Question

Papa Johns is considering updating their menu. In contention are two new mutually exclusive specialty pizzas. Each project is expected to produce the following cash flows What is the crossover rate for these two menu items? Enter all percentages as a decimal (ex. 25.7% as 0.257).

Nacho Cheese Pizza (Project A):
Yr 0: -$50,000
Yr 1: 31,000
Yr 2: 26,000
Yr 3: 27,000

Sour Cream and Guacamole Pizza (Project B):
Yr 0: -$50,000
Yr 1: 42,000
Yr 2: 21,000
Yr 3: 18,000

Answer 1

Crossover rate is where the NPV of both the projects is equal. This implies that the difference in their NPVs is 0. To find the rate, we can calculate the difference of the CF streams, and calculate the IRR of the differential stream. IRR is the rate where NPV =0 and the NPV of the differential stream should equal 0

Year	CF	Discount Factor	Discounted CF
0	$50000-50000=0	1/(1+0.159922022889605)^0=	1	1*0=	$                 -
1	31000-42000 = $-11,000.00	1/(1+0.159922022889605)^1=	0.862126919	0.862126919108574*-11000=	$   -9,483.40
2	26000-21000 = $5,000.00	1/(1+0.159922022889605)^2=	0.743262825	0.743262824651642*5000=	$    3,716.31
3	27000-18000 = $9,000.00	1/(1+0.159922022889605)^3=	0.640786889	0.640786889104856*9000=	$    5,767.08
				NPV = Sum of all Discounted CF	$             0.00

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 = 15.99% rounded to two decimal places

So the crossover rate is 15.99% or 16% rounded off.

To verify, we can calcualte the NPV at 16% for each project and see if the NPV is the same or not:

Nacho pizza NPV:

Year	CF	Discount Factor	Discounted CF
0	$   -50,000.00	1/(1+0.159922022889605)^0=	1	1*-50000=	$   -50,000.00
1	$     31,000.00	1/(1+0.159922022889605)^1=	0.862126919	0.862126919108574*31000=	$     26,725.93
2	$     26,000.00	1/(1+0.159922022889605)^2=	0.743262825	0.743262824651642*26000=	$     19,324.83
3	$     27,000.00	1/(1+0.159922022889605)^3=	0.640786889	0.640786889104856*27000=	$     17,301.25
				NPV = Sum of all Discounted CF	$     13,352.01

Sour Cream Pizza NPV:

Year	CF	Discount Factor	Discounted CF
0	$   -50,000.00	1/(1+0.159922022889605)^0=	1	1*-50000=	$   -50,000.00
1	$     42,000.00	1/(1+0.159922022889605)^1=	0.862126919	0.862126919108574*42000=	$     36,209.33
2	$     21,000.00	1/(1+0.159922022889605)^2=	0.743262825	0.743262824651642*21000=	$     15,608.52
3	$     18,000.00	1/(1+0.159922022889605)^3=	0.640786889	0.640786889104856*18000=	$     11,534.16
				NPV = Sum of all Discounted CF	$     13,352.01

So it is verified that the NPV is equal at this rate.