Rate of Return Analysis

1
Rate of Return Analysis

• Chapter 9 and Appendix A1

2
OUTLINE

• Definitions
• Computation of Breakeven Interest Rate
• Direct Solution
• Trial and Error
• Computer Solution
• IRR Criterion
• Mutually Exclusive Alternatives

3
Rate of Return (ROR)

• Internal Rate of Return(IRR) Marginal Efficiency
  of Capital Yield are same concept
• ROR is the interest rate earned on the unpaid
  balance of an amortized loan
• ROR is the break-even rate, i, which equates
  the PW of a projects cash outflows to the PW of
  its cash inflows
• Cost of capital or target return known as
  Minimum Acceptable Rate of Return (MARR)

4
N Service life of the project
n Year
i MARR Minimum Attractive Rate of
Return
A1 AN (1i) (1i)N
PW(i) A0

0
5

• IRR is the interest rate charged on the
  unrecovered project balance of the investment
  such that, when the project terminates, the
  unrecovered project balance is zero, let i 10

6

• 10 is earned on 10,000 during year 1
• 10 is earned on 6,979 during year 2
• 10 is earned on 3,656 during year 3
• the firm earns a 10 ROR on funds that remain
  internally invested in the project or
• if a project is financed with funds costing 10
  annually, the cash generated by the investment
  will be exactly sufficient to repay the principal
  and the annual int. in 3 years

7

• Observations
• Only one cash outflow at time 0 with a PW of
  10,000
• Three equal inflows of 4,021
• PW of these cash flows is 10,0004021(P/A,10,3)
  
• NPWPWinflows-PWoutflows0

8

• Simple Investments
• Initial cash flows are negative, and only one
  sign change occurs in the net cash flow series
  the opposite is Simple Borrowing
• Non-simple Investments
• More than one sign change occurs in the cash flow
  series

PW(i)
PW(i)
i1
i3
i
i2
i
non-simple
simple
9
Computation of IRR

• Direct Solution
• Trial and Error Solution
• Computer Solution

10
Direct Solution

• Example

11
Trial and Error Method

• Aiming for i that makes PW(i)0
• Guess a value of i
• Compute the PW of net cash flows
• Observe if PW is , -, or zero
• PW(i) is negative, lower the interest rate
• PW(i) is positive, raise the interest rate
• Continue until PW(i) is approximately zero

12
Example
2.8
(Units in millions)
After tax net cash flows
1.8
1.8
1.8
1.8
1.8
1.8
1.8
Guess i8
0
1
2
3
4
5
6
7
8
Sale value 1
Years
10
PW(8) -101.8(P/A, 8, 8)1(P/F, 8, 8)
0.88
Since PW is positive, raise the interest rate
Assume i12
PW(12) -101.8(P/A, 12, 8) 1(P/F, 12,
8) -0.65
Use interpolation
Check PW(i) with this i, iterate if necessary.
Computer value 10.18
13

• Graphical Method for i
• Create the NPW profile with i on horizontal axis
  and NPW on vertical axis
• For given cash flows calculate NPW at various i
  values
• Point at which the curve crosses i-axis
  approximates the the i

14
IRR Criterion

• Accept/Reject decision rule for simple
  investments
• i gt MARR, project acceptable (NPWMARR gt 0)
• i lt MARR, project is unacceptable (NPWMARR lt 0)
• For simple investments, i is the computed IRR
• Minimum Acceptable Rate of Return, MARR

15

• If IRR gt MARR, accept the project
• If IRR MARR, remain indifferent
• If IRR lt MARR, reject the project
• Nonsimple Investments
• multiple rates of return ambiguous situation
• use external interest rate on reinvested cash
  flows
• Use NPW criterion with MARR (much safer or graph
  NPW vs i across a long horizon)
• refer to Example 9.6 in the text

16
Example
MARR 15

i
10
20
Neither 10 nor 20 represents true IRR
17
Use NPW Criterion
Hence, the project is marginally acceptable
Mutually Exclusive Projects

• IRR and NPW may not give consistent results

IRR ignores the scale of the investment --
relative measure
18
Mutually Exclusive Projects

• Decision Rules
• For projects A and B where B is more costly
• Express B A (B-A)
• B has two CF components the same as A and the
  increment (B-A)
• B is preferred to A when ROR on the incremental
  component (B-A) gt MARR
• Compute the IRR on incremental Invest.

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• If IRRB-A gt MARR, select B
• If IRRB-A MARR, select either one
• If IRRB-A lt MARR, select A
• where B-A is an investment increment (-CF)

NOTE Want the first flow of the incremental CF
to be negative (investment flow) to be able to
calculate IRR
20
Example
MARR10
Solve and obtain iB2-B1 15 (simple
investment)
Since IRRB2-B1 gt MARR, we select B2
Alternatively could have measured for B1 and B2
the NPW at MARR and accepted the largest NPW in
excess of zero.
21
NPW Profiles
B2
Select B1
NPW
iB2-B115
B1
0
Interest Rate,
Select B2
PW(i)B2 gt PW(i)B1
22
IRR for Nonsimple Investments
Appendix A1

• Predicting Multiple is
• Net Cash Flow Rule Sign
• the number of real i (that are greater than
  100) is never greater than the number of sign
  changes in the sequence of the An. A zero cash
  flow is ignored
• Accumulated Cash Flow Sign Test
• If the accumulated net CF start negatively and
  changes sign only once, a unique positive i
  exists.