Pricing a Deferred Annuity-Immediate

A deferred annuity-immediate makes no payments during an initial waiting period, then pays a fixed amount at regular intervals. To price it at time 0, identify when the first payment occurs, value the level payment stream, and account for the time between time 0 and that stream. The key timing point is that a deferral of $m$ full payment periods places the first payment at time $m+1$, not at time $m$.

This tutorial considers level payments made at the end of each period and a constant effective interest rate per payment period. The same method applies whether the periods are years, months, or another regular unit, provided the interest rate is effective for that same period.

Definition: A deferred annuity-immediate with payment $R$, deferral $m$ periods, and $n$ payments has payments of $R$ at times $m+1,m+2,\ldots,m+n$. Here $m$ is a nonnegative integer, $n$ is a positive integer, and payments are made at the end of each period. Its present value is measured at time 0.

Present-Value Formula

Assume the effective interest rate is $i$ per payment period, with $i>-1$, and define the discount factor $v=1/(1+i)$. The ordinary annuity-immediate factor $a_{\overline{n}|}$ is the present value at the date immediately before its first payment of $n$ level payments of 1, each made at the end of a period. With these assumptions, the deferred annuity’s value at time 0 is:

PV_0=R v^m a_{\overline{n}|} =R v^m\frac{1-v^n}{i}, \qquad i\ne 0.

The formula also has a direct cash-flow form:

PV_0=R\left(v^{m+1}+v^{m+2}+\cdots+v^{m+n}\right).

The factor form is convenient for calculation; the sum makes the payment dates explicit. When $i=0$, every payment has discount factor 1, so the value is $nR$. The displayed quotient $(1-v^n)/i$ should not be used by substituting $i=0$; use the direct sum or its zero-interest limit instead.

Timing check: For a deferral of $m$ full periods, the first payment is at $m+1$. The factor $v^m$ shifts the value of an $n$-payment immediate annuity back $m$ periods; it does not mean that the first payment is made at time $m$.

Why the Formula Works

The annuity factor values the payments relative to the date one period before the first payment. The deferred stream’s first payment is at $m+1$, so that reference date is time $m$. At time $m$, the $n$ payments are one, two, through $n$ periods away, giving a value of $R a_{\overline{n}|}$. Discounting that value for $m$ periods to time 0 multiplies it by $v^m$.

Equivalently, each payment at time $m+k$, for $k=1,\ldots,n$, is discounted by $v^{m+k}$. Factoring $v^m$ out of that sum gives the ordinary annuity factor:

R\sum_{k=1}^{n}v^{m+k} =R v^m\sum_{k=1}^{n}v^k =R v^m a_{\overline{n}|}.

This also explains why the formula depends on the stated timing assumptions. If payments are made at the beginning of each period, or if the deferral is not a whole number of payment periods, the payment dates differ and this particular timing formula cannot be applied without adjustment.

Worked Valuations

Worked Example: Three Payments After a Two-Year Deferral

A hypothetical fund will pay $1,000 at the end of each year for three years, with the first payment due at the end of year 3. Find the present value at time 0 if the effective annual interest rate is 10%.

Method. The first payment is at year 3, so there are two full years with no payments: $m=2$. There are $n=3$ payments, and the annual effective rate is $i=0.10$. Thus $v=1/1.10$.

Using the factor method, calculate the annuity factor and the deferral discount factor from the exact rate:

a_{\overline{3}|} =\frac{1-(1.10)^{-3}}{0.10} \approx 2.486851991, \qquad v^2=(1.10)^{-2}\approx 0.826446281.

The displayed factors are rounded for readability; the multiplication uses their exact expressions in terms of 1.10. Therefore:

PV_0 =1000(1.10)^{-2} \frac{1-(1.10)^{-3}}{0.10} \approx \$2{,}055.2496 \approx \boxed{\$2{,}055.25}.

Check by cash flows. The payments occur at years 3, 4, and 5. Their discount factors are approximately $0.751314801$, $0.683013455$, and $0.620921323$, respectively. Their sum is approximately $2.055249579$, so the discounted payments total about $2,055.2496, again $2,055.25 to the nearest cent.

Common mistake avoided. Discounting the annuity factor by $v^3$ would add an extra year of deferral. Since the first payment is at year 3, the annuity’s reference date is year 2, and the correct multiplier is $v^2$.

Worked Example: Reading the Deferral from the First Payment

A hypothetical contract pays $800 at the end of each of two consecutive years. The first payment is due at the end of year 2. Find its present value if the effective annual interest rate is 5%.

Method. Since the first payment is at time 2, one full year has no payment, so $m=1$. There are $n=2$ payments. The cash flows occur at times 2 and 3, not at times 1 and 2.

