Deriving the Deferred Annuity-Immediate Value

Introduction

A deferred annuity-immediate is a level stream of payments that begins after a delay, with each payment made at the end of a payment period. Its present value depends on two separate pieces: the value of the payment stream at the point where it starts, and the discount needed to bring that value back to the valuation date. Keeping those pieces separate makes the timing easier to see and the formula easier to derive.

This tutorial derives the present value at time zero when the deferral is a whole number of periods. The key is to identify the annuity’s start date correctly: if the first payment is at the end of period $m+1$, the annuity-immediate begins at time $m$.

Definition: Consider level payments of amount $R$, made at the ends of $n$ consecutive periods. If the first payment is at time $m+1$, where $m$ is a nonnegative integer, the payment dates are $m+1,m+2,\ldots,m+n$. The stream is a deferred annuity-immediate with a deferral of $m$ periods. The payment amount $R$ is paid at each listed date, and no payments are made before time $m+1$.

The Formula and Its Timing

Assume a constant effective interest rate $i$ per payment period, with $i>0$, and let $v=1/(1+i)$ be the discount factor per period. The value of an ordinary annuity-immediate of $n$ level payments of 1, valued one period before its first payment, is $a_{\overline{n}|}$. For payments of amount $R$, the value at that same date is $R a_{\overline{n}|}$.

Theorem: Under a constant effective rate $i>0$ per payment period, the present value at time zero of $n$ level payments $R$ at times $m+1,\ldots,m+n$ is $$ PV_0=R v^m a_{\overline{n}|}. $$ Equivalently, since $a_{\overline{n}|}=(1-v^n)/i$, $$ PV_0=R v^m\frac{1-v^n}{i}. $$ Here $m$ counts the complete periods from time zero to the date immediately before the first payment. It is not the number of periods from time zero to the first payment.

The factor $a_{\overline{n}|}$ values the payments at time $m$, not at time zero. The factor $v^m$ then discounts that value across the $m$ periods from time $m$ to time zero. If $m=0$, the first payment is at time 1 and the formula reduces to the ordinary annuity-immediate value $R a_{\overline{n}|}$.

Why the Formula Holds

Each payment at time $m+k$, for $k=1,\ldots,n$, has present value $R v^{m+k}$. Adding those values gives the present value of the entire stream:

PV_0=R\sum_{k=1}^{n}v^{m+k} =R v^m\sum_{k=1}^{n}v^k.

The remaining sum is the ordinary annuity-immediate factor. To see its usual closed form, use the finite geometric series:

a_{\overline{n}|} =\sum_{k=1}^{n}v^k =\frac{v(1-v^n)}{1-v} =\frac{1-v^n}{i}.

The final equality follows because $1-v=i/(1+i)=iv$, so $v/(1-v)=1/i$. Substituting the annuity factor into the deferred sum proves $PV_0=R v^m a_{\overline{n}|}$. This derivation also shows why the extra discount is $v^m$: every payment is shifted $m$ periods later than the corresponding payment in an immediate annuity.

Locate the first payment.
If it is at the end of period $m+1$, then the annuity’s starting valuation date is time $m$.

Value the level stream at that date.
For $n$ payments of amount $R$, its value at time $m$ is $R a_{\overline{n}|}$.

Discount to the requested date.
Move the value from time $m$ to time zero by multiplying by $v^m$.

Worked Applications

Worked Example: Three Payments After a Two-Period Deferral

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

Method. Value the three-payment annuity at the date immediately before its first payment, then discount that value two years. This avoids treating the first payment date as the deferral length.

The payment dates are years 3, 4, and 5. Therefore $m=2$, $n=3$, $R=1000$, and $i=0.10$. The discount factor is

v=\frac{1}{1.10}=\frac{10}{11}.

First calculate the annuity factor:

v^3=\left(\frac{10}{11}\right)^3 =\frac{1000}{1331} \approx 0.75131480, $$ $$ a_{\overline{3}|} =\frac{1-v^3}{0.10} =\frac{1-0.75131480}{0.10} \approx 2.48685199.

The deferral discount is

v^2=\left(\frac{10}{11}\right)^2 \approx 0.82644628.

Thus

PV_0=1000(0.82644628)(2.48685199) \approx \$2{,}055.25.

Check by individual payments. The present value is also $1000(v^3+v^4+v^5)$. The factors are $v^3\approx0.75131480$, $v^4\approx0.68301346$, and $v^5\approx0.62092132$, so their sum is $2.05524958$. Multiplying by $1,000 gives $2,055.25, as required. The common mistake avoided is discounting by $v^3$: the first payment is at year 3, but the annuity value is located at year 2.

