Question

An investor wishes to accumulate 200 PLN. How many months must the investor pay 40 PLN at the beginning of each month if the interest rate is 12% per annum, compounding annually. Solve the problem of non-integer value of the term of annuities. Rework example with deposits at the end of each month. Please solve in details.

Answer 1

Target Accumulation = 200 PLN, Monthly Payments = 40 PLN, Interest Rate = 12 % per annum or 1 % per month

Let the number of months be N for beginning of month deposits.

As per the laws of time value of money:

200 = 40 x (1.01)^(N) + 40 x (1.01)^(N-1) + .................+ 40 x (1.01)

200 = { [(1.01)^(N) - 1] / [0.01] } x 40 x (1.01)

[(1.01)^(N) - 1] = [200/40] x [1/(1.01)] x [0.01]

(1.01)^(N) = 1+ 0.049505

N = $\log_{1.01}{1.049505}$ = 4.856 month ~ 5 months

If deposits are made at the end of the month, then the equation would loook like as shown below:

200 = 40 x (1.01)^(N-1) + 40 x (1.01)^(N-2) + .................+ 40 x (1.01) + 40

200 = { [(1.01)^(N) - 1] / [0.01] } x 40

[(1.01)^(N) - 1] = [200/40] x [0.01]

(1.01)^(N) = 1.05

N = $\log_{1.01}{1.05}$ = 4,90 months ~ 5 months