4. A loan of R120 454 is repaid by means of fourteen equal monthly payments, starting four years after the granting of the loan. Interest is \( 15 \% \) per annum compounded monthly. Find the value of the payments. A loan is repaid, starting in five years' time, by means of 12 quarterly payments of R7 000 . What is the amount of the loan, if the interest is \( 24 \% \)

We wish to answer two separate problems. ────────────────────────────── Problem (a): A loan of R120 454 is to be repaid by 14 equal monthly payments. The first payment is not made until 4 years after the loan is made (that is, at the end of month 48) and the interest rate is 15% per annum compounded monthly. Step 1. Find the monthly interest rate.   Nominal annual rate = 15% compounded monthly, so    i = 15%/12 = 0.15/12 = 0.0125 per month. Step 2. Write the relation between the loan and the payments. Since the 14 payments (each of amount X) form an annuity‐immediate starting at time 48 months, their value at time 48 is   X × a₍₁₄₎  with  a₍₁₄₎ = [1 – (1 + i)^(–14)]/i. However, these payments are made starting at month 48, so their value at time 0 is the value at time 48, discounted back 48 months:   Loan = X × a₍₁₄₎ × (1 + i)^(–48). Thus the payment is given by   X = Loan × (1 + i)^(48) / a₍₁₄₎. Step 3. Compute a₍₁₄₎. Using i = 0.0125,   a₍₁₄₎ = [1 – (1.0125)^(–14)] / 0.0125. First, compute (1.0125)^(14). One may use logarithms:   ln(1.0125) ≈ 0.0124228 ⇒ ln(1.0125)×14 ≈ 0.17392,   so (1.0125)^(14) ≈ exp(0.17392) ≈ 1.19082. Thus, (1.0125)^(–14) ≈ 1/1.19082 ≈ 0.8396. Then,   a₍₁₄₎ ≈ (1 – 0.8396)/0.0125 ≈ 0.1604/0.0125 ≈ 12.832. Step 4. Compute (1 + i)^(48).   (1.0125)^(48) = exp(48 × ln(1.0125)) ≈ exp(48 × 0.0124228) = exp(0.5969) ≈ 1.817. Step 5. Now compute the payment X.   X = 120 454 × 1.817 / 12.832. First, the numerator:   120 454 × 1.817 ≈ 218 865. Then,   X ≈ 218 865 / 12.832 ≈ 17 050. So, the monthly payment is approximately R17 050. ────────────────────────────── Problem (b): A loan is to be repaid by 12 equal quarterly payments of R7 000. The payments begin in 5 years’ time and the interest rate is 24% per annum compounded quarterly. Step 1. Find the quarterly interest rate. Since 24% per annum compounded quarterly gives   i = 24%/4 = 0.24/4 = 0.06 per quarter. Step 2. Write the relation between the loan and the payments. The 12 quarterly payments starting in 5 years form an annuity‐immediate whose first payment is at time 5 years. (Note that 5 years equals 20 quarters.) The present value at time 20 (in quarters) is   7000 × a₍₁₂₎  with  a₍₁₂₎ = [1 – (1.06)^(–12)]/0.06. To get the value of the loan at time 0, discount this back 20 quarters:   Loan = [7000 × a₍₁₂₎] / (1.06)^(20). Step 3. Compute a₍₁₂₎. First, compute (1.06)^(12). Using logarithms:   ln(1.06) ≈ 0.05827 ⇒ 12×0.05827 ≈ 0.69924,   so (1.06)^(12) ≈ exp(0.69924) ≈ 2.012. Thus, (1.06)^(–12) ≈ 1/2.012 ≈ 0.4970. Then,   a₍₁₂₎ ≈ (1 – 0.4970)/0.06 ≈ 0.5030/0.06 ≈ 8.383. Step 4. Compute (1.06)^(20).   (1.06)^(20) = exp(20 × ln(1.06)) ≈ exp(20 × 0.05827) = exp(1.1654) ≈ 3.207. Step 5. Now compute the loan amount.   Loan ≈ [7000 × 8.383] / 3.207. Calculate the numerator:   7000 × 8.383 ≈ 58 681. Then,   Loan ≈ 58 681 / 3.207 ≈ 18 300. Thus, the amount of the loan is approximately R18 300. ────────────────────────────── Final Answers: (a) The monthly repayment is about R17 050. (b) The loan amount is about R18 300.