Question

Derive the formula of the rate of interest (r) from the formula of compound interest (P(1+r/n)^nt

P = Principal

R = Rate of Interest

N = No. of times compounded annually

T = No. of years

Answer 1

The basic compound interest formula is this:

V(t) = V(0) X (1 + (r/k) )^ [ kt ]

Here

V(t) is the value at time t (and t is measured in years)

V(0) is the starting value

X is the multiplication symbol

r is the interest rate (using r = 0.04 to be 4% interest)

k is the number of compoundings per year

^ is the exponentiation operation (so 2^3 = 8, for example)

[ ] is the greatest integer function

As a simple example, consider quarterly compounding (k = 4) with interest rate 5% over 2 years, 4 months (t = 2.3333). The formula is

V(t) = V(0) (1 + 0.05/4) ^ [ 4X2.3333 ] = V(0) 1.0125 ^ [9.3333]

= V(0) 1.0125 ^ 9 = V(0) X 1.1183

This is just the rate 1.25% compounded 9 times.

If you were really interested in quarterly compounding (k = 12), this would be V(t) = V(0) (1 + 0.05/12) ^ [ 12 X 2.3333 ] = V(0) X 1.1235. This assumes that you kept the money in place until just after the final (28th) compounding at 2 years, 4 months.

I’d really recommend use of continuous compounding, which is so much more useful in financial math. The continuous compound interest formula is this:

V(t) = V(0) X exp( rt )

Here

V(t) is the value at time t (and t is measured in years)

V(0) is the starting value

r is the interest rate (using r = 0.04 to be 4% interest)

For the example above, this works out to V(t) = V(0) X 1.1237.