In Figure 9.8, the NPV profiles cross at about 11 percent. How can we determine just what this crossover point is? The crossover rate

Question

In Figure 9.8, the NPV profiles cross at about 11 percent. How can we determine just what this crossover point is? The crossover rate, by definition, is the discount rate that makes the NPVs of two projects equal. To illustrate, suppose we have the following two mutually exclusive investments:

Investment B	Investment A	Year
-$500	-$400	0
320	250	1
340	280	2

What’s the crossover rate?

Investment B	Investment A	Year
-$500	-$400	0
320	250	1
340	280	2

What’s the crossover rate?

Accepted Answer

To find the crossover, first consider moving out of Investment A and into Investment B.
If you make the move, you’ll have to invest an extra $100 ($500-400). For this $100 investment, you’ll get an extra $70 ($320-250) in the first year and an extra $60 ($340-280) in the second year. Is this a good move? In other words, is it worth investing the extra $100?
Based on our discussion, the NPV of the switch, NVP(B-A), is:

NPV(B-A)=-$[latex]100+[70/(1+R)]+[60/(1+R)^{2}][/latex]

We can calculate the return on this investment by setting the NPV equal to zero and solving for the IRR:

NPV(B-A)=0=$[latex]100+[70/(1+R)]+[60/(1+R)^{2}][/latex]

If you go through this calculation, you will find the IRR is exactly 20 percent. What this tells us is that at a 20 percent discount rate, we are indifferent between the two investments because the NPV of the difference in their cash flows is zero. As a consequence, the two investments have the same value, so this 20 percent is the crossover rate. Check to see that the NPV at 20 percent is $2.78 for both investments.
In general, you can find the crossover rate by taking the difference in the cash flows and calculating the IRR using the difference. It doesn’t make any difference which one you subtract from which. To see this, find the IRR for (A-B); you’ll see it’s the same number. Also, for practice, you might want to find the exact crossover in Figure 9.8 (hint: it’s 11.0704 percent).

Question 9.7: In Figure 9.8, the NPV profiles cross at about 11 percent. H...

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