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CHAPTER 6

CHAPTER 6. DISCOUNTING. 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. Discounting.

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CHAPTER 6

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  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

  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

  4. Mechanics of Discounting I PV = FV in yeart / [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 year1/ [1+r]^1) + (FV in year2/ [1+r]^2) + (FV in year3/ [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.

  7. What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 3 10% 100 PV = ?

  8. 3 1    PV = $100    1.10   = $100 0.7513 = $75.13.

  9. Spreadsheet Solution • 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? 4 0 1 2 3 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08= PV

  11. Spreadsheet Solution A B C D E 1 0 1 2 3 4 2 100 300 300 -50 3 530.09 Excel Formula in cell A3: =NPV(10%,B2:E2)

  12. Perpetuities 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

  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 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

  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

  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.

  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.

  19. Steps in Monte Carlo Simulation with Excel • 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) • Copy the entire row N times (spreadsheets up to 10K -- use logic functions or macros to replicate) • Chart array of outcomes (the results in last cells), plot as histogram, calculate  and 

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

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

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