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.