Todd can afford to pay $440 per month for the next 5 years in order to purchase a new car. The interest rate is 7.8 percent compounded monthly. What is the most he can afford to pay for a new car today?

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BlackTom AI · Accepted Answer

We start by restating the problem to ground our analysis: Todd can pay 440 dollars per month for 60 months (5 years) and the loan must be financed at 7.8% annual interest compounded monthly. We are asked for the maximum amount he can borrow today, i.e., the present value of the 60 monthly payments.

Key formula: present value of an ordinary annuity (payments at the end of each period) is PV = PMT × [1 − (1 + i)^−n] / i, where PMT is the monthly payment, i is the monthly interest rate, and n is the number of payments.

Step 1 — compute the monthly interest rate: annual rate is 7.8%, so i = 0.078 / 12 = 0.0065 (0.65% per month).

Step 2 — determine the number of payments: n = 5 years × 12 months/year = 60 months.

Step 3 — calculate the discount factor: (1 + i)^−n = (1.0065)^−60. This exponentiation is best approximated with logarithms or a calculator. Using a reasonable approximation, (1.0065)^60 is around 1.47, so (1.0065)^−60 is about 0.68. More precise estimates typically yield a value in the neighborhood of 0.678–0.679.

Step 4 — compute [1 − (1 + i)^−n]: with (1 + i)^−n ≈ 0.678, we get 1 − 0.678 ≈ 0.322.

Step 5 — divide by i: 0.322 / 0.0065 ≈ 49.5 (give or take a small amount depending on the exact (1.0065)^−60 value).

Step 6 — multiply by the monthly payment: PV ≈ 440 × 49.5 ≈ 21,780. This is the rough magnitude of the maximum amount Tod could borrow today.

Now compare this to the provided answer options by converting each back to an implied present value factor and seeing which is closest to our calculation:

- Option A: 21,802.89 implies a PV factor of 21,802.89 / 440 ≈ 49.56. This is very close to the 49.5 range we computed, suggesting this option is consistent with the standard present-value calculation.
- Option B: 32,161.83 implies a PV factor of ≈ 73.14, which would require a much larger sum of discounted payments and is not consistent with a 60-month, 0.65% monthly rate.
- Option C: 21,504.61 implies a PV factor of ≈ 48.92, which is somewhat lower than our estimate but could occur if the exact (1.0065)^−60 value were a bit larger than 0.678.
- Option D: 20,712.74 implies a PV factor of ≈ 47.06, which is noticeably lower than expected for these inputs and would require a notably smaller discount factor.

Between these, the factor around 49.5 aligns with the standard PV calculation for 60 payments at i = 0.0065. The options that are very far from this region (especially the 32,161.83 one) can be dismissed as inconsistent with the given payment stream and rate.

It is important to note that small differences in the (1.0065)^−60 value due to rounding can shift the exact present value by a few dollars, which is why choices in the vicinity of 21,7xx are plausible, while much larger or much smaller numbers are not.

In summary, after computing the present value of the 60 monthly payments at the given monthly rate and comparing the implied PV factors, the option that best matches the standard present-value calculation for these terms is the one whose PV factor is closest to 49.5, with the corresponding amount in the mid-20k range.

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Todd can afford to pay $440 per month for the next 5 years in order to purchase a new car. The interest rate is 7.8 percent compounded monthly. What is the most he can afford to pay for a new car today?

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