27108094-Chapter-7x

27108094-Chapter-7x - ENGINEERING ECONOMY, Sixth Edition by...

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Unformatted text preview: ENGINEERING ECONOMY, Sixth Edition by Blank and Tarquin CHAPTER 7 Mc Gra Hill w Rate of Return Analysis: Single Alternative Chapter 7 Learning Objectives 1. Definition of Rate of Return (ROR) 2. ROR using PW and AW 3. Calculations about ROR 4. Multiple RORs 5. Composite ROR 6. ROR of Bonds 02/19/10 Authored by Don Smith, Texas A&M University 2004 2 Section 7.1 INTERPRETATION OF A RATE OF RETURN VALUE •The IRR method is one of the popular time-discounted measures of investment worth that is related to the NPV approach. DEFINITION follows 02/19/10 Authored by Don Smith, Texas A&M University 2004 3 7.1 INTERPRETATION OF A RATE OF RETURN VALUE 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 equal “0”. 02/19/10 Authored by Don Smith, Texas A&M University 2004 4 7.1 INTERPRETATION OF A RATE OF RETURN VALUE In rate of return problems you seek an unknown interest rate (i*) that satisfies the following: PWi* (+ cash flows) – PWi* ( - cash flows) =0 This means that the interest rate – i*, is an unknown parameter and must be solved or approximated. 02/19/10 Authored by Don Smith, Texas A&M University 2004 5 7.1 Unrecovered Investment Balance ROR is the interest rate earned/charged on the unrecovered investment balance of a loan or investment project ROR is not the interest rate earned on the original loan amount or investment amount 02/19/10 Authored by Don Smith, Texas A&M University 2004 6 7.1 Unrecovered Investment Balance Consider the following loan You borrow $1000 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=$1000(A/P,10%,4) = $315.47 02/19/10 Authored by Don Smith, Texas A&M University 2004 7 7.1 The Loan Schedule Year BOY Bal Payment Interest Amount Prin. Red. Amount UnPaid Balance 0 $1,000 0 -- --- $1,000 1 1,000 315.47 100.00 215.47 784.53 2 784.53 315.47 78.45 237.02 547.51 3 547.51 315.47 54.75 260.72 286.79 4 286.79 315.47 28.68 286.79 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 8 7.1 Unrecovered Investment Balance For this loan the unpaid loan balances at the end of each year are: 0 $1,000 1 784.53 2 547.51 3 4 02/19/10 286.79 0 Unpaid loan balance is now “0” at the end of the life of the loan Authored by Don Smith, Texas A&M University 2004 9 7.1 Unrecovered Investment Balance The URBt=4 is exactly 0 at a 10% rate 0 1 784.53 2 547.51 3 286.79 4 02/19/10 $1,000 0 Authored by Don Smith, Texas A&M University 2004 10 7.1 Reconsider the following Assume you invest $1000 over 4 years The investment generates $315.47/year A= Draw the315.47flow diagram + cash 0 1 2 3 4 P=$-1,000 02/19/10 Authored by Don Smith, Texas A&M University 2004 11 7.1 Investment Problem What interest rate equates the future positive cash flows to the initial investment? We can state: -$1000= 315.47(P/A, i*,4) Where i* is the unknown interest rate that makes the PW(+) = PW(-) 02/19/10 Authored by Don Smith, Texas A&M University 2004 12 7.1 Investment Problem $1000= 315.47(P/A, i*,4) Solve the above for the i* rate (P/A,i*,4) = 1000/315.47 = 3.16987 Given n = 4 what value of i* yields a P/A factor value = 3.16987? Interest Table search yields i*=10% 02/19/10 Authored by Don Smith, Texas A&M University 2004 13 7.1 Investment Problem i*=10% per year Just like the loan problem, we can calculate the unrecovered investment balances that are similar to the unpaid loan balances See the next slide…. 