaranteed payments. At $i=0\%$, ${}_2p_x=0.90$, ${}_4p_x=0.70$, and $a_{x+4}=4.5$. Find its expected present value at issue. (Answer: $\$4{,}950.00$)
A hypothetical annuity-due pays $\$1{,}000$ annually after a two-year deferral, with one guaranteed payment. The effective annual interest rate is $5\%$, ${}_2p_x=0.95$, ${}_3p_x=0.90$, and $\ddot{a}_{x+3}=6.0$. Find the expected present value at issue, rounded to the nearest cent. (Answer: $\$5{,}526.40$)
A hypothetical annuity-due pays $\$800$ annually after a one-year deferral, with three guaranteed payments. At $i=0\%$, ${}_1p_x=0.97$, ${}_4p_x=0.85$, and $\ddot{a}_{x+4}=5.2$. Find the guaranteed-segment value, the tail value, and their total. (Answer: guaranteed segment $\$2{,}328.00$; tail $\$3{,}536.00$; total $\$5{,}864.00$)

What You'll Learn

Introduction

Payment Timing and Valuation

Why the Formula Separates into Two Parts

Worked Valuations

Worked Example: Deferred Annuity-Due with Two Guaranteed Payments

Worked Example: Discounting Both Segments of a Deferred Annuity-Due

Worked Example: Deferred Annuity-Immediate with a Guarantee

Common Exam Trap

Check Your Understanding

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