NPV and IRR are commonly used methods for evaluating investment decisions, but each has limitations. NPV compares the present value of cash inflows to the investment cost, accepting projects where NPV is positive. IRR is the discount rate that sets NPV to zero, accepting projects where IRR exceeds the opportunity cost of capital. However, IRR can indicate acceptance of projects with negative NPV. Managers also consider payback period and accounting rate of return, but these methods ignore timing of cash flows. Cash flows rather than accounting profits drive NPV analysis. Overall, NPV is preferred but multiple methods are often examined when evaluating large capital investments.

1. Lecture 3
NPV &
Alternative Investment
Decisions Rules
Financial Management(N12403)
Lecturer:
Farzad Javidanrad (Autumn 2014-2015)

2. NPV (Revisited)
• Shareholders expect companies to invest in all projects that their benefits are more than their costs
because any investment of this sort increase the net worth of the company and increase the value of
each share kept by shareholders.
• Companies can assure their shareholders that they will invest in those projects that their Net Present
Value (NPV) is positive; i.e. the discounted (or present value) of their cash inflow is bigger than the
cost of investment:
𝑁𝑃𝑉 = 𝑃𝑉 − 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 > 0
Case 1: Just one investment (𝐼0) in the beginning:
𝑁𝑃𝑉 =
𝑖=1
𝑡
𝐶𝑖
1 + 𝑟 𝑖
− 𝐼0 > 0
Case 2: Series of investment during the life time of the project:
𝑁𝑃𝑉 =
𝑖=1
𝑡
𝐶𝑖
1 + 𝑟 𝑖
−
𝑗=0
𝑚
𝐼𝑗
1 + 𝑟 𝑗
> 0 (𝑚 ≤ 𝑡)
Note: If 𝑁𝑃𝑉 = 0 it means that the project may content the shareholders’ expectations .
According to the
NPV rule, a project
is acceptable if its
NPV is positive
(investing in it adds
to the net worth of
the company) and
should be rejected if
its NPV is negative.

3. Problems of the NPV Rule
• Some problems of Using the NPV rule are:
a) The rule is sensitive to the choice of the rate of return. If the rate is over-estimated
by the financial manager, some potentially good projects fail to satisfy the rule and
if it is under-estimated some potentially bad projects will get the positive result
from the rule. If the rate, is the hurdle rate (requested as the minimum rate of
return by the shareholders and the NPV is zero (or positive), the project satisfies
(over satisfies) the shareholders expectations.
b) The cash flows are just a prediction, so, the risk of using wrong predictions are
always need to be taken into account, specifically for long-term projects.
c) In an unstable economic and business environment (change of inflation, change of
profitability of different projects, change of costs, technological changes and etc.)
the power of the rule decreases.

4. Strengths of the NPV Rule
Uses Cash
Flows
• Cash Flows
are better
than
Earnings
Uses all Cash
Flows
• Other
approaches
ignore cash
flows
beyond a
certain date
Discounts
Cash Flows
• Fully
incorporates
the Time
Value of
Money
• The difference between Cash Flow and Accounting Flow:
Midland plc is an Irish firm that refines and trades gold. At the end of the year,
it sold 2,500 ounces of gold for €1.67 million. The company had acquired the
gold for €1 million at the beginning of the year. The company paid cash for the gold
when it was purchased. Unfortunately it has yet to collect from the customer to whom
the gold was sold. Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

5. Cash Flow & Accounting Flow
Accounting Flow
The Midland plc
Accounting View
Income Statement
Year End December 31
Sales €1,670,000
Costs €1,000,000
Profit € 670,000
Cash Flow
The Midland plc
Financial View
Income Statement
Year Ended December 31
Cash inflow € 0
Cash outflow -€1,000,000
€1,000,000
Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

