Part II: Evaluating business & engineering assets Ch 5: Present worth analysis

Ch 6: Annual equivalence analysis

Ch 7: Rate-of-return analysis – – Methods for finding rate of return – Internal rate-of-return criterion – Incremental analysis for comparing mutually exclusive alternatives

Rate of return (RoR): a relative percentage method that measures the as a percent of over the life of a project Comparing mutually exclusive alternatives based on IRR

n Project A1 Project A2 0 -$1,000 -$5,000 Cash flow 1 $2,000 $7,000 RoR 100% 40% PW(10%) $818 $1,364

Question: Assume that you have $5,000 in your investment pool. Should you invest in project A1, which provides a higher RoR or in project A2, which has a higher PW? Incremental investment

Incremental investment n Project A1 Project A2 (A2 – A1)

0 -$1,000 -$5,000 -$4,000 1 $2,000 $7,000 $5,000

ROR 100% 40% 25% PW(10%) $818 $1,364 $546

Assuming a MARR of 10%, you can always earn that rate from other investment source, i.e., $4,400 balance at the end of one year for $4,000 investment. By investing the additional $4,000 in A2, you would make an additional $5,000, which is equivalent to earning at the rate of 25%. Therefore, the incremental investment in A2 is justified. Incremental analysis (procedure)

Step 1: Compute the cash flow for the difference between the projects (A,B) by subtracting the cash flow of the lower investment cost project (A) from that of the higher investment cost project (B).

Step 2: Compute the IRR on this incremental investment

(IRRB-A).

Step 3:Accept the investment B if and only if

IRRB-A > MARR

NOTE: Make sure that both IRRA and IRRB are greater than MARR. Ex 7.7 - Incremental rate of return

n B1 B2 B2 - B1

0 -$3,000 -$12,000 -$9,000 1 1,350 4,200 2,850 2 1,800 6,225 4,425 3 1,500 6,330 4,830 IRR 25% 17.43% 15%

Given MARR = 10%, which project is a better choice?

Since IRRB2-B1=15% > 10%, and also IRRB2 > 10%, select B2. IRR on increment investment: three alternatives

n D1 D2 D3 Step 1: Examine the IRR for each project to eliminate 0 -$2,000 -$1,000 -$3,000 any project that fails to meet the MARR. 1 1,500 800 1,500 Step 2: Compare D1 & D2: 2 1,000 500 2,000 IRRD1-D2= 27.61% > 15%, so D1 is better 3 800 500 1,000 Step 3: Compare D1 & D3: IRR 34.37% 40.76% 24.81% IRRD3-D1= 8.8% < 15%, so D1 is best Practice problem

You are considering four types of engineering A B C D designs. Initial cost $150 $220 $300 $340 The project lasts 10 years with the estimated cash Revenues/yr $115 $125 $160 $185 flows at the right. $70 $65 $60 $80 The rate (MARR) is Expenses/yr 10%. Which of the four is more attractive? Incremental analysis for cost-only projects

Items CMS Option FMS Option Annual O&M costs: labor $1,169,600 $707,200 material 832,320 598,400 overhead 3,150,000 1,950,000 tooling 470,000 300,000 inventory 141,000 31,500 income taxes 1,650,000 1,917,000 Total annual costs $7,412,920 $5,504,100 Investment $4,500,000 $12,500,000 Net salvage value $500,000 $1,000,000

Since we assume revenues would be the same for each project, these are “cost-only” projects Cannot calculate IRR unless revenue given Ex. 7.9 – Incremental cash flow (FMS – CMS)

Incremental n CMS Option FMS Option (FMS-CMS) 0 -$4,500,000 -$12,500,000 -$8,000,000 1 -7,412,920 -5,504,100 1,908,820 2 -7,412,920 -5,504,100 1,908,820 3 -7,412,920 -5,504,100 1,908,820 4 -7,412,920 -5,504,100 1,908,820 5 -7,412,920 -5,504,100 1,908,820 6 -7,412,920 -5,504,100 $2,408,820 Salvage + $500,000 + $1,000,000 Ex. 7.9 – IncrementalSolution: cash flow (FMS – CMS)

PW(i)FMS-CMS = –$8,000,000 + $1,908,820(P/A,i,5) + $2,408,820(P/F,i,6) = 0

IRRFMS-CMS = 12.43% < 15%

Select CMS Ultimate decision rule:

If IRR > MARR, accept

This rule works for any investment situation

In many situations, IRR = RoR but this relationship does not hold for an investment with multiple ROR’s. Resolution of multiple RoR’s (Chapter 7A)

