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By Assoc. Prof. Dr. Ahmet ÖZTAŞ

CE 533 - ECONOMIC DECISION ANALYSIS IN CONSTRUCTION. By Assoc. Prof. Dr. Ahmet ÖZTAŞ. CHP 6 . Rate of Return (ROR) Analysis. Gaziantep University Department of Civil Engineering. CHP 6 . Rate of Return (ROR) Analysis. Topics. Interpretation of ROR Values ROR Calculations

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By Assoc. Prof. Dr. Ahmet ÖZTAŞ

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  1. CE 533 - ECONOMIC DECISION ANALYSIS IN CONSTRUCTION ByAssoc. Prof. Dr. Ahmet ÖZTAŞ CHP 6. Rate of Return (ROR) Analysis Gaziantep University Department of Civil Engineering

  2. CHP 6. Rate of Return (ROR) Analysis Topics • Interpretation of ROR Values • RORCalculations • Cautions When Using ROR Method • Understanding Incremental ROR Analysis • ROR Evaluation • Multiple ROR Values • Removing Multiple ROR Values • Using Excel for ROR Analysis

  3. Introduction • This chapter presents the methods to evaluate alternatives using Rate of Return (ROR) procedure. • Other names of ROR: IRR, ROI, PI • Interest rate--- when discussing borrowed money; ROR when dealing with investments. • More than one ROR may exist. • How to recognize multiple ROR, if exists? • A unique ROR value can be obtained by using an external investment rate.

  4. 6.1 Interpretation of RORValues DEFINITION • Rate of return (ROR) is; • either the interest rate paid on the unpaid balance of barrowed money (a loan), • or the interest rate earned on the unrecovered balance of an investment • so that the final payment or receipt brings the balance to exactly “0” with interest considered.

  5. 6.1 Interpretation of RORValues • ROR is expressed as a % per period …i=10% • Mathematically, i rate must be: • If an i <= -100% entire amount is lost.. • One can have a negative i value (feasible), but not less than -100%! • Note: Definition does not state that ROR is on initial amount of investment; rather it is on the unrecovered balance, which changes each time period.

  6. 6.1 Unrecovered Investment Balance Example 6.1: Consider the following loan • You borrow $1,000 at 10% per year for 4 years • You are to make 4 equal end-of-year payments to pay off this loan • Your payments are: • A=$1,000(A/P,10%,4) = $315.47 • Compute ROR on unrecovered balance • Compute ROR on initial investment

  7. 6.1 The Loan Schedule a) Compute ROR on unrecovered balance

  8. 6.1 Unrecovered Investment Balance • For this loan the unpaid loan balances at the end of each year are: Unpaid loan balance is now “0” at the end of the life of the loan

  9. 0 1 2 3 4 P=$-1,000 6.1 Reconsider the Following • Assume you invest $1,000 over 4 years • The investment generates $315.47/year • Draw the cash-flow diagram A = +315.47

  10. Example 6.1 (b) Table 6.2 400 861.88

  11. Example 6.1 (b) The Loan Schedule • Table 6.2 shows unrecovered balance if 10% return is always figured on the initial $1000. Unrecovered amount: $138.12, because only $861.88 is recovered in 4 years.

  12. 6.1 Investment Problem • What interest rate equates the future positive cash flows to the initial investment? • We can state: • -$1,000= 315.47(P/A, i*,4) • Where i* is the unknown interest rate that makes the PW(+) = PW(-)

  13. 6.1 Investment Problem • $1,000= 315.47(P/A, i*,4) • Solve the above for the i* rate • (P/A,i*,4) = 1,000/315.47 = 3.16987 • Given n = 4, what value of i* yields a P/A factor value = 3.16987? • Interest Table search yields i*=10%

  14. 6.1 ROR – Explained • ROR is the interest rate earned on the unrecovered investment balances throughout the life of the investment. • ROR is not the interest rate earned on the original investment • ROR (i*) rate will also cause the NPV(i*) of the cash flow to = “0”.

  15. 6.2ROR Calculation • Objective is interest rate represented as i* • Calculations are reverse of previous methods (PW, AW), where i was known. • In rate of return problems you seek an unknown interest rate (i*) that satisfies the following: 0= - PWD+ PWR • PWD = present worth of costs • PWR = present worth of incomes

  16. 6.2 ROR using Present Worth • Finding the ROR for most cash flows is a trial-and-error effort. • The interest rate, i*, is the unknown • Solution is generally an approximation effort • May require numerical analysis approaches

  17. +$1,500 +$500 0 1 2 3 4 5 -$1,000 6.2 ROR using Present Worth • Example: • See Figure 6.1 • Assume you invest $1,000 at t = 0: Receive $500 @ t=3 and $1,500 at t = 5. • What is the ROR of this project?

