1. **Problem Statement:** Derek received a loan of 1300 at 6.50% annual interest compounded quarterly. He makes equal payments at the end of each quarter for 1 year (4 payments) to fully repay the loan. We need to find the size of each quarterly payment and complete the amortization schedule. 2. **Formula for Quarterly Payment:** The loan is an ordinary annuity with quarterly compounding. The payment $P$ is given by: $$P = \frac{r \times PV}{1 - (1 + r)^{-n}}$$ where: - $PV = 1300$ (present value or loan amount) - $r = \frac{6.50\%}{4} = 0.065/4 = 0.01625$ (quarterly interest rate) - $n = 4$ (number of payments) 3. **Calculate the payment:** $$P = \frac{0.01625 \times 1300}{1 - (1 + 0.01625)^{-4}}$$ Calculate denominator: $$1 + 0.01625 = 1.01625$$ $$1.01625^{-4} = \frac{1}{1.01625^4} \approx \frac{1}{1.067} = 0.937$$ So denominator: $$1 - 0.937 = 0.063$$ Calculate numerator: $$0.01625 \times 1300 = 21.125$$ Therefore: $$P = \frac{21.125}{0.063} \approx 335.32$$ 4. **Quarterly payment is approximately 335.32.** 5. **Amortization Schedule:** Start with principal balance 1300. | Payment # | Amount Paid | Interest Portion | Principal Portion | Principal Balance | |---|---|---|---|---| | 0 | - | - | - | 1300.00 | Calculate each payment's interest and principal portions and update balance: - Payment 1: - Interest = $1300 \times 0.01625 = 21.13$ - Principal = $335.32 - 21.13 = 314.19$ - New balance = $1300 - 314.19 = 985.81$ - Payment 2: - Interest = $985.81 \times 0.01625 = 16.02$ - Principal = $335.32 - 16.02 = 319.30$ - New balance = $985.81 - 319.30 = 666.51$ - Payment 3: - Interest = $666.51 \times 0.01625 = 10.83$ - Principal = $335.32 - 10.83 = 324.49$ - New balance = $666.51 - 324.49 = 342.02$ - Payment 4: - Interest = $342.02 \times 0.01625 = 5.56$ - Principal = $335.32 - 5.56 = 329.76$ - New balance = $342.02 - 329.76 = 12.26$ (small rounding difference) 6. **Adjust last payment principal portion to fully pay off loan:** - Final principal portion = $342.02$ - Final payment = Interest + Principal = $5.56 + 342.02 = 347.58$ 7. **Final amortization schedule:** | Payment # | Amount Paid | Interest Portion | Principal Portion | Principal Balance | |---|---|---|---|---| | 0 | - | - | - | 1300.00 | | 1 | 335.32 | 21.13 | 314.19 | 985.81 | | 2 | 335.32 | 16.02 | 319.30 | 666.51 | | 3 | 335.32 | 10.83 | 324.49 | 342.02 | | 4 | 347.58 | 5.56 | 342.02 | 0.00 | **Answer:** - Quarterly payment for first 3 payments: 335.32 - Final payment: 347.58