To determine the size of each annual payment for a loan that will be repaid in equal installments at the end of each year, we can use the formula for the annuity payment for a fixed loan. The formula to calculate the annual payment (PMT) is given by:
\[ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} \]
Where:
- \(P\) is the principal amount (the initial loan amount),
- \(r\) is the interest rate per period (as a decimal),
- \(n\) is the total number of payments.
In this case:
- \(P = 100,000\)
- \(r = 10% = 0.10\)
- \(n = 5\)
Now, we can substitute the values into the formula:
\[ PMT = \frac{100,000 \cdot 0.10}{1 - (1 + 0.10)^{-5}} \]
First, calculate \((1 + r)^{-n}\):
\[ 1 + 0.10 = 1.10 \] \[ (1 + 0.10)^{-5} = 1.10^{-5} \approx 0.62092 \]
Now, we calculate \(1 - (1 + r)^{-n}\):
\[ 1 - 0.62092 \approx 0.37908 \]
Now substitute this value back into the payment formula:
\[ PMT = \frac{100,000 \cdot 0.10}{0.37908} \approx \frac{10,000}{0.37908} \approx 26,417.41 \]
Thus, each annual payment must be approximately Br. 26,417.41.