CORPORATE FINANCE # 2: INTERNAL RATE OF RETURN

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1 CORPORATE FINANCE # 2: INTERNAL RATE OF RETURN Professor Ethel Silverstein Mathematics by Dr. Sharon Petrushka Introduction How do you compare investments with different initial costs ( such as $50,000 investment and a $75,000 investment; different life spans (such as a 5-year life and a 7- year life); and different intermediate payments. One way business people compare investments is by looking at their internal rate of returns (IRRs), the percentage that describe their annual return. The internal rate of return is the single interest rate that forces the present value of all future cash flows to exactly equal the purchase price. Corporate Finance 1

spans (such as a 5-year life and a 7- year life); and different intermediate payments.

2 Mathematics To be supplied For instruction on vectors and how to use them with the TI-83, see the mathematics section of the problem set QANT-301: Salaries One Variable Statistics. Corporate Finance 2

3 Using the TI-83 The TI-83 contains a function that will calculate the internal rate of return when the cash flow amounts come at equal time periods. The syntax of the function is: Where Note: Irr(P, L1). P= purchase price L1 is the vector containing the cash flow amounts, excluding the purchase price. The purchase price and cash flow amounts are entered as negative numbers if the cash is paid, and positive numbers if the cash is received. The internal rate of return is calculated by an iteration routine, similar to the one described in the mathematics section for the problem set. Example: Suppose you are considering lending $10,000 to a small business. In return, you will be paid back $3,000 at the end of the first year, $3,300 at the end of the second year, and $3,750 at the end of the third year. What is the internal rate of return? Solution First put the three payments into vector L1. Putting Numbers into a Vector: The TI-83 denotes the nth component of the vector L1 with the notation L1(n). Vector L1 may already have other numbers in it from earlier calculations. Therefore, first clear the vector L1 by pressing STAT [ClrList] [2 nd ] [L1] [ENTER] Next, enter the numbers into the corresponding components of vector L1. Corporate Finance 3

The purchase price and cash flow amounts are entered as negative numbers if the cash is paid, and positive numbers if the cash is received.

4 STAT [Edit] [if necessary, use < or > to move to vector L1] 3000 [ENTER] 3300 [ENTER] 3750 [ENTER] To return to the home screen, press [2 nd ] [Quit] Using the irr( function: To calculate the internal rate of return, go to the financial function irr( and enter the price paid as a negative number, , and the vector, L1, for the cash received. [2 nd ] [Finance] [irr( ] [-10000] [, ] [2 nd ] [L1)] [ENTER] The answer returned is.24 (rounded to the nearest hundredth) or 24%. Answer: The internal rate of return for that investment is 24%. Corporate Finance 4

negative number, -10000, and the vector, L1, for the cash received.

5 Business Application Let s assume that you are running a small business and that you are considering buying a new machine for $100,000. The cash inflow from this investment is given below: Year Cash Inflow 1 $60,000 2 $50,000 3 $40,000 4 $30,000 5 $ 0 after the fourth year, the machine become obsolete. The money to pay for the machine will be borrowed at 9% interest. Should you make this investment? Solution: All other things being equal, you should invest in the machine if the internal rate of return of its cash flow exceeds the cost of borrowing, 9%. For this problem, P=-$100,000(negative because the cash is being paid out). L1= [60000] [50000] [40000] [30000] Enter the numbers into vector L1, then press, [2 nd ] [Finance] [irr( ] [ ] [, ] [2 nd ] [L1)] [ENTER] The answer is 33%. Thus, the internal rate of return is 33%. Since 33% > 9%, the cost of borrowing the money, we should purchase the machinery. Commentary: When we speak of the internal rate of return for any asset (also referred to as the Yield to Maturity), we refer to a single interest rate that is used to describe the rate of return of cash flows over a period of time. The internal rate of return is important in comparing the returns of assets, and we do this very frequently. For instance, suppose we calculate that the internal rate of return a (IRR) of investment A is 9% a year, and the IRR of investment B is 10% a year. If we assume that the risk in the investments A and B is the same, then the higher rate of return would be preferable, (that is, choose investment B). Corporate Finance 5

The money to pay for the machine will be borrowed at 9% interest. Should you make this investment?

6 We can also say, however, that if A is riskier than B, but both have an IRR of 10% a year, then B would be a preferable investment because it would have the same percentage return but for a smaller amount of risk. Business people like to use the IRR as a way to compare the comparative advantage that one investment may have over another. Internal rate of return values are also linked to risk so that higher returns usually go with higher risks, and lower returns with lower risks. Internal Rate of Return compared with Net Present Value Theoretically, the IRR is not as good as using the net present value (see problem 8 in Finance, by Professor Harris), which shows the exact amount contributed by the investment to the value of the firm over the lifetime of the investment. The internal rate of return (percent annual return) on an investment does not deal with the fact that once the earnings come in, they should be reinvested by the firm in other investments which should also create a return at least equal to the cost of raising log term funds (cost of capital). Even with this weakness, however, businesses often prefer to use the IRR because it is easy to calculate. Corporate Finance 6

Internal rate of return values are also linked to risk so that higher returns usually go with higher risks, and lower returns with lower risks.

7 Additional Problems 1. Compare two investments for their annual internal rate of return. Investment ABC costs $50 and yields $6 a year for 7 years. Investment XYZ costs $55 and yields $6.50 per year for 8 years. Which investment should you choose? 2. You want a return of 12%. You have been offered an annuity for $50,000 that will return $15,000, $16,000, $17,000, $10,000 and $12,000 for each of the next 5 years. Should you accept it? 3. Suppose you have $1 million to invest for 3 years. One investment will pay $100,000 at the end of one year, $115,000 at the end of two years, and $130,000 plus your $1 million investment at the end of the third year. A second possible investment will pay $150,000 at the end of one year, $120,000 at the end of the second year, and $85,000 (plus your $1 million principal) at the end of the third year. Which investment should you choose? 4. Research Problem To be supplied. Corporate Finance 7

You have been offered an annuity for $50,000 that will return $15,000, $16,000, $17,000, $10,000 and $12,000 for each of the next 5 years. Should you accept it? 3.

NPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV

NPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV NPV Versus IRR W.L. Silber I. Our favorite project A has the following cash flows: -1 + +6 +9 1 2 We know that if the cost of capital is 18 percent we reject the project because the net present value is