Computing the value of an annuity for its first few years

Same annuity formula as Q6, but evaluated at the end of year 1, year 2 and year 3. Watch where the first payment's interest comes from — and where it doesn't.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Mai's retirement plan

Mai deposits $400 at the end of each year in an ordinary annuity paying 6% compounded annually.

Find the total value at the end of the 3rd year. Type just the number.

Do not round any intermediate computations.

The Okonkwos' fund

The Okonkwos deposit $2,500 at the end of each year at 4% compounded annually.

What is the value at the end of the 3rd year? Type just the number.

Do not round any intermediate computations.

Reading the pattern

An ordinary annuity pays 8% compounded annually. After one year the account holds $550.

What is in the account at the end of the 2nd year? Type just the number.

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

$FV = PMT \times \dfrac{(1 + i)^{k} - 1}{i}$ k is how many payments have been made so far

ordinary annuity → payments at the end of each period so year 1's value is exactly one payment

The only thing changing between the three parts is k. Everything else stays put. The surprise is year 1: because the payment arrives at the end of the year, it has had no time to earn anything, so the account holds exactly one payment.

Common slips

(1) Added interest to year 1. An ordinary annuity pays at period end. The first deposit earns nothing in its own year. (2) Multiplied year 1 by the number of years. Year 2 is not twice year 1 — the first payment has now earned a year of interest. (3) Used k = the year number when payments are monthly. Here they are annual, so k and the year number happen to match. They will not always.

04 Practice — Step by Step

Build it year by year

Nadia deposits $1,200 at the end of each year into an ordinary annuity paying 5% compounded annually.

Find the value at the end of years 1, 2 and 3.

Step 1

End of year 1

One payment has been made, at the very end of the year. How much is in the account?

Step 2

End of year 2

Year 1's $1,200 earns 5% over year 2, then the second $1,200 is deposited. Total?

Step 3

End of year 3

Same move once more. Round to the nearest cent.

Q5 Q7

Computing the value of an annuity for its first few years

Where the formula comes from