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## Discounting Overview

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**Discounting Overview**H. Scott Matthews 12-706 / 19-702 / 73-359 Lecture 3**Announcements**• HW 1 Returned • Comments from TA’s (good, need to do better) • Solutions / “best answers” posted this afternoon • Pipeline Case (for next Monday) is posted • Chris Hendrickson will do that case**Project Financing**• Goal - common monetary units • Recall - will only be skimming this material in lecture - it is straightforward and mechanical • Especially with excel, calculators, etc. • Should know theory regardless • Should look at problems in Chapter and ensure you can do them all on your own by hand**General Terms and Definitions**• Three methods: PV, FV, NPV • Future Value: F = $P (1+i)n • P: present value, i:interest rate and n is number of periods (e.g., years) of interest • i is discount rate, MARR, opportunity cost, etc. • Present Value: • NPV=NPV(B) - NPV(C) (over time) • Assume flows at end of period unless stated**Notes on Notation**• But [(1+i)-n ] is only function of i,n • $1, i=5%, n=5, [1/(1.05)5 ]= 0.784 = (P|F,i,n) • As shorthand: • Future value of Present: (P|F,i,n) • So PV of $500, 5%,5 yrs = $500*0.784 = $392 • Present value of Future: (F|P,i,n) • And similar notations for other types**Timing of Future Values**• Normally assume ‘end of period’ values • What is relative difference? • Consider comparative case: • $1000/yr Benefit for 5 years @ 5% • Assume case 1: received beginning • Assume case 2: received end**Timing of Benefits**• Draw 2 cash flow diagrams • NPV1 =1000 + 952 + 907 + 864 + 823 = $4,545 • NPV2 = 952 + 907 + 864 + 823 + 784 = $4,329 • NPV1 - NPV2 ~ $216 • Note on Notation: use U for Uniform $1000 value (a.k.a. “A” for annual) so (P|U,i,n) = (P|A,i,n)**Finding: Relative NPV Analysis**• If comparing, can just find ‘relative’ NPV compared to a single option • E.g. beginning/end timing problem • Net difference was $216 • Alternatively consider ‘net amounts’ • NPV1 =1000 + 952 + 907 + 864 + 823 = $4,545 • NPV2 = 952 + 907 + 864 + 823 + 784 = $4,329 • ‘Cancel out’ intermediates, just find ends • NPV1 is $216 greater than NPV2**Internal Rate of Return**• Defined as discount rate where NPV=0 • Literally, solving for breakeven discount rate • Graphically it is between 8-9% • But we could solve otherwise • E.g. • 1+i = 1.5, i=50% • Plug back into original equation<=> -66.67+66.67**Decision Making**• Choose project if discount rate < IRR • Reject if discount rate > IRR • Only works if unique IRR (which only happens if cash flow changes signs ONCE) • Can get quadratic, other NPV eqns**Another Analysis Tool**• Assume 2 projects (power plants) • Equal capacities, but different lifetimes • 70 years vs. 35 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? • Two methods**Rolling Over (back to back)**• Assume after first 35 yrs could rebuild • Makes them comparable - Option 1 is best • There is another way - consider “annualized” net benefits • Note effect of “last 35 yrs” is very small!**Recall: Annuities**• Consider the PV (aka P) of getting the same amount ($1) for many years • Lottery pays $A / yr for n yrs at i=5% • ----- Subtract above 2 equations.. ------- • a.k.a “annuity factor”; usually listed as (P|A,i,n)**Equivalent Annual Benefit - “Annualizing” cash flows**• Annuity factor (i=5%,n=70) = 19.343 • Ann. Factor (i=5%,n=35) = 16.374 • Of course, still higher for option 1 • Note we assumed end of period pays**Annualizing Example**• You have various options for reducing cost of energy in your house. • Upgrade equipment • Install local power generation equipment • Efficiency / conservation**Residential solar panels: Phoenix versus Pittsburgh**• Phoenix: NPV is -$72,000 • Pittsburgh: -$48,000 • But these do not mean much. • Annuity factor @5%, 20 years (~12.5) • EANC = $5800 (PHX), $3800 (PIT) • This is a more “useful” metric for decision making because it is easier to compare this project with other yearly costs (e.g. electricity)**Benefit-Cost Ratio**• BCR = NPVB/NPVC • Look out - gives odd results. Only very useful if constraints on B, C exist.**Question 2.4 from Boardman**• 3 projects being considered R, F, W • Recreational, forest preserve, wilderness • Which should be selected?**Question 2.4**Project “R with Road” has highest NB**Beyond Annual Discounting**• We generally use annual compounding of interest and rates (i.e., i is “5% per year”) • Generally, • Where i is periodic rate, k is frequency of compounding, n is number of years • For k=1/year, i=annual rate: F=P*(1+i)n • See similar effects for quarterly, monthly**Various Results**• $1000 compounded annually at 8%, • FV=$1000*(1+0.08) = $1080 • $1000 quarterly at 8%: • FV=$1000(1+(0.08/4))4 = $1082.43 • $1000 daily at 8%: • FV = $1000(1 + (0.08/365))365 = $1083.27 • (1 + i/k)kn term is the effective rate, or APR • APRs above are 8%, 8.243%, 8.327% • What about as k keeps increasing? • k -> infinity?**Continuous Discounting**• (Waving big calculus wand) • As k->infinity, PV*(1 + i/k)kn --> PV*ein • $1083.29 using our previous example • What types of problems might find this equation useful? • Where benefits/costs do not accrue just at end/beginning of period**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**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**Real Discount Rates (Campbell notation)**• 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: • Example: If i=10%, m=4% • Simple: r=6%, Precise: r=5.77%**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 “RealNominal” spreadsheet for nominal values