How much must be deposited at the end of each quarter at 8%… Step by Step

Step by step

The question

Question: How much must be deposited at the end of each quarter at 8% compounded quarterly to have $8000 after 4 years?

1 Read the numbers

FV is the money we want at the end: $8000.

We need PMT, each deposit.

r is the yearly rate: 8% is 0.08.

FV = $8,000 goal, r = 8% = 0.08

n is how many times a year: 4.

t is the time: 4 years.

n = 4 a year, t = 4 years

2 Per payment

Interest is added each quarter.

The rate for one quarter is i: r over n.

0.08 divided by 4 is 0.02.

i = rn = 0.084 = 0.02

i = the rate each quarter

And N is how many deposits: 4 times 4 is 16.

N = n × t = 4 × 4 = 16

3 Use the formula

The savings formula: FV = PMT times, (1 + i) to the power N, minus 1, over i.

FV = PMT × (1 + i)^N − 1i

FV = future value: the money at the end

We put the numbers in.

PMT stays a letter: we want it.

8000 = PMT × 1.02¹⁶ − 10.02

4 Work it out

With a calculator: 1.02 to the power 16 is about 1.37279.

Minus 1, divided by 0.02: about 18.6393.

1.02¹⁶ ≈ 1.37279

(1.37279 − 1) ÷ 0.02 ≈ 18.6393

So 8000 is PMT times 18.6393.

8000 ≈ PMT × 18.6393

5 Get PMT alone

To remove times 18.6393, we divide both sides by 18.6393.

PMT ≈ 800018.6393

8000 divided by 18.6393 is 429.2010 and a bit.

PMT ≈ 429.2010…

We round up to the next cent, so the goal is reached: 429 dollars and 21 cents.

PMT = $429.21

Careful!

Rounding down would leave the account a little short.

Careful: round UP, or the goal is not reached

6 Check

Let's check: 429 dollars and 21 cents times 18.6393 is about 8000 dollars and 17 cents.

That reaches the goal.

So the answer is right!

$429.21 × 18.6393 ≈ $8,000.17 ≥ $8,000

Let's look back.

First: read the numbers.

Then: per payment.

Then: use the formula.

Then: work it out.

Then: get PMT alone.

Last: check.

Answer