1. Discounting, Real, Nominal Values Costa Samaras 12-706 / 19-702

2. Agenda zNet Present Value zDiscounting and decision making zReal and nominal interest rates

3. Why are we learning this?

4. “The most powerful force in the universe is compound interest” - Albert Einstein

5. Why are we learning this? “You will use these methods on the EPP Part B exam (and probably throughout your life)” - EPP faculty

6. Project Financing zGoal - common monetary units zRecall - will only be skimming this material in lecture - it is straightforward and mechanical yEspecially with excel, calculators, etc. yShould know theory regardless yShould look at sample problems and ensure you can do them all on your own by hand

7. General Terms and Definitions zThree methods: PV, FV, NPV zFuture Value: F = $P (1+i) n y P: present value, i:interest rate and n is number of periods (e.g., years) of interest zi is discount rate, MARR, opportunity cost, etc. zPresent Value: zNPV=NPV(B) - NPV(C) (over time) zAssume flows at end of period unless stated

8. Notes on Notation yBut [(1+i) -n ] is only function of i,n y$1, i=5%, n=5, [1/(1.05) 5 ]= 0.784 = (P|F,i,n) zAs shorthand: yPresent value of Future : (P|F,i,n) xSo PV of $500, 5%,5 yrs = $500*0.784 = $392 yFuture value of Present : (F|P,i,n) yAnd similar notations for other types

9. Timing of Future Values zNormally assume ‘end of period’ values zWhat is relative difference? zConsider comparative case: y$1000/yr Benefit for 5 years @ 5% yAssume case 1: received beginning yAssume case 2: received end

10. Timing of Benefits zDraw 2 cash flow diagrams @ 5% yNPV 1 = 5 annual payments of $1000- beginning of period yNPV 2 = 5 annual payments of $1000- end of period zNPV 1 - NPV 2 ~ ?

11. Finding: Relative NPV Analysis zIf comparing, can just find ‘relative’ NPV compared to a single option yE.g. beginning/end timing problem yNet difference was -- zAlternatively consider ‘net amounts’ yNPV 1 yNPV 2 y‘Cancel out’ intermediates, just find ends yNPV 1 is $X greater than NPV 2

12. Internal Rate of Return zDefined as discount rate where NPV=0 yLiterally, solving for “breakeven” discount rate zIf we graphed IRR, it might be between 8-9% zBut we could solve otherwise yE.g. y y1+i = 1.5, i=50% y yPlug back into original equation -66.67+66.67

13. Decision Making zChoose project if discount rate < IRR zReject if discount rate > IRR zOnly works if unique IRR (which only happens if cash flow changes signs ONCE) zCan get quadratic, other NPV eqns

14. Another Analysis Tool zAssume 2 projects (power plants) yEqual capacities, but different lifetimes x70 years vs. 35 years yCapital costs(1) = $100M, Cap(2) = $50M yNet Ann. Benefits(1)=$6.5M, NB(2)=$4.2M zHow to compare? yCan we just find NPV of each? yTwo methods

15. Rolling Over (back to back) zAssume after first 35 yrs could rebuild yMakes them comparable - Option 1 is best yThere is another way - consider “annualized” net benefits yNote effect of “last 35 yrs” is very small ($3.5 M)!

16. Recall: Annuities zConsider the PV (aka P) of getting the same amount ($1) for many years yLottery pays $A / yr for n yrs at i=5% y----- Subtract above 2 equations.. ------- ya.k.a “annuity factor”; usually listed as (P|A,i,n)

17. Equivalent Annual Benefit - “Annualizing” cash flows zAnnuity factor (i=5%,n=70) = 19.343 yAnn. Factor (i=5%,n=35) = 16.374 yOf course, still higher for option 1 zNote we assumed end of period pays

18. Annualizing Example zYou have various options for reducing cost of energy in your house. yUpgrade equipment yInstall local power generation equipment yEfficiency / conservation

19. Residential solar panels: Phoenix versus Pittsburgh zPhoenix: NPV is -$72,000 zPittsburgh: -$48,000 zBut these do not mean much. zAnnuity factor @5%, 20 years (~12.5) zEANC = $5800 (PHX), $3800 (PIT) zThis is a more “useful” metric for decision making because it is easier to compare this project with other yearly costs (e.g. electricity)