Worked Example: Reading a Payment Description Carefully

A hypothetical research grant pays $250 at the end of each year for four years, with its first payment at the end of year 2. Find the present value at time zero at an effective annual rate of 4%.

Method. Translate the first payment date into the valuation date for the immediate annuity, then apply the deferral factor. The detail to watch is that “first payment at the end of year 2” means one full period of deferral, not two.

The payment dates are years 2, 3, 4, and 5. Thus $m=1$, $n=4$, $R=250$, and $i=0.04$, with $v=1/1.04$. The needed factors are

v\approx0.96153846,\qquad v^4\approx0.85480419, $$ $$ a_{\overline{4}|} =\frac{1-v^4}{0.04} =\frac{1-0.85480419}{0.04} \approx3.62989523.

Discount the value at time 1 back one period:

PV_0=250(0.96153846)(3.62989523) \approx \$872.57.

Check by individual payments. The four payment dates give $250(v^2+v^3+v^4+v^5)$. The factors are approximately $0.92455621$, $0.88899636$, $0.85480419$, and $0.82192711$. Their sum is $3.49028387$, and $250(3.49028387)=\$872.57$. This check confirms that the formula discounts each of the actual payment dates the correct number of periods.

Worked Example: Avoiding an Extra Period of Discount

A hypothetical conservation trust will make three annual payments of $1,100, with the first payment at the end of year 6. Calculate the present value at time zero at an effective annual rate of 10%.

Method. Identify the time immediately before the first payment. Since that date is year 5, the deferral is $m=5$, not 6. Then value the three-payment stream there and discount five periods.

Here $R=1100$, $n=3$, $m=5$, $i=0.10$, and $v=10/11$. The annuity factor is the same three-payment factor calculated from the rate:

a_{\overline{3}|} =\frac{1-(10/11)^3}{0.10} \approx2.48685199,\qquad v^5=\left(\frac{10}{11}\right)^5 \approx0.62092132.

Therefore,

PV_0=1100(0.62092132)(2.48685199) \approx \$1{,}698.55.

Check by individual payments. The payments are at years 6, 7, and 8, so the direct value is $1100(v^6+v^7+v^8)$. The factors are approximately $0.56447393$, $0.51315812$, and $0.46650738$, which sum to $1.54413943$. Multiplication by $1,100 gives $1,698.55. An extra discount by $v$ would incorrectly place the annuity value at year 6 rather than year 5, producing a value that is too small by a factor of $v$.

Assumptions and Common Traps

The formula uses a level payment amount and a constant effective rate per payment period. The payment frequency and rate period must match: for annual payments, $i$ must be an effective annual rate; for quarterly payments, it must be an effective rate per quarter. If a quoted rate is on a different basis, convert it before applying the formula. The formula also assumes that the payments occur at the ends of periods, as required for an annuity-immediate.

A reliable timeline check is to list the first and last payment dates before substituting any factors. For a deferral of $m$ periods followed by $n$ payments, the dates must run from $m+1$ through $m+n$. In particular, the first payment date is one period after the date at which $a_{\overline{n}|}$ is valued.

Common mistake: If the first payment is at time $m+1$, using $v^{m+1}a_{\overline{n}|}$ discounts the annuity value one period too far. The annuity factor already places the first payment one period after its valuation date. Place that valuation date at time $m$, then use $v^m$ to move it to time zero.

There is one special rate case. If $i=0$, the expression $(1-v^n)/i$ is undefined, so use the payment sum directly: each payment has the same value at time zero, and $PV_0=Rn$. A zero rate does not make deferred payments disappear or change their number; it only removes the effect of discounting.

The factor method is efficient, while the direct-sum check is useful when a payment date is easy to misread. Both describe the same cash flows. Their agreement is a strong check that the deferral has been counted from the correct date.

Check Your Understanding

For each problem, identify the deferral $m$ and payment count $n$ before calculating.

A hypothetical scholarship fund pays $400 at the end of each year for four years, beginning at the end of year 3. Find its value at time zero at an effective annual rate of 5%. (Answer: $1,286.51)
A project pays $900 at each year-end for three years, with the first payment at the end of year 5. Find the present value at 8% effective annually. (Answer: $1,704.82)
A hypothetical grant makes three payments of $100 at the ends of years 4, 5, and 6. Find its value at time zero at 10% effective annually. (Answer: $186.84)
At 10% effective annually, find the combined present value of $500 payments at the ends of years 3, 4, and 5, and $300 payments at the ends of years 5 and 6. (Answer: $1,383.24)
A fund pays $250 at the ends of years 4, 5, and 6. What is its value at time zero if the effective annual interest rate is 0%? (Answer: $750.00)

What You'll Learn