Question: Zandile borrowed money from a bank and she is repaying the loan by making monthly payments of $1091$ at the end of every month, each payment includes interest and repayment of the principle. The loan will be paid off in $8$ years and interest is calculated at a rate of $12.408\%$ per year, compounded monthly. What is the principle repayment component of the $6$th payment of the $5$th year?

1. **State the problem:** We need to find the principal repayment component of the 6th payment in the 5th year of a loan with monthly payments of $1091$, an annual interest rate of $12.408\%$ compounded monthly, and a loan term of 8 years. 2. **Identify given data:** - Monthly payment, $PMT = 1091$ - Annual interest rate, $r = 12.408\% = 0.12408$ - Monthly interest rate, $i = \frac{0.12408}{12} = 0.01034$ - Number of years, $n = 8$ - Total number of payments, $N = 8 \times 12 = 96$ - Payment number to analyze: $k = (5-1) \times 12 + 6 = 54$ (6th payment of 5th year) 3. **Formula for loan amortization:** The loan amount $P$ can be found using the annuity formula: $$ P = PMT \times \frac{1 - (1+i)^{-N}}{i} $$ 4. **Calculate the loan amount $P$:** $$ P = 1091 \times \frac{1 - (1+0.01034)^{-96}}{0.01034} $$ Calculate $(1+0.01034)^{-96}$: $$ (1.01034)^{-96} = \frac{1}{(1.01034)^{96}} \approx \frac{1}{2.718} = 0.368 $$ So, $$ P = 1091 \times \frac{1 - 0.368}{0.01034} = 1091 \times \frac{0.632}{0.01034} = 1091 \times 61.11 = 66630.01 $$ 5. **Calculate the outstanding balance after $k-1=53$ payments:** The outstanding balance after $m$ payments is: $$ B_m = P(1+i)^m - PMT \times \frac{(1+i)^m - 1}{i} $$ For $m=53$: $$ B_{53} = 66630.01 \times (1.01034)^{53} - 1091 \times \frac{(1.01034)^{53} - 1}{0.01034} $$ Calculate $(1.01034)^{53} \approx 1.718$ Calculate numerator for second term: $$ 1.718 - 1 = 0.718 $$ Calculate second term: $$ 1091 \times \frac{0.718}{0.01034} = 1091 \times 69.42 = 75764.22 $$ Calculate first term: $$ 66630.01 \times 1.718 = 114441.5 $$ So, $$ B_{53} = 114441.5 - 75764.22 = 38677.28 $$ 6. **Calculate interest component of the 54th payment:** $$ I_{54} = B_{53} \times i = 38677.28 \times 0.01034 = 399.8 $$ 7. **Calculate principal component of the 54th payment:** $$ Principal = PMT - I_{54} = 1091 - 399.8 = 691.2 $$ **Final answer:** The principal repayment component of the 6th payment of the 5th year is approximately **$691.20$**.