1. Chapter 5 Rate of Return Analysis: Single Alternative 5-1

2. LEARNING OBJECTIVES 1. Definition of ROR 2. ROR using PW and AW 3. Calculations about ROR 4. Multiple RORs 5. ROR of bonds ROR = Rate of Return 5-2

3. Sct. 1 Rate of Return - Introduction  Referred to as ROR or IRR (Internal Rate of Return) method  It is one of the popular measures of investment worth  DEFINITION -- ROR is either the interest rate paid on the unpaid balance of a loan, or the interest rate earned on the unrecovered investment balance of an investment such that the final payment or receipt brings the terminal value to exactly equal “0”  The ROR of found using a PW or AW relation. The rate determined is called i* 5-3

4. Unrecovered Investment Balance  ROR is the interest rate earned/charged on the unrecovered balance of a loan or investment project  ROR is not the interest rate earned on the original loan amount or investment amount (P)  The i* value is compared to the MARR --  If i* > MARR, investment is justified  If i* = MARR, investment is justified (indifferent decision)  If i* < MARR, investment is not justified 5-4

5. Valid Ranges for usable i* rates Mathematically, i* rates must be: 1.An i* = -100% signals total and complete loss of capital 2.One can have a negative i* value (feasible) but not less than –100% 3.All values above i* = 0 indicate a positive return on the investment 5-5

6. Sct.2 Calculation of i* using PW or AW Relations Set up an ROR equation using either PW or AW relations and equate to zero  0 = - PW of disbursements + PW receipts = - PW D + PW R  0 = - AW of disbursements + AW receipts = - AW D + AW R 5-6

7. i* by Trial and Error by Hand Using a PW Relation i* by Trial and Error by Hand Using a PW Relation 1.Draw a cash flow diagram 2.Set up the appropriate PW equivalence equation and set equal to 0 3.Select values of i and solve the PW equation 4.Repeat for values of i until “0” is bracketed, i.e., the equation is balanced 5.May have to interpolate to find the approximate i* value 5-7

8. ROR using Present Worth 0 1 2 3 4 5 -$1,000 +$500 +$1,500 Consider (Figure 5.2): 1000 = 500(P/F, i*,3) +1500(P/F, i*,5) Assume you invest $1,000 at t = 0; receive $500 @ t = 3 and $1500 at t = 5. What is the ROR of this project? The above PW expression must be solved by trial and error Guess at a rate and try it Adjust accordingly Bracket Interpolate i* approximately 16.9% per year on the unrecovered investment balances 5-8

9. Sct.3 Cautions When Using ROR  When applied correctly, ROR method will always result in a good decision and should be consistent with PW, AW, or FW methods.  However, for some types of cash flows the ROR method can be computationally difficult and/or lead to erroneous decisions  Reinvestment assumption is at i* for ROR method; not the MARR. If MARR is far from i*, must use composite rate (Sct 7.5)  Some cash flows will result in multiple i* values. Raises questions as to which, if any, i* value is proper value 5-9

10. Special ROR Procedure for Multiple Alternatives  For analysis of two or more alternatives using ROR, resort to a different analysis approach as opposed to regular PW or AW method  Must apply an incremental analysis approach to guarantee a correct decision, i.e., same as PW or AW 5-10

11. Sct.4 Multiple Rates of Return  A class of ROR problems exist that will possess multiple i* values  Capability to predict the potential for multiple i* values  Two tests can be applied prior to the analysis 5-11

12. Tests for Multiple i* values 1. Cash Flow Rule of Signs  The total number of real value i*’s is always less than or equal to the number of sign changes in the original cash flow series 2. Cumulative Cash Flow Rule of Signs  Form the cumulative cash flow of the investment and count the number of sign changes in the cumulative cash flow series  Must perform both tests to be sure of one i* > 0 Predicting the likelihood of multiple i* values 5-12

13. Test 1 -- Cash Flow Rule of Signs  Examples of sign test for maximum i* values  Signs on cash flows by year 123456 Max i* values - +++ -- 2 + - + - ++4 - ++++ + 1 5-13

14. Test 2 -- Cumulative Cash Flow (CCF) Signs  A sufficient, but not necessary, condition for a single positive i* value is:  Initial cash flow has negative sign  The CCF value at year n is > 0  and there is exactly one sign change in the CCF series 5-14

15. Typical Bond Cash Flow From the issuing company’s perspective P 0 is invested Net proceeds to company from sale of a bond A = the periodic bond interest payments from the firm to bond holders n periods F n is payment to bondholder to redeem the bond P 0 = A(P/A,i%,n) + F n (P/F i%,n) 5-15

16. ROR for Bond Investment: Ex 5.1  Purchase Price: P = $800/bond  Bond interest at 4% paid semiannually for $1,000 face value  Life = 20 years  Question: If you pay the $800 per bond, what is the ROR (yield) on this investment? 5-16

17. Ex. 5.1 -- Cash Flow Diagram …. …. …. 0 1 2 3 4 39 40 $800 F 40 = $1000 A= $1000(0.04/2) = $20.00 every 6 months for 20 years A = +$20/6 months From the bond purchaser’s perspective Pay $800 per bond to receive the $20each 6-months in interest cash flow plus $1,000 at the end of 40 time periods. What is the ROR of this cash flow? 5-17

18. Ex 5.1 -- Closed Form Setup Setup is:  0 = -$800 +20(P/A,i*,40 + $1000(P/F,i*,40)  Solve for i*  Manual or computer solution yields: i*=2.87%/6 months (intermediate answer)  Nominal ROR/year = (2.87%)(2) = 5.74%/yr  Effective ROR/year: (1.0287) 2 – 1 = 5.82%/yr 5-18