Please show your work and write solutions in details, including formulas and how
ID: 2651475 • Letter: P
Question
Please show your work and write solutions in details, including formulas and how you plug numbers, given in each question, into these formulas. Please, write short VERBAL conclusion at the end of each problem summarizing your answer in words. Verbal answers are not substitute for the quantitative solution.
Question 1
Consider the following mutually exclusive investments
T=0
1
2
Investment A:
-100
20
120
Investment B:
-100
100
31.25
a) Find IRRs for both projects
b) Draw a graph, where you will show the NPV of each project as a function of its discount rate (i.e NPV on the vertical axis and r on the horizontal axis). Both NPVs should be on the same graph.
c) Find the cross over rate
d) Please describe as fully as possible which project is the best.
T=0
1
2
Investment A:
-100
20
120
Investment B:
-100
100
31.25
Explanation / Answer
Internal rate of return (IRR) is the discount rate at which the net present value of an investment becomes zero. In other words, IRR is the discount rate which equates the present value of the future cash flows of an investment with the initial investment.
IRR Calculation
The calculation of IRR is a bit complex than other capital budgeting techniques. We know that at IRR, Net Present Value (NPV) is zero, thus:
NPV = 0; or
PV of future cash flows Initial Investment = 0; or
Where,
r is the internal rate of return;
CF1 is the period one net cash inflow;
CF2 is the period two net cash inflow,
CF3 is the period three net cash inflow, and so on ...
But the problem is, we cannot isolate the variable r (=internal rate of return) on one side of the above equation. However, there are alternative procedures which can be followed to find IRR. The simplest of them is described below:
Solution:
W
Assume that r is 10%.
NPV at 10% discount rate =17.3(Investment A), 16.7(Investment B)
Since NPV is greater than zero we have to increase discount rate, thus we take the discount rate at 12%
NPV at 12% discount rate = .86(Investment A), .45(Investment B)
Since NPV is less that zero here, IRR must be lower than 12% and greater than 10%, i.e.11% for both the projects.
NPV at 11% discount rate = -.01(Investment A), .05(Investment B)
So IRR for both the projects is 10.95%.
I am not able to paste graph here due to some technical issue, you shoud take discount rate at horizontal axis and NPV onon vertical axis and find out the cross over rate.
Crossover rate is the cost of capital at which the net present values of two projects are equal. It is the point at which the net present value profile of one project crosses over (intersects) the net present value profile of the other project.
Since crossover rate is the rate at which NPVs of two projects are equal, we can find it by equating NPV equation for the first project with NPV equation for the second project and then solving it for r.
In the above equation, we have CF1, CF2, CF3, A, CFi, CFii and B which means we can solve it for r. r—the cost of capital is the crossover rate.
Subtracting all the cash flows of ivestment A from Investmet B gives us the following cash flows stream:
Initial cash outflows = 100-100 = 0
Cash flows at end of year 1= 20-100= -80
Cash flows at end of year 2 = 120-31.25 =88.75
An IRR equation can be developed from the given stream. It would look like:
Solving the above equation we get the value of r. This is the crossover rate.
Investment B is considered best when the cost of capital is below crossover rate.And investment A is considered best if cost of capital is above crossover rate.
CF1 + CF2 + CF3 + ... Initial Investment = 0 ( 1 + r )1 ( 1 + r )2 ( 1 + r )3