To calculate the present value of a loan that will amount to a future value of birr 5,000 in four years with a 10% interest rate compounded semiannually, we can use the formula for present value (PV):
\[ PV = \frac{FV}{(1 + r/n)^{nt}} \]
Where:
- \(FV\) is the future value (Birr 5000)
- \(r\) is the annual interest rate (0.10)
- \(n\) is the number of compounding periods per year (2 for semiannual)
- \(t\) is the number of years (4)
Step 1: Substitute the values into the formula.
- Future Value (FV): 5000
- Annual Interest Rate (r): 0.10
- Compounding periods per year (n): 2
- Number of years (t): 4
Step 2: Calculate the present value.
\[ PV = \frac{5000}{(1 + 0.10/2)^{2 \cdot 4}} \]
Step 3: Calculate the components.
-
Calculate \(r/n\): \[ r/n = 0.10/2 = 0.05 \]
-
Calculate the exponent \(nt\): \[ nt = 2 \cdot 4 = 8 \]
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Calculate \((1 + r/n)^{nt}\): \[ (1 + 0.05)^8 = (1.05)^8 \]
Calculating \((1.05)^8\): \[ (1.05)^8 \approx 1.432364654 \]
Step 4: Substitute back to find PV:
\[ PV = \frac{5000}{1.432364654} \approx 3497.35 \]
Thus, the present value of the loan is approximately Birr 3497.35.