02/19/10 Authored by Don Smith, Texas A&M University 2004 14 7.1 Unrecovered Investment Balances (UIB) We set up the following table t C.F(t) Future Value for 1 year UIBt 0 -1,000 ---- -1,000 1 +315.47 -1000(1.10)+315.47= -784.53 2 +315.47 -784.53(1.10)+315.47= -547.51 3 +315.47 -547.51(1.10)+315.47= -286.79 4 +315.47 -286.79(1.10)+315.47= 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 15 7.1 UIB’s for the Example See Figure 7.1 t C.F(t) UIBt 0 -1,000 -1,000 1 +315.47 -784.53 2 +315.47 -547.51 3 +315.47 -286.79 4 +315.47 0 02/19/10 •The unrecovered investment balances have been calculated at a 10% interest rate. •Note, the investment is fully recovered at the end of year 4 •The UIB = 0 at a 10% interest rate Authored by Don Smith, Texas A&M University 2004 16 7.1 UIB’s for the Example See Figure 7.1 t C.F(t) UIBt 0 -1,000 -1,000 1 +315.47 -784.53 2 +315.47 -547.51 3 +315.47 -286.79 4 +315.47 0 02/19/10 •The 10% rate is the only interest rate that will cause the UIB at the end of the project’s life to equal exactly “0” Authored by Don Smith, Texas A&M University 2004 17 7.1 UIB’s for the Example See Figure 7.1 t C.F(t) UIBt 0 -1,000 -1,000 1 +315.47 -784.53 2 +315.47 -547.51 3 +315.47 -286.79 4 +315.47 0 02/19/10 •Note, all of UIB’s are negative at the 10% rate. •This means that the investment is unrecovered throughout the life. Authored by Don Smith, Texas A&M University 2004 18 7.1 Pure Investment The basic definition of ROR is the interest rate that will cause the investment balance at the end of the project to exactly equal “0” If there is only one such interest rate that will cause this, the investment is said to be a “PURE” investment or, Conventional investment 02/19/10 Authored by Don Smith, Texas A&M University 2004 19 7.1 UIB’s for the Example “Unrecovered” means that the investment balance for a given year is negative. If a project’s UIB’s are all negative (under-recovered) then that investment will possess one unique interest rate to cause the UIBt= to n equal “0” 02/19/10 Authored by Don Smith, Texas A&M University 2004 20 7.1 UIB’s for the Example See Figure 7.1 This diagram depicts the unpaid loan balance At the ROR, the URB will be exactly “0” at the end of the life of the project 02/19/10 Authored by Don Smith, Texas A&M University 2004 21 7.1 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”. 02/19/10 Authored by Don Smith, Texas A&M University 2004 22 Section 7.2 ROR using Present Worth •PW definition of ROR •PW(-CF’s) = PW(+CF’s) •PW(-CF’s) - PW(+CF’s) =0 •AW definition of ROR •AW(-CF’s) = AW(+CF’s) •AW(-CF’s) - AW(+CF’s) = 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 23 7.2 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 02/19/10 Authored by Don Smith, Texas A&M University 2004 24 7.2 ROR using Present Worth +$1,500 •See Figure 7.2 0 1 +$500 2 3 4 5 •$1,000 Assume you invest $1,000 at t = 0: Receive $500 @ t=3 and $1500 at t = 5. What is the ROR of this project? 02/19/10 Authored by Don Smith, Texas A&M University 2004 25 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. 1000 = 500(P/F, i*,3) +1500(P/F, i*,5) •Can you solve this directly for the value of i*? •NO! •Resort to trial and error approaches 02/19/10 Authored by Don Smith, Texas A&M University 2004 26 7.2 ROR using Present Worth 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. 