Use $v=1/1.05$ and the deferred annuity-immediate formula:

PV_0 =800(1.05)^{-1} \frac{1-(1.05)^{-2}}{0.05} \approx 800(1.770867076989) \approx \$1{,}416.6937 \approx \boxed{\$1{,}416.69}.

Check by cash flows. The two discount factors are $(1.05)^{-2}\approx0.907029478458$ and $(1.05)^{-3}\approx0.863837598531$. Their sum is approximately $1.770867076989$. Multiplying by $800 gives the same present value, $1,416.69 to the nearest cent.

Common mistake avoided. A one-period deferral does not mean the first payment is at the end of year 1. The annuity-immediate payments begin one period after the date at which the annuity is valued; here the first payment is at year 2.

Worked Example: A Longer Deferral and a Direct-Sum Check

A hypothetical benefit pays $600 at the end of each year for three years, with the first payment at the end of year 5. Find its present value at time 0 using an effective annual interest rate of 8%.

Method. The first payment at year 5 implies $m=4$, and the three annual payments imply $n=3$. The effective rate per annual payment period is $i=0.08$, so the present value is $600v^4a_{\overline{3}|}$.

a_{\overline{3}|} =\frac{1-(1.08)^{-3}}{0.08} \approx 2.577096987, \qquad v^4=(1.08)^{-4}\approx 0.735029853.

Retaining the exact powers of 1.08 for the multiplication gives:

PV_0 =600(1.08)^{-4} \frac{1-(1.08)^{-3}}{0.08} \approx \$1{,}136.5459 \approx \boxed{\$1{,}136.55}.

Check by cash flows. The payments are at years 5, 6, and 7. The corresponding discount factors are approximately $0.680583197$, $0.630169627$, and $0.583490395$. Their sum is approximately $1.894243219$; multiplying by $600 gives approximately $1,136.5459, or $1,136.55 to the nearest cent.

Common mistake avoided. The exponent on the first payment’s discount factor is 5, but the exponent on the annuity factor’s separate deferral multiplier is 4. These are consistent: the annuity factor already discounts its first payment by one period, so $v^4a_{\overline{3}|}$ includes $v^5$ for the first payment.

Practical Checks and Common Traps

A timeline is often the fastest way to catch an error. Mark time 0, the end of the deferral period, and every payment date before substituting into a formula. Then check that the first term of the direct sum is the discount factor for the first actual payment.

Match rate period to payment period.
Use an effective rate per payment period. If payments are annual, $i$ must be an effective annual rate; do not insert a rate for a different period without converting it.

Count the payment dates.
If there are $m$ full periods without payments, the first payment is at time $m+1$, and the final one is at time $m+n$.

Apply the deferral once.
Use $v^m a_{\overline{n}|}$. The annuity factor already accounts for the one-period wait until its first payment.

Verify the value.
Write out $R(v^{m+1}+\cdots+v^{m+n})$. Agreement with the factor method checks the deferral and payment count.

A frequent incorrect calculation is $R v^{m+1}a_{\overline{n}|}$. That expression discounts the annuity factor one period too far: because $a_{\overline{n}|}$ already includes the first payment one period after its reference date, the deferral multiplier should be $v^m$, not $v^{m+1}$. Another error is to use $v^m$ as the first payment’s discount factor. It is only the shift applied to the annuity factor; the first actual payment is discounted by $v^{m+1}$.

The annuity factor also has limits. The formula assumes a finite number of equal payments, equally spaced payment dates, and a constant effective rate per period. It does not directly price a stream with unequal payments or irregular dates. For zero interest, use $nR$; for beginning-of-period payments, payment timing differs from an annuity-immediate.

Key takeaway: For $n$ level end-of-period payments of $R$, with $m$ full payment periods before the first payment and effective rate $i$ per period, the time-0 value is $R v^m a_{\overline{n}|}$. Confirm the timeline: its payments must be at $m+1$ through $m+n$.

Check Your Understanding

For each problem, identify the first payment date before calculating. Assume annual end-of-year payments and an effective annual interest rate unless stated otherwise.

A hypothetical fund pays $900 at the end of each year for three years, with the first payment at the end of year 3. Find its present value at 5%. (Answer: $2,223.06)
A hypothetical award pays $400 at the end of each of two years, with the first payment at the end of year 4. Find its present value at 10%. (Answer: $521.57)
A hypothetical account pays $1,200 at the end of each of four years, with the first payment at the end of year 2. Find its present value at 4%. (Answer: $4,188.34)
A hypothetical benefit pays $1,500 at the end of each of two years, with its first payment at the end of year 6. Find its present value at 8%. (Answer: $1,820.49)
A hypothetical contract pays $2,500 at the end of each of six years, with the first payment at the end of year 1. Find its present value at 10%. (Answer: $10,888.15)

Pricing a Deferred Annuity-Immediate

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