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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 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 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) 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) 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. 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 PV = ? 10% 1 2 3 100 PV = FVn 1 n = FVn 1+ i 1+ i 1 PV = $100 1.10 n 3 = $100 0.7513 = $75.13. Spreadsheet Solution • Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, -100) = 75.13 What is the PV of this uneven benefit stream? 0 10% 1 2 3 4 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV Spreadsheet Solution 1 A B C D E 0 1 2 3 4 100 300 300 -50 2 3 530.09 Excel Formula in cell A3: =NPV(10%,B2:E2) Perpetuities PV = NBF / r Where NBF = a specified annual netbenefit flow For example: $186k / .03 = $6.2m 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 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, generalpurpose bonds of similar maturities In Project analysis 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 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 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. 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. 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 Monte Carlo Setup LNG Navigation Safety Factor 1.00 0.20 0.04 NORMINV Probability Mean Standard Deviation =NORMINV(RAND(),C$10,(C$9-C$11)/3.29) Monte Carlo Setup Probability of a Disaster Given a Massive Spill 10% =IF(RAND()<F$10,1,0) IF Logical Test Value if true Value if false