A company is considering whether to rent or to buy a Plebney machine. the machine would cost $100,000 to buy, or $40,000 a year to rent. 1he company will need the machine for another three years, after which it will have a salvage value of $20,000. 1he company's pre-tax MARR is 10%, and

Question

A company is considering whether to rent or to buy a Plebney machine. the machine would cost [latex]\$100,000[/latex] to buy, or [latex]\$40,000[/latex] a year to rent. 1he company will need the machine for another three years, after which it will have a salvage value of [latex]\$20,000[/latex]. 1he company's pre-tax MARR is [latex]10\%[/latex], and the lease charges would be paid at the end of every year. 1he company is taxed at [latex]50\%[/latex], the machine can be depreciated at [latex]30\%[/latex] per year, and inflation is expected to be [latex]15\%[/latex] over the next three years.

Accepted Answer

Simplest Analysis: No Tax, No Inflation

this is the level of analysis we were doing in Chapter 3. We calculate the present worth of each option:
Buy: PW = -100,000 + 20,000(P/F, 0.1, 3)
= -100,000 + 20,000(0.7513)
= -84,974
Lease: PW = -40,000(P/A, 0.1, 3)
= -40,000(2.487)
= -99,480

1he decision at this level of analysis seems straightforward: buying saves us about [latex]\$15,000[/latex], so that's what we should do. (We are not disturbed by the fact that both PWs are negative; we suppose that the company needs the machine to stay in business.)

Second Analysis: Tax, but No Inflation

We now perform an after-tax analysis, using the methods developed in Chapter 12. We first calculate an after-tax MARR:

[latex]MARR_{after-tu} = MARR_{pre-tax} × 0.5 = 5\%[/latex]

To evaluate the "Buy" option, use this to calculate the capital salvage factor (CSF) and the capital tax factor (CTF):

CSF[latex] = 1 - td/(i + d) = 1 - 0.5 × 0.3/0.35 = 0.57[/latex]

CTF = 0.58
Buy: PW = -100,000CTF + 20,000(P/F, 0.05, 3) × CSF
= -58,000 + 20,000(0.8639)(0.86)
= -48,282

To evaluate the "Lease" option, we note that the lease payments can be deducted from pre-tax cash flow.

Lease: PW = -40,000(P/A, 0.05, 3)(1 - 0.5)
= -40,000(2.72)(0.5)
= -54,400

Despite the increased sophistication of our analysis, the conclusion hasn't changed: buying is still the better option, though the margin in its favour has dropped from about [latex]\$15,000[/latex] to about [latex]\$6,000[/latex].

Third Analysis: Tax plus Inflation

We first use Equation 14-1 to calculate the inflated after-tax rate of return (MARR*):

1 + i = (1 + i')(1 + f) or i= i' +f+ i′f (14-1)

MARR* = (1 + MARR)(l + F) - 1 = 1.05 × 1.15 - 1 = 0.21

We must re-calculate the CSF and the CTF based on MARR*:

CSF = 1 - td/(i + d) = 1 - 0.5 × 0.3/0.51 = 0.706
CTF = 0.73

We've been told that the salvage price of the machine in three years time will be [latex]\$20,000[/latex]. Is this the price in actual dollars or real dollars? Let's first make the pessimistic assumption that this number is in actual dollars. Then the present worth of buying is

Buy (actual salvage cost): PW = -100,000CTF + 20,000(P/F, 0.21, 3) × CSF
= -73,000 + 20,000(0.5645)(0.706)
= -[latex]\$65,153[/latex]

Note that the factor of (P/F, 0.21, 3) does two things: it deflates the [latex]\$20,000[/latex] we receive for salvage to its equivalent in real dollars, then brings this real-dollar sum back to its equivalent value today. If, on the other hand, we make the optimistic assumption that the salvage price is in real dollars, then to find its present value, we just need to move it back through time at the real after-tax MARR of [latex]5\%[/latex]:

Buy (real salvage cost): PW = -100,000CTF + 20,000(P/F, 0.05, 3) × CSF
= -73,000 + 20,000(0.8638)(0.706)
= -[latex]\$60,857[/latex]

We have a similar choice with respect to the leasing costs. We can either make the pessimistic assumption that the leasing company will put the lease cost up every year to keep pace with inflation (in which case the actual-dollar costs will increase while the real-dollar costs remain constant); or, more optimistically, we can assume that the lease cost has been written into a contract that will remain in effect for the next three years (in which case the actual-dollar costs will remain constant while the real-dollar costs decline). We find the PW based on each assumption:

Lease (payments rise with inflation): PW = -40,000(P/A, 0.05, 3)(1 - 0.5)
= -40,000(2.72)(0.5)
=[latex] -\$54,400[/latex]

Note that in this case, the annual [latex]\$40,000[/latex] is in real dollars, so we use the factor (P/A, 0.05, 3) to find the equivalent present value.

Lease (payments fixed by contract): PW = -40,000(P/A, 0.21, 3)(1 - 0.5)

= -40,000(2.07)(0.5)

= [latex]-\$41,400[/latex]

Note that in this case, the annual [latex]\$40,000[/latex] is in actual dollars, so we use the factor (P/A, 0.21, 3), which both deflates the lease payments and moves them back to the present.
This final level of analysis changes our conclusion: leasing is now the better option. And this conclusion is robust; it remains true whether we assume that the lease costs and salvage costs are real or actual.
Including inflation changes the conclusion since it reduces the present value of the tax savings associated with a capital equipment acquisition.

Question 14.11: A company is considering whether to rent or to buy a Plebney...

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