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Exam FM · Annuities Immediate and Due · Tutorial 128 of 998

Comparing Immediate and Due Values

Learn to compare equal immediate and due annuities by relating their values at the same valuation date.

Beginner 10 min read

What You'll Learn

  • Identify the payment dates of an annuity-immediate and an annuity-due
  • Compare their present values for equal payments, counts, and effective rates
  • Compare their accumulated values at the same focal date
  • Derive the one-period value relationship from the payment timelines
  • Avoid treating a due annuity as an immediate annuity with an extra payment

Why Payment Timing Changes an Annuity's Value

Two annuities can have the same payment amount and the same number of payments, yet have different values because the payments occur at different times. An annuity-due pays at the beginning of each period; an annuity-immediate pays at the end. If the payment amount, payment count, and interest rate are unchanged, each due payment occurs exactly one period earlier than its corresponding immediate payment.

That one-period shift affects every payment, not just the first one. At a present-value date, each due payment is discounted for one fewer period. At a common accumulated-value date, each due payment earns interest for one additional period. The resulting relationship is especially useful for checking calculations and comparing payment schedules.

Definition: An annuity-immediate with \(n\) payments of amount \(R\) has payments at times \(1,2,\ldots,n\). An annuity-due with the same \(n\) payments has payments at times \(0,1,\ldots,n-1\). The payment amount \(R\) is made at each listed time.

Assume payments are annual, the effective annual interest rate is \(i>-1\), and values are measured using that same rate. The usual exam setting has \(i>0\). Write \(v=1/(1+i)\). The present-value annuity factors are \(a_{\overline{n}|}\) for an annuity-immediate and \(\ddot{a}_{\overline{n}|}\) for an annuity-due. The corresponding accumulated-value factors at time \(n\) are \(s_{\overline{n}|}\) and \(\ddot{s}_{\overline{n}|}\).

The Comparison Formulas

For equal payment amounts and counts, the two present values are

$$ \operatorname{PV}_{\text{immediate}}=R a_{\overline{n}|} =R(v+v^2+\cdots+v^n) $$
$$ \operatorname{PV}_{\text{due}}=R\ddot{a}_{\overline{n}|} =R(1+v+\cdots+v^{n-1}) =(1+i)R a_{\overline{n}|} $$

At time \(n\), the accumulated values are

$$ \operatorname{AV}_{\text{immediate}}=R s_{\overline{n}|} =R\big((1+i)^{n-1}+\cdots+(1+i)+1\big) $$
$$ \operatorname{AV}_{\text{due}}=R\ddot{s}_{\overline{n}|} =R\big((1+i)^n+\cdots+(1+i)\big) =(1+i)R s_{\overline{n}|} $$

The formulas compare values at the same date: time 0 for present values, and time \(n\) for accumulated values. For \(i>0\), the due value is greater by the factor \(1+i\). If \(i=0\), the values are equal. The algebraic relationships still hold when \(-1<i<0\), but then the immediate value is greater because \(1+i<1\).

Theorem: For the same positive payment \(R\), payment count \(n\), and effective rate \(i>-1\), an annuity-due value is \(1+i\) times the corresponding annuity-immediate value, when both are evaluated at time 0 or both at time \(n\). In particular, when \(i>0\), the due value is larger.

Why the Factor Is \(1+i\)

The identity follows directly from the payment dates. At time 0, an immediate annuity has payments at times \(1\) through \(n\), so its present-value terms are \(Rv,Rv^2,\ldots,Rv^n\). The due annuity has payments at times \(0\) through \(n-1\), so its terms are \(R,Rv,\ldots,Rv^{n-1}\). Each due term is the corresponding immediate term divided by \(v\). Since \(1/v=1+i\), the due present value is \(1+i\) times the immediate present value.

At time \(n\), the immediate payment at time \(k\) accumulates for \(n-k\) periods, while the corresponding due payment at time \(k-1\) accumulates for \(n-k+1\) periods. Thus every due payment earns one additional period of interest. The accumulated value of each due payment is \(1+i\) times the corresponding immediate payment's accumulated value, so the same factor applies to their sums.

