Start With the Payment Dates
Deferred-annuity errors often come from choosing a formula before drawing the timeline. The word “deferred” alone does not tell you which power of \(v\) to use. First identify the date of the first payment, then count the payments and the intervals between the valuation date and that payment.
Throughout the annual examples, assume a constant effective annual interest rate \(i>0\), with \(v=1/(1+i)\), and payments at the stated year-ends. A payment at time 0 is immediate; a payment at time 1 is one year from now. For payments at another frequency, use an interest rate appropriate to that payment period. A constant positive rate is also necessary for the standard perpetuity value \(R/i\) to converge.
The exponent \(k-1\) is intentional. The annuity-immediate factor \(a_{\overline{n}|}\) values payments at times 1 through \(n\) as of time 0. To shift those payments so the first is at time \(k\), shift the whole stream by \(k-1\) periods. Likewise, a perpetuity-immediate is worth \(R/i\) one period before its first payment, not at the date of that payment.
A Timeline Check Before You Calculate
When a problem says that payments begin after a delay, translate that wording into dates. “Three years from now” means a first payment at time 3; “after three full years without payments” also means a first payment at time 4 if payments occur at year-ends. These descriptions differ by one period, so do not replace the stated timing with an assumed convention.
Write the actual time of the first payment as \(k\); do not use the number of empty periods without checking the dates.
For a finite annuity, list all payment dates, including the last. For a perpetuity, state that the same spacing continues indefinitely.
An annuity-immediate factor is valued one payment period before its first payment. Discount from that date to time 0.
Use an effective rate per payment period in the annuity or perpetuity factor, and use that same period when counting discount factors.
Why the Deferral Exponent Is One Less Than the First Payment Time
The formula follows directly from the dates of the payments. For an annuity-immediate whose first payment is at time \(k\), the payment dates are \(k\) through \(k+n-1\). Discount each payment to time 0:
The factor \(a_{\overline{n}|}\) is the sum \(v+v^2+\cdots+v^n\). The extra \(v^{k-1}\) shifts its first payment from time 1 to time \(k\). For a perpetuity, the same reasoning gives a geometric series:
because \(v/(1-v)=1/i\). The infinite series converges when \(0<v<1\), equivalently when \(i>0\). This also explains the perpetuity timing rule: \(R/i\) is the value one period before the first payment, so it must be discounted from time \(k-1\), not time \(k\).
Worked Example: A Deferred Finite Annuity
A fund will pay \$1,000 at the end of each year for three years, with the first payment at the end of year 3. The effective annual interest rate is 20%. Find the present value at time 0.
Method. The payment dates are 3, 4, and 5. Thus \(k=3\), \(n=3\), and the valuation date for the three-payment annuity-immediate factor is time 2. Discount that factor two years. This avoids treating the first payment date as the number of years to discount the factor.
The discount factor is \(v=1/1.20=0.8333333333\). The relevant powers are \(v^2=0.6944444444\), \(v^3=0.5787037037\), \(v^4=0.4822530864\), and \(v^5=0.4018775720\). Therefore,
and the present value is
Check. Discounting the payments separately gives \(1000(v^3+v^4+v^5)=1000(0.5787037037+0.4822530864+0.4018775720)=\$1{,}462.83\), confirming the factor method. The common mistake is to use \(v^3a_{\overline{3}|}\); that places the first payment at time 4 instead of time 3.
Perpetuity Timing and the Boundary Payment
For a perpetuity-immediate, the first payment is one period after the date at which the perpetuity value \(R/i\) is located. This matters when interpreting phrases such as “the first payment is at time 6.” In that case, the value \(R/i\) is at time 5, and it is discounted five years to time 0.
A perpetuity-due has a different first-payment pattern: its first payment occurs at the valuation date of the due factor. If a perpetuity-due begins with a payment at time \(k\), its value at time \(k\) is \(R\ddot{a}_{\overline{\infty}|}=R/d\), where \(d=i/(1+i)\) is the effective annual discount rate. The time-0 value is \(Rv^k/d\). Do not apply the immediate-perpetuity exponent to a due stream without first accounting for its payment at the factor’s valuation date.
Worked Example: A Perpetuity-Immediate First Paying at Time 6
A hypothetical endowment makes annual payments of \$1,200 forever, with the first payment at time 6. The effective annual interest rate is 4%. Find the present value at time 0.
Method. Since this is a perpetuity-immediate, the value \(R/i\) is one year before its first payment. The first payment is at time 6, so the perpetuity value is at time 5. Discount five years to time 0.
