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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. - PowerPoint PPT Presentation
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Discounting Overview
H. Scott Matthews12-706 / 19-702 / 73-359Lecture 3
Announcements
HW 1 Returned Comments from TA’s (good, need to do
better) Solutions / “best answers” posted this
afternoonPipeline Case (for next Monday) is
posted Chris Hendrickson will do that case
Project Financing
Goal - common monetary unitsRecall - 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, NPVFuture 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
€
P = F(1+i)n
= F(1+ i)−n
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
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P = F(1+i)n
= F(1+ i)−n PF =
1(1+i )n
=(1+ i)−n
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)
€
NPV1 = $1000 + 10001.05 + 1000
1.052 + 10001.053 + 1000
1.054
€
NPV2 = 10001.05 + 1000
1.052 + 10001.053 + 1000
1.054 + 10001.055
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€
0 = −$100k1+i + $150k
(1+i)2
€
$100k1+i = $150k
(1+i)2
€
$100k = $150k1+i
€
$100k1+0.5 = $150k
(1+0.5)2
Decision Making
Choose project if discount rate < IRRReject if discount rate > IRROnly 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!
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NPV1 = −$100M + 6.5M1.05 + 6.5M
1.052 + ...+ 6.5M1.0570 = $25.73M
€
NPV2R = $18.77M + 18.77M1.0535 = $22.17M
NPV2 =−$50M + 4.2M1.05 + 4.2M
1.052+ ...+ 4.2M
1.0535=$18.77M
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)
P = A1+i +
A(1+i )2
+ A(1+i )3
+ ..+ A(1+i )n
P *(1+ i) =A+ A(1+i )
+ A(1+i )2
+ ..+ A(1+i )n−1
P * (1+ i)−P =A− A(1+i )n
P * (i) =A(1− 1(1+i )n
) =A(1−(1+ i)−n)P = A(1−(1+i )−n )
i ;P / A= (1−(1+i )−n )i
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 1Note we assumed end of period pays
€
EANB = NPVannuity _ factor
€
recall : annuity _ factor = (1−(1+i)−n )i
€
EANB1 = $25.73M19.343 = $1.33M
€
EANB2 = $18.77M16.374 = $1.15M
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 -$72,000Pittsburgh: -$48,000
But these do not mean much. Annuity factor @5%, 20 years (~12.5)
EANC = $5800 (PHX), $3800 (PIT)This is a more “useful” metric for decision
making because it is easier to compare this project with other yearly costs (e.g. electricity)
Benefit-Cost Ratio
BCR = NPVB/NPVC
Look out - gives odd results. Only very useful if constraints on B, C exist.
Question 2.4 from Boardman
3 projects being considered R, F, W Recreational, forest preserve, wilderness Which should be selected?
Alternative Benefits($)
Costs($)
B/CRatio
NetBenefits ($)
R 10 8 1.25 2R w/ Road 18 12 1.5 6F 13 10 1.3 3F w/ Road 18 14 1.29 4W 5 1 5 4W w/ Road 4 5 0.8 -1Road only 2 4 0.5 -2
Question 2.4
Base Case Net Benefits ($)
-4 -2 0 2 4 6 8
R
R w/ Road
F
F w/ Road
W
W w/ Road
Road only
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
€
F = P(1+i
k)kn
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 exampleWhat 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 dataReal: ‘constant’ or adjusted data
Use deflator or price index for realFor 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%
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r = (i−m )1+m
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 valuesSee “RealNominal” spreadsheet for
nominal values