Financial Management MGT201
SOME SPECIAL AREAS OF CAPITAL BUDGETING
In this lecture, we will discuss some special areas of capital budgeting in which the calculation
of NPV & IRR is a bit more difficult. These concepts will be explained to you with help of numerical
As it is mentioned in the previous lectures that we are studying the area of capital budgeting as
it relates to projects, which means investments in real assets (land, property etc.) The major difficulty in
the NPV calculation is your ability to forecast the cash flows. Therefore, it is necessary that one should
spent time on this so that the cash flow forecast is accurate.
We, have a simple formula to calculate the cash flows. The way we define the net Incremental
after tax cash flows for the purposes of this course is
Net After-tax Cash Flows = Net Operating Income + Depreciation + Tax Savings from Depreciation +
Net Working Capital required for this project + Other Cash Flows
The things we left out from the formula given above are certain incidental cash flows (Include
Opportunity Costs and Externalities but Exclude Sunken Costs.)
Two Major Criteria of Capital Budgeting:
1. Net Present Value (NPV)
2. Internal Rate of Return (IRR)
a. Combined View: NPV Profile (NPV vs i Graph)
The NPV is the most important because it has a direct link with shareholders wealth maximization.
Let us discuss in detail about the difficulties faced in NPV & IRR with the help of certain
numerical examples and explanations.
First, we would discuss the case of Multiple IRRs.
In this case, you have a project with certain cash flows that are not normal and when
you try to calculate IRR you obtain more than one IRR answer. This is the case where you have more
than one sign change taking place in your cash flow diagram. Sign change means that you have two
adjacent arrows one of them is downward pointing & the other one is upward pointing. In general, our
cash flow diagram starts with down ward pointing arrow (Investment) and it is followed with series of
upward pointing arrows (net incoming cash) during the life of project. However, during the life of
project if you have any net cash outflow or downward pointing arrow then that would be second sign
change and you can expect to have multiple answer for IRR.
In this particular case, calculating the NPV and setting it equal to zero to calculate IRR will give
you two answers & both of them would be wrong.
The alternative is to use Modified IRR or MIRR approach.
The logic behind MIRR is that instead of looking at net cash flows you look at cash inflows and
outflows separately for each point in time. Discount all the Outflows during the life to the present and
Compound all the Inflows to the termination date. Assume reinvestment at a Cost of Capital or Discount
Factor (or Required Return) such as the risk free interest rate.
The MIRR represents the discount rate, which will equate the Future Value of cash inflows to
Present Value of cash outflows.
(1+MIRR) n = CF in * (1+k) n-t
CF out /(1+k) t
Modified Internal Rate of Return (MIRR) would provide us with an answer, which is entirely different
from our previous IRR calculations
A project with the following cash flows: Initial Investment = -Rs100, Year 1 = +Rs500, Year 2 = -
If we use standard NPV equation to calculate the IRR
IRR Equation: NPV = 0 = -100 + 500/ (1+IRR) - 500/ (1+IRR) 2
You would come up with 2 answers
Financial Management MGT201
IRR = 38% and 260%
Both of these answers are incorrect. Therefore, we will use the modified IRR approach to calculate the
actual IRR for this project.
MIRR Approach (Assume Cost of Capital k = 10%):
(1+MIRR) n =
CF in * (1+k) n-t
CF out / (1+k) t
We use 1.1 as compound factor because we assume "i"=10% = Risk free rate return. Here`t' refers to
the time in which a particular cash flow occurs, while `n' is the total life span of the project.
(1+MIRR) 2 = 500 * (1+0.1)2-1
(100 / 1.1) + (500 / (1+0.1)2
(1+MIRR) 2 = 550 / 513 = 1.07
MIRR = 0.0344 = 3.44%
This answer is entirely different from the previous answers that we got from calculating the IRR.
However, MIRR gives you the best possible answer and the most realistic too.
Now, let us talk about the case of comparing projects with different lives.
NPV of Projects with Different Lives:
Suppose that you have two projects having different life spans. It is not entirely accurate to
calculate NPV's in simple manner and to compare them and pick the project with higher NPV. Because
you are comparing a certain project that is generating cash flows for a short period of time with another
project that is yielding cash flows over a longer time. We use following two approaches to rank these
kinds of projects.
Common Life Approach:
In this approach, the idea is quite simple. You need to bring all the projects to the same
length in time. In other words, you are required to convert all the projects to the identical life
span. You can do that by finding least common multiple for common life. For example, if you
are comparing two projects one has life of 4 years and the other, which has a life of 5 years, the
least common multiple is 20 years. Sketch out the cash flow diagram and repeat the cash flow
for each of the project such that they fit in exact number of time in 20 years. In case of project
with a life of 4 years, you can replicate the cash flows 5 times in a period of 20 years. . In case
of project with a life of 5 years, you can replicate the cash flows 4 times in a period of 20 years.
