Question

Determine the accumulated amount of an annuity consisting of 10 payments of P20,000 each.
The payment is made at the beginning of each month. Money is worth 10% compounded
semi-annually.

*CASH FLOW Diagram Needed

Answer 1

First calculate the Effective annual rate (EAR) with semiannual compounding

EAR = (1 + r/m) ^m – 1

Where, nominal annual interest rate, r = 10%

Number of compounding per year, m = 2 (semiannual)

Therefore

EAR = (1 + 10%/2) ^2 - 1 = 0.1025 or 10.25%

But the payment is monthly; therefore we have to calculate monthly interest rate with annual effective rate of 10.25% in following manner

Effective annual rate (EAR) = (1 + i) ^m – 1

Where, Effective annual rate (EAR) = 10.25%

Monthly interest rate i =?

Number of payments per year, m = 12

Therefore

10.25% = (1 + i) ^12 - 1

Or i = (1.1025) ^ (1/12) – 1 = 0.008165 or 0.8165%

We can use Future value (FV) of an Annuity due formula to find out the accumulated Future value from the payments of P 20, 000 per month (as the payments are made at the starting of the month)

FV = PMT*(1+i) *{(1+i) ^n−1} / i

Where FV =?

PMT = monthly payment = P 20,000

Annual interest rate = 10% per year; Monthly interest rate i =0.8165%

n = N = number of payments =10

Therefore,

FV = P 20,000 * (1+0.8165%)* [(1+0.8165%) ^10 -1]/ 0.8165%

FV = P 209,205.14

The accumulated amount of an annuity is P 209,205.14

Cash Flow diagram -