Two Truck Loans: The Cheaper Rate That Isn't
A caterer wants to buy a second food truck priced at USD 48 000. Dealer A offers 6.0% per year compounded monthly, repaid over 60 equal monthly instalments with no other charges. Dealer B advertises a lower rate, 5.4% per year compounded monthly over the same 60 months, but adds a USD 1 200 documentation fee to the amount borrowed and requires a balloon payment of USD 8 000 at the end of month 60. The caterer's accountant argues that B must be cheaper because the quoted rate is lower, and notes that B also has the smaller monthly payment. Both dealers quote nominal annual rates compounded monthly, and both require the first payment one month after the loan is taken out. The caterer can afford either monthly payment but has no separate savings, so the balloon would have to be funded from the truck's revenue in month 60. She wants a recommendation she can defend to her bank.
1. Calculate the monthly repayment and the total interest paid under Dealer A's offer.
Read the worked answer
The monthly rate is the nominal rate divided by 12: i = 0.06/12 = 0.005, with n = 60 payments.
Using the amortisation formula (R = \dfrac{P,i}{1-(1+i)^{-n}}):
((1.005)^{60} = 1.34885), so ((1.005)^{-60} = 0.74137).
(R = \dfrac{48000 \times 0.005}{1-0.74137} = \dfrac{240}{0.25863} \approx 927.99).
Monthly repayment ≈ USD 928.00.
Total paid = (60 \times 927.99 = 55,679.40), so total interest ≈ USD 7 679 (about 16% of the amount borrowed).
The key judgement here is that the nominal rate alone does not describe the cost; the total cash paid does, and it is this figure that Dealer B must be compared against.
2. For Dealer B, explain why the amount borrowed is USD 49 200, then find the monthly repayment given the USD 8 000 balloon.
Read the worked answer
The fee is added to the loan, so the caterer borrows 48 000 + 1 200 = USD 49 200, while only 48 000 of value goes to the truck. The monthly rate is 0.054/12 = 0.0045.
The 60 payments do not repay the whole loan, because 8 000 is still owed at month 60. The payments must therefore amortise the loan minus the present value of the balloon:
(PV(\text{balloon}) = 8000(1.0045)^{-60} = 8000 \times 0.76386 = 6,110.88).
Amount to amortise = 49 200 − 6 110.88 = 43 089.12.
(R = \dfrac{43,089.12 \times 0.0045}{1-0.76386} = \dfrac{193.90}{0.23614} \approx 821.13).
Monthly repayment ≈ USD 821.13.
Total cash out = (60 \times 821.13 + 8000 = 57,267.80). The fee is not added again: it was financed, so it is already inside the payments. Note that B's lower monthly payment is bought partly by deferring 8 000 of principal to month 60.
3. Compare the two offers using effective annual rates and recommend one, stating clearly what the recommendation depends on.
Read the worked answer
Dealer A: (\text{EAR} = (1.005)^{12}-1 = 6.17%).
Dealer B cannot be judged by its quoted 5.4%, because the caterer receives only 48 000 while repaying 821.13 for 60 months plus 8 000 at month 60. Find the monthly rate r satisfying
(48,000 = 821.13,a_{\overline{60}|r} + 8000(1+r)^{-60}).
Numerically r ≈ 0.00526, so (\text{EAR} = (1.00526)^{12}-1 \approx 6.49%).
So B, the 'cheaper' loan, costs about 0.3 percentage points more per year, roughly USD 1 590 more in nominal payments. Recommend A, since the caterer can afford the higher monthly payment and has no savings to meet the balloon.
The recommendation depends on three things: (i) that all payments are made on time; (ii) that the caterer can genuinely carry USD 928 per month, because B's advantage is only cash-flow timing; (iii) that the USD 8 000 is not sitting in an account earning more than about 6.5% after tax, in which case retaining cash and taking B could be rational. If any of these fail, the ranking changes.