Compute the NPV of each project over the common life and choose the project with the highest
Equivalent Annual ANNUITY (EAA) Approach:
In this case, our logic is to find out that for a particular project of limited life giving you
the certain net present value calculated in a simple way, what kind of yearly annuity gives the same
NPV. You can then compare annual annuity of each project and choose the highest. You are comparing
cash flow of two projects both of which are taking place in a period of one year only. You can also
convert the cash flows of the project to the perpetuity, which is infinite, and then you can compare the
NPV's like of different projects. That is also correct since life spans are identically infinite.
We have 2 Projects with following Cash Flows:
Project A: Io= - Rs100, Yr 1 = +Rs200
Project B: Io= - Rs200, Yr1= +Rs200, Yr2= +Rs200
Simple NPV Computation (assuming i=10%):
NPV Project A = -100 + 200/1.1 = +Rs 82
NPV Project B = -200 + 200/1.1 + 200/ (1.1)2 = +Rs 147
Conclusion from Simple (or Normal) NPV Calculation is that Project B is better. It is incorrect because
here we are comparing apples to the oranges since the project lives are different!
Common Life Approach:
Common Life Span=Least common multiple = 2 Years (because this is the shortest cycle in which both
project lives can exactly be replicated back to back).
Financial Management MGT201
In this Cash Flow Pattern of A is repeated exactly 2 times to cover the life of the longer Project
B. The project A's outflow 100 & inflow of 200 then we replicate it with down ward pointing arrow
with 100 and upward pointing arrow with 200 amount in the 2nd year. Project B remains unchanged
Common Life (C.L.) NPV's:
Project A C.L. NPV = -100 + [(200-100)/1.1] + 200/ (1.1)2 = +Rs 156
Project B C.L. NPV = Same as before = +Rs 147
Now our conclusion has changed! After doing the Common Life NPV, Project A looks better. The
Simple NPV of Project A was + Rs 82 but after increasing its life to match Project B's, the NPV of
Project A increased. It is the correct answer. Also, note that how the NPV of A increased from 82 to 156
(almost double) because you double the life of the project.
Now we solve this problem with Equivalent Annual Annuity Approach
Equivalent Annual Annuity Approach:
In this we are explaining that how we can achieve same NPV value from an annuity stream.
Here, we are doing a back calculation that we knew the NPV's but which annuity stream they are
representing with in the life span of the project. Then we compare the annual annuity of both projects.
The life span remains same
Start with the Simple (or Normal) NPV's calculated earlier (at i = 10%):
Project A Simple NPV = + Rs 82
Project B Simple NPV = + Rs 147
To find EAA
Multiply the Simple NPV of each project by the EAA Factor
EAA FACTOR = (1+ i) n / [(1+i) n -1] where n = life of project & i=discount rate
Project A's EAA Factor = 1.1 / (1.1-1) = 11
Project B's EAA Factor = 1.12 / (1.12-1) = 5.76
EAA for each project
Project A's EAA = Simple NPV * EAA Factor = 82*11= + Rs 902
Project B's EAA = 147*5.76 = + Rs 847
Conclusion: Project A is better. Same conclusion as Common Life Approach but of course the
numbers for EAA and NPV are different.
Companies and individuals running different types of businesses have to make the choice of the asset
according to the life span of the project. For instance, a tailor shop owner would have to decide whether
to invest in a sewing machine that has a useful life of ten years or to invest in another machine with a
Financial Management MGT201
useful life of three years. These decisions are important since they involve major cash outflows of the
business. There are advantages & disadvantages associated with different life span.
Different Lives & Budget Constraint:
Companies and individuals running different types of businesses have to make the choice of the
asset according to the life span of the project.
Advantages of asset with a long life:
The advantage of a longer asset life is that the cash flows from the project become more
predictable, since there are lesser cash outflows occurring during the life of the project.
Disadvantage of asset with very long life:
It does not give you the opportunity (or option) to extract full value of asset and replace the
equipment quickly in order to keep pace with technology, better quality, and lower costs.
Advantages of asset with short life
The advantage of a short life asset is that the investor, by making reinvestment in the asset of a
superior quality, lowers down the costs and updates the project to the new technological requirements.
Disadvantage of assets with very short life:
The disadvantage is that the money will have to be reinvested in some other project with an
uncertain NPV and return so it is risky. If a good project is not available, the money will earn only a
minimal return at the risk free interest rate.
While exercising the option of different project timing, the projects can be compared by applying
Common Life and EAA Techniques to quantitatively.
We have been addressing the issue of capital budgeting with very idealistic assumptions. In
practical life, individuals and companies have a limited amount of money and limited human resources
in terms of either skill or numbers. It can be argued that the firm can also meet their requirements by
borrowing. IN real life, managers may avoid borrowing to limit their risk exposure. This prevents them
from undertaking projects with high positive NPVs that would have added to the firm's value and
maximized shareholder wealth!
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