1. Problem 4: Calculate the size of quarterly payments to settle a $15,000 loan at 5.25% annual interest compounded monthly over 10 years. 2. Formula: Use the annuity payment formula for compound interest payments: $$PMT = \frac{PV \times r}{1 - (1 + r)^{-n}}$$ where $PV$ is the present value (loan amount), $r$ is the interest rate per payment period, and $n$ is the total number of payments. 3. Important rules: - Convert annual interest rate to the rate per payment period. - Number of payments $n$ equals total years times payments per year. 4. Given: - $PV = 15000$ - Annual interest rate $= 5.25\% = 0.0525$ - Compounded monthly, but payments every 3 months (quarterly), so $r = \frac{0.0525}{12} \times 3 = 0.013125$ - Number of payments $n = 10 \times 4 = 40$ 5. Calculate payment: $$PMT = \frac{15000 \times 0.013125}{1 - (1 + 0.013125)^{-40}}$$ 6. Calculate denominator: $$1 - (1 + 0.013125)^{-40} = 1 - (1.013125)^{-40}$$ 7. Calculate $(1.013125)^{-40}$: $$= \frac{1}{(1.013125)^{40}} \approx \frac{1}{1.665} = 0.6006$$ 8. So denominator: $$1 - 0.6006 = 0.3994$$ 9. Calculate numerator: $$15000 \times 0.013125 = 196.875$$ 10. Final payment: $$PMT = \frac{196.875}{0.3994} \approx 492.8$$ --- 11. Problem 5a: Calculate monthly finance payments for Toyota Prius V costing 34975 with residual value 17693.75, 3.90% annual interest compounded monthly, 4-year term, including 13% HST. 12. Adjust price for HST: $$PV = 34975 \times (1 + 0.13) = 39521.75$$ 13. Given: - $PV = 39521.75$ - $FV = 17693.75$ - Annual interest rate $= 3.90\% = 0.039$ - Monthly interest rate $r = \frac{0.039}{12} = 0.00325$ - Number of payments $n = 4 \times 12 = 48$ 14. Use annuity payment formula with future value: $$PMT = \frac{PV \times r - FV \times r / (1 + r)^n}{1 - (1 + r)^{-n}}$$ 15. Calculate $(1 + r)^n$: $$1.00325^{48} \approx 1.1717$$ 16. Calculate numerator: $$39521.75 \times 0.00325 - \frac{17693.75 \times 0.00325}{1.1717} = 128.423 - 49.07 = 79.353$$ 17. Calculate denominator: $$1 - (1.00325)^{-48} = 1 - \frac{1}{1.1717} = 1 - 0.8539 = 0.1461$$ 18. Final payment: $$PMT = \frac{79.353}{0.1461} \approx 542.9$$ --- 19. Problem 5b: Calculate monthly lease payments for the same car, assuming lease payments cover depreciation plus interest. 20. Lease payment formula: $$PMT = \frac{(PV - FV) + (PV + FV) \times r \times n / 2}{n}$$ 21. Given: - $PV = 39521.75$ - $FV = 17693.75$ - $r = 0.00325$ - $n = 48$ 22. Calculate depreciation: $$PV - FV = 39521.75 - 17693.75 = 21828$$ 23. Calculate interest portion: $$(PV + FV) \times r \times \frac{n}{2} = (39521.75 + 17693.75) \times 0.00325 \times 24 = 57215.5 \times 0.00325 \times 24 = 4461.9$$ 24. Total amount to be paid: $$21828 + 4461.9 = 26289.9$$ 25. Monthly lease payment: $$PMT = \frac{26289.9}{48} \approx 547.7$$ --- Filled MODE sheets: Problem 4: MODE: 1 (Payment mode) N: 40 I/Y: 5.25 PV: 15000 PMT: ? FV: 0 P/Y: 4 C/Y: 12 Problem 5a: MODE: 1 N: 48 I/Y: 3.90 PV: 39521.75 PMT: ? FV: 17693.75 P/Y: 12 C/Y: 12 Problem 5b: MODE: 1 N: 48 I/Y: 3.90 PV: 39521.75 PMT: ? FV: 17693.75 P/Y: 12 C/Y: 12