Question

You are deciding between two mutually exclusive investment opportunities. Both require the same initial investment of

$ 10.4 million. Investment A will generate $1.99 million per year​ (starting at the end of the first​ year) in perpetuity. Investment B will generate

$ 1.55 million at the end of the first​ year, and its revenues will grow at 2.2 % per year for every year after that.

a. Which investment has the higher​ IRR?

b. Which investment has the higher NPV when the cost of capital is 6.8 %

c. In this​ case, for what values of the cost of capital does picking the higher IRR give the correct answer as to which investment is the best​ opportunity?

Answer 1

a. The equation for IRR of first and second investment will be respectively-

-10.44 + 1.99/(1+r) + 1.99/(1+r)^2 +.....=0 ----------------------------------A.

-10.44 + 1.55/(1+r) + 1.55*1.022/(1+r)^2 + 1.55*1.022^2/(1+r)^3 + .......=0 ---------------B.

These two equations will be solved by the formula of geometric progression.

-10.44 + 1.99/(1+r) * 1/(1-1/(1+r))=0

10.44 = 1.99/r. Hence r = 1.99/10.44 = 19.06%

Similarly for the second investment, the equation will become-

10.44 = 1.55/(1+r) * 1/(1-1.022/(1+r)) = 1.55/(r-0.022).

Hence r = 17.046%.

Therefore, first investment has a higher IRR.

b. In the equations A and B above written above, we substitute 6.8% in the place of r to find NPV.

Hence, for first investment NPV = 18.824

and for second investment NPV = 23.255.

Hence, the second investment has the higher NPV.

c. For this we calculate the rate at which both options have same NPV.

NPV (1st) = NPV (2nd)

10.44-1.99/r = 10.44-1.55/(r-0.022)

(r-0.022)/r = 1.55/1.99

r-0.022 = 0.7788r

r=9.95%.

Hence, for cost of capital values above 9.95%, picking the higher IRR gives the correct answer