Rate of Return Analysis Lecture No. 23 Chapter 7 Contemporary Engineering Economics Copyright © 2016

Rate of Return Investopedia® says: IRRs can also be compared against prevailing rates of return in the securities market. If a firm can't find any projects with IRRs greater than the returns that can be generated in the financial markets, it may simply choose to invest its retained earnings in the market. What it is: Interest earned on your invested capital, or commonly known as internal rate of return (IRR) ASimple Example: The interest earned on your savings account is the rate of return on your deposits.

Wal-Mart Investment Problem Given: P = $1,650 F = $15,384576 N = 44 years Find: i Formula to Use F = P(1 + i)N $15,384,576 = $1,650(1 + i)44 i = 23.09% Cash Flow Diagram $15,384,576 1970 2014 $1,650

How Good Was the Wal-Mart Investment and What Can It Be Compared with? If you did not invest $1,650 in Wal-Mart stock, what could you use your money for? If you took out $1,650 from your savings account and invested in Wal-Mart stock, you could have If the best you could do was to leave the money in a savings account to earn 6% interest over 44 years, you would have $21,426. What is the meaning of 6% interest? This will be your opportunity cost rate or minimum return required for any investment. $15,384,576 Or the equivalent to earning 23.09% interest each year on your savings account over 44 years.

Is This a Good Investment? In 1970, as long as you could earn more than a 6% interest in another investment opportunity, you would take that investment. Therefore, that 6% is viewed as a minimum attractive rate of return (or required rate of return). This is the interest rate commonly used in NPW analysis. So to see if the proposed investment is a good one, you adopt the following decision rule: ROR (23.09%) > MARR(6%)

Why is ROR measure so popular? This project will bring in a 15% rate of return on the investment. This project will result in a net surplus of $10,000in NPW. Which statement is easier to understand? Neither statement describes the nature of the investment project in any complete sense. However, the rate of return is somewhat easier to understand, because many of us are so familiar with savings-and-loan interest rates, which are, in fact, rates of return on your deposits.

Definition 1: Interest Earned on Loan Balance Rate of return (ROR) is defined as the interest rate earned on the unpaid (outstanding) balanceof an installment loan. Example: A bank lends $10,000 and receives an annual repayment of $4,021 over 3 years. The bank is said to earn a return of 10%on its loan of $10,000.

Loan Balance Calculation: A = $10,000 (A/P, 10%, 3) = $4,021 Unpaid Loan Balance at Beginning of Year Return on Unpaid Balance (10%) Payment Received from Borrower Unpaid Loan Balance at End of Year n 1 2 3 $0 -$10,000 -$6,979 -$3,656 $0 -$10,000 -$10,000 -$6,979 -$3,656 $0 -$1,000 -$698 -$366 +$4,021 Suppose that you borrow $10,000 from a bank with a repayment schedule of $4,021 each year for three years. How would you determine the interest rate that the bank charges on this transaction? As we learned in Chapter 3, you would set up the equivalence equation $10,000 = $4,021(P/A, i, 3) and solve for i. It turns out that i = 10%. In this situation, the bank will earn a return of 10% on its investment of $10,000 made at time 0. The bank calculates the balances over the life of the loan as shown in Table. A negative balance indicates an unpaid balance. In other words, you still owe money to the bank. Observe that, for the repayment schedule shown, the 10% interest is calculated only on each year’s outstanding balance. In this situation, only part of the $4,021 annual payment represents interest; the remainder goes toward repaying the principal. Thus, the three annual payments repay the loan itself and additionally provide a return of 10% on the amount still outstanding each year. Note that when the last payment is made, the outstanding principal is eventually reduced to zero.2 If we calculate the NPW of the loan transaction at its rate of return (10%), we see that PW(10%) = -$10,000 + $4,021(P/A, 10%, 3) = 0 which indicates that the bank can break even at a 10% rate of interest. If the customer pays back more than $4,021, the bank will earn more than 10% return. If the customer pays less than $4,021, the bank earns less than 10% return on its investment. In other words, the rate of return becomes the rate of interest that equates the present value of future cash repayments to the amount of the loan. This observation prompts the second definition of rate of return. A return of 10% on the amount still outstanding at the beginning of each year

Definition 2: Break-Even Interest Rate Rate of return (ROR) is the break-even interest rate, i*, which equates the present worth of a project’s cash outflows to the present worth of its cash inflows. Mathematical Relation: Example:

Definition 3: Return on Invested Capital: Internal Rate of Return The internal rate of return (IRR) is the interest rate earned on the unrecovered project balance of the investment such that when the project terminates, the unrecovered project balance will be zero. Example: A company invests $10,000 in a computer system, which results in equivalent annual labor savings of $4,021 over 3 years. The company is said to earn a return of 10% on its investment of $10,000.

Return on Invested Capital The firm earns a 10% rate of return on funds that remain internally invested in the project. Since the return is internal to the project, we call it internal rate of return.