Internal Rate of Return

EXERCISE 01

Project Alfa costs €120 now and pays €150 one year later.

A.Calculate IRR.

A.

0=-I_0+\frac{CF_1}{1+IRR}\quad\Rightarrow\quad IRR=\frac{CF_1}{I_0}-1

IRR=\frac{150}{120}-1

25%.

Internal Rate of Return

EXERCISE 02

Year	0	1	2	3
Project A	-5000	2000	2500	3000
Project B	-3000	2500	1000	500

A.Solve for each project’s IRR.

A.

0=-5000+\frac{2000}{1+IRR_A}+\frac{2500}{(1+IRR_A)^2}+\frac{3000}{(1+IRR_A)^3}

0=-3000+\frac{2500}{1+IRR_B}+\frac{1000}{(1+IRR_B)^2}+\frac{500}{(1+IRR_B)^3}

IRR(A) = 21.64779%; IRR(B) = 21.89591%.

These are the positive roots, found numerically. Ranking by IRR alone is not enough to choose between mutually exclusive projects of different scale.

Internal Rate of Return

EXERCISE 03

A.Does a positive IRR guarantee a positive NPV?

A.

No. For conventional cash flows with a unique IRR and a decreasing NPV profile, NPV is positive only when IRR exceeds the required return, zero when the two are equal, and negative when IRR is lower. Merely exceeding zero is insufficient. The equivalence is not universal for non-conventional cash flows.