How much must be deposited at the end of each month at 4% compounded… Step by Step

Step by step

The question

Question: How much must be deposited at the end of each month at 4% compounded monthly to have $10000 after 5 years?

1 Read the numbers

FV is the money we want at the end: $10000.

We need PMT, each deposit.

r is the yearly rate: 4% is 0.04.

FV = $10,000 goal, r = 4% = 0.04

n is how many times a year: 12.

t is the time: 5 years.

n = 12 a year, t = 5 years

2 Per payment

Interest is added each month.

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

0.04 divided by 12 is 0.003333.

i = rn = 0.0412 = 0.003333

i = the rate each month

And N is how many deposits: 12 times 5 is 60.

N = n × t = 12 × 5 = 60

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.

10000 = PMT × 1.00333⁶⁰ − 10.003333

4 Work it out

With a calculator: 1.00333 to the power 60 is about 1.221.

Minus 1, divided by 0.003333: about 66.299.

1.00333⁶⁰ ≈ 1.221

(1.221 − 1) ÷ 0.003333 ≈ 66.299

So 10000 is PMT times 66.299.

10000 ≈ PMT × 66.299

5 Get PMT alone

To remove times 66.299, we divide both sides by 66.299.

PMT ≈ 1000066.299

10000 divided by 66.299 is 150.8319 and a bit.

PMT ≈ 150.8319…

We round up to the next cent, so the goal is reached: 150 dollars and 84 cents.

PMT = $150.84

Careful!

Rounding down would leave the account a little short.

Careful: round UP, or the goal is not reached

6 Check

Let's check: 150 dollars and 84 cents times 66.299 is about 10000 dollars and 54 cents.

That reaches the goal.

So the answer is right!

$150.84 × 66.299 ≈ $10,000.54 ≥ $10,000

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