Net investment: project balances (PB’s) computed at i* are

all < 0 throughout investment, with A0 = 0 Also called pure investment, i.e. firm does not overdraw on its return & “borrow” from the project A positive balance at some time during the project indicates that the firm acts as a borrower, i.e. mixed investment

nABCD 0 -$1,000 -$1,000 -$1,000 -$1,000

1 -$1,000 $1,600 $500 $3,900 2 $2,000 -$300 -$500 -$5,030 3 $1,500 -$200 $2,000 $2,145 i*, % 33.64 21.95 29.95 10, 30, 50 Ex. 7A.1 Project balances

A: pure B: mixed * * nAn PB(i ) nAn PB(i ) 0 -$1,000 -$1,000 0 -$1,000 -$1,000 1 -$1,000 -$2336 1 $1,600 -$381 2 $2,000 -$1122 2 $300 $164 3 $1,500 0 3 $200 0

C: pure D: mixed * * nAn PB(i ) nAn PB(i ) 0 -$1,000 -$1,000 0 -$1,000 -$1,000 1 $500 -$800 1 -$3,900 $2,400 2 $500 -$1,539 2 -$5,030 -$1,430 3 $2,000 0 3 $2,145 0 Need for an external

In prior analyses, case borrowed (released) from a project was assumed to earn i* This may not be possible, since external may earn less than i* That is, the rate of return on the project is generally higher than that from external investments Thus it may be necessary to calculate project balances for a project’s cash flow at 2 rates of interest: one on internal & one on external investments By separating these interest rates, can compute the true rate of return (true IRR) on internal investments, or the return on invested capital (RIC) Steps to calculate IRR for a mixed investment

1. Identify MARR, or external rate

2. Calculate PB(i, MARR)n or simply PBn

PB(i, MARR)0 = A0

PB0(1+i) + Ai if PB0<0 PB(i, MARR)i = { PB0(1+MARR) + Ai if PB0>0

PBn-1(1+i) + An if PBn-1<0 PB(i, MARR)n-1 = { PBn-1(1+MARR) + An if PBn-1>0

3. Determine i by solving he terminal project balance equation

Accept a project if IRR > MARR Ex. 7A.2 – reconsider Ex 7.6

i* =20% n = 0 n = 1 n = 2 Beg. balance -$1,000 +$1,100 Interest -$200 +$220 Payment -$1,000 +$2,300 -$1,320 End balance -$1,000 +$1,100 $0

PB(i, 15%)0 = –$1,000,000

PB(i, 15%)1 = –$1,000,000(1 + i) + $2,300,000 = $1,300,000 – $1,000,000i = $1,000,000(1.3 – i)

If i < 1.3 -> PB(I,15%)1 > 0 PB(i, 15%)2 = $1,000,000(1.3 – i)(1 + 0.15) – $1,320,000 = $175,000 – $1,150,000i = 0 IRR = 0.1522 Æ 15.22%

If i < 1.3 -> PB(I,15%)1 > 0 PB(i, 15%)2 = $1,000,000(1.3 – i)(1 + i) – $1,320,000 = -$20,000 + $305,000i – $1,000,000i2 = 0 IRR = 0.1 or 0.2, which violates the assumption that Æ i > 1.3 Summary

Rate of return (ROR) is the interest rate earned on unrecovered project balances such that an investment’s cash receipts make the terminal project balance zero. Rate of return is an intuitively familiar & understandable measure of project profitability that many managers prefer to NPW or other equivalence measures. Mathematically we can determine the rate of return for a given project cash flow series by locating an interest rate that equates the net present worth of its cash flows to zero. This break-even interest rate is denoted by i*. Summary (cont.) Internal rate of return (IRR) is another term for RoR that stresses the fact that we are concerned with the interest earned on the portion of the project that is internally invested, not those portions that are released by (borrowed from) the project. To apply rate of return analysis correctly, we need to classify an investment into either a simple or a non- simple investment. A simple investment is defined as one in which the initial cash flows are negative and only one sign change occurs in the net cash flow, whereas a non-simple investment is one for which more than one sign change occurs in the net cash flow series. Multiple i*’s occur only in non-simple investments. However, not all non-simple investments will have multiple i*’s either. Summary (cont.) For a simple investment, the solving rate of return (i*) is the rate of return internal to the project; so the decision rule is: If IRR > MARR, accept the project. If IRR = MARR, remain indifferent. If IRR < MARR, reject the project. IRR analysis yields results consistent with NPW and other equivalence methods. For a nonsimple investment, because of the possibility of having multiple rates of return, we need to calculate the true IRR, or known as “return on invested capital.” However, if your objective is to make an accept or reject decision, it is recommended either the NPW or AE analysis be used. When properly selecting among alternative projects by IRR analysis, incremental investment must be used.