NPV and IRR
Dan Saunders
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1.1
Net Present Value
Why “Net”?
The word “net” in N P V is a reminder that there is a cash flow at time t = 0. Typically, we think
of N P V when evaluating the undertaking of a project. Usually, that requires a large outlay of
capital now in exchange for a series of positive cash flows in the future. Hence, “net” means “net
of the cost now ”. We can think of many applications for the N P V that do not necessarily conform
to this story. Nevertheless, this simple model says that:
N P V = CF0 + P V
1.2
Basics
When applying the N P V to a real-world problem, the bulk of the work will be figuring out the
series of cash flows. However, for the sake of understanding the theoretical aspects of N P V , we
will abstract from this process, and take as given the sequence of cash flows:
{CF0 , CF1 , ..., CFT }
Then the only thing left is to know the cost of capital of the firm, r, so that these cash flows
may be properly discounted:
T
X
CFt
N P V = CF0 +
(1 + r)t
t=1
So long as N P V > 0, the project creates wealth for the firm. Specifically, the N P V is the present
value of that wealth, which allows the expected value-added to be absorbed into today’s share price.
We can also compare two projects by their net present values. We will always prefer project a to
project b if N P Va > N P Vb , so long as no other criteria affect the decision.
Eg 1. The cost of capital is r = 0.1. Suppose a new project costs CF0 = −$1, 000, but it will
produce an infinite stream of $90 payments. Is it worthwhile?
∞
X
$90
$90
= −$1, 000 +
= −$1, 000 + $900 = −$100
N P V = −$1, 000 +
1.1t
0.1
t=1
This project has a negative N P V and, therefore, should not be undertaken.
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1.3
Politics of NPV
That seems simple, so why bother with anything else? Well, the cost of capital for a firm is
valuable, private information. Any N P V that uses the wrong discount factor will be worthless. On
the other hand, management will always try to minimize the number of people who have access to
this knowledge. Therefore, a blunt tool, called the “internal rate of return”, is often used in order
to preserve the private information.
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2.1
Internal Rate of Return
Basics
Definition. The IRR is that rate which solves equates the net present value with zero:
0 = CF0 +
T
X
t=1
CFt
(1 + IRR)t
Many basic rules of intuition turn out to be true. So long as the problem is about undertaking
a project, the IRR will be greater than the cost of capital if-and-only-if the net present value is
positive. On the other hand, if the problem is one of lending, then the rules must be reversed.
One major limitation of the IRR is that it cannot be compared across projects, i.e., IRRa >
IRRb tells us nothing about N P Va or N P Vb .
Eg 2. Suppose a firm has a cost of capital of r = 0.2. It has to decide between two projects.
Project one costs $1, 000 now, and it will generate $1, 400 in revenue next year. Project two costs
$500 now, and it will generate $800 in revenue next year. Both share the same net present value:
N P V1 = N P V2 = $166.67
However, project one has a significantly lower internal rate of return:
IRR1 = 0.4 < 0.6 = IRR2
To see it’s usefullness, notice that:
IRR1 > r, N P V1 > 0 and IRR2 > r, N P V2 > 0
Thus, the IRR is a blunt indicator of whether or not to undertake any individual project, but it is
meaningless to compare across projects.
2.2
Calculating IRR
We are tasked with solving the equation from the definition above. First, we will move CF0 inside
of the summation:
T
X
CFt
0=
(1 + IRR)t
t=0
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Since each cash flow may be different, this is not an annuity, and we cannot simplify it using
a formula. Suppose we attempt to express the right-hand-side of this equation with a common
denominator:
PT
CFt (1 + IRR)T −t
0 = t=0
(1 + IRR)T
We can then cross-multiply to get rid of the denominator:
0=
T
X
CFt (1 + IRR)T −t
t=0
Let’s replace our index with T = N and t = N − n, so that the problem is now:
0=
N
X
CFN −n (1 + IRR)n
n=0
Define the coefficients by (a0 , a1 , ..., aN ) = (CFN , CFN −1 , ..., CF0 ), i.e. let an = CFN −n ∀n, and
let x = 1 + IRR. Then we are solving an N th degree polynomial, in the hopes that there is at least
one (ideally, only one) real-valued solution to the problem:
0=
N
X
an xn
n=0
Thus, the difficulty associated with calculating the internal rate of return is just a modern manifestation of an ancient mathematical problem which cannot be solved analytically, but whose solution
is easily approximated using modern computing.
Eg 3. Suppose a project costs $1, 000 now, and it will pay $1, 000 tomorrow and the next day.
What is the internal rate of return? Well, this can be solved analytically:
0 = −$1, 000 +
$1, 000
$1, 000
+
1 + IRR (1 + IRR)2
Multiplying through by (1 + IRR)2 , we see that we have to solve a quadratic equation:
0 = −$1, 000 (1 + IRR)2 + $1, 000 (1 + IRR) + $1, 000
| {z } |
{z
} | {z } | {z } | {z }
a
+b
x2
x
+c
Using the quadratic formula to solve in terms of (1 + IRR) yields:
p
−1000 ± 10002 − 4(−1000)(1000)
1 + IRR =
2(−1000)
Therefore, we have two choices, and it is clear which is correct:
IRR = −1.62 or IRR = 0.62
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