6. NPV (Revisited)
• Positive NPV is not the only criterion for prioritising investment projects.
Chief Financial Officers (CFOs) usually consider other measurements such as
1) Project’s internal rate of return (IRR)
2) Project’s payback
3) Project’s book rate of return (Average accounting return)
4) Profitability index
• Among them NPV and IRR
are more popular.
Responses from 392 CFOs
across Canada and the US
(Source: Graham and Harvey, Journal of Financial Economics, 2001)

7. Internal Rate of Return (IRR)
• In lecture 1, we noted that the NPV rule can be expressed in terms of rate of return. The rule
is: “accept investment opportunities offering rates of return above their opportunity cost of
capital” (Brealy et. al, p.111)
• Rate of return is a discount rate that makes NPV=0.
• Rate of return for a single payoff is defined as:
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝑃𝑎𝑦𝑜𝑓𝑓
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
− 1
And the discount rate that makes NPV=0 (for simplicity, for just one period cash flow) can be
calculated as:
𝑁𝑃𝑉 =
𝐶1
1 + 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑅𝑎𝑡𝑒
− 𝐼0 = 0
So;
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑅𝑎𝑡𝑒 =
𝐶1
𝐼0
− 1
This discount rate is known as Internal Rate of Return (IRR).

8. Internal Rate of Return
• IRR for multi-period cash inflow can be calculated by solving the following equation
(either through trial and error or through specific software):
𝑁𝑃𝑉 =
𝑖=1
𝑡
𝐶𝑖
1 + 𝐼𝑅𝑅 𝑖
− 𝐼0 = 0
• IRR Rule: If IRR is bigger than the opportunity cost of capital 𝑟 the project is
profitable, i.e. 𝐼𝑅𝑅 > 𝑟
• Note that the IRR, as a measure of profitability, depends on the amount of cash inflow for a
project and timing of that but the opportunity cost of capital depends on the rate of return of
other similar assets in the market.
IRR
Discount Rate %
NPV
r
There should be a negative
relationship between IRR and NPV.
In Microsoft Excel we
can use the IRR
command to calculate
this rate.

9. Issues with IRR
a) IRR does not differentiate between borrowing and lending.
Project B should not be accepted as it has negative NPV but for both projects IRR=50%
because:
𝐴: −1000 +
+1500
1 + 𝐼𝑅𝑅
= 0 → 𝐼𝑅𝑅 = 50%
𝐵: +1000 −
1500
1 + 𝐼𝑅𝑅
= 0 → 𝐼𝑅𝑅 = 50%
But in project B there is a positive relationship between IRR and NPV.
Project Lending(-)
Borrowing(+)
Cash flow
(1st year)
IRR NPV at 10%
A -1000 +1500 +50% +364
B +1000 -1500 +50% -364
Adopted from Brealy et al. , p.113

10. Issues with IRR
b) For some projects, if there is a negative
cash flow (such as the cost of
decommissioning/cleaning) after the end
of the project’s life, there could be a double
rates of return, which makes financial
managers confused.
c) The IRR rule can be misleading if there are mutually exclusive projects (cannot be
invested simultaneously) with different cash flows and initial investment.
d) If there is more than one opportunity cost for a project at different years we do not
know which one need to be compared with IRR.
Adopted from http://cfatutor.files.wordpress.com/2013/07/multipleirr.png
Project 𝑰 𝟎
Cash flow
(1st year)
IRR NPV at
10%
D -10,000 +20,000 100% +8,182
E -20,000 +35,000 75% +11,818
Adopted from Brealy et al. , p.114

11. Payback Rule
• Is it reasonable to buy a new car for £8,000 if the average daily cost of
transport for a family of three is £10? By Saving £10 daily, how long does it
take to cover £8000?
• The cost of the car can be covered within 2 years and 3 months through £10
saving per day (how?). This is payback period for the above investment.
• A project’s payback period is the number of years required to compensate the
initial cost of investment by accumulating cash inflow of the project (or in our
example, accumulating of savings).
• If a manager decides to use the payback rule he/she needs to define a cut-off
period which is the time-limit for reimbursing the initial cost of investment.

