1. 1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: 12-706 and 73-359 Lecture 8 - 9/23/2002

2. 12-706 and 73-3592 PS #2 notes  Please ‘show your work’ if using Excel by printing cell formulas

3. 12-706 and 73-3593 Final Notes on Uncertainty  It is inherent to everything we do  Our goal then is to best understand and model its existence, make better results  We ‘internalize’ the uncertainty by making ranges or distributions of variables  We see the effects by performing sensitivity analysis (one of three methods)

4. 12-706 and 73-3594 Common Monetary Units  Often face problems where benefits and costs occur at different times  Need to adjust values to common units to compare them  Recall photo sensor example from last lecture - could look at values over several years...

5. 12-706 and 73-3595 Example - Compounding  Buy painting for $10,000  Will be worth $11,000 in one year (sure)  Need to consider ‘opportunity cost’  Make table or diagram of streams of benefits and costs over time  Have several analysis options  Put $10,000 in savings, would earn interest (8%), or 10,000(1.08)=10,800  So should you buy the painting?

6. 12-706 and 73-3596 Decision Rules  As always, should choose option that maximizes net benefits  Now we are using that same rule with values adjusted for time value of money  Still choose option that gives us the highest value  In this case it is ‘buying the painting’  Called ‘future value’ when you compound current value to future

7. 12-706 and 73-3597 Alternative - Present Value  Do the problem in reverse  Time - line representation  How much money you would need to invest in savings to get $11,000 in 1 year  FV=$10,000(1+i) : $10,000 was ‘present’  PV=FV/(1+i); PV=$11,000/(1.08)=$10,185  Since greater, should buy painting  Last option - convert all to present value

8. 12-706 and 73-3598 Net Present Value Method  Investment of $10,000 now for painting represented as -$10,000  Receipt of $11,000 in a year as $11,000/(1.08)  So NPV= -$10,000 + $10,185 = $185  Since NPV positive, should buy painting (it has positive net benefits)  Relevance to Kaldor-Hicks, BCA rule?

9. 12-706 and 73-3599 General Terms  Three methods: PV, FV, NPV  FV = $X (1+i) n  X : present value, i:interest rate and n is number of periods (eg years) of interest  Rule of 72  PV = $X / (1+i) n  NPV=NPV(B) - NPV(C) (over time)  Real vs. Nominal values

10. 12-706 and 73-35910 Real and Nominal  Nominal: ‘current’ or historical data  Real: ‘constant’ or adjusted data  Use deflator or price index for real  For investment problems:  If B&C in real dollars, use real disc rate  If in nominal dollars, use nominal rate  Both methods will give the same answer

11. 12-706 and 73-35911 Real Discount Rates  Market interest rates are nominal  They reflect inflation to ensure value  Real rate r, nominal i, inflation m  Simple method: r ~ i-m r+m~i  More precise: r=(i-m)/(1+m)  Example: If i=10%, m=4%  Simple: r=6%, Precise: r=5.77%

12. 12-706 and 73-35912 Rates to Use for Analysis  In example, investments vs. savings  We assumed an actual option for rate  But can use any rate to discount FV!  Called a discount rate- may be set for us  MARR: opportunity cost of funds  Assume all values ‘real’ unless stated otherwise

13. 12-706 and 73-35913 Garbage Truck Example  City: bigger trucks to reduce disposal $$  They cost $500k now  Save $100k 1st year, equivalent for 4 yrs  Can get $200k for them after 4 yrs  MARR 10%, E[inflation] = 4%  All these are real values  See spreadsheet for nominal values

14. 12-706 and 73-35914 Sensitivity Analysis  How do NPV results change with i?  Back to our garbage trucks example  Vary the real discount rate from 4-10%  NPV declines as rate i increases  Future benefits ‘discounted more’  See updated RealNominal.xls

15. 12-706 and 73-35915 Other Issues  Inflation hard to predict  Tend to use historical trends/estimates  Terminal or residual values  Value of equipment at end of investment  Timing - typically assume beginning