20. Benefit-Cost Ratio zBCR = NPV B /NPV C zLook out - gives odd results. Only very useful if constraints on B, C exist.

21. Example z3 projects being considered R, F, W yRecreational, forest preserve, wilderness yWhich should be selected?

22. Example Project “R with Road” has highest NB

23. Beyond Annual Discounting zWe generally use annual compounding of interest and rates (i.e., i is “5% per year”) zGenerally, yWhere i is periodic rate, k is frequency of compounding, n is number of years yFor k=1/year, i=annual rate: F=P*(1+i) n ySee similar effects for quarterly, monthly

24. Various Results z$1000 compounded annually at 8%, yFV=$1000*(1+0.08) = $1080 z$1000 quarterly at 8%: yFV=$1000(1+(0.08/4)) 4 = $1082.43 z$1000 daily at 8%: yFV = $1000(1 + (0.08/365)) 365 = $1083.27 z(1 + i/k) kn term is the effective rate, or APR yAPRs above are 8%, 8.243%, 8.327% zWhat about as k keeps increasing? yk -> infinity?

25. Continuous Discounting z(Waving big calculus wand) zAs k->infinity, P*(1 + i/k) kn --> P*e in y$1,083.29 continuing our previous example zWhat types of problems might find this equation useful? yWhere benefits/costs do not accrue just at end/beginning of period

26. IRA example zWhile thinking about careers.. zGovernment allows you to invest $5k per year in a retirement account zStart doing this ASAP after you get a job. zSee ‘IRA worksheet’ in RealNominal

27. US Household Income (1967-90) Income in current and 2005 CPI-U-RS adjusted dollars

28. Real and Nominal zNominal: ‘current’ or historical data zReal: ‘constant’ or adjusted data yUse inflation deflator or price index for real

29. US Gasoline Prices (1970-2008) Income in current and 2007 CPI-U-RS adjusted dollars

30. Adjusting to Real Values zPrice Index (CPI, PPI) - need base year yMarket baskets of goods, tracks price changes yE.g., http://www.minneapolisfed.org/research/data/us/calc/E.g., http://www.minneapolisfed.org/research/data/us/calc/ yCPI-U-RS 1990 =198.0; CPI 2005 =286.7 ySo $30,757 1990$ * (286.7/198.0) = $44,536 2005$ zPrice Deflators (GDP Deflator, etc.) yWork in similar ways but based on output of economy not prices

31. Other Real and Nominal Values zExample: real vs. nominal GDP yIf GDP is $990B in $2000.. (this is nominal) yand GDP is $1,730B in $2001 (also nominal) yThen nominal GDP growth = 75% yIf 2001 GDP equal to $1450B “in $2000”, then that is a real value and real growth = 46% xThen we call 2000 a “base year” yUse this “GDP deflator” to adjust nominal to real yGDP deflator = 100 * Nominal GDP / Real GDP y=100*(1730/1450) = 119.3 (changed by 19.3%)

32. Nominal Discount Rates zMarket interest rates are nominal yThey ideally reflect inflation to ensure value zBuy $100 certificate of deposit (CD) paying 6% after 1 year (get $106 at the end). Thus the bond pays an interest rate of 6%. This is nominal. yWhenever people speak of the “interest rate” they're talking about the nominal interest rate, unless they state otherwise.

33. Real Discount Rates zSuppose inflation rate is 3% for that year yi.e., if we can buy a “basket of goods” today for $100, then we can buy that basket next year and it will cost $103. zIf buy the $100 CD at 6% nominal interest rate.. ySell it after a year and get $106, buy the basket of goods at then- current cost of $103, we will have $3 left over. ySo after factoring in inflation, our $100 bond will earn us $3 in net income; a real interest rate of 3%.

34. Real / Discount Rates zMarket interest rates are nominal yThey reflect inflation to ensure value zReal rate r, nominal i, inflation m y“Real rates take inflation into account” ySimple method: r ~ i-m r+m~i yMore precise: yExample: If i=10%, m=4% ySimple: r=6%, Precise: r=5.77%

35. Discount Rates - Similar zFor investment problems: yIf B & C in real dollars, use real disc rate yIf in nominal dollars, use nominal rate yBoth methods will give the same answer zUnless told otherwise, assume we are using (or are given!) real rates.

36. Garbage Truck Example zCity: bigger trucks to reduce disposal $$ yThey cost $500k now ySave $100k 1st year, equivalent for 4 yrs yCan get $200k for them after 4 yrs yMARR 10%, E[inflation] = 4% zAll these are real values zSee “RealNominal” spreadsheet

37. Summary and Take Home Messages zThree methods for getting common units yPV, FV, NPV zProjects with unequal lifetimes require “annualizing” flows of costs and benefits zKeep nominal with nominal and real with real