Rate of Return
TOPIC 4
Rate of Return One Project
Rate of Return
Definition:
A relative percentage method which measures the yield as a percentage of investment
over the life of a project.
Example:
𝑃
𝑃
𝑃𝑊 = −100 + 60 , 𝑖%, 1 + 60 , 𝑖%, 2
𝐹
𝐹
𝒊 % = ???
Rate on Investment
Definition 1:
Rate of Return is defined as the interest earned on the unpaid balance of an installment loan.
Rate on Investment
Definition 2:
Rate of return is the break-even interest rate, i* which equates the present worth
of a project’s cash outflows to the present worth of its cash inflows.
Mathematical relation:
𝑃𝑊 𝑖 ∗ = 0
𝑃𝑊 𝑖 ∗
𝑐𝑎𝑠ℎ 𝑖𝑛𝑓𝑙𝑜𝑤𝑠
= 𝑃𝑊 𝑖 ∗
𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑓𝑙𝑜𝑤𝑠
𝐴𝑊 𝑖 ∗ = 0
𝐴𝑊 𝑖 ∗
𝑐𝑎𝑠ℎ 𝑖𝑛𝑓𝑙𝑜𝑤𝑠
= 𝐴𝑊 𝑖 ∗
𝑐𝑎𝑠ℎ 𝑜𝑢𝑡𝑓𝑙𝑜𝑤𝑠
Example
Investment Classification
Simple Investment
Non-simple Investment
▪ Initial cash flows are negative, and
only one sign change occurs in the
net cash flows series
▪ Initial cash flows are negative, but
more than one sign changes in the
remaining cash flow series
▪ ROR: A unique ROR
▪ ROR: A possibility of multiple RORs
Multiple ROR Values
Test 1: Descarte’s Rule of Signs
Test 2: Norstrom’s Criteria
Total number of real i* values is ≤ the
If the cumulative cash flow starts off
number of sign changes in net cash
negatively and has only one sign
flow series.
change, there is only one positive root.
Rate of Return Multiple Alternatives
Example:
Why Incremental Analysis is Necessary
Incremental Cash Flow for ROR Analysis
𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 = 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑩 − 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑨
Where alternative B is the larger initial investment
A
B
Incremental Cash Flow
B-A
First Cost, $
-40,000
-60,000
-20,000
Annual Cost, $/year
-25,000
-19,000
+6,000
8,000
10,000
+2,000
Salvage Value, $
This incremental ROR is
identified as ∆𝒊∗
The ROR on the extra
$20,000 investment
in B determines which
alternative to select
Equal-service life
are necessary when
performing ROR
analysis.
Therefore, the LCM
must be used.
−5000 + 1900
𝑃
𝑃
𝑃
, ∆𝑖 ∗ , 10 − 11000 , ∆𝑖 ∗ , 5 + 2000 , ∆𝑖 ∗ , 10 = 0
𝐴
𝐹
𝐹
𝐴𝑠𝑠𝑢𝑚𝑒 𝑖 ∗ = 12%
−5000 + 1900 5.6502 − 11000 0.5674 + 2000 0.3220 = 137.98 ≠ 0
𝐴𝑠𝑠𝑢𝑚𝑒 𝑖 ∗ = 14%
−5000 + 1900 5.2161 − 11000 0.5194 + 2000 0.2697 = −263.41 ≠ 0
PW will be zero between 12% and 14%.
Use straight-line interpolation:
Vendor B is selected.
∆𝑖 ∗ = 12.68%
>
MARR