# Present Value, Break-Even Analysis, Leverage

A firm has three investment opportunities. Each costs $1,000, and the firm's cost of capital is 10 percent. The cash inflow of each investment is as follows:

Cash Inflow A B C

Year

1 300 500 100

2 300 400 200

3 300 200 400

4 300 100 500

a. If the net present value method is used, which investment(s) should the firm make?

b. What is the internal rate of return of investment A? The internal rate of return of investment B is 10.22% and 6.15% for investment C. Which investment(s) should the firm make?

c. What is the payback period for each investment?

A firm needs $100 to start and expects:

Sales $200

Expenses $185

Tax rate 33% of earnings

a. What are earnings if the owners put up the $100?

b. If the firm borrows $40 of the initial at 10%, what are the profits received by the owner?

c. What is the return on the owners' investment in each case? Why do the returns differ?

d. If expenses rise to $194, what will be the returns in each case?

e. In which case did the returns decline more?

f. What generalization can you draw form the above?

#### Solution Preview

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Here are your answers.

Question 1

a. The NPV of a project is calculated as:

C0 + C1/(1+r) + C2/(1+r)^2 + ...

where

C0, C1, C2... are the cash flows from period 0, period 1, period 2, etc.

r is the interest rate in decimal form (in this case, 0.10, because the cost of capital is 10%)

Let's calculate it for project A:

NPV(A) = -1000 + 300/1.10 + 300/1.10^2 + 300/1.10^3 + 300/1.10^4 = -49.04

Notice that the cash flow in perdiod 0 is -$1,000, because that's the cost of the investment, which is paid "today". We've thus found that the NPV of project A is -$49.04.

Using the same formula for the other projects, we get:

NPV(B) = $3.69

NPV(C) = -$101.77

Since project B has the highest NPV, then the firm should make this investment.

b. The IRR of an investment is the discount rate that makes its NPV equal to zero. We thus have to solve:

0 = -1000 + 300/(1+r) + 300/(1+r)^2 + 300/(1+r)^3 + 300/(1+r)^4

Usually, the IRR cannot be found analytically, so some software must ...

#### Solution Summary

Benefits of leverage are noted. Present Value, Break-Even Analysis, and Leverage notes are also assessed.