02/19/10 Authored by Don Smith, Texas A&M University 2004 27 7.2 Trial and Error Approach •Iterative procedures require an initial guess for i* •One makes an educated first guess and calculates the resultant PV at the guess rate. 02/19/10 Authored by Don Smith, Texas A&M University 2004 28 7.2 Trial and Error Approach •If the NPV is not = 0 then another i* value is evaluated…. Until NPV “close” to “0” •The objective is to obtain a negative PV for an i* guess value then. •Adjust the i* value to obtain a positive PV given the adjusted i* value •Then interpolate between the two i* values 02/19/10 Authored by Don Smith, Texas A&M University 2004 29 7.2 Trial and Error Approach – Bracket “0” •If the NPV is not = 0 then another i* value is evaluated •A negative NPV generally indicates the i* value is too high •A positive NPV suggests that the i* value was too low 02/19/10 Authored by Don Smith, Texas A&M University 2004 30 7.2ROR Criteria •Determine the i* rate •If i* => MARR, accept the project •If i* < MARR, reject the project 02/19/10 Authored by Don Smith, Texas A&M University 2004 31 7.2 Spreadsheet Methods Excel supports ROR analysis •RATE(n,A,P,F) can be used when a time t = 0 investment (P) is made followed by “n” equal, end of period cash flows (A) •This is a special case for annuities only 02/19/10 Authored by Don Smith, Texas A&M University 2004 32 7.2 Example 7.3 – In Excel MARR= Life Year 0 1 2 3 4 5 6 7 8 9 10 ROR-Guess ROR NPV= 02/19/10 10.00% 10 Cash Flow •Excel Setup for ROR -$500,000 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 $710,000 $300,000 =IRR(D6:D16,D19) D19 = Guess Value 0% 5.16% -$168,674.03 Authored by Don Smith, Texas A&M University 2004 33 7.2 Example 7.2 Investment Balances •Investment balances at the i* rate •The time t = 10 balance = 0 at i* •As it should! 02/19/10 Authored by Don Smith, Texas A&M University 2004 34 ENGINEERING ECONOMY Fifth Edition Blank and Tarquin Mc Gra Hill w CHAPTER VII Section 7.3 Cautions when using the ROR Method Section 7.3 Cautions when using the ROR Method •Important Cautions to remember when using the ROR method…… 02/19/10 Authored by Don Smith, Texas A&M University 2004 36 7.3 Cautions when using the ROR Method No.1 •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*??? 02/19/10 Authored by Don Smith, Texas A&M University 2004 37 7.3 Cautions when using the ROR Method. 2. 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. PV and AW 02/19/10 Authored by Don Smith, Texas A&M University 2004 38 7.3 Cautions when using the ROR Method: 3. 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 02/19/10 Authored by Don Smith, Texas A&M University 2004 39 7.3 Cautions when using the ROR Method: 4. Special Procedure for Multiple Alternatives •For analysis of two or more alternatives using ROR one must resort to a different analysis approach as opposed to the PW/AW methods •For ROR analysis of multiple alternatives must apply an incremental analysis approach 02/19/10 Authored by Don Smith, Texas A&M University 2004 40 7.3 Cautions when using the ROR Method: ROR is more difficult! •ROR is computationally more difficult •But is a popular method with financial managers •ROR is used internally by a substantial number of firms •Suggest using PW/AW methods where possible 02/19/10 Authored by Don Smith, Texas A&M University 2004 41 7.3 Valid Ranges for usable i* rates •Mathematically, i* rates must be: −100% < i ≤ +∞ * • If an i* <= -100% this signals total and complete loss of capital. •i*’s < -100% are not feasible and not considered •One can have a negative i* value (feasible) but not less than –100%! 