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

1 answer

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.