Practice Maths

Topic Review — Loans, Investments and Annuities — Solutions

← Loans, Investments and Annuities

This review covers compound interest, reducing balance loans, annuities, and comparing financial options. Click each answer to reveal the worked solution.

Review Questions

  1. $8 000 is invested at 5% p.a. compounded annually. What is the value after 4 years?
    A = 8 000 × 1.054 = 8 000 × 1.2155 = $9 724
  2. $15 000 is invested at 4.8% p.a. compounded monthly. Find the value after 3 years.
    r = 4.8% / 12 = 0.4% per month = 0.004; n = 36 months
    A = 15 000 × (1.004)36 = 15 000 × 1.15397 = $17 309.55
  3. Write the recurrence relation for a $25 000 reducing balance loan at a monthly interest rate of 0.6% and monthly repayment of $800. Find the balance after month 1.
    Vn+1 = Vn(1.006) − 800,   V0 = 25 000
    V1 = 25 000 × 1.006 − 800 = 25 150 − 800 = $24 350
  4. In month 1 of a $50 000 loan at 6% p.a. (0.5% monthly), a $400 repayment is made. How much of this payment goes to interest and how much reduces the principal?
    Interest = 50 000 × 0.005 = $250
    Principal reduction = 400 − 250 = $150

    Note: since the repayment ($400) barely exceeds the interest ($250), this loan would take a very long time to repay.
  5. Complete the repayment table for a $10 000 loan at 0.5% monthly, $350 repayments, for 3 months.
    Month Opening Interest Principal Paid Closing
    1$10 000.00$50.00$300.00$9 700.00
    2$9 700.00$48.50$301.50$9 398.50
    3$9 398.50$46.99$303.01$9 095.49
  6. A loan balance is $3 500. Monthly interest rate is 0.8%, and the monthly repayment is $300. How many more months until the loan is paid off?
    Vn+1 = Vn(1.008) − 300:
    V1 = 3 500 × 1.008 − 300 = 3 528 − 300 = 3 228
    V2 = 3 228 × 1.008 − 300 = 3 253.82 − 300 = 2 953.82
    V3 = 2 953.82 × 1.008 − 300 = 2 977.45 − 300 = 2 677.45
    V4 = 2 677.45 × 1.008 − 300 = 2 698.87 − 300 = 2 398.87
    V5 = 2 398.87 × 1.008 − 300 = 2 418.06 − 300 = 2 118.06
    V6 = 2 118.06 × 1.008 − 300 = 2 135.01 − 300 = 1 835.01
    V7 = 1 835.01 × 1.008 − 300 = 1 849.69 − 300 = 1 549.69
    V8 = 1 549.69 × 1.008 − 300 = 1 562.09 − 300 = 1 262.09
    V9 = 1 262.09 × 1.008 − 300 = 1 272.19 − 300 = 972.19
    V10 = 972.19 × 1.008 − 300 = 979.97 − 300 = 679.97
    V11 = 679.97 × 1.008 − 300 = 685.41 − 300 = 385.41
    V12 = 385.41 × 1.008 − 300 = 388.49 − 300 = 88.49
    V13: 88.49 × 1.008 = 89.20 → final payment ≈ $89.20

    The loan is paid off in 13 more months (with a smaller final payment in month 13).
  7. Write the recurrence relation for a future value annuity where $250 is deposited monthly at 0.4% monthly interest. Find the balance after 3 months.
    Vn+1 = Vn(1.004) + 250,   V0 = 0
    V1 = 0 + 250 = 250.00
    V2 = 250 × 1.004 + 250 = 251 + 250 = 501.00
    V3 = 501 × 1.004 + 250 = 503.00 + 250 = $753.00
  8. Use the future value formula to find how much $400 monthly deposits at 5.4% p.a. (0.45% monthly) accumulate to after 18 months.
    FV = 400 × (1.004518 − 1) / 0.0045
    1.004518 ≈ 1.0836
    FV = 400 × 0.0836 / 0.0045 = 400 × 18.578 = $7 431.20
  9. A retiree has $120 000 and withdraws $900 per month. The monthly interest rate is 0.4%. Write the recurrence relation and find the balance after 2 months.
    Vn+1 = Vn(1.004) − 900,   V0 = 120 000
    V1 = 120 000 × 1.004 − 900 = 120 480 − 900 = 119 580
    V2 = 119 580 × 1.004 − 900 = 120 058.32 − 900 = $119 158.32
  10. A $60 000 annuity pays out $800 per month at 0.35% monthly interest. Approximately how many months will the fund last? (Use the PV formula.)
    60 000 = 800 × (1 − 1.0035−n) / 0.0035
    60 000 × 0.0035 / 800 = 1 − 1.0035−n
    0.2625 = 1 − 1.0035−n
    1.0035−n = 0.7375
    n = ln(0.7375) / (−ln(1.0035)) ≈ 0.3046 / 0.003494 ≈ 87 months (7 years 3 months)
  11. A $6 000 flat-rate loan charges 9% p.a. over 2 years. Find the total interest and monthly repayment.
    Total interest = 6 000 × 0.09 × 2 = $1 080
    Monthly repayment = (6 000 + 1 080) / 24 = 7 080 / 24 = $295
  12. A $6 000 reducing balance loan at 9% p.a. (0.75% monthly) has a monthly repayment of $272.48 over 24 months. Compare the total interest with the flat-rate loan in Question 11.
    Total paid = 272.48 × 24 = $6 539.52
    Total interest = 6 539.52 − 6 000 = $539.52

    The reducing balance loan saves $1 080 − $539.52 = $540.48 in interest — almost exactly half — for the same stated rate over the same term.
  13. How much must be deposited monthly into an account earning 4.8% p.a. (0.4% monthly) to accumulate $20 000 in 36 months?
    FV = D × (1.00436 − 1) / 0.004
    1.00436 ≈ 1.15397
    20 000 = D × 0.15397 / 0.004 = D × 38.493
    D = 20 000 / 38.493 = $519.57 per month
  14. A borrower can afford $350 per month. They take out a $15 000 loan at 6% p.a. (0.5% monthly). Will they pay off the loan in 4 years (48 months)? Use the PV formula to check.
    PV for $350/month for 48 months at 0.5%:
    PV = 350 × (1 − 1.005−48) / 0.005
    1.00548 ≈ 1.2705, so 1.005−48 ≈ 0.7871
    PV = 350 × (1 − 0.7871) / 0.005 = 350 × 42.58 = $14 903

    Since $14 903 < $15 000, a $350 repayment is not quite enough to pay off a $15 000 loan in 48 months (about $97 short). The required monthly repayment would be approximately $352.
  15. A home loan of $380 000 is offered at either 5.0% p.a. over 30 years or 5.4% p.a. over 25 years. Monthly repayments are $2 039.81 and $2 295.62 respectively. Compare the total interest paid and explain which you would recommend.
    Option A (5.0%, 30 years):
    Total paid = 2 039.81 × 360 = $734 331.60  →  Interest = $354 331.60

    Option B (5.4%, 25 years):
    Total paid = 2 295.62 × 300 = $688 686  →  Interest = $308 686

    Option B saves $45 645.60 in total interest despite a higher stated rate and higher monthly repayments. Recommendation: Option B — the shorter term dramatically reduces total interest paid. If the borrower can afford the higher monthly payment, Option B is clearly better financially. The extra $255.81/month for 25 years (≈ $76 743) is far less than the $45 645 interest saving, making the shorter term the better choice.