How can I determine the interest rate, given increasing monthly deposits and a final amount?

Question

I need help to determine the interest rate of this investment.

• The initial value is nothing
• The monthly investment is R500.
• It stays R500 a month until the year is over and increases by R50. Thus, R550 per month the 2nd year, and R600 the 3rd year, etc.
• Total period is 10 Years
• Total invested: R87000 over the 10 years.
• Total returned (deposits plus interest): R149028

can someone who knows the formula please help.

Answer 1

I'm assuming that compound interest is paid monthly. In that case, the interest rate is ~0.9162% per month (i.e. an APR of 11.57%).

The following formula is the one you are interested in:

sum_{y=0}^9 (sum_{m=0}^11 (500+50*y)*(1+T)^(120-12*y-m))=149028

I didn't attempt to solve this, but instead used trial and error. The value T=0.0091624 works nicely; you can verify this in Wolfram Alpha.

Getting at APR/EAR is simply (1+T)^12 - 1.

Answer 2

A method to work this out can be found by using a simpler quarterly example over two years. Using an example rate, r = 0.01, this is the example calculation for the first year

y1q1 = 0 + 500
y1q2 = y1q1 (1 + r) + 500
y1q3 = y1q2 (1 + r) + 500
y1q4 = y1q3 (1 + r) + 500
y1q4 (1 + r) =  2050.502505

Equivalent to the summation

Continuing, this is the calculation for two years

y2q1 = y1q4 (1 + r) + 500 + 50
y2q2 = y2q1 (1 + r) + 500 + 50
y2q3 = y2q2 (1 + r) + 500 + 50
y2q4 = y2q3 (1 + r) + 500 + 50
y2q4 (1 + r) = 4389.313885

Equivalent to this summation

To create a general formula this needs to be re-expressed as a double summation, where n is the total number of periods, n = 8

This can be generalised, where

y is the number of years
m is the number of months or quarters (or days)
p is the initial regular deposit
d is the annual deposit increase

By induction, this can be reduced to a formula

Checking

r = 0.01
p = 500
d = 50
y = 2
m = 4
n = 8

((1 + r)^(1 + n) (d + p (-1 + (1 + r)^m) +
     (1 + r)^(-m y) (-d + p + d y -
        (1 + r)^m (p + d y))))/(r (-1 + (1 + r)^m)) = 4389.313885

This can be used to solve for the OP's values

fv = 149028
p  = 500
d  = 50
y  = 10
m  = 12
n = 120

Plot of future value for a range of r also showing the target fv

Solving exactly yields r = 0.009162396432

Giving an annual effective rate of

(1 + 0.009162396432)^12 - 1 = 0.115662 = 11.5662 %