PPA 723: Managerial Economics Lecture 20: Benefit/Cost Analysis 1, Present Value and Discounting The Maxwell School, Syracuse University Professor John

PPA 723: Managerial Economics

Lecture 20:

Benefit/Cost Analysis 1,

Present Value and Discounting

The Maxwell School, Syracuse UniversityProfessor John Yinger

Managerial Econ., Lecture 20: Present Value & Discounting

OutlineIntroduction to Benefit/Cost Analysis

Discounting

Mechanics of Discounting

Discounting in Benefit/Cost Analysis


Introduction

Today we start a section on benefit/cost analysis (or B/C).

B/C analysis is a set of tools to help make decisions about implementing public programs with complex impacts.

Despite the claims of some, B/C cannot lead to purely objective decisions.


What is Benefit/Cost Analysis?

B/C is a set of tools to help decide whether the world with a project is preferable to the world without the project.

B/C is a general framework of decision-making The only serious restriction of B/C is that it requires a focus

on the preferences of affected people. B/C also involves value judgments about relative weights

to give to preferences of various groups.


Criticisms of B/C

Some people criticize B/C because it cannot place a clear value on all of a program’s impacts.

But one cannot blame B/C for the complexity of the world In fact, B/C is a very flexible tool that helps decisions makers deal

with this complexity.

B/C is often taught as narrow tool with that yields an objective valuation of a program.

I reject this view. B/C is very flexible. B/C cannot eliminate the need for value judgments!


A Framework for B/C

Potential government projects generally have a wide range of impacts across time, across markets, and across groups.

B/C provides a way to collapse these complex impacts into a more manageable form: Impacts across time are collapsed using discounting. Impacts across markets are collapsed using a

willingness to pay metric Impacts across groups are either collapsed using

(implicit or explicit) equity judgments or are simply highlighted so that equity impact can be seen and debated.


A Framework for B/C

One way to think about B/C therefore is that it starts with a large three-dimensional problem.

The dimensions are time, market, and group.

And then it collapses this problem as much as possible using discounting, willingness to pay, and equity judgments.


Outcomes

People

Year

A Framework for B/C


Discounting

Discounting is an analytical tool that makes it possible to compare dollars received or spent in the future with those received or spent in the present.

These dollars are not automatically equivalent because there is an opportunity cost associated to waiting to receive a payment (or not waiting to make a payment).


Present and Future Values

Future value (FV) depends on the present value (PV), the interest rate, and the number of years.

Put PV dollars in bank today and allow interest to compound for t years:

FV = PV (1 + i)t



Present Value

To understand PV, we can ask two equivalent questions:How much is $1 in the future worth today?How much money, PV, must we put in bank

today at i to get a specific FV at some future time?

The answer:

PV = FV/(1 + i)t


Example

The general formula isPV = FV/(1 + i)t

For example, to find FV = $100 at end of year with i = 4%

PV = $100/1.04 = $96.15


PV of a Stream of Payments

You agree to pay $10 at end of each year for 3 years to repay a debt with i = 10%.

PV = $10/1.11 + $10/1.12 + $10/1.13 $24.87


PV of a Stream of Payments, 2

More generally, for a payment of f every year for t years,

1 2

1 1 1...

(1 ) (1 ) (1 )tPV f

i i i



Figure 16.1Present Value of a Dollar in the Future

Present value,

PV, of $1

20

10

40

50

60

70

80

90

$1

t , Years

0 10 20 30 40 50 60 70 80 90 100

i = 0%

i = 5%i = 10%i = 20%

30


Adjusting for Inflation

Nominal amount you pay next year is

Future debt in today's dollars is

If = 10%, a nominal payment of next year is in today’s (real) dollars.

/(1 )f f

f /1.1 0.909f f f

f


Nominal vs. Real Interest Rates

Banks pay a nominal interest rate, If the real discount rate is i, banks'

nominal interest rate is such that a dollar today pays (1 + i)(1 + ) in next year’s dollars

BecauseThe nominal interest rate is

i

1 (1 )(1 ) 1i i i i

i i i


Real Interest Rate

The above equation implies that

If inflation is low ( ), then we can closely approximate the real rate by

So: real interest rate = nominal interest rate minus anticipated inflation.

1

ii

i i

0


Discounting and Life

This may the most useful class in this course, because you will all benefit from knowing about discounting.

Discounting is fundamental to the logic of mortgages, pensions, and many other aspects of modern life.


Mortgages

A mortgage is an agreement in which the borrower receives a check from a lender in exchange for a promise to make a series of payments in the future.

The mortgage amount equals the present value of the stream of monthly payments.


Mortgages, 2

Here, as a special bonus, is the formula for a mortgage.

In this formula, P = monthly payment, M = mortgage amount, m = monthly interest rate, N = length of mortgage in months,

1 (1 ) N

mP M

m


Discounting in B/C

• A program is said to be worth doing if the present value of net benefits is positive.

• If there is a budget constraint, one picks the set of affordable programs that yield the highest present value of net benefits.


Net Present Value

The PV of net benefits is:

1 20 1 2

1 20 1 2

( ) ( ) ( )

...(1 ) (1 ) (1 )

...(1 ) (1 ) (1 )

TT

TT

PV NB PV B PV C

B B BB

i i i

C C CC

i i i


Key Issues in Discounting

The first key issue is how to select the discount rate.

Most analysts say to pick a low-risk long-term rate, such as a U.S. Treasury Bill.Low-risk is appropriate because governments have a

diverse portfolioLong-term is appropriate because government projects

are meant to stay in place a while (and some are long-lived anyway).


Key Issues in Discounting, 2

The second key issue is to be consistent.

The numerator (benefits and costs) and the denominator (the discount rate) should both be either in real terms or in nominal terms. Nominal/real or real/nominal calculations are not

correct.

The problem is that either real/real or nominal/nominal calculations require an estimate of inflation, which makes analysts uncomfortable.


Real/Real Calculations

Here the numerator is easy. Benefits and costs in each year are entered in real terms with no inflation adjustment. So a $1,000 benefit today that is expected to continue just stays at $1,000.

But the denominator is hard. Any observed interest rate is nominal because it recognizes that money will be paid back in the future in dollars that are not worth as much. This is part of the opportunity cost of receiving money in the future. So to get a real rate, anticipated inflation must be subtracted from a market rate. The trouble is that anticipated inflation is not directly observed.


Nominal/Nominal Calculations

In this case, the numerator is hard. The $1,000 in the first year must be inflated with an expected inflation rate.

The denominator is easy because market rates are already in nominal terms.


Consistent Calculations

Note that the real/real and nominal/nominal approaches are equivalent.

If the numerator is inflated at rate a per year and the project has a long life, then the present value expression collapses to one with real benefits in the numerator and r-a, the real interest rate, in the denominator.

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PPA 723: Managerial Economics Lecture 20: Benefit/Cost Analysis 1, Present Value and Discounting The Maxwell School, Syracuse University Professor John