ECON 4120
Applied Welfare Econ & Cost Benefit Analysis
Memorial University of Newfoundland
Chapter 6
Discounting Future Benefits and Costs
Discounting
Purpose: Some practical issues one must know in order to
compute the net present value (NPV) of a project
It assumes the social discount rate is given, which is reasonable
as the rate is often set by an public agency
The chapter covers: the basics of discounting (two-periods);
compounding and discounting over multiple periods (years);
the timing of benefits and costs; horizon (terminal) values;
comparing projects with different time frames; inflation and
the difference between nominal and real dollars; relative price
changes; and sensitivity analysis in discounting
Discounting
Appendix 6A provides shortcut formulas for
calculating the present value of annuities and
perpetuities
You are responsible for going through this
appendix yourselves
BASICS OF DISCOUNTING
Projects with Lives of One Year
 Discounting takes place over periods not
years. However, for simplicity, assume that
each period is a year. First we discuss projects
that last one year.
 There are three possible methods to evaluate
potential projects: future value analysis,
present value analysis and net present value
analysis. Each gives the same answer.
PV =
FV
(1+ i)
BASICS OF DISCOUNTING

Future Value Analysis – Choose the project with the largest future value,
FV, where the future value in one year of an amount X invested at interest
rate i is:
FV = X (1 + i)
(6.1)

Present Value Analysis – Choose the project with the largest present
value, PV, where the present value of an amount Y received in one year is:
PV = Y/(1 + i)
(6.2)

If the PV of a project equals X, and the FV of a project equals Y, both
equations (6.1) and (6.2) imply: PV=FV /(1 + i)

This equation shows that discounting (the process of calculating the
present value of future amounts) is simply the opposite of compounding
(the process of calculating future values).
BASICS OF DISCOUNTING

Net Present Value Analysis – Choose the
project with the largest net present value,
which calculates the sum of the present values
of all the benefits and costs of a project
(including the initial investment):

NPV = PV(benefits) – PV(costs)
(6.3)
BASICS OF DISCOUNTING
Usually projects are evaluated relative to the status quo:
 If there is only one new potential project and its
impacts are calculated relative to the status quo, it
should be selected if its NPV > 0, and should not be
selected if its NPV < 0.
 If the impacts of multiple, mutually exclusive
alternative projects are calculated relative to the
status quo, one should choose the project with the
highest NPV, as long as this project’s NPV > 0.
 If the NPV < 0 for all projects, one should maintain
the status quo.
COMPOUNDING AND DISCOUNTING OVER
MULTIPLE YEARS



Interest is compounded when an amount is invested
for a number of years and the interest earned each
period is reinvested.
Interest on reinvested interest is called compound
interest.
The future value, FV, of an amount X invested for n
years with interest compounded annually at rate i is:
FV= X (1+i)n
(6.4)
Present value again




Present value: the amount a specified
money sum received at a future date is
worth today
$1 today can turn into $1(1+r) in a year’s
time right?
Therefore, we know that the present value
(PV) of $1(1+r) of next year is $1 of today
This is because $1 is equal to
1+
r
$ 1+r
Present value vs Future Value


When we translate the future into the
present, we discount the future value (we
make it look smaller to get the PV)
When we translate the present into the
future, we compound the present value (we
make it look bigger to get the FV)
Present value vs Future Value



But we do have a translating device to
make values smaller to the right extent,
when calculating the PV of a sum received
in the following period
The discount factor 1/(1+r)
which is given by the discount rate r
Discount rate



Discount rate: the value placed upon current
consumption, relative to future consumption
It measures how strongly we prefer money now
to later given our possibilities to transform
money today into money tomorrow
If this rate is high, we will only trade off money
now only for lots of money later, since we know
how much we lose by waiting
Discount factor


Discount Factor - The factor that translates
expected benefits or costs in any given
future year into present value terms
The discount factor is equal to 1/(1 + r)t
where r is the interest rate and t is the
number of years from the current year until
the future year
Discount factor


The discount factor is equal to 1/(1 + r)t
where r is the interest rate and t is the
number of years from the current year until
the future year
The higher r and or the higher t the smaller
the discount factor, so the bigger the
difference between the future value and the
present value
Discounting




So what is the present value of $100
received in 10 years time from now if the
discount rate is 3% (not much higher than
the real bank interest rates)? And $10 at a
5%?
$10/(1.05)10 =$6.14
what about these $10 in 5 years?
$10/(1.05)5 =$7.84
NET PRESENT VALUE


The net present value of an investment is
equal to the difference between benefits
received in several periods and costs faced
in in several periods...
when all expressed in their Present Value
NPV = >
t
Bt
1 Ý
1+
rÞt?1
Ct
?
Ý
1+
rÞt?1
NET PRESENT VALUE




The net present value of an investment is equal to the
difference between benefits received in several periods
and costs faced in in several periods...
Usually, there is a natural choice for t -- the “useful” life
of the project, such as when or an asset undergoes a major
refurbishment or the assets are sold.
Sometimes we use variations of the formula with infinite t
See textbook for alternative ways to calculate horizon
values
COMPARING PROJECTS WITH DIFFERENT
TIME FRAMES



