Question

A firm is evaluating whether to build a new factory. The proposed investment will cost the firm $1 million to construct and provide cash savings of $200,000 per year over the next 10 years.

(1) What rate of return does the investment offer?

(2) What is the maximum risk-free rate under which the firm is willing to make this investment?

(3) If the firm were to invest another $500,000 in the facility at the end of 5 years, it would extend the life of the project for 4 years, during which time it would continue receiving cash savings of $150,000. What is the internal rate of return for this additional investment?

Answer 1

1. The rate of return of the investment is the IRR of this project. The IRR (r) is the discount rate at which NPV = 0

So,

NPV = -1000000+ 200000/(1+r) +200000/(1+r)^2+ ....+ 200000/(1+r)^10 = 0

=> 200000/r*(1-1/(1+r)^10) = 1000000

=> 1/r*(1-1/(1+r)^10) = 5

Using hit and trial method

Putting r =0.1 , the Left hand side of above equation = 6.14456

Putting r =0.15 , the Left hand side of above equation = 5.018769

Putting r =0.155 , the Left hand side of above equation = 4.92458

Putting r =0.152 , the Left hand side of above equation = 4.980746

Putting r =0.151 , the Left hand side of above equation = 4.999699

The rate of 15.10% is close to the true value of r

So, the investment provides a return of 15.10%

2. If the project cashflows are certain and free of default, the maximum value of the riskfree rate can be upto the required rate of return i.e. 15.10%

So, the maximum risk-free rate under which the firm is willing to make this investment is 15.10%

3. Additional Investment of $500000 at the end of 5 years gives additional revenue of $150000 from year 11-14. So, the IRR (r) is given as

So, -500000/(1+r)^5+ 150000/(1+r)^11 + ... + 150000/(1+r)^14 = 0

=>  -500000+ 150000/(1+r)^6 + ... + 150000/(1+r)^9 = 0

=> 1/(1+r)^5 * ( 1/(1+r)+ ... + 1/(1+r)^4 ) = 500000/150000

=> 1/(1+r)^5*(1-1/(1+r)^4)/r= 3.333

Using hit and trial method

Putting r =0.1 , the Left hand side of above equation = 1.9682

Putting r =0.05 , the Left hand side of above equation = 2.7783

Putting r =0.04 , the Left hand side of above equation = 2.983509

Putting r =0.03 , the Left hand side of above equation = 3.2064

Putting r =0.025 , the Left hand side of above equation = 3.325037

Putting r =0.024 , the Left hand side of above equation = 3.349369

So, r ;lies between 0.024 and 0.025, Using linear approximation method

r= 0.024+ (3.349369-3.3333)/(3.349369-3.325037)*(0.025-0.024) = 0.02466

which is the correct value of r to two decimal places

So, the investment provides a return of 2.466%