To find the annual payment for a loan of 20,000 to be repaid over 5 years with an annual interest rate of 12%, we can use the annuity payment formula. The annual payment (PMT) for an ordinary annuity can be calculated using the following formula:

[ PMT = \frac{PMT}{[(1 - (1 + i)^{-n}] / i} ]

Where: - PMT is the payment amount we want to find. - ( i ) is the interest rate per period (in this case, annually). - n is the number of payments, which is 5 years or 5 periods in this case.

Plugging in the numbers, we get:

[ PMT = \frac{PMT}{[(1 + 0.12)^{-5}] / 0.12} ]

We need to solve for PMT, the annual payment amount. To do this, we can rearrange the formula to isolate PMT:

[ PMT = \frac{PMT}{[(1 + 0.12)^{-5}] / 0.12} ]

[ PMT = \frac{PMT}{[(1 + 0.12)^{-5}] / 0.12} \times [(1 + 0.12)^5] \times 0.12 ]

[ PMT = PMT \times (1 + 0.12)^5 \times 0.12 ]

Since PMT appears on both sides of the equation, we can't directly solve for it algebraically. However, we can use a financial calculator or a spreadsheet function like Excel's PMT function to solve for the payment.

[ PMT ≈ \frac{20,000}{[(1 + 0.12)^{-5}] / 0.12} \times [(1 + 0.12)^5] \times 0.12 ]

[ PMT ≈ 4,215.52 ]

So, the annual payment would be approximately $4,215.52.