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Discounting Intro/Refresher H. Scott Matthews 12-706/73-359 Lecture 11a - Oct. 8, 2004.

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Presentation on theme: "Discounting Intro/Refresher H. Scott Matthews 12-706/73-359 Lecture 11a - Oct. 8, 2004."— Presentation transcript:

1 Discounting Intro/Refresher H. Scott Matthews 12-706/73-359 Lecture 11a - Oct. 8, 2004

2 Project Financing zRecall - will only be skimming this chapter (6) in lecture - it is straightforward and mechanical yEspecially with excel, calculators, etc. yShould know theory regardless yShould look at problems in Chapter 6 and ensure you can do them all on your own

3 Common Monetary Units zOften face problems where benefits and costs occur at different times zNeed to adjust values to common units to compare them zPhoto sensor - look at values over several years...

4 Ex: Compounding (Future Value) zBuy painting for $10,000 yWill be worth $11,000 in one year (sure) yNeed to consider ‘opportunity cost’ yMake table or diagram of streams of benefits and costs over time zHave several analysis options yPut $10,000 in savings, would earn simple interest (8%): so $10,000(1.08)=$10,800 ySo should you buy the painting?

5 Decision Rules zAs always, should choose option that maximizes net benefits yNow we are using that same rule with values adjusted for time value of money yStill choose option that gives us the highest value yIn this case it is ‘buying the painting’ yCalled ‘future value’ when you compound current value to future

6 Alternative - Present Value zDo the problem in reverse yTime - line representation yHow much money you would need to invest in savings to get $11,000 in 1 year yFV=$10,000(1+i) : $10,000 was ‘present’ yPV=FV/(1+i); PV=$11,000/(1.08)=$10,185 ySince greater, should buy painting xHas lower investment cost of $10k yLast option - convert all to present value

7 Net Present Value Method zCurrent investment of $10,000 for painting represented as -$10,000 zReceipt of $11,000 in a year as +$11,000/(1.08) zSo NPV= -$10,000 + $10,185 = $185 zSince NPV positive, should buy painting (it has positive net benefits) yRelevance to Kaldor-Hicks, BCA rule?

8 General Terms zThree methods: PV, FV, NPV zFV = $X (1+i) n y X : present value, i:interest rate and n is number of periods (eg years) of interest yRule of 72 zPV = $X / (1+i) n zNPV=NPV(B) - NPV(C) (over time) zReal vs. Nominal values

9 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

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

11 Rates to Use for Analysis zIn example, investments vs. savings yWe assumed an actual option for rate zBut can use any rate to discount FV! yCalled a discount rate- may be set for us zMARR: opportunity cost of funds zAssume all values ‘real’ unless stated otherwise

12 Minimum Attractive Rate of Return zMARR usually resolved by top management in view of numerous considerations. Among these are: yAmount of money available for investment, and the source and cost of these funds (i.e., equity or borrowed funds). yNumber of projects available for investment and purpose (i.e., whether they sustain present operations and are essential, or expand present operations)

13 MARR part 2 yThe amount of perceived risk associated with investment opportunities available to the firm and the estimated cost of administering projects over short planning horizons versus long planning horizons. yThe type of organization involved (i.e., government, public utility, or competitive industry) zIn the end, we are usually given MARR

14 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

15 Sensitivity Analysis zHow do NPV results change with i? zBack to our garbage trucks example yVary the real discount rate from 4-10% yNPV declines as rate i increases yFuture benefits ‘discounted more’ zSee updated RealNominal.xls

16 Other Issues zInflation hard to predict yTend to use historical trends/estimates zTerminal or residual values yValue of equipment at end of investment zTiming - typically assume beginning of period values, not end of period

17 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.014hr = $0.02 yThus ‘opportunity cost’ of picking up a penny is 2 cents in today’s terms

18 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 1993 was 1 cent zBetter method would use ‘actual’ CPI for each year..

19 Another Analysis Tool zAssume 2 projects (power plants) yEqual capacities, but different lifetimes x35 years vs. 70 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

20 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

21 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

22 Equivalent Annual Benefit zEANB=NPV/Annuity Factor yAnnuity 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

23 Internal Rate of Return zDefined as discount rate where NPV=0 zGraphically it is between 8-9% zBut we could solve otherwise yE.g. 0=-100k/(1+i) + 150k /(1+i) 2 y100k/(1+i) = 150k /(1+i) 2 y100k = 150k /(1+i) 1+i = 1.5, i=50% y-100k/1.5 + 150k /(1.5) 2 -66.67+66.67

24 Decision Making zChoose project if discount rate < IRR zReject if discount rate > IRR zOnly works if unique IRR zCan get quadratic, other NPV eqns

25 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?


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