Calculation of crossover rate?

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I don’t know what I’m obviously missing here, but how do we calculate the crossover rate (specifically looking at Question 13 of Reading 44 in Cap Budgeting in CFAI)? Looked up the answer and cannot figure out how they reached it. Thanks!
 
Here’s a quick and dirty way of solving. Point where the two NPV profiles intersect will give you the crossover rate. To plot any curve you need at least 3 points. So we will work towards approximately sketching the two NPV profiles and you need “3 points” to complete a curve.
From the given question this is how you get those 3 points:
1. sum of undiscounted cash flows
project 1: (0,44) —–> (discount rate, NPV)
project 2: (0,75)
2. From the IRR information
project 1: (16.37,0)
project 2: (15.02,0)
3. From NPV @ 10% required rate of return
project 1: (10,14.12)
project 2: (10,19.53)
roughly plot these points and draw the curve for 2 projects. You will see they intersect at a point between 10% and 15.02%.
The explanation looks lengthy but when you do it, its fairly quick.
 
Actually it’s very simple.
In the same problem, you’re given the cashflows of the two projects. Take the difference of every two cashflows and input the difference as if it’s a new project. Calculate its IRR and that is your crossover rate.
This is how you solve the problem:
-100 - (-100) = 0 <- Year 0 (difference)
36 - 0 = 36 <- Year 1 (difference)
36 - 0 = 36 <- Year 2 (difference)
36 - 0 = 36 <- Year 3 (difference)
36 - 175 = -139 <- Year 4 (difference)
CF0 = 0
CF1 = 36
CF2 = 36
CF3 = 36
CF4 = -139
Solve for IRR. It will return 13.16%, which is the ratio provided by the key answer.
 
I know I did not post the original question, but-
Thanks Thunderanalyst and Zenji for two very effective and quick ways to tackle this type of problem! That is what makes this forum such a powerful resource.
 
zenji Wrote:
——————————————————-
> Actually it’s very simple.
>
> In the same problem, you’re given the cashflows of
> the two projects. Take the difference of every two
> cashflows and input the difference as if it’s a
> new project. Calculate its IRR and that is your
> crossover rate.
>
> This is how you solve the problem:
>
> -100 - (-100) = 0 <- Year 0 (difference)
> 36 - 0 = 36 <- Year 1 (difference)
> 36 - 0 = 36 <- Year 2 (difference)
> 36 - 0 = 36 <- Year 3 (difference)
> 36 - 175 = -139 <- Year 4 (difference)
>
> CF0 = 0
> CF1 = 36
> CF2 = 36
> CF3 = 36
> CF4 = -139
> Solve for IRR. It will return 13.16%, which is the
> ratio provided by the key answer.
++++++++++++++++++++++++++++++++++++++++=
Zenji, where did you learn this? it is amazing.
I also tried project 2 each year CF minus project 1 one, got same result. thanks!
 
> ++++++++++++++++++++++++++++++++++++++++=
> Zenji, where did you learn this? it is amazing.
> I also tried project 2 each year CF minus project
> 1 one, got same result. thanks!
You know this is actually something they teach you in one of your first finance courses in your undergrad. Funny, I just covered this topic using the schweser notes today and completely forgot about how to calculate the crossover.
 
my undergrad is not in economie , finance ,or accounting. so …
I am glad I found this site , very helpful.
 
In case you want to see why zenji’s solution works, use the following concepts:
1) The IRR is the discount rate that results in an NPV of zeor
2) NPVs are “additive”
So, by (1), the IRR of the “difference project” (i.e. the synthetic project you created by taking the differences in cash flows) is the discount rate that forces the NPV of that project to zero.
By (2) the NPV of the “difference” is equal to the difference in the NPVs. In other words, at that rate, NPV1`-NPV2 = ), so NPV1 = NPV2.
When I first saw this, I also though it was very cool (but then I’m the alpha nerd, so it’s not surprising).
 
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