02/19/10 Authored by Don Smith, Texas A&M University 2004 42 Section 7.4 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 02/19/10 Authored by Don Smith, Texas A&M University 2004 43 7.4 Tests for Multiple i* values 1. Cash Flow Rule of Signs 2. Cumulative Cash Flow Rule of Signs test • Example follows 02/19/10 Authored by Don Smith, Texas A&M University 2004 44 7.4 Cash Flow Rule of Signs Test •The total number of real values i*’s is always less than or equal to the number of sign changes in the original cash flow series. •Follows from the analysis of a n-th degree polynomial •A “0” value does not count as a sign change •Example follows… 02/19/10 Authored by Don Smith, Texas A&M University 2004 45 7.4 C.F. Rule of Signs example •Consider Example 7.4 Year 0 1 2 3 Cash Flow $2,000 -$500 -$8,100 $6,800 + + Result: 2 sign changes in the Cash Flow 02/19/10 Authored by Don Smith, Texas A&M University 2004 46 7.4 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 02/19/10 Authored by Don Smith, Texas A&M University 2004 47 7.4 Norstrom’s Test •Norstrom’s Test1972 works with the accumulated cash flow •For the example problem the accumulated cash flow (ACF) is…. 02/19/10 Authored by Don Smith, Texas A&M University 2004 48 7.4 Accumulated CF Sign Test (ACF) •For the problem form the accumulated cash flow from the original cash flow. Year 0 1 2 3 Cash Flow $2,000 -$500 -$8,100 $6,800 Accum. C.F $2,000 $1,500 -$6,600 $200 Count sign changes here 02/19/10 Authored by Don Smith, Texas A&M University 2004 49 7.4 Accumulated CF signs Example A.C.F Year 0 1 2 3 Cash Flow $2,000 -$500 -$8,100 $6,800 Accum. C.F $2,000 $1,500 -$6,600 $200 + + + 2 sign changes in the ACF. 02/19/10 Authored by Don Smith, Texas A&M University 2004 50 7.4 ACF Sign Test States: •A sufficient but not necessary condition for a single positive i* value is: •The ACF value at year “N” is > 0 •There is exactly one sign change in the ACF 02/19/10 Authored by Don Smith, Texas A&M University 2004 51 7.4 ACF Test - continued •If the value of the ACF for year “N” is “0” then an i* of 0% exists •If the value of ACF for year “N” is > 0, this suggests an i* > 0 •If ACF for year N is < 0 there may exist one or more negative i* values – but not always 02/19/10 Authored by Don Smith, Texas A&M University 2004 52 7.4 ACF Test - continued •Alternatively: •If the ACF in year “0” < 0 •And…. •One sign change in the ACF series then •Have a unique i* value! 02/19/10 Authored by Don Smith, Texas A&M University 2004 53 7.4 ACF Test - continued •If the number of sign changes in the ACF is 2 or greater this strongly suggest that multiple rates of return exist. 02/19/10 Authored by Don Smith, Texas A&M University 2004 54 7.4 Example 7.4 – ACF Sign Test Year 0 1 2 3 Cash Flow $2,000 -$500 -$8,100 $6,800 Accum. C.F $2,000 $1,500 -$6,600 $200 2 Sign Changes here •Strong evidence that we have multiple i* values •ACF(t=3) = $200 > 0 suggests positive i* (s) 02/19/10 Authored by Don Smith, Texas A&M University 2004 55 7.4 Excel Analysis of Ex. 7.4 Year 0 1 2 3 Sum Cash Flow $2,000 -$500 -$8,100 $6,800 $200 ROR-Guess ROR NPV= 0% 7.47% $200.00 ROR-Guess ROR 30% 41.35% 02/19/10 •We find two i* values: •{7.47%,41.35%} First i* using a guess value of 0% Second i* value using guess value of 30% Authored by Don Smith, Texas A&M University 2004 56 7.4 Investment (Project) Balance •Examine the Investment or Project balances at each i* rate Inv Bal i-rate Year 0 1 2 3 02/19/10 7.47% Project Balances @ i* Rate $2,000.00 $1,649.36 -$6,327.47 $0.00 Inv Bal i-rate Year 0 1 2 3 41.35% Project Balances @ i* Rate $2,000.00 $2,327.04 -$4,810.69 $0.00 Authored by Don Smith, Texas A&M University 2004 57 7.4 Investment Balances at both i*’s Inv Bal i-rate Year 0 1 2 3 7.47% Project Balances @ i* Rate $2,000.00 $1,649.36 -$6,327.47 $0.00 •Terminal IB(7.47%) = 0 •Terminal IB(41.35%)=0 02/19/10 Inv Bal i-rate Year 0 1 2 3 41.35% Project Balances @ i* Rate $2,000.00 $2,327.04 -$4,810.69 $0.00 •Both i*’s yield terminal IB’s equal to 0! Authored by Don Smith, Texas A&M University 2004 58 7.4 Investment Balances at both i*’s •Important Observations •The IB’s for the terminal year (3) both equal 0 •Means that the two i* values are valid ROR’s for this problem •Note: The IB amounts are not all the same for the two i* values. •IB amounts are a function of the interest rate used to calculate the investment balances. 