Loan Repayment and the Cost of a Longer Term
A family borrows $18,000 for a car. The bank offers two options. Option 1: 3-year term at 7.2% per annum, reducing-balance interest, monthly repayments. Option 2: 5-year term at the same 7.2% per annum. The family's monthly budget is tight, so they are tempted by Option 2 because the monthly repayment is lower. A relative warns that "a longer loan always costs far more." The family wants to know exactly how much more Option 2 costs in total and whether the lower monthly figure is worth it. They can afford either monthly repayment but prefer to keep payments low.
1. Using the reducing-balance repayment formula, calculate the monthly repayment for each option and the total interest paid.
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Monthly rate i = 0.072 ÷ 12 = 0.006. Repayment formula: R = P·i ÷ (1 − (1+i)^(−n)), with P = 18,000.
Option 1 (n = 36): (1.006)^36: ln(1.006) ≈ 0.005982; ×36 = 0.21535; e^0.21535 ≈ 1.2403. (1.006)^(−36) = 1 ÷ 1.2403 ≈ 0.8063. 1 − 0.8063 = 0.1937. R = 18,000 × 0.006 ÷ 0.1937 = 108 ÷ 0.1937 ≈ $557.6 per month. Total paid = 557.6 × 36 ≈ $20,074. Total interest ≈ $2,074.
Option 2 (n = 60): (1.006)^60: 0.005982 × 60 = 0.35892; e^0.35892 ≈ 1.4319. (1.006)^(−60) = 1 ÷ 1.4319 ≈ 0.6984. 1 − 0.6984 = 0.3016. R = 108 ÷ 0.3016 ≈ $358.1 per month. Total paid = 358.1 × 60 ≈ $21,486. Total interest ≈ $3,486.
Comparison: Option 2 lowers the monthly payment by about $200 (from ~$558 to ~$358) but adds about $1,412 in total interest ($3,486 − $2,074).
2. Evaluate the relative's claim that a longer loan "always costs far more," and advise the family.
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The claim is directionally right but overstated as stated.
- It is true that Option 2 costs more in total interest: about $1,412 extra, roughly a 68% increase in interest ($1,412 ÷ $2,074 ≈ 0.68).
- But "far more" needs context. $1,412 over five years is about $24 per month of extra cost spread across the term. Relative to an $18,000 loan, the extra interest is about 7.8% of the principal - noticeable but not catastrophic.
- The claim "always" is too strong: whether a longer term is worth it depends on the borrower's cash-flow needs and on the interest rate. At low rates the extra interest is smaller; at high rates it grows.
Advice: The trade-off is clear - Option 2 buys about $200/month of breathing room at a total cost of about $1,412. Since the family can afford either payment, the financially efficient choice is Option 1: they save $1,412 and finish three years earlier, freeing the money for other uses. Option 2 only makes sense if the $200/month is genuinely needed for essentials or to avoid higher-cost debt elsewhere. Recommend Option 1, with Option 2 as a fallback if cash flow tightens.
3. The family could instead make Option 2's repayment but over a shorter effective term. Explain how an extra payment changes the outcome.
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If the family takes Option 2 (5-year term) but pays extra each month, the loan clears sooner and total interest falls.
Illustration: Suppose they take Option 2 and pay $450/month instead of $358.1.
Using the balance formula B = P(1+i)^n − R·[(1+i)^n − 1] ÷ i, we find when B reaches 0. Equivalently, solve for n: n = −ln(1 − P·i ÷ R) ÷ ln(1+i). P·i = 108. P·i ÷ R = 108 ÷ 450 = 0.24. 1 − 0.24 = 0.76. ln(0.76) = −0.2744. ln(1.006) = 0.005982. n = 0.2744 ÷ 0.005982 ≈ 45.9 months.
So paying $450/month clears the loan in about 46 months instead of 60. Total paid ≈ 450 × 45.9 ≈ $20,655, giving interest ≈ $2,655 - about $831 less than the $3,486 under the standard Option 2 schedule, though still more than Option 1's $2,074.
Key point: the term is not fixed by the contract; the repayment drives the payoff date. A borrower who can pay more than the minimum shortens the effective term and cuts interest, so the "long loan is expensive" problem is partly within the borrower's control. The best outcome remains Option 1 (lowest total interest), but a disciplined extra-payment strategy on Option 2 narrows the gap substantially.