Continous growth and the number e?

Question

Which is better, an account paying 5.3% interest compounded continuously or an account paying 5.5% interest compounded annually? Justify your answer

Answer 1

In one year, you will have 1.055 times your principal if you place it in the account compounded annually. If you place it in the account compounded continuously, you will have e^0.053 times your principal, which is approximately 1.0544. This is less than 1.055, so you will make strictly more money if you place it in the account compounded annually. Therefore, 5.5% compounded annually is better than 5.3% compounded continuously.

Answer 2

In n years, 5.3% compounded continuously gives a growth factor of e^(0.053y) = 1.05443^y, but 5.5% compounded annually gives a growth factor of 1.055^y, and the latter is greater.

Answer 3

An interest rate of 5.3% compounded over a number of n periods gives a return rate of (1+0.053/n)^n. As n increases toward infinity the return approaches a limit of exp(0.053) = 1.05443. So the continuous rate of 5.3% represents an equivalent APR of 5.44%