Question

find the present value of a loan that will amount to birr 5000 in four years if money is worth 10% compounded semiannually?

Answer 1

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.

1. **Future Value (FV)**: 5000
2. **Annual Interest Rate (r)**: 0.10
3. **Compounding periods per year (n)**: 2
4. **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.

1. Calculate \(r/n\):
\[
r/n = 0.10/2 = 0.05
\]

2. Calculate the exponent \(nt\):
\[
nt = 2 \cdot 4 = 8
\]

3. 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**.