The five-year accumulation factor is \(1.04^5=1.2166529024\), so \(v^5=1/1.2166529024\). Hence,
Check. The first payment contributes \(1200v^6\) to the present value. The full stream is \(1200(v^6+v^7+\cdots)\), which equals \(1200v^5/i\), or \$24,657.81. The common mistake is to discount the \(R/i\) value six years, giving \(1200v^6/i=\$23{,}709.40\). That shifts the whole stream one extra year and makes its first payment occur at time 7.
Payment-Period Rates Must Match Payment Dates
An annuity factor counts payment periods, so its interest rate must also be expressed per payment period. If the quoted rate is nominal \(i^{(m)}\), convertible \(m\) times per year, the effective rate per conversion period is \(i^{(m)}/m\). When payments occur at that same frequency, use this periodic rate and count periods, not years.
If instead the given rate is an effective annual rate \(i\), the equivalent effective rate per quarter is \((1+i)^{1/4}-1\), not generally \(i/4\). Dividing an effective annual rate by four does not preserve the annual accumulation factor. Always identify whether the quoted annual rate is nominal or effective before converting it.
Worked Example: Quarterly Payments and a Nominal Rate
A contract pays \$100 at the end of each quarter for three quarters. The first payment is at the end of quarter 3. The nominal annual interest rate is 8% convertible quarterly. Find the present value.
Method. The rate per quarter is \(0.08/4=0.02\). The payments occur at quarters 3, 4, and 5, so the annuity-immediate factor is valued at quarter 2 and discounted two quarters to time 0.
Let \(v_q=1/1.02=0.9803921569\). Then \(v_q^2=0.9611687812\), \(v_q^3=0.9423223345\), \(v_q^4=0.9238454260\), and \(v_q^5=0.9057308100\). The three-payment factor is
Discounting this factor by two quarters gives
Check. Directly, \(100(v_q^3+v_q^4+v_q^5)=100(0.9423223345+0.9238454260+0.9057308100)=\$277.19\). The common mistake is to count three years of deferral or to use the annual rate 8% in a quarterly factor. The timeline contains two quarters before the annuity-immediate factor’s valuation date, and the payment-period rate is 2%.
Common Timing Errors to Catch
A quick independent check is often enough to catch an off-by-one error. Write the first few discounted terms and compare their exponents with the payment dates. If a payment occurs at time \(k\), its present-value term must contain \(v^k\). The formula should reproduce that first term.
| Payment pattern | Correct time-0 value | Timing check |
|---|---|---|
| Finite annuity-immediate, \(n\) payments, first at time \(k\) | \(Rv^{k-1}a_{\overline{n}|}\) | The first term is \(Rv^k\). |
| Perpetuity-immediate, first payment at time \(k\) | \(Rv^{k-1}/i\) | The first term in the infinite sum is \(Rv^k\). |
| Perpetuity-due beginning at time \(k\) | \(Rv^k/d\) | The due value includes a payment at time \(k\). |
The formulas assume level payments, regular payment spacing, and a constant interest rate per payment period. They do not apply unchanged if payment dates are irregular, payments vary, or rates vary by period. In those cases, value each payment using its actual date and the appropriate accumulation or discount factors.
- Using the first payment time as the deferral exponent. For an immediate annuity or perpetuity, the exponent is one less than the first payment time because the factor’s value is located one period before the first payment.
- Counting empty periods rather than payment dates. “Three years with no payments” may mean the first payment is at time 4, not time 3. Use the wording and payment convention in the problem.
- Discounting a due payment as if it were delayed. A due stream includes a payment at the valuation date of its factor. Mark that payment explicitly before applying a discount factor.
- Mixing annual and payment-period rates. Convert the rate to the payment period, then use the same period unit for the exponent and the annuity factor.
Check Your Understanding
For each problem, identify the first payment date, match the rate to the payment period, and check the first discounted payment term.
- An annual annuity-immediate pays \$500 three times, with the first payment at time 3. The effective annual rate is 10%. Find the present value. (Answer: \$1,027.62)
- A perpetuity-immediate pays \$900 annually, first at time 8. The effective annual rate is 5%. Find its present value. (Answer: \$12,792.26)
- A perpetuity-immediate pays \$250 at the end of every quarter, first at quarter 4. The nominal annual rate is 12% convertible quarterly. Find the present value. (Answer: \$7,626.18)
- An annual annuity-immediate pays \$200 at times 3, 4, and 5. The effective annual rate is 25%. Find the present value. (Answer: \$249.86)