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Solutions — Reducing Balance Loans

  1. Fluency $15 000 loan at 4% p.a. (monthly rate 0.333%), $400 repayments. Balance after 2 months.
    r = 0.00333, V0 = 15 000, P = 400

    V1 = 15 000 × 1.00333 − 400 = 15 049.95 − 400 = 14 649.95
    V2 = 14 649.95 × 1.00333 − 400 = 14 698.73 − 400 = $14 298.73
  2. Fluency Balance $8 200, monthly rate 0.5%, payment $350.
    Vnew = 8 200 × 1.005 − 350 = 8 241 − 350 = $7 891
  3. Fluency V0 = 10 000, monthly rate 0.4%, P = 300. Write recurrence relation and find V1.
    Recurrence relation: Vn+1 = Vn(1.004) − 300

    V1 = 10 000 × 1.004 − 300 = 10 040 − 300 = $9 740
  4. Fluency $25 000 loan at 6% p.a. monthly, $600 repayment in month 1. Interest vs principal split.
    Monthly rate = 6% ÷ 12 = 0.5% = 0.005

    Interest component: I = 25 000 × 0.005 = $125
    Principal component: 600 − 125 = $475
  5. Understanding $30 000 car loan at 7.2% p.a. (0.6% monthly), $700 repayments. Repayment table for 4 months.
    r = 0.006, P = 700

    Month Opening Balance Interest Charged Principal Paid Closing Balance
    1 $30 000.00 $180.00 $520.00 $29 480.00
    2 $29 480.00 $176.88 $523.12 $28 956.88
    3 $28 956.88 $173.74 $526.26 $28 430.62
    4 $28 430.62 $170.58 $529.42 $27 901.20

    Notice that each month the interest decreases and the principal paid increases, even though the repayment stays constant.
  6. Understanding Loan A (5% p.a.) vs Loan B (8% p.a.), same $10 000 principal and 5-year term. Which costs more in total interest?
    Loan B costs more in total interest.

    Why: A higher interest rate means more interest is charged on the outstanding balance each month. This leaves less of each repayment to reduce the principal, so the balance stays higher for longer — which in turn generates even more interest. This compounding effect means the higher-rate loan pays significantly more total interest over its life, even though the principal and term are identical.

    For the same repayment amount: Loan B repayments would need to be higher than Loan A’s to clear the debt in 5 years, increasing total cost further.
  7. Understanding Balance $5 000, monthly rate 0.75%, $400 repayments. How many more months until paid off?
    r = 0.0075, P = 400. Iterate Vn+1 = Vn(1.0075) − 400:

    V0 = 5 000.00
    V1 = 5 000 × 1.0075 − 400 = 5 037.50 − 400 = 4 637.50
    V2 = 4 637.50 × 1.0075 − 400 = 4 672.28 − 400 = 4 272.28
    V3 = 4 272.28 × 1.0075 − 400 = 4 304.32 − 400 = 3 904.32
    V4 = 3 904.32 × 1.0075 − 400 = 3 533.60
    V5 = 3 533.60 × 1.0075 − 400 = 3 160.10
    V6 = 3 160.10 × 1.0075 − 400 = 2 783.80
    V7 = 2 783.80 × 1.0075 − 400 = 2 404.68
    V8 = 2 404.68 × 1.0075 − 400 = 2 022.72
    V9 = 2 022.72 × 1.0075 − 400 = 1 637.89
    V10 = 1 637.89 × 1.0075 − 400 = 1 250.17
    V11 = 1 250.17 × 1.0075 − 400 = 859.55
    V12 = 859.55 × 1.0075 − 400 = 466.00
    V13 = 466.00 × 1.0075 − 400 = 69.50
    V14: 69.50 × 1.0075 = 70.02 → final payment of $70.02

    The loan is paid off in 14 more months (with a smaller final payment of approximately $70 in month 14).
  8. Understanding $400 000 home loan at 4.8% p.a. (0.4% monthly), $2 100 repayments. Balance after 3 months and total interest.
    r = 0.004, P = 2 100, V0 = 400 000

    Month 1: Interest = 400 000 × 0.004 = $1 600    End: 400 000 + 1 600 − 2 100 = $399 500
    Month 2: Interest = 399 500 × 0.004 = $1 598    End: 399 500 + 1 598 − 2 100 = $398 998
    Month 3: Interest = 398 998 × 0.004 = $1 595.99    End: 398 998 + 1 595.99 − 2 100 = $398 493.99

    Balance after 3 months: $398 493.99
    Total interest paid: 1 600 + 1 598 + 1 595.99 = $4 793.99

    Note: After 3 months of $2 100 payments ($6 300 total), only $1 506.01 has gone to reducing the principal. The vast majority of early repayments is interest.
  9. Problem Solving $12 000 loan at 9% p.a. (0.75% monthly), $350 repayments. Table for months 1–5, then $500 repayments from month 6.
    r = 0.0075, P = 350

    (a) Repayment table, months 1–5:
    Month Opening Balance Interest Principal Paid Closing Balance
    1 $12 000.00 $90.00 $260.00 $11 740.00
    2 $11 740.00 $88.05 $261.95 $11 478.05
    3 $11 478.05 $86.09 $263.91 $11 214.14
    4 $11 214.14 $84.11 $265.89 $10 948.25
    5 $10 948.25 $82.11 $267.89 $10 680.36

    (b) Month 6 with P = $500:
    Opening balance = $10 680.36
    Interest = 10 680.36 × 0.0075 = $80.10
    Principal paid = 500 − 80.10 = $419.90
    Closing balance = $10 260.46

    The increased repayment cuts $419.90 from the principal in one month, compared to only ~$267 in the previous months.
  10. Problem Solving $20 000 loan at 6% p.a. (0.5% monthly). Find monthly repayment for V48 ≈ 0, then total interest.
    (a) Finding the repayment amount:
    Try P = 470 (as suggested). Using the annuity formula to check:
    P = PV × r / (1 − (1 + r)−n)
    = 20 000 × 0.005 / (1 − 1.005−48)
    = 100 / (1 − 0.7871)
    = 100 / 0.2129
    ≈ $469.70 per month

    Rounding up to $470 ensures the loan is fully cleared by month 48 (with a slightly smaller final payment).

    Iterative check with P = 470: After 48 months, V48 ≈ $14 (near zero) — confirming P ≈ $470 is correct.

    (b) Total interest paid:
    Total paid = $470 × 48 = $22 560
    Total interest = $22 560 − $20 000 = $2 560

    Using exact P = $469.70: Total paid = $22 545.60, interest = $2 545.60.

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