Chapter 7 - Rate of
Return Analysis
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EGR 403 Capital Allocation Theory
Dr. Phillip R. Rosenkrantz
Industrial & Manufacturing Engineering Department
Cal Poly Pomona
EGR 403 - The Big Picture
• Framework: Accounting & Breakeven Analysis
• “Time-value of money” concepts - Ch. 3, 4
• Analysis methods
–
–
–
–
Ch. 5 - Present Worth
Ch. 6 - Annual Worth
Ch. 7, 8 - Rate of Return (incremental analysis)
Ch. 9 - Benefit Cost Ratio & other techniques
• Refining the analysis
– Ch. 10, 11 - Depreciation & Taxes
– Ch. 12 - Replacement Analysis
EGR 403 - Cal Poly Pomona - SA9
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Three Major Methods of
Economic Analysis
• PW - Present Worth
• AW - Annual Worth
• IRR - Internal Rate of Return
If P = A(P/A, i, n)
Then (P/A, i, n) = P/A
Solve for (P/A, i, n) and look up
interest in Compound Interest Tables
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Internal Rate of Return (IRR)
• The interest rate paid on the unpaid balance of a
loan such that the payment schedule makes the
unpaid loan balance equal to zero when the final
payment is made. Ex: P = $5000, i = 10%, n = 5
Year
1
2
3
4
5
6
Principal Prin. Paid
5000.00
818.99
4181.01
900.89
3280.13
990.97
2289.15 1090.07
1199.08 1199.08
0.00
Int Paid
500.00
418.10
328.01
228.92
119.91
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Payment
1318.99
1318.99
1318.99
1318.99
1318.99
0.00
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Calculating Rate of Return
• The IRR is the interest rate at which the
benefits equal the costs. IRR = i*
PW Benefit - PW Cost = 0
PW Benefit/PW Cost = 1
NPW = 0
EUAB - EUAC = 0
PW Benefit = PW Cost
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Calculating IRR - Example 7-1
• PWB/PWC = 1
• 2000(P/A, i, 5)/8200 = 1
• (P/A, i, 5) = 8200/2000 =
4.1
• From Table, IRR =
7%
From Compound Interest Tables
Interest rate
(P/A,i,5)
6%
7%
8%
4.212
4.100
3.993
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Calculating IRR - Example 7-2
Sometimes we have more than one factor in our equation.
When that happens we cannot solve for just one factor.
If we use: EUAB - EUAC = 0
100 + 75(A/G, i, 4) - 700(A/P, i, 4) = 0
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Calculating IRR - Example 7-2 (cont’d)
• No direct method for calculating. Use trial and error
and iterate to get answer.
• Try i = 5%:
100 + 75(A/G, 5%, 4) - 700(A/P, 5%, 4) = + 11
+ 11 is too high. The interest rate was too low
• Try i = 8%
100 + 75(A/G, 8%, 4) - 700(A/P, 8%, 4) = - 6
- 6 is too low. The interest rate was too high
• Try i = 7%
100 + 75(A/G, 8%, 4) - 700(A/P, 8%, 4) = 0
Therefore IRR = 7%
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Calculating IRR - Example 7-3
• Example 7-3 shows a series of cash flows that does not match any
of our known patterns. We must use trial and error.
• Using NPW = 0, suppose we start with i = 10% . NPW = + 10.16,
which is too high.
• Using i = 15%, NPW = - 4.02. IRR is between 10% & 15%
• The iterations may be graphed and the true IRR will be indicated
at the point where the NPW curve = 0.
Yr CF
0 - 100
1 + 20
2 + 20
3 + 30
4 + 40
5 + 40
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Calculating IRR - Example 7-3 (Cont’d)
• We can use linear interpolation to find estimate
the point where the curve crosses 0.
• IRR = i* = 10% + (15%-10%)[10.16/(10.16 +
4.02)] = 13.5%
• This is a linear interpolation of a non-linear
function so the answer is slightly inaccurate,
but good enough for decision making here
(after all, the guesswork in our future cash
flows introduces uncertainty in the analysis).
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Calculating IRR - Example 7-3 (Cont’d)
• To get an exact answer, we can use the IRR function in EXCEL
• Select the IRR function from the fx icon.
