Chapter 5 Rate of Return Analysis: Single Alternative

1. Chapter 5 Rate of Return Analysis: Single Alternative

2. LEARNING OBJECTIVES ROR = Rate of Return • Definition of ROR • ROR using PW and AW • Calculations about ROR • Multiple RORs • ROR of bonds

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” • TheROR of found using a PW or AW relation. The rate determined is called i*

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

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 = - PWD+ PWR • 0 = - AW of disbursements + AW receipts = - AWD+ AWR

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

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

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

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

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

12. Tests for Multiple i* values Predicting the likelihood of 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

13. Test 1 -- Cash Flow Rule of Signs • Examples of sign test for maximum i* values • Signs on cash flows by year

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

15. Typical Bond Cash Flow From the issuing company’s perspective P0 is invested Net proceeds to company from sale of a bond n periods A = the periodic bond interest payments from the firm to bond holders P0 = A(P/A,i%,n) + Fn(P/F i%,n) Fn is payment to bondholder to redeem the bond

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?

17. F40 = $1000 0 1 2 3 4 39 40 …. …. …. $800 Ex. 5.1 -- Cash Flow Diagram From the bond purchaser’s perspective A = +$20/6 months 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? A= $1000(0.04/2) = $20.00 every 6 months for 20 years

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