Question

You are analyzing the following two mutually exclusive projects and have developed the following information. What is the crossover rate? Year Project A Cash Flow Project B Cash Flow $75,000 $26,300 $29,500 $45,300 $75,000 $24,000 $26,900 $51,300 14.32 percent 13.33 percent O 15.20 percent O 14.60 percent 13.17 percent

Answer 1

Crossover rate is the rate at which the Net Present Value of both the projects is equal.

We know that NPV is equal to $Cash Outflow + \frac{Cash Inflow Year 1}{(1+r)^1} \frac{Cash Inflow Year 2}{(1+r)^2}+.....\frac{Cash Inflow Year n}{(1+r)^n}$

where n is the life of the project

r is the discounting rate

We have to determine r at which NPV of both projects is equal. So,

$-75,000 + \frac{26,300}{(1+r)^1} \frac{29,500}{(1+r)^2}+\frac{45,300}{(1+r)^3}$= $-75,000 + \frac{24,000}{(1+r)^1} \frac{26,900}{(1+r)^2}+\frac{51,300}{(1+r)^3}$

solving both sides, we get

$\frac{2,300}{(1+r)^1} \frac{2,600}{(1+r)^2}+\frac{-6,000}{(1+r)^3}$ = 0

In order to calculate r, we have to apply trial and error method.

That, is calculate NPV at two rates and then use a formula to calculate r.

So, assuming r = 15%, we get

	Cash Flow	Present Value Factor	Present Value
1	2300	0.8696	2000
2	2600	0.7561	1965.97
3	-6000	0.6575	-3945.10
	NPV		20.88

assuming r = 14%, we get

	Cash Flow	Present Value Factor	Present Value
1	2300	0.8772	2017.54
2	2600	0.7695	2000.62
3	-6000	0.6750	-4049.83
	NPV		-31.67

Formula to calculate r = $Lower rate + \frac{NPV at Lower Rate}{NPV at Lower Rate - NPV at Higher Rate} * (Higher Rate - Lower Rate)$

$14 +\frac{-31.67}{-31.67-20.88}* (15-14)$

$14 +\frac{-31.67}{-52.55}* (1)$

r = 14.60%

Note: Present Value Factors are calculated as

$\frac{1}{1.15^1}$ =0.8696

$\frac{1}{1.15^2}$= 0.7561

$\frac{1}{1.15^3}$=0.6575

$\frac{1}{1.14^1}$=0.8772

$\frac{1}{1.14^2}$=0.7695

$\frac{1}{1.14^3}$= 0.6750