Mountain Environmental Consulting may buy some field equipment for $40,000 or lease it for $2500 per month. In either case, the equipment will be replaced in 2 years. The salvage value of the equipment after 2 years is $6000. What is the IRR or cost of the lease?

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Accepted Answer

TABLE SOLUTION

The first step is to summarize the cash flows in a table. We must use monthly periods and remember that lease fees (an annuity due) are paid at the start of the period.

Buy – Lease	Buy	Lease	Month
−$37,500	−$40,000	−$2500	0
2,500		−2500	1-23
6,000	6,000	0	24

The easiest way to analyze this example is to simply set the present worths of buying and leasing equal to each other.

[latex]-\$2500-\$2500(P/A,i_{mon},23)=-\$40,000+\$6000(P/F,i_{mon},24)[/latex]

[latex]0=\$37,500-\$2500(P/A,i_{mon},23)-\$6000(P/F,i_{mon},24)[/latex]

The cash flow table was arranged so that the alternative with the lower cost now (leasing) was listed first. This ensures that the column of incremental cash flows (buy − lease) is an investment—the negative cash flows come first. An additional $37,500 must be paid or invested at time 0 to buy rather than lease.

The order of the two alternatives can be reversed. If Mountain Environmental leases the equipment, it avoids the expenditure of $37,500 at time 0. It incurs a cost of $2500 at the ends of Months 1 through 23, and it gives up the salvage value of $6000 at the end of Month 24. This cash flow pattern (positive at time 0 and negative in later years) occurs because leasing is a way to borrow money.

The column of incremental difference describes the decision to buy rather than lease, which invests $37,500 more now to avoid the later lease payments.

This equation is easier to solve if n = 24 for both factors. There is a total cash flow of −$6000 at the end of Month 24. This cash flow can be split into −$2500 and −$3500. Then both the uniform and single payment periods continue to the end of Month 24.

[latex]0=\$37,500-\$2500(P/A,i_{mon},24)-\$3500(P/F,i_{mon},24)[/latex]

[latex]PW_{0\%}=\$37,500-\$2500(24)-\$3500(1)=-\$26,000[/latex]

[latex]PW_{5\%}=\$37,500-\$2500(13.799)-\$3500(0.3101)=-\$1917[/latex]

The answer is clearly between 0% and 5%, and closer to 5%. Try 4% next.

[latex]PW_{4\%}=\$37,500-\$2500(15.247)-\$3500(0.3901)=-\$1983[/latex]

[latex]i_{mon}=4\%+(5\%-4\%)(1983)/(1983+1917)=4.51\%[/latex]

The effective annual rate is

[latex]i_{a}=1.0451^{12}-1=0.6975=69.8\%[/latex]

5-BUTTON SOLUTION

As noted above, using n = 24 makes things easier. The resulting interest rate is shown in the first calculation of the 5-BUTTON SOLUTION. It is 4.49% per month, which is a very high rate.

The second calculation identifies the payments as beginning-of-period cash flows by using BEGIN on a financial calculator or Type=1 in a spreadsheet. Then the initial investment is −$40,000 and the final salvage value is $6000.

Compounding the monthly interest rate for 12 periods with a PV = −1 leads to an FV that equals 1 + effective interest rate. Subtracting 1 from the FV value of 1.694 shows the annual rate is 69.4%, which means the firm should buy (not lease) the equipment.

H	G	F	E	D	C	B	A
Answer	Solve for	FV	PV	PMT	n	i	Problem	1
4.49%	[latex]i_{qtr}[/latex]	3500	-37,500	2,500	24		Exp. 7-10	2
4.49%	[latex]i_{qtr}[/latex]	6000	-40,000	2,500	24	using Type=1 or Begin		3
1.694	FV		-1	0	12	4.49%		4
69.4%	effective annual rate							5

In this case leasing is an extremely expensive way to obtain the equipment.

Question 7.10: Mountain Environmental Consulting may buy some field equipme...

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