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

Managerial Econ., Lecture 20: Present Value & Discounting Outline  Introduction to Benefit/Cost Analysis  Discounting  Mechanics of Discounting  Discounting in Benefit/Cost Analysis

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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!

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting Outcomes People Year A Framework for B/C

Managerial Econ., Lecture 20: Present Value & Discounting 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).

Managerial Econ., Lecture 20: Present Value & Discounting 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

Managerial Econ., Lecture 20: Present Value & Discounting

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

Managerial Econ., Lecture 20: Present Value & Discounting Example  The general formula is  PV = FV/(1 + i) t  For example, to find FV = $100 at end of year with i = 4%  PV = $100/1.04 = $96.15

Managerial Econ., Lecture 20: Present Value & Discounting 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/ $10/ $10/1.1 3  $24.87

Managerial Econ., Lecture 20: Present Value & Discounting PV of a Stream of Payments, 2  More generally, for a payment of f every year for t years,

Managerial Econ., Lecture 20: Present Value & Discounting

Figure 16.1 Present Value of a Dollar in the Future Present value, PV, of $ $1 t, Years i = 0% i = 5% i = 10% i = 20% 30

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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  Because  The nominal interest rate is

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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,

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting Net Present Value  The PV of net benefits is:

Managerial Econ., Lecture 20: Present Value & Discounting 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 portfolio  Long-term is appropriate because government projects are meant to stay in place a while (and some are long- lived anyway).

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.

Managerial Econ., Lecture 20: Present Value & Discounting 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.