1. **Problem Statement:** Calculate the level payment amount and complete the loan amortization schedule for a loan of 90000 with payments every 4 months over 20 months at a quarterly interest rate of 3%. The payments are made at the end of each period. 2. **Given:** - Principal $P = 90000$ - Number of months = 20 - Payment interval = 4 months - Number of payments $n = \frac{20}{4} = 5$ - Quarterly interest rate $i = 3\% = 0.03$ 3. **Formula for level payment $R$ for an amortizing loan:** $$ R = P \times \frac{i}{1 - (1+i)^{-n}} $$ This formula calculates the fixed payment amount that fully amortizes the loan over $n$ periods. 4. **Calculate the payment $R$:** $$ R = 90000 \times \frac{0.03}{1 - (1+0.03)^{-5}} = 90000 \times \frac{0.03}{1 - (1.03)^{-5}} $$ Calculate $(1.03)^{-5}$: $$ (1.03)^{-5} = \frac{1}{(1.03)^5} = \frac{1}{1.159274} \approx 0.862609 $$ So, $$ R = 90000 \times \frac{0.03}{1 - 0.862609} = 90000 \times \frac{0.03}{0.137391} = 90000 \times 0.21834 = 19650.6 $$ 5. **Amortization schedule calculations:** - Period 0: Outstanding balance = 90000 For each period $k$ from 1 to 5: - Interest paid = Outstanding balance at previous period $\times i$ - Principal repaid = Payment $R$ - Interest paid - New outstanding balance = Previous outstanding balance - Principal repaid 6. **Period 1:** - Interest paid = $90000 \times 0.03 = 2700$ - Principal repaid = $19650.6 - 2700 = 16950.6$ - Outstanding balance = $90000 - 16950.6 = 73049.4$ 7. **Period 2:** - Interest paid = $73049.4 \times 0.03 = 2191.48$ - Principal repaid = $19650.6 - 2191.48 = 17459.12$ - Outstanding balance = $73049.4 - 17459.12 = 55590.28$ 8. **Period 3:** - Interest paid = $55590.28 \times 0.03 = 1667.71$ - Principal repaid = $19650.6 - 1667.71 = 17982.89$ - Outstanding balance = $55590.28 - 17982.89 = 37607.39$ 9. **Period 4:** - Interest paid = $37607.39 \times 0.03 = 1128.22$ - Principal repaid = $19650.6 - 1128.22 = 18522.38$ - Outstanding balance = $37607.39 - 18522.38 = 19085.01$ 10. **Period 5:** - Interest paid = $19085.01 \times 0.03 = 572.55$ - Principal repaid = $19650.6 - 572.55 = 19078.05$ - Outstanding balance = $19085.01 - 19078.05 = 6.96 \approx 0.00$ (rounded) **Final amortization schedule:** | Period | Payment Amount | Interest Paid | Principal Repaid | Outstanding Balance | |--------|----------------|---------------|------------------|---------------------| | 0 | | | | 90000.00 | | 1 | 19650.60 | 2700.00 | 16950.60 | 73049.40 | | 2 | 19650.60 | 2191.48 | 17459.12 | 55590.28 | | 3 | 19650.60 | 1667.71 | 17982.89 | 37607.39 | | 4 | 19650.60 | 1128.22 | 18522.38 | 19085.01 | | 5 | 19650.60 | 572.55 | 19078.05 | 0.00 | This completes the loan amortization schedule with level payments.