what is the amount of an annuity if the size of each payment is birr 100 payable at the end of each quarte for one year at an interest rate of 4% compounded quarterly?

1 answer

To calculate the present value of an annuity where payments are made at the end of each period, we can use the present value of an annuity formula:

\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

Where:

  • \(PV\) = Present value of the annuity
  • \(P\) = Payment amount (Birr 100)
  • \(r\) = Interest rate per period
  • \(n\) = Total number of payments

Given:

  • Payment amount (\(P\)) = 100 Birr
  • Annual interest rate = 4% = 0.04
  • Compounded quarterly, so the interest rate per quarter (\(r\)) = \(0.04 / 4 = 0.01\)
  • Total duration = 1 year, so the total number of payments (\(n\)) = \(4\) (quarterly payments)

Now plug in the values into the formula:

\[ PV = 100 \times \left( \frac{1 - (1 + 0.01)^{-4}}{0.01} \right) \]

First, calculate \((1 + r)^{-n}\):

\[ (1 + 0.01)^{-4} = (1.01)^{-4} \approx 0.9608 \]

Now calculate the rest:

\[ PV = 100 \times \left( \frac{1 - 0.9608}{0.01} \right) \]

\[ PV = 100 \times \left( \frac{0.0392}{0.01} \right) \]

\[ PV = 100 \times 3.92 \]

\[ PV = 392 \]

Thus, the present value of the annuity is approximately 392 Birr.