1
Match the streams.
Confirm that the payment amount and count are equal, then write the payment dates for both streams.
2
Choose one valuation date.
Use time 0 for present value or a common later date, such as time \(n\), for accumulated value.
3
Apply the timing relationship.
For the due value, multiply the corresponding immediate value by \(1+i\). Check that the payment schedule really is shifted by one full period.

Worked Comparisons

Worked Example: Present and Accumulated Values at 10%

A fund receives three annual payments of $1,000. Compare an annuity-immediate with an annuity-due at an effective annual interest rate of 10%. Find both present values at time 0 and both accumulated values at time 3.

Method. First identify the schedules. The immediate payments occur at times 1, 2, and 3; the due payments occur at times 0, 1, and 2. Use the annuity factors and the one-period relationship.

For the immediate present value,

$$ a_{\overline{3}|} =\frac{1-v^3}{i} =\frac{1-(1.10)^{-3}}{0.10} =\frac{1-0.7513148009}{0.10} =2.486851991 $$

Therefore, the immediate present value is \(1{,}000(2.486851991)=\$2{,}486.851991\). The due present value is

$$ 1.10(\$2{,}486.851991)=\$2{,}735.537190 $$

For accumulated value at time 3,

$$ s_{\overline{3}|} =\frac{(1.10)^3-1}{0.10} =\frac{1.331-1}{0.10} =3.31 $$

The immediate accumulated value is \(1{,}000(3.31)=\$3{,}310\). The due accumulated value is \(1.10(\$3{,}310)=\$3{,}641\). As a check, the immediate payments accumulate to \(1{,}000(1.10)^2+1{,}000(1.10)+1{,}000=1{,}210+1{,}100+1{,}000=\$3{,}310\). The due payments accumulate to \(1{,}000(1.10)^3+1{,}000(1.10)^2+1{,}000(1.10)=1{,}331+1{,}210+1{,}100=\$3{,}641\).

Check and trap avoided. The due present value exceeds the immediate present value, and their ratio is \(1.10\). The due stream has three payments, not four: it begins at time 0 and ends at time 2. Do not add a payment to the immediate stream while leaving all its original payment dates unchanged.

Worked Example: Comparing Equal Due Payments with an Equivalent Immediate Payment

A proposed schedule pays $800 at the beginning of each year for four years. The effective annual interest rate is 5%. Find its present and accumulated values, then determine the amount of each of four end-of-year payments that would have the same values.

Method. The proposed schedule is an annuity-due with \(R=\$800\) and \(n=4\). First calculate the immediate factor; then use the factor identity to find both the due values and the equivalent immediate payment.

The immediate present-value factor is

$$ a_{\overline{4}|} =v+v^2+v^3+v^4 =0.9523809524+0.9070294785+0.8638375985+0.8227024749 =3.5459505043 $$

Thus the present value of the due schedule is

$$ \$800(1.05)(3.5459505043) =\$800(3.7232480295) =\$2{,}978.598424 $$

The accumulated-value factor for the immediate annuity is

$$ s_{\overline{4}|} =\frac{(1.05)^4-1}{0.05} =\frac{1.21550625-1}{0.05} =4.310125 $$

The due accumulated value at time 4 is

$$ \$800(1.05)(4.310125) =\$800(4.52563125) =\$3{,}620.505 $$

An immediate payment amount \(Q\) with the same payment count and rate must satisfy \(Q a_{\overline{4}|}=\$2{,}978.598424\). Since the due factor is \(1.05\) times the immediate factor, \(Q=\$800(1.05)=\$840\). This also matches the accumulated value: \(\$840(4.310125)=\$3{,}620.505\).

Check and trap avoided. The equivalent end-of-year payment is greater than $800 because it arrives one period later. The calculation preserves the four-payment count; it does not change the schedule into five payments.

Worked Example: Reading the First Payment Date Correctly

A hypothetical service contract makes three equal annual payments of $1,200, with the first payment made immediately and the next two made one and two years later. At an effective annual interest rate of 5%, find the accumulated value at time 3 and compare it with the value of three $1,200 payments made at the ends of years 1, 2, and 3.

