I'm playing around with the compound interest formula and I'm not getting the answers I expect. Wondered why??

Question

Below is what I was reading:

If you deposit $£1,000$ in a bank account which is paying $3\%$ compound interest per year. How much interest would be earnt over $3$ years?

The formula used to solve this was:

$$1000 \times 1.03^3 = £1,092.73$$

Then you subtract for the answer:

$$1,092.73 – 1000 = £92.73$$

I then started playing around with the formula, just to get a better understanding.

I did this:

$1,000 \times 0.03^3$ and expected the answer to be $92.73$ but instead I got $0.027$

Why is the answer coming out as $0.027$ and not $92.73$?

Answer 1

According to your expectations, we have that:

$$1000 \times 1.03^3 - 1000 = 1000 \times 0.03^3$$

Which gives us

$$1000 \times (1.03^3-1) = 1000 \times 0.03^3$$

And so

$$1.03^3-1=0.03^3$$

But expanding it out,

$$1.03^3 - 1^3 = 0.03^3$$ $$0.03(1.03^2+1.03+1) = 0.03^3$$ $$1.03^2+1.03+1 = 0.03^2$$

We can see that the RHS is smaller than $1$, but the LHS is bigger than one, so this is not true.

Answer 2

You just did A minor yet critical mistake, Which is easy to identify, $1000((1.03)^3-1)\neq1000((0.03)^3)$

Because $(1.03)^3-1\neq(1.03-1)^3$