Chapter 5 Rate of Return Analysis: Single Alternative

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## Chapter 5 Rate of Return Analysis: Single Alternative

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**Chapter 5**Rate of Return Analysis: Single Alternative**LEARNING OBJECTIVES**ROR = Rate of Return • Definition of ROR • ROR using PW and AW • Calculations about ROR • Multiple RORs • ROR of bonds**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***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**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**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**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**+$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**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**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**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**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**Test 1 -- Cash Flow Rule of Signs**• Examples of sign test for maximum i* values • Signs on cash flows by year**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**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**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?**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**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