12. Payback Rule
• If the payback period for a project is more than cut-off period, the project cannot be
accepted. This means the cumulative returns after t years (cut-off period) is less than
the initial investment: 𝑖=1
𝑡
𝐶𝑖 < 𝐼0
• Consider three different projects A, B and C as following. At a 10% opportunity cost
of capital which project will be selected if the cut-off period is 2 years?
• 𝑁𝑃𝑉 𝐴 = −3000 +
800
1.10
+
1500
1.102 +
4000
1.103 ≈ 1972.2
• 𝑁𝑃𝑉 𝐵 = −3000 +
1500
1.10
+
2300
1.102 +
1000
1.103 ≈ 15.8
• 𝑁𝑃𝑉 𝐶 = −3000 +
1800
1.10
+
2200
1.102 +
1000
1.103 ≈ 288.5
Project 𝑰 𝟎 𝑪 𝟏 𝑪 𝟐 𝑪 𝟑
According to the payback rule,
projects B & C are acceptable but
project A should be rejected but
based on their NPV, project A is
the best.

13. Issues with Payback Rule
a) The result of the payback rule depends on the date is chosen as the cut-off date because the
rule ignores all the cash inflow after the that date. In our example, if the cut-off date was
the third year, all projects would be profitable.
b) The rule does not consider the time value of money as it gives equal weight to all cash
inflows before the cut-off date.
c) If we do not consider the life of projects, using this rule could be very misleading and it may
lead the managers to accept very poor projects. Different projects are equally attractive
as long as the rule approves them.
d) In the projects with multi-period investments (e.g. investment in the middle of the period)
this rule cannot be applied.
e) It does not take the shareholders’ expected rate of return into the account.
• A modified version of this method is the Discounted Payback Method, which brings the PV of
the cash inflows (discounted cash inflow) into the calculation before applying the cut-off
period. It is simple and uses time value of money.

14. Book Rate of Return (Average Accounting Return)
• According to this method a project is acceptable if its average accounting return is greater
than or equal to the target return and it will be rejected if it is less than that.
•Consider a company that is evaluating
whether to buy a store in a new shopping
centre. The purchase price is £500,000. We
will assume that the store has an estimated
life of five years and will need to be
completely scrapped or rebuilt at the end of
that time. For simplicity sake, the asset will
depreciated using straight line depreciation
(this does not occur in countries that use
IFRS but suits to illustrate the method).
The Target Return on new Investments is
15 percent.
Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

15. AAR Rule
Step 1
• Determine Average Net Income
Step 2
• Determine Average Investment
Step 3
• Determine Average Accounting Return
Step 3: Determine Average Accounting Return
AAR=
£50,000
£250,000
= 20%
Step 2: Determine Average Investment
(£500,000+400,000+300,000+200,000+100,000+0)/6=£250,000
Step 1: Determine Average Net Income
[£100,000 + 150,000 + 50,000 + 0 + (-50,000)]/5 = £50,000
Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

16. AAR Rule
Average
Accounting
Return is 20%
Target
Accounting
Return is 15%
Accept
Strengths
• Simple return based
measure
Weaknesses
• Does not use cash flows
• Does not use time value
of money
• Arbitrary target rate
Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

17. Profitability Index Rule
• In mathematics we have:
𝑎 > 𝑏 ↔
𝑎 − 𝑏 > 0
𝑜𝑟
𝑎
𝑏
> 1
• NPV rule states that a project is profitable if 𝑖=1
𝑡 𝐶 𝑖
1+𝑟 𝑖 − 𝐼0 > 0. the profitability
index rule states a project is profitable if the present value of all cash inflows (during
the project’s life) is bigger than the investment:
𝑖=1
𝑡 𝐶𝑖
1 + 𝑟 𝑖
𝐼0
> 1
• Similar to NPV rule, this rule would be misleading if the expectation regarding the
future returns could not be fulfilled.

