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## Discounting Overview

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**Discounting Overview**H. Scott Matthews 12-706 / 19-702 / 73-359 Lecture 3**Book Progress Update**• They’ve been shipped - not sure when they’ll arrive. Paperback price. • First 20 get them, I think I have 18 so far.**Project Financing**• Goal - common monetary units • Recall - will only be skimming this chapter 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 • FV = $PV (1+i)n • PV: present value, i:interest rate and n is number of periods (e.g., years) of interest • Rule of 72 • i is discount rate, MARR, opportunity cost, etc. • PV = $FV / (1+i)n • NPV=NPV(B) - NPV(C) (over time) • Other methods: IRR (rate i at which NPV=0) • All methods give same qualitative answer. • Assume flows at end of period unless stated**Notes on Notation**• PV = $FV / (1+i)n = $FV * [1 / (1+i)n ] • But [1 / (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**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 .0014hr = $0.02 • Thus ‘opportunity cost’ of picking up a penny is 2 cents in today’s terms**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 1984 was 1 cent • Better method would use ‘actual’ CPI for each year..**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 + 1000/1.05 + 1000/1.052 + 1000/1.053 + 1000/1.054 • NPV1 =1000 + 952 + 907 + 864 + 823 = $4,545 • NPV2 = 1000/1.05 + 1000/1.052 + 1000/1.053 + 1000/1.054 + 1000/1.055 • NPV2 = 952 + 907 + 864 + 823 + 784 = $4,329 • NPV1 - NPV2 ~ $216 • Note on Notation: use U for uniform $1000 value (or A for annual) so (P|U,i,n) or (P|A,i,n)**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**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 (using Cambpell notation)**• Market interest rates are nominal • They reflect inflation to ensure value • Real rate r, inflation i, nominal rate m • Simple method: r ~ m-i <-> r + i ~ m • More precise: r=(m - i)/(1+i) • Example: If m=10%, i=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 spreadsheet for nominal values**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”**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?**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 • NPV(1)=-100+(6.5/1.05)+..+6.5/1.0570=25.73 • NPV(2)=-50+(4.2/1.05)+..+4.2/1.0535=18.77 • NPV(2R)=18.77+(18.77/1.0535)=22.17 • Makes them comparable - Option 1 is best • There is another way - consider “annualized” net benefits**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**Benefit-Cost Ratio**• BCR = NPVB/NPVC • Look out - gives odd results. Only very useful if constraints on B, C exist.**Beyond Annual Discounting**• We generally use annual compounding of interest and rates (i.e., i is “5% per year”) • Generally, FV = PV (1 + i/k)kn • Where i is periodic rate, k is frequency of compounding, n is number of years • For k=1/year, i=annual rate: FV=PV(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?**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