Analysts should not choose one project
over another solely based on the NPV of
each project if the time spans are different
Such projects are not directly comparable.
So what can the analyst do? Two options…
Rolling over the Shorter Project


If project A spans n times the number of years as project B, then
assume that project B is repeated n times and compare the NPV of n
repeated project Bs to the NPV of (one) project A.
For example, if project A lasts 30 years and project B lasts 15 years,
compare the NPV of project A to the NPV of 2 back-to-back project
B’s, where the latter is computed:
NPV = x + x/(1+i)15
where, x = NPV of one15-year project B.
Equivalent Annual Net Benefits (EANB) Method



The EANB is the amount received each year for the life
of the project that has the same NPV as the project itself.
The EANB of a project is computed by dividing the NPV
by the appropriate annuity factor, ain:
EANB = NPV/ ain
The appropriate annuity factor ain is the present value of
an annuity of $1 for the life of the project (n years), where
i = interest rate used to compute the NPV. Obviously, one
would choose the project with the highest EANB.
Other Considerations


Shorter projects also have an additional benefit
(not included in EANB) because one does not
necessarily have to roll-over the shorter project
when it is finished.
A better option might be available at that time.
This additional benefit is called quasi-option
value and is discussed further in Chapter 7.
NET PRESENT VALUE



If the NPV of a prospective project is
positive, it should be accepted, but if it is
negative then the project should be rejected
OK, I will believe it…
but then, what should I do with my
savings?
NET PRESENT VALUE




what should I do with my savings?
Invest them in any project that promises
returns at a rate r
That is the key: your project was not good
if you used a discount rate r
Because you would lose money in PV if
the discount rate was to be r
NET PRESENT VALUE


In that case, instead of going for the
project, put you money into something that
pays an interest rate r
like a bank account
Internal rate of return


Another way to look at it is to find the
particular value of the discount rate that
would make the NPV=0 for your project
That is called the internal rate of return of
your project
Internal rate of return




Mathematically:
IRR = r*
such that
NPV(r*)=0
NPVÝ
r Þ= >
D
t
Bt
1 Ý
1+
r DÞt?1
Ct
? D t?1
Ý
1+
r Þ
=0
Internal rate of return


Therefore, if you can find an alternative
project that pays more than r*, you should
put your money there!!! Not in your
original project!
Think about it: if the IRR (r*) is lower than
what you can get anywhere else, then you
are not getting a benefit from the project to
cover the opportunity cost of your money!
Internal rate of return

Think about it in a different way: if the IRR
(r*) is lower than the rate of interest at
which you need to repay your loan for the
funds for the project, you should not
undertake the project!
Example of the power of compounding




The Dutch allegedly bought Manhattan in
1626 for about $24 worth of beads and trinkets
if Native Americans had invested in tax-free
bonds 7% APR bonds, it would now be worth
over $2.0 trillion > assessed value of
Manhattan
if US had invested $7.2 million it paid Russia
in 1867 in tax-free 7% APR bonds
money worth only $50.9 billion < Alaska's
current value
REAL VERSUS NOMINAL DOLLARS



Conventional private sector financial analysis
measures monetary amounts in nominal dollars
(sometimes called current dollars).
But, due to inflation, one cannot buy as many goods
and services with a dollar today as one could one, two
or more years previously. It is important to control for
inflation
We control for inflation by converting nominal dollars
to real dollars (sometimes called constant dollars).
We usually use the consumer price index (CPI), but
sometimes use the gross national product (GNP).
Estimates of Inflation and Problems with CPI
The CPI is the most commonly used measure of inflation. Prior to 1998,
the CPI overstated inflation by about 0.8% to 1.6% per annum. Four
reasons for the overstatement are:

1)
Commodity substitution effect: The CPI did not accurately
reflect changes in consumer purchases, such as switching to lowerpriced substitutes;

2)
New goods: The “basket” of goods did not include some
new products, e.g., new (cheaper) generic drugs;

3)
Quality improvements: The CPI did not accurately reflect
changes in product quality, e.g. more safe or reliable cars;

4)
Discount store effect: Consumers are shopping more at
discount stores, which have less expensive products.
Real rates of interest





In a world with inflation...
What is the difference between the real and
the nominal interest rate?
Imagine that prices actually grow at a rate:E
We are going to label the real interest rate
i* and the nominal interest rate i
Real rates of interest





If we want to put next year’s prices in real
terms, we want to discount the inflation
we want to adjust for inflation, dividing by
1+ E
Using this idea, we can find the real rate of
interest
i* is more or less equal to i - E for low
levels of inflation, like the ones we
normally face
Real rates of interest


The proof is given in Varian, for example
Check that it makes sense, and then just
remember that the results is a good
approximation in most cases
NEXT
Existence Values
READ CHAPTER 9