6. Discounting Reading: BGVW, Chapters 6

1. Introduction Benefits and costs usually extend over many time periods, need means to compare “Discount ” benefits and costs in future time periods to get flows in a common metric. Treat future flows as worth less than current flows

2. Basics Future Value Future value of $10,000 in one year if interest rate is 8% Future value of $10,000 in two years if interest rate is 8% Compounding

Present Value Present Value of $10,000 received in one year if interest rate is 8%

Present Value of $10,000 received in 2 years if interest rate is 8% Larger n, smaller PV Larger i, smaller PV

3. Net Present Value PV Benefits

PV Costs Net PV

Example in Figure 6.4

5. Timing and Terminal Values Comments Higher the discount rate the lower the future counts If you go out far enough the flow values go to zero 5. Timing and Terminal Values NPV rise if benefits are realized earlier and costs are realized later Terminal Values National park (long lived, pertetuity values make sense) World’s fair (short lived, decommissioning makes sense) Salvage values (eg., buses) Simple Projections Depreciation --sensitivity analysis

6. Real vs. Nominal Accounting for inflation in the measures of benefits and costs Nominal dollars: current dollars, money as we see it Real dollars: adjusts for inflation, purchasing power held constant

Usually done w/ CPI Example: Convert $X received in 1999 to real 1990 dollars:

This also implies real discount rates and nominal discount rates This also implies real discount rates and nominal discount rates. That latter are not adjusted for inflation. If inflation is rising at 4% per year and interest is 7%, we say the real rate of interest (real economic growth) is 3%. Formula (Really 2.91%, not 3%) Can use either real or nominal, just be consistent

7. Internal Rates of Return The discount rate at which NPV is zero is the internal rate of return of the project. Sometimes compared w/ cost of return on government money if invested elsewhere Problems: same problem as b/c ratio can have multiple rates

8. Sensitivity Analysis over Discount Rates Adjust discount rate and see how project outcome varies Extremely common Calculator

9. Shortcuts in Discounting Annuities/Perpetuities/Growth Rates Pages 159-163 Value of Perpetuity (forever) Value of an Annuity that Grows or Declines (forever) – growth is g

Value of Perpetuity (n years) Value of an Annuity that Grows or Declines (forever) – grow is g