In our previous article (What Makes an Investment Attractive?) we
concluded that a business will be more interesting the more money it produces,
the quicker the money is received and when its return sufficiently compensates
for the risk taken. In this piece I will explain the Net Present Value method
which helps deciding whether or not an investment is attractive. This is the
most important and most complex article of this module so it must be read
carefully.
Suppose we have €100 to invest at a 12% yield over one year. This
means that at the end of a year we will get €112, that is, our initial
investment of €100 plus €12, which corresponds to the return on the initial investment
(12% of €100).
Just as €112 is the amount you receive by investing €100 for one year
with a yield of 12%, €100 is the amount you would have to invest at a 12% yield
in order to get €112 at the end of a year.
If we call C the money to be
received at the end of a year (€112), PV
the invested amount (€100)
and r the return (12%), the previous
expression can be generalized as:
PV = C/(1+r)
PV
(or €100 in our example) is known as the "Present Value"
resulting from "discounting" C (or €112) at r% (or
12%) for one year.
As stated before, PV is the
amount of money that must be invested to get the expected sum after a year given
an established return. But it can also be understood as the amount in today's
money that is equivalent to the amount to be received at the end of one year,
taking into account the risk that is being taken.
The return r (12% in our
case) is called the "discount rate". The discount rate depends
on the risk being taken. We can assume that in our example, the risk taken
demands a yield of at least 12%. In other words, the discount rate corresponding
to the risk of the investment is 12%. A riskier investment would have required
a higher discount rate and a less risky investment a lower discount rate.
Why the PV formula is
important?
Let´s call A our previous
investment opportunity. Imagine you are offered another investment opportunity,
which we call B. B also requires an initial investment of €100 but produces €120 at
the end of a year. However, B is
riskier than A and thus requires a
discount rate of 15%.
Applying the formula to calculate the present values of A and B we obtain,
PV(A) = €112/(1+0.12) = €100
PV(B) = €120/(1+0.15) = €104.35
Note that €104.35 is greater than €100. That is, the PV of B is greater than the PV
of A.
Given that the present value represents the amount in today's money
that is equivalent to the money that is to be received in the future (taking
into account the risk) and the present value of B is greater than the present value of A, then we can conclude B
is preferable to A.
In A, €100 are invested to
get the same €100 in today's money terms, so that the surplus was zero. This
means that return was exactly equal to the minimum required (12%).
In B, €100 are invested to
obtain €104.35 in today's money terms, so that there was a €4.35 surplus. The
fact that there is a surplus implies that the investment has yielded more than
the minimum required (over 15%). In fact, it is clear that the yield was 20%
(the €100 invested became €120 after a year).
Then we can also affirm that B
is better than A because B has produced more than the minimum
required while this was not the case of A.
The surplus is the result of subtracting the initial investment from
the present value. This surplus is called "Net Present Value".
The Net Present Value (NPV) is useful
for comparing investments when the the initial investment is not the same.
In our example, the result might have favored A if the amount to invest in B
would have been higher. For example, if the initial investment for B were €110 (instead of €100) its NPV would have been:
€104.35 - €110 = -€5.65,
This is worse than a zero NPV
for investment A. So now B is preferable.
Formally, the net present value is calculated as follows.
For investment A:
NPV(A) = -€100 + €112/1.12 = €0
And for investment B (when
investing € 110):
NPV (B) = -€110+€120/1.15 = -€110 + €104.35 = -€5.65
The general formula is:
NPV = -C(0) + C(1)/(1+r) ,
where we differentiate between C0
the amount invested (at time "zero") and C1 the amount received at the
end of the first year (at time "1").
The general rule is that when
comparing two investment opportunities, the preferable option will be the one
with that higher NPV.
But the story does not end here. An investment being “preferable” does
not necessarily mean that it must be acceptable.
To explain this let´s return to case B, when €110 were invested. NPVB
was negative (-€5.65) meaning that the PV
(€104.35) is less than the amount invested (€110). This means that the inverted
sum failed to produce the minimum required return, or what is the same, that
the yield was below 15%. In fact, if we make the computations we see that the
€110 invested produced only a yield of 9.1% since 9.1% of €110 is equal to €10
(for a total received of €120).
We would have had to invest €104.35 to get the minimum demanded return
of 15% since the difference between €120 and €104.35 corresponds exactly to 15%
of €104.34.
Also, remember that when the initial investment was €100, NPVB was positive. Applying
the NPV formula we get:
NPV(B) = -€100 + €120/1.15 = -€100 + €104.35 = +€4.35
That is, the PV (€104.35) is
now greater than the amount invested (€100). As we know, this time the yield
was 20%, well above the minimum requirement of 15%.
This allows us to conclude that when the NPV is zero, the investment produces exactly the required minimum
return (equal to the discount rate). When the NPV is negative, the investment produces less than the minimum
required return, and when the NPV is positive
the investment produces more than the minimum required.
An investment is acceptable only when it produces at least the minimum
required return. That is, when its NPV
is equal to or greater than zero. The investment must be rejected if the NPV is negative.
The NPV method is very
useful because it allows comparison of investment alternatives with different
risks and different amounts invested or received. In fact it is a method widely
used by analysts when making investment decisions.
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