Question

Consider the following two projects for Copper Mountain Sports. Both project are projected to produce cash flows for 5 years at which time the equipment will have become technologically obsolete. For these projects, calculate the "cross-over rate". The cross -over rate is the discount rate where both projects would have the same NPV.

Year	Snow Shoes	Snowmobiles
0	-$250,000	-$250,000
1	$25,000	$150,000
2	$80,000	$160,000
3	$200,000	$75,000
4	$190,000	$50,000
5	$150000	$5000

a) 33.67%

b) These two projects do not have a cross-over rate

c) 10.55%

d) 30.52%

e) 32.30%

Answer 1

There are two ways to solve this question. One is to calculate the NPV at each of the given rate choices for each project. The second is to calculate the difference of CFs for both the projects and then calculate the IRR of the resulting CFs because crossover rate is where the NPV of both the projects is equal so the difference between them should be 0 and that is exactly what the IRR does, it makes the NPV of the CF stream equal 0. The CF Stream will be the difference of the CFs of each project.

Let's look at the difference of CF stream and calculate the IRR and then verify the same by putting as a discount rate for each individual project:

Step 1 calculate the difference of the CFs of the two projects

Year	Snow Shoes	Snowmobiles	Difference
0	-250000	-250000	-250000-(-250000)=0
1	25000	150000	25000-150000=-125000
2	80000	160000	80000-160000=-80000
3	200000	75000	200000-75000=125000
4	190000	50000	190000-50000=140000
5	150000	5000	150000-5000=145000

Step 2) Calculate the IRR of the above CF stream:

IRR is the rate at which NPV = 0

IRR can be calculated using either a financial calculator or excel or through hit and trial:

Using Excel we get the IRR = 30.5232956623504 or 30.52% rounded to two decimal places

Step 3) Calculate the NPV of each of the projects using the discount rate of 30.52%

Snowshoe:

Year	CF	Discount Factor	Discounted CF
0	$ -250,000.00	1/(1+0.305232956623504)^0=	1	1*-250000=	$ -250,000.00
1	$     25,000.00	1/(1+0.305232956623504)^1=	0.766146759	0.766146759415952*25000=	$     19,153.67
2	$     80,000.00	1/(1+0.305232956623504)^2=	0.586980857	0.586980856963565*80000=	$     46,958.47
3	$   200,000.00	1/(1+0.305232956623504)^3=	0.449713481	0.449713481401834*200000=	$     89,942.70
4	$   190,000.00	1/(1+0.305232956623504)^4=	0.344546526	0.344546526441682*190000=	$     65,463.84
5	$   150,000.00	1/(1+0.305232956623504)^5=	0.263973205	0.263973204701317*150000=	$     39,595.98
				NPV = Sum of all Discounted CF	$     11,114.65

Snowmobile:

Year	CF	Discount Factor	Discounted CF
0	$ -250,000.00	1/(1+0.305232956623504)^0=	1	1*-250000=	$ -250,000.00
1	$   150,000.00	1/(1+0.305232956623504)^1=	0.766146759	0.766146759415952*150000=	$   114,922.01
2	$   160,000.00	1/(1+0.305232956623504)^2=	0.586980857	0.586980856963565*160000=	$     93,916.94
3	$     75,000.00	1/(1+0.305232956623504)^3=	0.449713481	0.449713481401834*75000=	$     33,728.51
4	$     50,000.00	1/(1+0.305232956623504)^4=	0.344546526	0.344546526441682*50000=	$     17,227.33
5	$        5,000.00	1/(1+0.305232956623504)^5=	0.263973205	0.263973204701317*5000=	$        1,319.87
				NPV = Sum of all Discounted CF	$     11,114.65

As the NPV of both the projects is the same, then 30.52% is definitely the crossover rate.

So the correct option is D