Question

Most savings banks advertise that they compound interest continuously, meaning that the amount S(t) in an account satisfies the differential equation dS/dt=rS, where r is the annual interest rate and t is time measured in years.

1. a) Show that an annual interest rate of 8% compounded continuously is the same as an annual interest rate of 8.33% compounded in years.
2. b) Show that an annual interest rate of r compounded continuously is the same as an annual interest rate of e^r-1 compounded annually.

Answer 1

Solution:

To calculate annual interest rate we use the following formula

Annual Percentage Rate
Formula
(1+i/m)^m - 1
m	365	Number of compounding per year
I	8.00%
i/m	0.000219		1.000
(1+i/m)^m	1.083
APR	8.33%

or we find out using financial calculator or excel formula

(Exponent of rate%) - 1

Exp(8%) = 1.0833

Exp(8%)-1 = 1.0833 - 1 = 0.0833 = 8.33%

Hence an annual interest rate of r compounded continuously is the same as an annual interest rate of er-1 compounded annually.

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