Compound Interest Calculations?

Question

For three years i deposit $890 at the start of each month into an account that earns me 5% pa, compounding monthly.

How much money do i have at the end of the three years. Please show all working.

Any Help in this matter would be much appreciated

Answer 1

I'm an actuary, so this is right up my alley.

First we need a periodic interest rate (rate per payment period).  You did not mention if the 5% was nominal or effective.  I'll assume it was effective.  To get the periodic monthly rate:

j = (1+i)^(1/12) -1 = 0.4074% (we'll just use j, and keep the accuracy by using a calculator)

(if it was nominal, then j = 5%/12 = 0.4167%, so yes, it does matter)

The formula for a periodic deposit of equal payments, at the periodic interest rate j, where payments are made at the start of each period is:

Future Value = P{[(1+j)^n - 1]/j}*(1+j)

where P = monthly payment amount, n=number of payments, and j is the periodic interest rate from above (either 0.4074% if the 5% was effective, or 0.4167% if the 5% was nominal).

In case you care, the actual actuarial symbol (which I can't type using this editor) is pronounced "s double-dot angle n", which is the accummulated value of a series of cash flows made at the beginning of each period.

FV = 890*{[(1+j)^36 - 1]/j}*(1+j) = $34,573.76 (effective)

or $34,634.18 (nominal)

Answer 2

Let:

S = sum after n periods

n = number of periods = 3 x 12 = 36

P = periodic deposit = 890

i = interest per period = 0.05/12

S = P(1 + i)((1 + i)^n -1)/i = 890(1 + 0.05/12)((1 + 0.05/12)^36 -1)/(0.05/12) = $34634.18