1. CHAPTER 6 DISCOUNTING

2. CONVERTING FUTURE VALUE TO PRESENT VALUE Making decisions having significant future benefits or costs means looking at consequences from where we are right now: converting future benefit/cost flows to PRESENT VALUES Making decisions having significant future benefits or costs means looking at consequences from where we are right now: converting future benefit/cost flows to PRESENT VALUES

3. Discounting Future values are converted to present values by means of a discount rate. That is, future nominal benefits are worth less than present benefits of equal magnitude -- the WIMPY principal -Inflation -Markets tell us that people demand compensation for forgoing current consumption Future values are converted to present values by means of a discount rate. That is, future nominal benefits are worth less than present benefits of equal magnitude -- the WIMPY principal -Inflation -Markets tell us that people demand compensation for forgoing current consumption

4. Mechanics of Discounting I PV = FV in year t / [1+r]^t Where PV = Present Value FV = Future Value (real or nominal) t = Year r = Discount Rate (real or nominal) PV = FV in year t / [1+r]^t Where PV = Present Value FV = Future Value (real or nominal) t = Year r = Discount Rate (real or nominal)

5. Mechanics of Discounting II For a Stream of Benefits from year 1 to year t, SUM [add up] all the present values for all net future values Where t = 3 PV = (FV in year 1 / [1+r]^1) + (FV in year 2 / [1+r]^2) + (FV in year 3 / [1+r]^3) For a Stream of Benefits from year 1 to year t, SUM [add up] all the present values for all net future values Where t = 3 PV = (FV in year 1 / [1+r]^1) + (FV in year 2 / [1+r]^2) + (FV in year 3 / [1+r]^3)

6. Three Ways to Find PVs Solve the equation with a regular calculator (or use FV tables from an accounting text). Use a financial calculator. Use a spreadsheet. Solve the equation with a regular calculator (or use FV tables from an accounting text). Use a financial calculator. Use a spreadsheet.

7. 10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0123 PV = ?

8.  PV= $100 1 1.10 = $1000.7513 = $75.13.       3

9. Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, -100) = 75.13 Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, -100) = 75.13

10. What is the PV of this uneven benefit stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

11. Spreadsheet Solution Excel Formula in cell A3: =NPV(10%,B2:E2) ABCDE 101234 2100300300-50 3530.09

12. Perpetuities PV = NBF / r Where NBF = a specified annual net- benefit flow For example: $186k /.03 = $6.2m PV = NBF / r Where NBF = a specified annual net- benefit flow For example: $186k /.03 = $6.2m

13. Alternative Discount Rates Market rate = r + i + b + y Where r = real, risk-free rate i = the expected rate of inflation b = project specific (nondiversifiable) risk y = income tax adjustment Nominal risk-free rate [n] = r + i Market rate = r + i + b + y Where r = real, risk-free rate i = the expected rate of inflation b = project specific (nondiversifiable) risk y = income tax adjustment Nominal risk-free rate [n] = r + i

14. Use of Alternative Discount Rates Use real rate [r] with real FVs -For example, where you are using current costs to estimate future costs Use nominal rate [n] with nominal FVs -For example, where you are making identical nominal principal and interest payments each year WHAT NOMINAL RATE SHOULD YOU USE? Borrowing rate on tax-exempt, general- purpose bonds of similar maturities Use real rate [r] with real FVs -For example, where you are using current costs to estimate future costs Use nominal rate [n] with nominal FVs -For example, where you are making identical nominal principal and interest payments each year WHAT NOMINAL RATE SHOULD YOU USE? Borrowing rate on tax-exempt, general- purpose bonds of similar maturities In Project analysis

15. Annualizing Capital Costs Since real government budgets are formulated one year at a time, the budget tends to be biased against delivery methods requiring up-front investments The proper solution is converting everything to PV However, there is a reasonable alternative, which is the annualizing capital costs Since real government budgets are formulated one year at a time, the budget tends to be biased against delivery methods requiring up-front investments The proper solution is converting everything to PV However, there is a reasonable alternative, which is the annualizing capital costs

16. Mechanics of Annualizing Annual Cost of a Capital Asset = P [r + d - a] Where P = Purchase Price [replacement cost] d = Depreciation rate [wear and tear + obsolescence] a = Appreciation rate Annual Cost of a Capital Asset = P [r + d - a] Where P = Purchase Price [replacement cost] d = Depreciation rate [wear and tear + obsolescence] a = Appreciation rate

17. DOES THE CHOICE OF DISCOUNT RATE MATTER? Yes – choice of rate can affect policy choices. Generally, low discount rates favor projects with the highest total benefits. High SDRs rates favor projects where the benefits are front-end loaded. Yes – choice of rate can affect policy choices. Generally, low discount rates favor projects with the highest total benefits. High SDRs rates favor projects where the benefits are front-end loaded.

18. Appendix: Monte Carlo Simulation with Excel Most spread sheets provide a function for generating random variables that are distributed uniformly from 0 to 1 [in Excel the function is RAND()] To generate uniform random variables with other ranges, one simply multiplies the draw from the uniformly distributed from 0 to 1 by the desired range and adds the minimum value [for SDRs with  = 2% and a range from 0 to 4%, use the following formula: RAND()*.04] Alternatively you can combine functions for the inverse of the cumulative normal distribution and the uniform distribution: NORMSINV(RAND()) The standardized normal distribution can be given any  and  through simple transformations: add a constant =  and multiply by the square root of the desired variance. Most spread sheets provide a function for generating random variables that are distributed uniformly from 0 to 1 [in Excel the function is RAND()] To generate uniform random variables with other ranges, one simply multiplies the draw from the uniformly distributed from 0 to 1 by the desired range and adds the minimum value [for SDRs with  = 2% and a range from 0 to 4%, use the following formula: RAND()*.04] Alternatively you can combine functions for the inverse of the cumulative normal distribution and the uniform distribution: NORMSINV(RAND()) The standardized normal distribution can be given any  and  through simple transformations: add a constant =  and multiply by the square root of the desired variance.

19. Steps in Monte Carlo Simulation with Excel 1.Construct a row of appropriate random variables and the formulas that use them to compute net benefits (the last cell in the row should contain net benefits) 2.Copy the entire row N times (spreadsheets up to 10K -- use logic functions or macros to replicate) 3. Chart array of outcomes (the results in last cells), plot as histogram, calculate  and  1.Construct a row of appropriate random variables and the formulas that use them to compute net benefits (the last cell in the row should contain net benefits) 2.Copy the entire row N times (spreadsheets up to 10K -- use logic functions or macros to replicate) 3. Chart array of outcomes (the results in last cells), plot as histogram, calculate  and 

20. Monte Carlo Setup =NORMINV(RAND(),C$10,(C$9-C$11)/3.29) NORMINV Probability Mean Standard Deviation

21. Monte Carlo Setup =IF(RAND()<F$10,1,0) IF Logical Test Value if true Value if false