  18. 7.2 ROR using Present Worth • Write a present worth expression, set equal to “0” and solve for the interest rate that satisfies the formulation. 1,000 = 500(P/F, i*,3) +1,500(P/F, i*,5) • Can you solve this directly for the value of i*? • NO! • Resort to trial-and-error approaches

  19. 7.2 ROR using Present Worth 1,000 = 500(P/F, i*,3) +1,500(P/F, i*,5) • Guess at a rate and try it • Adjust accordingly • Interpolate • i* approximately 16.9% per year on the unrecovered investment balances

  20. 6.2 Trial-and Error-Approach • General procedure for PW-based equation is; • 1- Draw a cash flow diagram • 2- Set up the rate of return equation in the form of • 0= PWD+ PWR • 3- Select values of i by trial and error until the equation is balanced.

  21. 6.2 Trial-and-Error Approach • If the NPV is not = 0, assume another i*. • If NPV > 0 increase “i* guess value” • If NPV < 0 reduce “i* guess value” • The objective is to obtain a negative PW and a positive PW value. • Then, interpolate between the two i* values

  22. 6.2 i* Spreatsheet Approach • Fastest way is to apply RATE function. • Format of function is: • RATE(n,A,P,F) • If CF vary over the years, IRR function is used to determine i*.

  23. 6.2 ROR Criteria • Determine the i* rate • If i* => MARR, accept the project • If i* < MARR, reject the project

  24. 6.3 Cautions when using the ROR Method • Important Cautions to remember when using the ROR method…… • There are some assumptions and difficulties with ROR analysis that must be considered when calculating i* and in interpreting its real world meaning for a particular project.

  25. 6.3 Cautions when using the ROR • 1-Computational Difficulties • ROR method is computationally more difficult than PW/AW • Can become a numerical analysis problem and the result is an approximation • Conceptually more difficult to understand • 2- Special Procedure for Multiple Alternatives • For ROR analysis of multiple alternatives, apply an incremental analysis approach.

  26. 6.3 Cautions when using the ROR • 3- Multiple i* values • Many real-world cash flows may possess multiple i* values • More than one i* value that will satisfy the definitions of ROR • If multiple i*’s exist, which one, if any, is the correct i*???

  27. 6.3 Cautions when using the ROR • 4- Reinvestment Assumptions • Reinvestment assumption for the ROR method is not the same as the reinvestment assumption for PW and AW • PW and AW assume reinvestment at the MARR rate • ROR assumes reinvestment at the i* rate • Can get conflicting rankings with ROR vs. PW and AW

  28. 6.4 Understanding Incremental ROR Analysis • From previous chapters, we know that using PW, AW or FW one mutually exclusive alternative can be identified. In this chapter We also learned that ROR can be used to identify the best alternative; However, it is not always as simple as selecting the highest rate of return (ROR) alternative. • Give an example: • Investment amount:90,000 MARR= 16% • Alt A: P=50,000, ROR=35%; • Alt B: P=85,000, ROR=29%;

  29. 6.4 Understanding Incremental ROR Analysis • Major dilemma of ROR method when comparing alternatives: Under some circumstances, alternative ROR (i*) values do not provide same ranking of alternatives as do PW and AW analyses. • To resolve this dilemma… conduct an incremental analysis between two alternatives at a time and base the alternative selection on the ROR of the incremental cash flow series. • A standardized format (A table, see Table 6.3) simplifies the incremental analysis.

  30. 6.4 Understanding Incremental ROR Analysis • Table shows a “format for incremental Cash Flow Tabulation” • If Equal lives: “Year” column will go from 0 to n. • If Equal lives: “Year” column will go from 0 to LCM of two lives. • Incremental ROR analysis requires equal-servis comparison between alternatives.

  31. 6.4 Understanding Incremental ROR Analysis • When LCM of lives is used, salvage value and reinvestment in each alternative are shown at their respective times. • If study period is used, the cash flow tabulation is for the specified period. • Simplification: Use convention that between 2 alternatives, the one with the larger initial investment will be regarded as Alternative B. Then, for each year in table: • Incremental CF = Cash Flow (B) – Cash Flow (A)

  32. 6.4 Understanding Incremental ROR Analysis • There are 2 types of alternatives: • Revenue alternative: There are both – and + CFs • Cost alternative: All CF estimates are negative (-). • In either case, use Equation Inc. CF = CF(B) – CF(A)to determine the incremental CF series with the sign of each cash flow carefully. See Examples 6.4 and 6.5. • Then, determine incremental rate of return (Δi*) on extra amount required by the larger investment alternative.

  33. 6.4 Understanding Incremental ROR Analysis • This rate, termed Δi*, represents the return over n years expected on optional extra investment in year 0. • Selection criteria: • If Δi*≥ MARR, select larger investment alt. (Alt. B) • Othervise, select lover investment alt. (Alt. A).

  34. 6.4 Understanding Incremental ROR Analysis • If analysis is between multiple revenue alternatives: • Determine each alternative’s i* value and remove those alternatives with i*≤ MARRsince, their return is too low. • Then, complete the incremental analysis for the remaining alternatives. • If all alternatives i*≤ MARRchoose “DN” alternative. • Note: this can not be done for cost alternatives since they have no positive CF.

