Question 5.2: Converting the APR to a Discount Rate Your firm is purchasin...

PLAN

The cost of leasing the system is a 48-month annuity of $4000 per month:

We can compute the present value of the lease cash flows using the annuity formula, but first we need to compute the discount rate that corresponds to a period length of one month. To do so, we convert the borrowing cost of 6% APR with monthly compounding to a monthly discount rate using Eq. 5.2

Interest Rate per Compounding Period =\frac{APR}{m} (m = number of compounding periods per year).              (5.2)

Once we have a monthly rate, we can use the present value of annuity formula Eq. 4.5

PV (Annuity of C for N Periods with Interest Rate r) = C\times \frac{1}{r} \left(1-\frac{1}{(1+r)^N}\right )                        (4.5)

to compute the present value of the monthly payments and compare it to the cost of buying the system.

Execute
As Eq. 5.2 shows, the 6% APR with monthly compounding really means 6%/12 = 0.5% every month. The 12 comes from the fact that there are 12 monthly compounding periods per year. Now that we have the true rate corresponding to the stated APR, we can use that discount rate in the annuity formula Eq. 4.5 to compute the present value of the monthly payments:

Using a financial calculator or spreadsheet:

Evaluate
Thus, paying $4000 per month for 48 months is equivalent to paying a present value of $170,321.27 today. This cost is $170,321.27 – $150,000 = $20,321.27 higher than the cost of purchasing the system, so it is better to pay $150,000 for the system rather than lease it. One way to interpret this result is as follows: At a 6% APR with monthly compounding, by promising to repay $4000 per month your firm can borrow $170,321 today.
With this loan it could purchase the phone system and have an additional $20,321 to use for other purposes.