Annuities-Due: Timing Differences and Key Formulas for Exam FM

Annuity-Due Definition

An annuity-due makes payments at the beginning of each period, one period earlier than the corresponding annuity-immediate. For n payments of 1 at the beginning of each period, the present value is a-double-dot-angle-n = (1 - v^n) / d, and the accumulated value is s-double-dot-angle-n = ((1+i)^n - 1) / d, where d = i/(1+i) is the annual effective discount rate.

The timing distinction is critical. An annuity-immediate of 3 annual payments has payments at times 1, 2, and 3. An annuity-due of 3 annual payments has payments at times 0, 1, and 2. The annuity-due starts one period earlier and ends one period earlier.

The key relationship connecting the two types is: a-double-dot-angle-n = (1+i) * a-angle-n, and similarly s-double-dot-angle-n = (1+i) * s-angle-n. Every annuity-due value is exactly (1+i) times the corresponding annuity-immediate value. This factor arises because each payment is received one period earlier, and receiving money one period earlier makes it worth (1+i) times as much.

Derivation

The present value of an annuity-due is the sum: a-double-dot-angle-n = 1 + v + v^2 + ... + v^(n-1). Notice that this starts at v^0 = 1 (the payment at time 0 requires no discounting) and ends at v^(n-1). Applying the geometric series formula: a-double-dot-angle-n = (1 - v^n)/(1 - v) = (1 - v^n)/d, since 1 - v = d.

The accumulated value counts forward from each payment: s-double-dot-angle-n = (1+i)^n + (1+i)^(n-1) + ... + (1+i). The first payment at time 0 grows for n full periods, and the last payment at time n-1 grows for 1 period. This gives s-double-dot-angle-n = ((1+i)^n - 1) * (1+i) / ((1+i) - 1) = ((1+i)^n - 1)(1+i)/i = ((1+i)^n - 1)/d.

Notice the structural parallel with annuity-immediate formulas: the numerators are identical, but the denominators use d instead of i. Since d = i/(1+i) is less than i, the annuity-due values are always larger than the corresponding annuity-immediate values, consistent with receiving each payment earlier.

Conversion Identities

Several equivalent expressions connect annuity-due and annuity-immediate values. For present values: a-double-dot-angle-n = 1 + a-angle-(n-1). This says the present value of n beginning-of-period payments equals the first payment (received immediately, worth 1) plus the present value of the remaining n-1 end-of-period payments.

For accumulated values: s-double-dot-angle-n = s-angle-(n+1) - 1. This says the accumulated value of n beginning-of-period payments equals the accumulated value of n+1 end-of-period payments minus 1 (because the annuity-immediate includes one extra payment at the end that the annuity-due does not).

These identities are not merely theoretical curiosities. On Exam FM, you may be given a-angle-n values in a table and asked to find a-double-dot values, or vice versa. Being able to convert quickly saves time. For example, if a-angle-10 at 5% = 7.7217, then a-double-dot-angle-10 at 5% = 1.05 * 7.7217 = 8.1078, or equivalently, 1 + a-angle-9 at 5% = 1 + 7.1078 = 8.1078.

Practical Applications

Annuities-due model situations where payments occur at the start of each period. Common examples include rent (due at the beginning of the month), insurance premiums (due at the start of each coverage period), and lease payments. In contrast, loan payments are typically made at the end of each period (annuity-immediate).

For a payment of R per period, PV = R * a-double-dot-angle-n and FV = R * s-double-dot-angle-n. Example: monthly rent of $1500 for 2 years at 6% compounded monthly. The monthly rate is 0.5%, and n = 24 months. PV = 1500 * a-double-dot-angle-24 at 0.5% = 1500 * (1.005) * a-angle-24 at 0.5% = 1500 * 1.005 * 22.5629 = $34,018.24.

Another application is retirement savings. If you deposit $500 at the beginning of each month into an account earning 8% compounded monthly for 30 years, the accumulated value is 500 * s-double-dot-angle-360 at 0.6667% = 500 * (1.006667) * s-angle-360 at 0.6667%. This is larger than the annuity-immediate accumulation by a factor of 1.006667 because each deposit has one extra month to earn interest.

When to Use Which on Exam FM

Read each problem carefully to determine when payments occur. Phrases like "beginning of month," "due at the start," "payable in advance," or "first payment today" signal an annuity-due. Phrases like "end of month," "payable in arrears," "first payment in one year," or "first payment at the end" signal an annuity-immediate.

If the problem is ambiguous, draw a timeline showing when each payment occurs relative to the valuation date. Mark time 0 and place each payment on the timeline. If the first payment is at time 0, it is an annuity-due. If the first payment is at time 1, it is an annuity-immediate.

The most common error on Exam FM annuity problems is using the wrong annuity type. This shifts the answer by a factor of (1+i), which is usually close to 1, making the error subtle and hard to catch. Always verify the timing before selecting your formula.

Study Tips

Memorize both the annuity-due formulas and the conversion relationships. On Exam FM, you should be able to go from annuity-immediate to annuity-due values (and back) in seconds. Practice problems that mix both types, such as comparing the cost of a lease with payments at the beginning versus end of month. Work through at least 20 annuity-due problems, paying special attention to the timing language in each problem statement. Build the habit of asking "when is the first payment?" before writing any formula. This single question will prevent the majority of annuity errors on exam day.