  35. 6.4 Understanding Incremental ROR Analysis • When independent alternatives compared: • No incremental analysis is necessary. • All alternatives i*≥ MARR are acceptable.

  36. 6.5ROR Evaluation Two or More Mutually Exclusive Alternatives • When selecting from two or more mutually exclusive alternatives on the basis of ROR, equal-service comparison is required, and an incremental ROR analysis must be used. • Incremental ROR value between two alternatives (B and A) is identified as Δi*B-A, (or shortly Δi*). • Select the alternative that: • 1- Requires the largest initial investmet. • 2- has a Δi*≥ MARR, indicating that extra initial investment is economically justified.

  37. 6.5ROR Evaluation Two or More Mutually Exclusive Alternatives • Before conducting incremental evaluation, classify alternatives as cost or revenue alternatives. • Cost: Evaluate alternatives only against each other. • Revenue: First evaluate against do-nothing (DN), against each other.

  38. 6.5ROR Evaluation Two or More Mutually Exclusive Alternatives • Procedure: (To compare multiple, mutually exclusive alternatives using PW-based equivalence relation) • 1- Order alternatives by increasing initial investment For revenue alternatives add DN as the first one. • 2- Determine incremental CF between the first two ordered alternatives (B-A). • 3- Set up a PW-based relation of this incr. CF series and determine Δi*, the incremental ROR. • 4- If Δi*≥ MARR, eliminate A; B is the survivor. Otherwise A is survivor. • 5- Compare survivor to the next alternative.

  39. Example 6.6 • … Alternative machines from manufacturers in Asia,America, Europe, and Africa are available with the cost estimates in Table 6.6. • Annual cost estimates are expected to be high to ensure readiness at any time.The company representatives have agreed to use the average of the corporateMARR values, which results in MARR = 13.5%. • Use incremental ROR analysisto determine which manufacturer offers the best economic choice.

  40. Example 6.6 - Solution • Follow the procedure for incremental ROR analysis. • 1. These are cost alternatives and are arranged by increasing first cost. • 2. The lives are all the same at n = 8 years. The B - A incremental cashflows are indicated in Table 6.7. The estimated salvage values are shownseparately in year 8.

  41. Example 6.6 - Solution • 3. Following PW relation for (B-A) results in Δi*= 14.57%. 0= -1500 + 300(P/A, Δi*,8) + 400(P/F,Δi*,8) • 4. Since this return exceeds MARR = 13.5%, A is eliminated and B is thesurvivor. • 5. The comparison of C-to-B results in the eliminatian of C based on Δi* = -18.77% from the incremental relatian 0= -3500 + 200(P/A,Δi*,8) - 200(P/F,Δi*,8) • The D-to-B incremental cash flow PW relation for the final evaIuatian is O = -8500 + 1800(P/A,Δi*,8) + lOO(P/F,Δi*,8) • With Δi* = 13.60%, machine D is the overall, though marginal, survivorof the evaluation; it should be purchased and located in the event of oil spillaccidents.

  42. 6.6 Multiple ROR Values • 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: • 1. Cash Flow Rule of Signs (Descartes’ rule) • 2. Cumulative Cash Flow Test (Norstom’s criterion)

  43. 6.6 Multiple ROR Values • 1) Cash Flow Rule of Signs Test (Descartes’ rule): • “The maximum number of i* values is equal to the number of sign changes in the cash flow series. • Follows from the analysis of a n-th degree polynomial. • A “0” value does not count as a sign change

  44. 6.6 Multiple ROR Values • 2) Cumulative Cash Flow Test (Norstrom Criterion): • “There is one nonnegative i* value if the cumulative cash flow series, S0, S1,… Sn, changes sign only once and Sn#0. • To perform the test, cont the number of sign changes in Sn seires. • Sn= C.C.F through period n.

  45. 6.6 C.F. Rule of Signs Example • Consider Example 6.8 (Page 140) + - - + Result: 2 sign changes in the Cash Flow

  46. 6.6 Results: CF Rule of Signs Test • Two sign changes in this example • Means we can have a maximum of 2 real potential i* values for this problem • Beware: This test is fairly weak and the second test must also be performed.

  47. 6.6Cumulative CF Sign Test (Norstrom Criterion) • For the problem form the cumulative cash flow from the original cash flow. Count sign changes here

  48. 6.6Cumulative CF Signs – Example C.C.F + + - + 2 sign changes in the CCF. • This indicates that there is not just 1 nonnegative root. • So, as many as two i* values can be found.

  49. 6.6CCF Test –Rules • If the value of the CCF for year “n” is “0”, then an i* of 0% exists • If the value of CCF for year “n” is > 0, this suggests an i* > 0 • If CCF for year n is < 0, there may exist one or more negative i* values – but not always. • If the number of sign changes in the CCF is 2 or greater, this strongly suggests that multiple rates of return exist.

  50. 6.6 Example 6.8 – CCF Sign Test 2 Sign Changes here • Strong evidence that we have multiple i* values • CCF(n=3) = $200 > 0 suggests positive i* (s)

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