Annuities-Immediate: Present and Accumulated Value Formulas

Annuity-Immediate Definition

An annuity-immediate (also called an ordinary annuity) makes payments at the end of each period. For n payments of 1 at the end of each period, the present value at time 0 is a-angle-n = (1 - v^n) / i, where v = 1/(1+i). The accumulated value at time n is s-angle-n = ((1+i)^n - 1) / i.

These are arguably the most important formulas on Exam FM. Nearly every financial calculation involving a series of equal payments uses these building blocks. For a payment of R per period, simply multiply: PV = R * a-angle-n and FV = R * s-angle-n.

The notation "a-angle-n" (sometimes written with an angle bracket, as in a with angle n) represents the present value of n payments of 1. The "s-angle-n" represents the accumulated (future) value. The interest rate i is implicit in the notation; when clarity requires it, you can write a-angle-n at rate i.

Derivation and Understanding

The present value is the sum of a geometric series. Each payment of 1 at time k has present value v^k, so the total is: a-angle-n = v + v^2 + ... + v^n. This is a geometric series with first term v, common ratio v, and n terms. Using the geometric series formula: a-angle-n = v(1 - v^n)/(1 - v). Since 1 - v = iv/(1+i) and v/(1-v) = 1/i, this simplifies to (1 - v^n)/i.

Similarly, the accumulated value at time n treats each payment's growth: the payment at time 1 grows for n-1 periods, the payment at time 2 grows for n-2 periods, and so on. The last payment at time n earns no interest. So s-angle-n = (1+i)^(n-1) + (1+i)^(n-2) + ... + (1+i) + 1 = ((1+i)^n - 1)/i.

A critical relationship connects the two: s-angle-n = a-angle-n * (1+i)^n. This says the accumulated value is just the present value grown forward n periods. Conversely, a-angle-n = s-angle-n * v^n. Either formula can be derived from the other by multiplying or dividing by (1+i)^n.

Key Properties and Special Values

Several properties of annuity values are worth memorizing for speed on Exam FM. First, a-angle-1 = v and s-angle-1 = 1. Second, as n increases, a-angle-n approaches 1/i (the perpetuity value) because v^n approaches 0. Third, s-angle-n grows without bound as n increases.

The reciprocal of a-angle-n gives the payment needed to repay a loan of 1: R = 1/a-angle-n. This is the basis of all loan amortization calculations. The reciprocal of s-angle-n gives the periodic deposit needed to accumulate 1: D = 1/s-angle-n. This is the basis of sinking fund calculations.

Another useful identity: 1/a-angle-n = 1/s-angle-n + i. This says the loan payment equals the sinking fund deposit plus interest on the loan. To see why, consider a loan of 1 at rate i: the borrower can either amortize (paying 1/a-angle-n per period) or pay interest only (i per period) while depositing into a sinking fund (1/s-angle-n per period). Both strategies repay the loan.

Exam FM Problem Types

Exam FM problems with annuities-immediate typically give three of the four quantities {payment amount, interest rate, number of periods, present or accumulated value} and ask for the fourth. Set up the equation and solve.

Finding the payment: A person wants to accumulate $50,000 in 10 years by making quarterly deposits into an account earning 8% compounded quarterly. Here i = 0.02 per quarter, n = 40 quarters. R * s-angle-40 at 2% = 50,000. Since s-angle-40 = ((1.02)^40 - 1)/0.02 = 60.402, we get R = 50,000/60.402 = $827.85 per quarter.

Finding the number of periods: A loan of $20,000 at 6% annual interest is repaid with annual payments of $3,000. How many full payments are needed? Solve 3000 * a-angle-n = 20,000, so a-angle-n = 6.6667. Using the formula (1 - 1.06^(-n))/0.06 = 6.6667, we get 1.06^(-n) = 0.60, so n = ln(0.60)/ln(1/1.06) = 8.77. This means 8 full payments plus a smaller final payment, or 9 payments with a reduced last payment.

Finding the interest rate: A machine costing $15,000 is purchased with 5 annual payments of $3,500. Solve 3500 * a-angle-5(i) = 15,000, so a-angle-5 = 4.2857. By trial: at i = 5%, a-angle-5 = 4.3295; at i = 6%, a-angle-5 = 4.2124. Interpolating: i is approximately 0.05 + (4.3295 - 4.2857)/(4.3295 - 4.2124) * 0.01 = 0.05 + 0.374 * 0.01 = 5.37%.

Non-Integer Periods and Irregular Payments

When the number of periods is not a whole number, special handling is required. Suppose a loan of $10,000 at 8% is repaid with annual payments of $2,000. The annuity runs out before the loan is fully repaid if n is not an integer. In practice, the borrower makes n full payments of $2,000 and a smaller final payment. The outstanding balance after the last full payment determines the size of the drop payment.

For annuities where the payment period differs from the interest period, you must adjust the interest rate. If the interest rate is annual but payments are monthly, convert to a monthly effective rate: j = (1+i)^(1/12) - 1, then use j as the rate and 12n as the number of periods.

Study Tips

Memorize the formulas for a-angle-n and s-angle-n along with the relationship s-angle-n = a-angle-n * (1+i)^n. Practice computing annuity values by hand for small n to build intuition, then use the formulas for larger problems. On Exam FM, draw a timeline for every annuity problem showing when payments occur relative to the valuation date. This simple step prevents the most common error: using the wrong annuity type or misaligning the valuation point. Work through at least 25 annuity-immediate problems covering all four types (finding the payment, rate, period count, and present or future value) before the exam.