16. 12-706 and 73-35916 Ex: The Value of Money (pt 1)  When did it stop becoming worth it for the avg American to pick up a penny?  Two parts: time to pick up money?  Assume 5 seconds to do this - what fraction of an hour is this? 1/12 of min =.0014 hr  And value of penny over time? Assume avg American makes $30,000 / yr  About $14.4 per hour, so.014hr = $0.02  Thus ‘opportunity cost’ of picking up a penny is 2 cents in today’s terms

17. 12-706 and 73-35917 Ex: The Value of Money (pt 2)  If ‘time value’ of 5 seconds is $0.02 now  Assuming 5% long-term inflation, we can work problem in reverse to determine when 5 seconds of work ‘cost’ less than a penny  Using Excel (penny.xls file):  Adjusting per year back by factor 1.05  Value of 5 seconds in 1985 was 1 cent  Better method would use ‘actual’ CPI for each year..

18. 12-706 and 73-35918 Another Analysis Tool  Assume 2 projects (power plants)  Equal capacities, but different lifetimes  35 years vs. 70 years  Capital costs(1) = $100M, Cap(2) = $50M  Net Ann. Benefits(1)=$6.5M, NB(2)=$4.2M  How to compare-  Can we just find NPV of each?

19. 12-706 and 73-35919 Rolling Over (back to back)  Assume after first 35 yrs could rebuild  NPV(1)=-100+(6.5/1.05)+..+6.5/1.05 70 =25.73  NPV(2)=-50+(4.2/1.05)+..+4.2/1.05 35 =18.77  NPV(2R)=18.77+(18.77/1.05 35 )=22.17  Makes them comparable - Option 1 is best  There is another way - consider annualized net benefits

20. 12-706 and 73-35920 Annuities  Consider the PV of getting the same amount ($1) for many years  Lottery pays $P / yr for n yrs at i=5%  PV=P/(1+i)+P/(1+i) 2 + P/(1+i) 3 +…+P/(1+i) n  PV(1+i)=P+P/(1+i) 1 + P/(1+i) 2 +…+P/(1+i) n-1  -------  PV(1+i)-PV=P- P/(1+i) n  PV(i)=P(1- (1+i) -n )  PV=P*[1- (1+i) -n ]/i : annuity factor

21. 12-706 and 73-35921 Equivalent Annual Benefit  EANB=NPV/Annuity Factor  Annuity factor (i=5%,n=70) = 19.343  Ann. Factor (i=5%,n=35) = 16.374  EANB(1)=$25.73/19.343=$1.330  EANB(2)=$18.77/16.374=$1.146  Still higher for option 1  Note we assumed end of period pays

22. 12-706 and 73-35922 Internal Rate of Return  Defined as the discount rate where NPV=0  Graphically it is between 8-9%  But we could solve otherwise  E.g. 0=-100k/(1+i) + 150k /(1+i) 2  100k/(1+i) = 150k /(1+i) 2  100k = 150k /(1+i) 1+i = 1.5, i=50%  -100k/1.5 + 150k /(1.5) 2 -66.67+66.67

23. 12-706 and 73-35923 Decision Making  Choose project if discount rate < IRR  Reject if discount rate > IRR  Only works if unique IRR  Can get quadratic, other NPV eqns

24. 12-706 and 73-35924 Perpetuity (money forever)  Can we calculate PV of $A received per year forever at i=5%?  PV=A/(1+i)+A/(1+i) 2 +…  PV(1+i)=A+A/(1+i) + …  PV(1+i)-PV=A  PV(i)=A, PV=A/i  E.g. PV of $2000/yr at 8% = $25,000  When can/should we use this?

25. 12-706 and 73-35925 IRA example  While thinking about careers..  Government allows you to invest $2k per year in a retirement account and deduct from your income tax  Investment values will rise to $5k soon  Start doing this ASAP after you get a job.  See ‘IRA worksheet’ in RealNominal