How much must be deposited at the end of each year at 5% compounded… Step by Step

Step by step

The question

Question: How much must be deposited at the end of each year at 5% compounded yearly to have $50000 after 15 years?

1 Read the numbers

FV is the money we want at the end: $50000.

We need PMT, each deposit.

r is the yearly rate: 5% is 0.05.

FV = $50,000 goal, r = 5% = 0.05

n is how many times a year: once.

t is the time: 15 years.

n = 1 a year, t = 15 years

2 Per payment

Interest is added each year.

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

0.05 divided by 1 is 0.05.

i = rn = 0.051 = 0.05

i = the rate each year

And N is how many deposits: 1 times 15 is 15.

N = n × t = 1 × 15 = 15

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.

50000 = PMT × 1.05¹⁵ − 10.05

4 Work it out

With a calculator: 1.05 to the power 15 is about 2.07893.

Minus 1, divided by 0.05: about 21.5786.

1.05¹⁵ ≈ 2.07893

(2.07893 − 1) ÷ 0.05 ≈ 21.5786

So 50000 is PMT times 21.5786.

50000 ≈ PMT × 21.5786

5 Get PMT alone

To remove times 21.5786, we divide both sides by 21.5786.

PMT ≈ 5000021.5786

50000 divided by 21.5786 is 2317.1144 and a bit.

PMT ≈ 2317.1144…

We round up to the next cent, so the goal is reached: 2317 dollars and 12 cents.

PMT = $2,317.12

Careful!

Rounding down would leave the account a little short.

Careful: round UP, or the goal is not reached

6 Check

Let's check: 2317 dollars and 12 cents times 21.5786 is about 50000 dollars and 12 cents.

That reaches the goal.

So the answer is right!

$2,317.12 × 21.5786 ≈ $50,000.12 ≥ $50,000

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