1. Discounting Overview H. Scott Matthews 12-706 / 19-702 / 73-359 Lecture 3

2. Book Progress Update zThey’ve been shipped - not sure when they’ll arrive. Paperback price. zFirst 20 get them, I think I have 18 so far.

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

4. General Terms and Definitions zThree methods: PV, FV, NPV zFV = $PV (1+i) n y PV: present value, i:interest rate and n is number of periods (e.g., years) of interest yRule of 72 zi is discount rate, MARR, opportunity cost, etc. zPV = $FV / (1+i) n zNPV=NPV(B) - NPV(C) (over time) zOther methods: IRR (rate i at which NPV=0) zAll methods give same qualitative answer. zAssume flows at end of period unless stated

5. Notes on Notation zPV = $FV / (1+i) n = $FV * [1 / (1+i) n ] yBut [1 / (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: yFuture value of Present: (P|F,i,n) xSo PV of $500, 5%,5 yrs = $500*0.784 = $392 yPresent value of Future: (F|P,i,n) yAnd similar notations for other types

6. Ex: The Value of Money (pt 1) zWhen did it stop becoming worth it for the avg American to pick up a penny? zTwo parts: time to pick up money? yAssume 5 seconds to do this - what fraction of an hour is this? 1/12 of min =.0014 hr zAnd value of penny over time? Assume avg American makes $30,000 / yr yAbout $14.4 per hour, so.0014hr = $0.02 yThus ‘opportunity cost’ of picking up a penny is 2 cents in today’s terms

7. Ex: The Value of Money (pt 2) zIf ‘time value’ of 5 seconds is $0.02 now yAssuming 5% long-term inflation, we can work problem in reverse to determine when 5 seconds of work ‘cost’ less than a penny zUsing Excel (penny.xls file): yAdjusting per year back by factor 1.05 yValue of 5 seconds in 1984 was 1 cent zBetter method would use ‘actual’ CPI for each year..

8. 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

9. Timing of Benefits zDraw 2 cash flow diagrams zNPV 1 = 1000 + 1000/1.05 + 1000/1.05 2 + 1000/1.05 3 + 1000/1.05 4 yNPV 1 =1000 + 952 + 907 + 864 + 823 = $4,545 zNPV 2 = 1000/1.05 + 1000/1.05 2 + 1000/1.05 3 + 1000/1.05 4 + 1000/1.05 5 yNPV 2 = 952 + 907 + 864 + 823 + 784 = $4,329 zNPV 1 - NPV 2 ~ $216 zNote on Notation: use U for uniform $1000 value (or A for annual) so (P|U,i,n) or (P|A,i,n)

10. Relative NPV Analysis zIf comparing, can just find ‘relative’ NPV compared to a single option yE.g. beginning/end timing problem yNet difference was $216 zAlternatively consider ‘net amounts’ yNPV 1 =1000 + 952 + 907 + 864 + 823 = $4,545 yNPV 2 = 952 + 907 + 864 + 823 + 784 = $4,329 y‘Cancel out’ intermediates, just find ends yNPV 1 is $216 greater than NPV 2

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

12. Real Discount Rates (using Cambpell notation) zMarket interest rates are nominal yThey reflect inflation to ensure value zReal rate r, inflation i, nominal rate m ySimple method: r ~ m-i r + i ~ m yMore precise: r=(m - i)/(1+i) zExample: If m=10%, i=4% ySimple: r=6%, Precise: r=5.77%

13. 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 spreadsheet for nominal values

14. Annuities zConsider the PV of getting the same amount ($1) for many years yLottery pays $P / yr for n yrs at i=5% yPV=P/(1+i)+P/(1+i) 2 + P/(1+i) 3 +…+P/(1+i) n yPV(1+i)=P+P/(1+i) 1 + P/(1+i) 2 +…+P/(1+i) n-1 y------- yPV(1+i)-PV=P- P/(1+i) n yPV(i)=P(1- (1+i) -n ) yPV=P*[1- (1+i) -n ]/i : “annuity factor”

15. Perpetuity (money forever) zCan we calculate PV of $A received per year forever at i=5%? zPV=A/(1+i)+A/(1+i) 2 +… zPV(1+i)=A+A/(1+i) + … zPV(1+i)-PV=A zPV(i)=A, PV=A/i zE.g. PV of $2000/yr at 8% = $25,000 zWhen can/should we use this?

16. 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

17. Rolling Over (back to back) zAssume after first 35 yrs could rebuild yNPV(1)=-100+(6.5/1.05)+..+6.5/1.05 70 =25.73 yNPV(2)=-50+(4.2/1.05)+..+4.2/1.05 35 =18.77 yNPV(2R)=18.77+(18.77/1.05 35 )=22.17 yMakes them comparable - Option 1 is best yThere is another way - consider “annualized” net benefits

18. Equivalent Annual Benefit zEANB=NPV/Annuity Factor zAnnuity factor (i=5%,n=70) = 19.343 yAnn. Factor (i=5%,n=35) = 16.374 zEANB(1)=$25.73/19.343=$1.330 zEANB(2)=$18.77/16.374=$1.146 yStill higher for option 1 zNote we assumed end of period pays

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

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

21. 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?

22. Continuous Discounting z(Waving big calculus wand) zAs k->infinity, PV*(1 + i/k) kn --> PV*e in y$1083.29 using our previous example zWhat types of problems might find this equation useful?

23. IRA example zWhile thinking about careers.. zGovernment allows you to invest $2k per year in a retirement account and deduct from your income tax yInvestment values will rise to $5k soon zStart doing this ASAP after you get a job. zSee ‘IRA worksheet’ in RealNominal

24. Examples (from Campbell)