18. Some Rules When Applying NPV
• Rule 1: Only cash flow is relevant*.
Cash flow is the difference between cash received and cash paid out and should not be confused
by accounting income (earning) (see slide No. 5). In accounting income, which shows how well
the company is operating, depreciation cost is deducted each year (as cash outflow) but in NPV
we need to record capital expenditure when they occur and not later as depreciation cost.
How to calculate Net Cash Flow? (for a good example see Brealy’s book, Chapter 6, section 6-2, IM&C’s Fertilizer Project, Page
137-139 or see Hillier’s book, Chapter 7, section 7.2, Whair Balls Ltd: An Example, Page 184-189)
• Net cash flows comes through operations or investments or through change in working
capitals, in fact:
Net Cash Flow=Cash flow from capital investment and disposal + Operating cash flow + Cash flow
from change in working capitals
• Remember that the working capital for a company is the set of short-term/current assets and
liabilities (see the lecture 1, the balance sheet example).

19. Some Rules When Applying NPV
• The first component of this sum is easy to calculate but there are different ways to calculate the
operating cash flow:
𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒏𝒈 𝑪𝒂𝒔𝒉 𝑭𝒍𝒐𝒘 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 – 𝐸𝑥𝑝𝑒𝑛𝑠𝑒𝑠 – 𝑇𝑎𝑥𝑒𝑠
(𝑂𝐶𝐹 = 𝑅 − 𝐸 − 𝑇)
Which is, in fact, the profit after tax deduction.
Or
𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒏𝒈 𝑪𝒂𝒔𝒉 𝑭𝒍𝒐𝒘 = 𝐴𝑓𝑡𝑒𝑟 − 𝑇𝑎𝑥 𝑃𝑟𝑜𝑓𝑖𝑡 + 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛
(𝑂𝐶𝐹 = 𝐴𝑇𝑃 + 𝐷)
Or
𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒏𝒈 𝑪𝒂𝒔𝒉 𝑭𝒍𝒐𝒘 = (𝑅𝑒𝑣𝑒𝑛𝑢𝑒 – 𝐸𝑥𝑝𝑒𝑛𝑠𝑒𝑠) × (1 – 𝑇𝑎𝑥 𝑅𝑎𝑡𝑒) + (𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 × 𝑇𝑎𝑥 𝑅𝑎𝑡𝑒)
(𝑂𝐶𝐹 = (𝑅 − 𝐸) × (1 − 𝑡) + (𝐷 × 𝑡))
Which has the same meaning but in this method, the tax is imposed as a percentage of profit and not as
a lump-sum.
In Hillier et al (2013) we have:
𝑅 − 𝐸 = 𝐸𝐵𝐼𝑇 (Earning Before
interest and taxes)
The Top-Down
Approach
The Bottom-up
Approach
Net Income
The Tax Shield
Approach

20. Some Rules When Applying NPV
• The third component of the sum (cash flow from changes in working capital), includes three
items: Inventory, Accounts Receivable and Accounts Payable, i.e.:
𝑊𝑜𝑟𝑘𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 = 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 + 𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑠 𝑅𝑒𝑐𝑒𝑖𝑣𝑎𝑏𝑙𝑒 − 𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑠 𝑃𝑎𝑦𝑎𝑏𝑙𝑒
(𝑊𝐶 = 𝐼𝑛𝑣 + 𝐴𝑅 − 𝐴𝑃)
• Net working capital is the amount of short-term liquid assets which a business should have
in order to pay for unexpected expenses or even planned and short-term obligations.
• On the other hand:
𝑊𝑜𝑟𝑘𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑅𝑎𝑡𝑖𝑜 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐴𝑠𝑠𝑒𝑡𝑠
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠
• If the ratio is below one, it means the working capital is negative and the business may run
into trouble if it continues to have more liabilities than assets. On the other hand, if the ratio
is very high and remains high for many years, it reflects that the financial manager does not
know how to use the assets to create more value for the shareholders.

