Your firm needs to invest in a new delivery truck. The life expectancy of the delivery truck is five years. You can purchase a new delivery truck for an upfront cost of $300,000, or you can lease a truck from the manufacturer for five years for a monthly lease payment of $6000 (paid at the end of each month). Your firm can borrow at 8.00% APR with quarterly compounding.

The monthly discount rate that you should use to evaluate the truck lease is closest to ________.

Question

Your firm needs to invest in a new delivery truck. The life expectancy of the delivery truck is five years. You can purchase a new delivery truck for an upfront cost of $300,000, or you can lease a truck from the manufacturer for five years for a monthly lease payment of $6000 (paid at the end of each month). Your firm can borrow at 8.00% APR with quarterly compounding.

The monthly discount rate that you should use to evaluate the truck lease is closest to ________.

BlackTom AI · Accepted Answer

First, identify what the question is asking: we need the monthly discount rate that is consistent with an 8.00% APR with quarterly compounding, to evaluate a monthly lease payment.
Next, determine the effective per-quarter rate implied by the given APR with quarterly compounding. An APR of 8% with quarterly compounding means the rate per quarter is 0.08 / 4 = 0.02, i.e., a 2.0% interest charge every three months.
Then translate this quarterly effect into a monthly rate. If the monthly rate is i, then after three months the growth factor is (1 + i)^3. This should equal the quarterly growth factor, which is 1 + 0.02 = 1.02, because three months at the monthly rate should compound to the same amount as one quarter at the quarterly rate.
Set up the equation: (1 + i)^3 = 1.02. Solve for i by taking the cube root: 1 + i = (1.02)^(1/3).
Compute the cube root: (1.02)^(1/3) ≈ 1.006624..., so i ≈ 0.006624, which is about 0.6624% per month.
Rounding considerations: depending on calculator precision, you may see 0.6623% or 0.6624% as the monthly rate. The key result is that the monthly discount rate is approximately 0.662% per month, reflecting the equivalence of 8% APR with quarterly compounding to a monthly rate via (1 + i)^3 = 1.02.
Thus, the monthly discount rate you should use to evaluate the lease payments is about 0.662% per month, aligning with the given compounding structure.

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