02/19/10 Authored by Don Smith, Texas A&M University 2004 59 7.4 PV Plot of 7.4 250.000 200.000 i* = 7.47% i*=41.35% PV - $$ 150.000 100.000 50.000 0.000 0.00 -50.000 0.20 0.40 Interest Rate 0.60 -100.000 -150.000 02/19/10 Authored by Don Smith, Texas A&M University 2004 60 7.4PV Plot - continued 250.000 200.000 PV < 0 150.000 100.000 50.000 0.000 0.00 -50.000 0.20 0.40 0.60 If the MARR is between the two i* values this investmen t would be rejected! -100.000 -150.000 02/19/10 Authored by Don Smith, Texas A&M University 2004 61 7.4 Comments on ROR •Multiple i* values lead to interpretation problems •If multiple i*’s – which one, if any is the “correct” one to use in an analysis? •Serves to illustrate the computational difficulties associated with ROR analysis •Section 7.5 provides an alternative ROR approach – Composite ROR (C02/19/10 Authored by Don Smith, Texas A&M University 2004 ROR) 62 Section 7.5 Composite ROR Approach •Consider the following investment +$9,000 +$8,000 0 5 -$10,000 02/19/10 1 2 3 4 Determine the ROR as…. Authored by Don Smith, Texas A&M University 2004 63 7.5 Composite ROR Approach •Analysis reveals: Year 0 1 2 3 4 5 Sum ROR-Guess ROR 02/19/10 Cash Flow -$10,000 $0 $8,000 $0 $0 $9,000 $7,000 i* = 16.82%/year on the unrecovered investment balances over 5 years 0% 16.82% Authored by Don Smith, Texas A&M University 2004 64 7.5 Composite ROR Approach: IB’s • The Investment Balances ar i* are: Year 0 1 2 3 4 5 Sum ROR-Guess ROR 02/19/10 Cash Flow -$10,000 $0 $8,000 $0 $0 $9,000 $7,000 0% 16.82% Project Balances @ i* Rate -$10,000.00 -$11,681.59 -$5,645.95 -$6,595.36 -$7,704.43 $0.00 •All IB’s are negative for t = 0 –4: IB(5) = 0 •Conventional (pure) investment Authored by Don Smith, Texas A&M University 2004 65 7.5 Composite ROR Approach • i* = 16.82% +$9,000 +$8,000 0 5 1 2 3 4 Question: Is it reasonable to -$10,000 assume that the +8,000 can be invested forward at 16.82%? 02/19/10 Authored by Don Smith, Texas A&M University 2004 66 7.5 Composite ROR Approach • Remember ……. •ROR assumes reinvestment at the calculated i* rate •What if it is not practical for the +$8,000 to be reinvested forward one year at 16.82%? 02/19/10 Authored by Don Smith, Texas A&M University 2004 67 7.5 Reinvestment Rates • Most firms can reinvest surplus funds at some conservative market rate of interest in effect at the time the surplus funds become available. •Often, the current market rate is less than a calculated ROR value •What then is the firm to do with the +$8,000 when it comes in to the firm? 02/19/10 Authored by Don Smith, Texas A&M University 2004 68 7.5 Reinvesting • Surplus funds must be put to good use by the firm •These funds belong to the owners – not to the firm! •Owners expect such funds to be put to work for benefit of future wealth of the owners 02/19/10 Authored by Don Smith, Texas A&M University 2004 69 7.5 Composite ROR Approach • Consider the following representation. Invested Funds The Firm Project Returns back 02/19/10 Authored by Don Smith, Texas A&M University 2004 70 7.5 Composite ROR Approach • Or, put in another context…. The Firm Project borrows from the firm Project Project Lends back to the firm 02/19/10 Authored by Don Smith, Texas A&M University 2004 71 7.5 Composite ROR Approach • Or, put in another context…. Project borrows from the firm The Firm At the i* rate (16.82%) Project