Coming now to section 4, investment decision criteria.
And this is the largest section in this reading. Analysts
use several important criteria to evaluate capital
investments. Some known metrics or some known
methods are given right here: NPV, IRR payback and
discounted payback, profitability index, and AAR.
We will talk about each of these. Let's start
with NPV. We have seen this in quantitative methods,
but we will cover it again over here. Net present value
is the present value of the future after-tax cash
flows minus the initial investment today. To come
up with an NPV we need a discount rate or an interest
rate. Let's see in our example that is 10% We are considering
a project where we invest 100 today and after
one year we will make 10, after two years we make 10 again,
and after three years we make 110. What is the NPV
of this project? If you look at the definition the
net present value is the present value of the future
after-tax cash flows minus the investment outlay.
What is the present value of all these numbers? The
present value of the cash flow at period 1 is 10 divided
by 1.1 because we have a discount rate of 10%. The present
value of the second cash flow is 10 / 1.1 squared because
this cash flow is happening at the end of two years.
The present value of the cash flow which happens
after 3 years is 110 / 1.1 cubed. The sum of these three
represents the present value of the future or after-tax
cash flows. From this, we subtract the initial outlay
of hundred and when you put all these four numbers
together, you have the NPV. If you do the calculation,
what you will actually get is 0. This particular project
has an NPV of 0 which means that the present value of
the future cash flows is equal to the initial cost.
If we had higher cash flows then the NPV would be positive.
If we had lower cash flows then the NPV would be negative.
This brings us to what's called the NPV decision
rule. For independent projects a positive NPV means
that you should accept the project. Positive NPV
means that if you consider all the future cash flows,
their present value is greater than the cost of the
initial outlay. And therefore it makes sense to
accept the project. A negative NPV means that the
present value of the future cash flows is less than
your initial outlay and there you clearly reject
a project. What do you do for mutually exclusive
projects? There clearly you accept the project
with the higher NPV, as long as the NPV is positive.
Let us consider this example. You are given a cost of
capital of 10%. Here is what you need to do. The initial
cost is thousand. So you say - 1000 + 500 divided
by 1.1 because 10% is the discount rate. So you do
1 + 10% which is 0.1 plus 400 divided by 1.1 squared plus
300 divided by 1.1 cubed plus 100 divided by 1.1 ^ 4. You do the calculation
and you will come up with the NPV and you can use the
same process for project B. As you can see this
is a very long-winded process and if you start doing
this on the exam chances are that you will not finish
on time. What you should do instead is learn how to
use the calculator and we've seen this in quantitative
methods also, therefore, you can consider this a review.
Here are the keystrokes that you need to enter. You
hit the cash flow button, which is right here second
clear work to make sure all the cash registers are cleared
and here is what you should then see. You enter thousand
plus minus that has a way of saying that the first
cash flow is -1000. Down arrow, 500, enter. This sets the first cash
flow to 500. Frequency: 1 because the 500 is only
happening once. You enter all the other cash flows
and then once all the cash flows have been entered
you hit the NPV button. The calculator will ask you
for the interest rate you plug in 10 and hit enter
and down arrow, compute. You should get the NPV of 78.82.
Make sure that you are very comfortable doing
this. Practice coming up with the NPV for project
B, and the answer is given on the previous slide.
Next, internal rate of return or IRR, IRR measures
the return for a given project. Let me go back to a
simple example that I discussed earlier. Let's
say you invest 100 and then you get 10 at the end
of year one,10 at the end of year two and 110
at the end of year three. What is the IRR for this project?
Notice that you are getting a 10% return. With a 100
investment you get 10, 10, and then 10 plus your 100 back. So
clearly the return is 10% which means that IRR is
equal to 10%. What I have done here is Illustrated
the fundamental idea behind IRR, which is the fact
that it is giving you a measure of return. The second
bullet point gives you a slightly more complicated
definition of IRR which you also need to understand
IRR is the discount rate that makes the present value
of the future cash flows equal to the investment
outlay. Let us see how this statement works in the
context of our example. What is the initial outlay? The initial
outlay is 100. What is the present value of the future
cash flows? That is equal to 10 / (1 + r) The r is what
we will be calculating. + 10 / (1 + r) squared. + 110 / (1 +r) cubed
The value of r which makes the present value
of the future cash flows equal to the initial investment
is the IRR. Notice that if you plug in 10% or 0.1 for r
you will actually get a hundred. So by plugging in
you recognize that the IRR is 0.1 or 10%. Clearly this
is very hard to do and we are going to learn how to use
the calculator but the point of going through this
exercise is to illustrate the fact that this statement
does indeed hold true. Another way of expressing
this statement is to say that IRR is the discount
rate which makes NPV equal to zero. Notice that what
I wrote here could also be written as follows.
-100 + 10/(1 + r) + 10/(1 + r) squared + 110/ (1 + r) cubed and
all this is equal to 0. What I've done on the right hand
side is simply rewritten this expression. The left
side of the equation is simply the NPV of the project.
So the discount rate which will make NPV equal to
zero is the IRR rate. Finally, what is the IRR decision
rule? If the IRR is greater than the required rate
of return then accept the project. Simplistically
right now think of the required rate of return as
the return that you require or think of it as the cost
of raising money. In other words, if this $100
can be raised at an 8% rate. It costs you 8%
to say borrow 100. Would you do a project which
gives you a return of 10%? The answer is yes, if your
IRR is greater than the required return of 8% then
you accept the project. On the other hand if your required
return was 12% because it costs you 12% to raise your
100 then it does not make sense to do the project
and that's the second sub-bullet over here. If IRR
is less than the required rate of return then you
should reject the project. What you are looking
at here is the decision rule for independent projects.
What about the IRR decision rule for mutually exclusive
projects? What you do there is select the project
with the higher IRR as long as the IRR is greater
than the required rate of return. Let us consider
this example. Given what you just learned how would
you come up with the IRR of project A and project B?
Here is what you would do. This is the long-winded
method. We said that the IRR is that rate which makes
the present value of the future cash flows equal
to the initial investment. The initial investment
is 1,000, present value of future cash flows would
be 500 / (1 + r) + 400 / (1 + r) squared + 300 / (1 + r) cubed + 100/ (1 + r) ^ 4.
This is extremely difficult to solve and you might plug in
some numbers for r to figure out which number will
give you 1,000. But the best way to solve this
problem is to use the calculator. For project A you
again enter the cash flow the way we entered cash
flows for the NPV example. Here we do not enter any
interest rate because the rate is what we are computing.
So once you've entered all the cash flow then hit IRR
and compute and the calculator gives you the IRR
Just once in your life, you can then go plug this rate
into the long-winded formula and convince yourself
that it works. For practice I want you to compute
the IRR for project B because you need to be very
comfortable using the calculator. The answer you
should get is 11.79%.