Question

You are deciding between two mutually exclusive investment opportunities. Both require the same initial investment of $9.8 million. Investment A will generate $2.11million per year​ (starting at the end of the first​ year) in perpetuity. Investment B will generate $1.43 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 6.6%​?

c. In this​ case, when does picking the higher IRR give the correct answer as to which investment is the best​ opportunity?

Answer 1

Part a:

Investment A:
Net present value (NPV)=Cash flow per year/Required return-Initial cash outflow
Given that, cash flow per year=$2.11 million
Initial cash outflow=$9.8 million
When NPV becomes zero, required rate becomes internal rate of return or IRR.
Equating NPV to zero we get,
Cash flow per year/Required rate-Initial cash outflow=0
=>$2.11/IRR-$9.8=0
=>2.11/IRR=9.8
=>2.11/9.8=IRR
=>IRR=0.215306122 or 21.53% (Rounded up to two decimal places)

Investment B:
NPV=[Cash flow at the end of the first year/(Required return-Growth rate)]-Initial cash outflow
Cash flow at the end of the first year=$1.43 million
Growth rate=2.2%
Initial cash outflow=$9.8 million
NPV=$1.43/(IRR-2.2%)-$9.8=0
=>$1.43/(IRR-2.2%)=$9.8
=>$1.43/$9.8=(IRR-2.2%)
=>0.145918367=IRR-0.022
=>0.145918367+0.022=IRR
=>0.167918367=IRR
=>IRR=16.79% (Rounded up to two decimal places)

Investment A has higher IRR.

Part b:

Investment A:
Net present value (NPV)=Cash flow per year/Required rate-Initial cash outflow
Given that, cash flow per year=$2.11 million
Initial cash outflow=$9.8 million
Required return=Cost of capital=6.6%
NPV=$2.11/6.6%-$9.8=$31.96969697-$9.8=$22.16969697
So, NPV=$22.17 million (Rounded up to two decimal places)

Investment B:
Cash flow at the end of the first year=$1.43 million
Growth rate=2.2%
Initial cash outflow=$9.8 million
For investment B, NPV=$1.43/(Required return-2.2%)-$9.8
Required return=Cost of capital=6.6%
NPV=$1.43/(6.6%-2.2%)-$9.8
=$1.43/(0.044)-$9.8
=$32.5-$9.8=$22.7
So, NPV=$22.7 million

Investment B has higher NPV when the cost of capital is 6.6%.

Part c:
In this case, when both projects have same NPV and cost of capital, picking the investment with higher IRR gives best​ investment opportunity.

Explanation for better understanding:
To answer this part, we need to calculate the cost of capital at which the NPV of both the investments will be equal.
For investment A: NPV=$2.11/Required return-$9.8
For investment B: NPV=$1.43/(Required return-2.2%)-$9.8
Taking required return as the cost of capital, we get:
$2.11/cost of capital-$9.8=$1.43/(cost of capital-2.2%)-$9.8
=>2.11/cost of capital=$1.43/(cost of capital-2.2%)
=>2.11*(cost of capital-2.2%)=1.43*(cost of capital)
=>2.11*cost of capital-2.11*2.2%=1.43*cost of capital
=>2.11*cost of capital-1.43*cost of capital=0.04642
=>0.68*cost of capital=0.04642
=>cost of capital=0.04642/0.68=0.068264706 or 6.83% (Rounded up to two decimal places)

Given that the projects are mutually exclusive.
Mutually exclusive projects refers to a set of projects out of which only one project can be selected.
Now, cost of capital=6.83% when both projects have same NPV.
The project with higher IRR should be considered for investment.
So, here investment A should be accepted because IRR of investment A is higher than investment B.