21. Some Rules When Applying NPV
• Rule 2: Estimate cash flows on an incremental basis.
Any additional cash flows after the acceptance of the project should be considered.
Furthermore, all elements that have impacts on the cash flow should be considered such as
taxes, side effects (positive or negative), the age and the state of the project considering the
history behind that and its perspective, opportunity costs, salvage value, overhead costs (rent,
heat, light, admin costs, management and supervisory salaries) and etc..
• The side effects could be positive (synergy) or negative (erosion). For example, if a company
decides to launch a new version of its product, decrease in the demand for the older version
of the product should be considered into the incremental cash flow.
• Many investments projects generate incremental cash flow long time after the initial
investment such as services and spare parts sale. These need to be considered into the cash
flow.
• At the end of the project it is possible to sell or re-use many real assets (such as equipment)
and this salvage value (after tax) is a positive inflow to the company. Any extra benefit or cost
should be considered.
• Sunk cost has already happened in the past and they cannot be changed by accepting or
rejecting the project and it should be ignored.

22. Some Rules When Applying NPV
• Rule 3: Inflation should be always part of calculation.
Interest rates and discount rates are usually quoted in nominal terms and the real rates should
be adjusted for inflation, using Fisher’s equation:
𝑅𝑒𝑎𝑙 𝑅𝑎𝑡𝑒 =
1 + 𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑅𝑎𝑡𝑒
1 + 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒
− 1 ⟹ 𝑟 =
1 + 𝑖
1 + 𝜋 𝑒 − 1
We can re-write this formula as:
1 + 𝑟 =
1 + 𝑖
1 + 𝜋 𝑒
• The cash flows should also be adjusted for inflation, because £100 in 2008 does not have the
purchasing power of £100 in 2014 . We need to remember that inflation makes forecasting
future cash flow very complicated. Inflation also change the hurdle rate and force share
holders to demand more return.

23. Some Rules When Applying NPV
• Nominal cash flow should be discounted by nominal discount rate and real cash flow should
be discounted by real discount rate. In each case, the PV should be the same, because:
𝐶 𝑁 = 1 + 𝜋 𝑒 𝑡 × 𝐶 𝑅
Where 𝐶 𝑁 is a nominal value and 𝐶 𝑅 is the real value, adjusted for the expected inflation rate
𝜋 𝑒, at year 𝑡. Now, if we try to find the PV of a value at year 𝑡, using nominal and real values, we
will have:
𝑃𝑉 =
𝐶 𝑁
1 + 𝑖 𝑡
=
1 + 𝜋 𝑒 𝑡 × 𝐶 𝑅
1 + 𝑖 𝑡
=
𝐶 𝑅
1 + 𝑖
1 + 𝜋 𝑒
𝑡 =
𝐶 𝑅
1 + 𝑟 𝑡
• If we want to have the real purchasing of 𝐶 𝑅 in year 𝑡, we need to calculate the correct
nominal value 𝐶 𝑁 at that year as the shareholders (like other people) always get happy with
the nominal values (they might have different expectations about the inflation rate!!!).

24. Example on Real & Nominal Discounting
o Shields Electric Forecasts the Following cash flows on a particular project: (Example 7.7,
Hillier et all 2013, p.192)
The nominal discount rate is 14 percent, and the inflation rate is forecast to be 5
percent. What is the value of the project?
• Nominal Cash Flows:
2
£600 £650
£26.47 £1, 000
1.14 (1.14)
   
Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

25. Example on Real & Nominal Discounting
£600
1.05
 
 
  2
£650
(1.05)
 
 
 
Real Discount Rate: (1.14/1.05) – 1 = 8.57143%
NPV:
2
£571.43 £589.57
£26.47 £1, 000
1.0857143 (1.0857143)
   
Adopted from Hillier’s PPT , The McGraw-Hill Companies, 2012

26. Some Rules When Applying NPV
• Rule 4: Investment and financing decisions should be always separated.
The project cash flow does not depend on how you finance it. If the project is
financed through borrowing the amount of debt (original and interest) should
not be considered as cash outflow and deducted from each year return. A project
should be treated as fully equity-financed by stockholders.