• Block the column on the spreadsheet that has the cash flows for
all years.
• The function returns the IRR.
-100
20
30
20
40
40
13.47% =IRR(A1:A6)
The IRR function in
EXCEL allows you to
evaluate the return of
investments very easily
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Calculating IRR for a Bond - Example 7-4a
Bond Costs and Benefits:
Purchase price = $1000
Dividends = $40 every six months
Sold after one year for $950
Calculation of Periodic interest rate & IRR:
m = 2 compounding periods/year
1000 = 40(P/A, i, 2) + 950(P/F, i, 2)
By trial and error and interpolation i*  1.5%
IRR Nominal rate = 2 x 0.015 = 0.03 (3%)
IRR Effective rate = (1 + 0.015)2 - 1 = 0.0302 (3.02%)
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Example 7-4a EXCEL Solution
• Use IRR function to find periodic IRR (i)
• Find nominal using r = i * m
• Use EFFECT function to find effective interest rate
Period
0
1
2
Buy/sell Dividend
-1000
40
950
40
Total
-1000
40
990
1.52%
3.04%
3.06%
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periodic
nominal
effective
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Rate Of Return (ROR) Analysis
• Most frequently used measure of merit in
industry.
• More accurately called Internal Rate of
Return (IRR).
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Calculating ROR
• Where two mutually exclusive alternatives will
provide the same benefit, ROR is performed using an
incremental rate of return (DROR) on the difference
between the alternatives.
• You cannot simply choose the higher IRR alternative.
Two-alternative
situation
Decision
DROR  MARR
Choose higher-cost
alternative
DROR < MARR
Choose lower-cost
alternative
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The Minimum Attractive Rate of
Return (MARR)
• The MARR is a minimum return the
company will accept on the money it invests
• The MARR is usually calculated by
financial analysts in the company and
provided to those who evaluate projects
• It is the same as the interest rate used for
Present Worth and Annual Worth analysis.
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ROR on Alternatives With Equivalent Benefits
Example 7-5: Consider the lease vs. buy situation. MARR = 10%
• Leasco: Lease for five years for 3 annual payments of $1000 each
• Saleco: Purchase up front for $2783
• Both alternatives have a $1200/year benefit for 5 years
Year
0
1
2
3
4
5
IRR/period
Cash flow - Cash flow - Cash flow alternative alternative alternative
A (Leaseco) B (Saleco)
B-A
-$1,000.00
$200.00
$200.00
$1,200.00
$1,200.00
$1,200.00
-$2,783.00
$1,200.00
$1,200.00
$1,200.00
$1,200.00
$1,200.00
-$1,783.00
$1,000.00
$1,000.00
$0.00
$0.00
$0.00
48.72%
32.60%
8.01%
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Example 7-5 (Cont’d)
• Cannot simply pick the highest IRR if alternatives have
different investment costs
• Must examine the incremental cash flows!!
• Subtract the cash flows for the “Lower First Cost”
alternative from the cash flows of the “Higher First Cost”
alternative to obtain the “Incremental Cash Flow” or D.
• Compute the IRR on the incremental cash flow. This is
the DROR.
• For this problem the DROR is 8.01% which is <
MARR, therefore choose the lower cost alternative.
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Example 7-5 (Cont’d)
• Q. Why did we do this?
• A. Both alternatives were acceptable compared only to
the MARR. Since either alternative will work, the
question is whether we want to spend the additional
$1783 to go from the lower cost to the higher cost
alternative. The benefit for doing so is the savings of
two years of $1000 lease payments. Essentially we are
getting an 8.01% return on that $1783 investment. The
company can get 10% ROR on its money elsewhere, so
reject the increment. That is, spend $1000 now on
Leaseco and invest the other $1783 for a higher return.
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Analysis Period
• Just as in PW and AW analysis the analysis period
must be considered:
– Useful life of the alternative equals the analysis period.
– Alternatives have useful lives different from the
analysis period.
– The analysis period is infinite, n = .
For an example of that uses a
common multiple of the
alternate service lives, see
Example 7-10. EXCEL would
be useful here because of the
irregularity of the cash flows.
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7-10
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