Method. The stated first payment is immediate, so the first schedule is an annuity-due at times 0, 1, and 2. The comparison schedule is an annuity-immediate at times 1, 2, and 3. Both are accumulated to time 3.

For the immediate schedule,

$$ s_{\overline{3}|} =\frac{(1.05)^3-1}{0.05} =\frac{1.157625-1}{0.05} =3.1525 $$

Its accumulated value is \(\$1{,}200(3.1525)=\$3{,}783\). The due schedule is one period earlier, so its accumulated value is

$$ 1.05(\$3{,}783)=\$3{,}972.15 $$

The difference is \(\$3{,}972.15-\$3{,}783=\$189.15\), which is \(0.05(\$3{,}783)\), as the one-period relationship requires. To check directly, the due payments accumulate to \(\$1{,}200(1.05)^3+\$1{,}200(1.05)^2+\$1{,}200(1.05)=\$1{,}389.15+\$1{,}323+\$1{,}260=\$3{,}972.15\).

Check and trap avoided. “First payment immediately” means time 0, so this is due, even though the contract lasts three years. Counting payment dates rather than relying on the phrase “three years” prevents shifting the first payment to time 1.

Common Mistakes and When the Comparison Applies

A frequent error is to treat an annuity-due as an annuity-immediate plus one extra payment. That changes the number of payments and does not represent a one-period shift. For example, in the first worked example the incorrect calculation \(\$2{,}486.851991+\$1{,}000=\$3{,}486.851991\) adds a payment at time 0 but leaves the original payments at times 1, 2, and 3. It therefore describes four payments, not the three-payment due annuity. The correct due present value is \(\$2{,}735.537190\): the due payments occur at times 0, 1, and 2, so each corresponding payment has shifted earlier.

The factor relationship also depends on comparing like with like. Confirm all of the following before using it:

  • The payment amounts are equal in the two schedules.
  • The payment counts are equal.
  • Each due payment is exactly one payment period earlier than its corresponding immediate payment.
  • The same effective rate per payment period is used to value both schedules.
  • The values being compared are at the same focal date.

If payment intervals are not annual, use the effective rate for one payment interval, not automatically the effective annual rate. If the schedules have different payment amounts, counts, or intervals, the simple factor comparison does not directly apply; value the cash flows at a common date using their actual dates and amounts. For ordinary positive interest rates, the practical check is straightforward: with otherwise identical schedules, the annuity-due should have the greater present and accumulated values.

Key takeaway: An annuity-due is the same number of equal payments as an annuity-immediate, shifted one period earlier. With effective rate \(i\) per payment period, its present value and its accumulated value at the corresponding common date are each \(1+i\) times the immediate value.

Check Your Understanding

For each problem, identify the payment timing before applying a factor. Round dollar answers to the nearest cent where needed.

  1. Five payments of $400 are made at the ends of five successive years. The effective annual rate is 5%. How much greater is the present value of five $400 payments made at the beginnings of those years? (Answer: $86.59)
  2. At an effective annual rate of 10%, compare the present values of two schedules, each with two payments: the first schedule pays $100 at the ends of years 1 and 2; the second pays $100 at times 0 and 1. Give the due value and the ratio of due value to immediate value. (Answer: $190.91; ratio 1.10)
  3. Six annual deposits of $300 are made at the ends of years 1 through 6, earning 4% effective annually. Find their accumulated value at time 6 and the accumulated value of six equal deposits made at times 0 through 5. (Answer: $1,989.89 and $2,069.49)
  4. A schedule makes eight payments of $500 at the beginning of each year. The effective annual interest rate is 3%. What payment at each year-end for eight years has the same present value? (Answer: $515)
  5. At an effective annual rate of 7%, compare the present values at time 0 and accumulated values at time 3 for three payments of $2,000, first made at times 1, 2, and 3, and then made at times 0, 1, and 2. (Answer: Present values are $5,248.63 and $5,616.04; accumulated values are $6,429.80 and $6,879.89.)

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