Project Lends back to the firm But can the firm reinvest these funds at 16.82%? Probably not! 02/19/10 Authored by Don Smith, Texas A&M University 2004 72 7.5 Composite ROR Approach • So, we may have to consider a reinvestment rate that is closer to the current market rate for reinvestment of the $8,000 for the next time period(s) •Assume a reasonable market rate is say, 8% per year. •Call this rate an external rate - c 02/19/10 Authored by Don Smith, Texas A&M University 2004 73 7.5 The external rate - c • The external interest rate – c, is a rate that the firm can reinvest surplus funds for at least one time period at a time. •c is often set to equal the firm’s current MARR rate 02/19/10 Authored by Don Smith, Texas A&M University 2004 74 7.5 Composite ROR Approach • Thus, a procedure has been developed that will determine the following: •Find i* given c – if multiple ROR’s exist. The “/” is read “given” •For multiple i*’s in a problem, the i.e., i* given a analysis determines a single value for “c” c i* given •Denoted i*/c or, i’ •i’ is called the composite University 2004i*/c 75 rate = 02/19/10 Authored by Don Smith, Texas A&M 7.5 Composite ROR Approach • Finding i’ is a much more involved process •Prior to digital computers, only very small (N <= say 4-5 time periods) could be manually evaluated •Requires a recursive analysis best left to a computer program and spreadsheet •Example 7.6 illustrates a manual 02/19/10 approach Authored by Don Smith, Texas A&M University 2004 76 7.5 Example 7.6 • Cash Flow is : Year 0 1 2 3 02/19/10 Cash Flow $2,000 -$500 -$8,100 $6,800 Authored by Don Smith, Texas A&M University 2004 77 7.5 Ex. 7-5 Multiple i*’s • i*1 = 7.468; i*2 = 41.35% Year 0 1 2 3 02/19/10 Cash Flow ROR-Guess ROR NPV= $2,000 -$500 -$8,100 ROR-Guess ROR $6,800 0% 7.468% $200.00 30% 41.35% Authored by Don Smith, Texas A&M University 2004 78 7.5 Investment Balances are: • IB(7.468%) Inv Bal i-rate Year 0 1 2 3 02/19/10 7.468% Project Balances @ i* Rate $2,000.00 $1,649.36 -$6,327.47 $0.00 • IB(41.352%) Inv Bal i-rate Year 0 1 2 3 41.352% Project Balances @ i* Rate $2,000.00 $2,327.04 -$4,810.69 $0.00 Authored by Don Smith, Texas A&M University 2004 79 7.5 Project Balances @ 41.35% PB's - $ Project Balances $3,000 $2,000 $1,000 $0 -$1,000 -$2,000 -$3,000 -$4,000 -$5,000 -$6,000 0 1 2 Years 02/19/10 Overrecovered 3 Underrecovered Authored by Don Smith, Texas A&M University 2004 80 7.5 Project Balances @ 7.468%% Overrecovered Project Balances $4,000 PB's - $ $2,000 $0 -$2,000 0 1 2 3 -$4,000 -$6,000 -$8,000 Years 02/19/10 Underrecovered Authored by Don Smith, Texas A&M University 2004 81 7.5 Assume you Reinvest at 20% • c = 20% •Positive IB’s are reinvested at 20% •Not at the computed i* rate •Now, what is the modified ROR – i’? •Must perform a recursive analysis 02/19/10 Authored by Don Smith, Texas A&M University 2004 82 7.5 Recursive IB’s are…. •Let Ibj = Fj •F0 = +2000 (> 0; invest at 20%) •F1 = 2000(1.20) – 500 = +1900 •+1900 > 0; invest at 20% Underrecovered •F2 = 1900(1.20) – 8100 = -5820 •-5820 < 0; invest at i’ rate 02/19/10 Authored by Don Smith, Texas A&M University 2004 83 7.5 Recursive IB’s are…. Repeated from the previous slide for clarity •F2 = 1900(1.20) – 8100 = -5820 •-5820 < 0; invest at i’ rate • F3 =-5820(1+i’) + 6800 •Since N = 3, F3 = 0 •Solve; -5820(1+i’) + 6800 = 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 84 7.5 Single conditional ROR value • -5820(1+i) + 6800 = 0 • i’ = [6800/5820] –1 • i’ = 16.84% given c = 20% •If MARR = 20% and i’ = 16.84% this investment would be rejected •Reduced the problem to a single conditional ROR value. 02/19/10 Authored by Don Smith, Texas A&M University 2004 85 7.5 What if c = i*1 (7.47%)? • Here, we examine what happens IF we assume the reinvestment rate, c, equals one of the computed i* values. •This approach assumes that the reinvestment rate, c, is one of the i* rates •Recursive calculations follow…. 02/19/10 Authored by Don Smith, Texas A&M University 2004 86 7.5 Recursive IB’s are…. (@7.47%) •Let IBj = Fj •F0 = +2000 (> 0; invest at 7.47%) •F1 = 2000(1.0747) – 500 = +1649.40 •>0: Over-recovered balance •F2 = +1649.40(1.074) – 8100 = -6327.39 •Now, under-recovered balance Authored by Don Smith, Texas A&M University 2004 02/19/10 87 7.5 Recursive IB’s are…. •F2 = +1649.40(1.074) – 8100 = -6327.39 •F3 =-6327(1+i’) + 6800 •Since N = 3, F3 must = 0 (terminal IB condition) •Solve; -6327(1+i’) + 6800 = 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 88 7.5 Single conditional ROR value •Must solve: • -6327.39(1+i’) + 6800 = 0 •(1+i’) = 6800/6327.39 •i’ = 0.0747 = 7.47% •Given c = one of the i*’s, the i’ will equal the i* used as the reinvestment rate! 02/19/10 Authored by Don Smith, Texas A&M University 2004 89 Section 7.6 Rate of Return on a Bond Investment • Review Chapter 5 Section on Bonds •Bond problems represent a classical ROR type problem •Bond problems represent a conventional investment with a unique ROR •Example Follows: 02/19/10 Authored by Don Smith, Texas A&M University 2004 90 7.6 Example 7.8 • Purchase Price: $800/bond •4%, $1000 bond •Life = 20 years •Interest paid semiannually •Question: If you pay $800 per bond what is the ROR (yield) on this investment? 02/19/10 Authored by Don Smith, Texas A&M University 2004 91 7.6 Ex. 7.8: Cash Flow Diagram F40 = $1000 A = +$20/6 months 0 39 1 40 2 3 4 …. …. …. $800 $I/6 mos = $1000(0.04/2) = +$20.00 (every 6 months) 02/19/10 Authored by Don Smith, Texas A&M University 2004 92 7.6 Ex. 7-8: 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 •Nominal ROR/year = (2.87%)(2) = 5.74%/yr, C.S.A. •Effective ROR/year = (1.0287)2 – 1 = 5.82%/yr 02/19/10 Authored by Don Smith, Texas A&M University 2004 93 Chapter 7 Summary • ROR is a popular analysis method •Understood in part by practitioners •Presents computational difficulties •No ROR may exist •Multiple ROR’s may exist 02/19/10 Authored by Don Smith, Texas A&M University 2004 94 Chapter 7 Summary cont. • For cash flows where multiple i* values exist, one must decide to: •Go with one of the i* rates or, •Calculate the composite rate, i’ •If an exact ROR is not required, it is suggested to go with PW or AW 02/19/10 Authored by Don Smith, Texas A&M University 2004 95 Chapter 7 Summary cont. • For cash flows where multiple i* values exist, one must decide to: •Go with one of the i* rates or, •Calculate the composite rate, i’ •If an exact ROR is not required, it is suggested to go with PW or AW 02/19/10 Authored by Don Smith, Texas A&M University 2004 96 Chapter 7 Summary cont. • To have a possible i* value with a given cash flow, there must be at least one negative CF amount in the series •IROR can be demanding and is best accomplished using a computer algorithm •IROR is often misunderstood by practitioners but is a popular method of analysis 02/19/10 Authored by Don Smith, Texas A&M University 2004 97 Chapter 7 Summary cont. • In real world applications one may encounter a cash flow that possess no usable i* value! •These types of problems are difficult to explain to uninformed decision makers. 02/19/10 Authored by Don Smith, Texas A&M University 2004 98 ENGINEERING ECONOMY Sixth Edition Blank and Tarquin END OF SLIDE SET Mc Gra Hill w ...
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