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Discounting, Real, Nominal Values
Costa Samaras12-706 / 19-702
Agenda
Net Present ValueDiscounting and decision makingReal and nominal interest rates
Why are we learning this?
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Why are we learning this?
“The most powerful force in the universe is compound interest” - Albert Einstein
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Why are we learning this?
“You will use these methods on the EPP Part B exam (and probably throughout your life)” - EPP faculty
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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 sample problems 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
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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: Present value of Future : (P|F,i,n)
So PV of $500, 5%,5 yrs = $500*0.784 = $392
Future value of Present : (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 @ 5%
NPV1 = 5 annual payments of $1000- beginning of period
NPV2 = 5 annual payments of $1000- end of period
NPV1 - NPV2 ~ ?
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 --
Alternatively consider ‘net amounts’ NPV1 NPV2 ‘Cancel out’ intermediates, just find ends NPV1 is $X greater than NPV2
Internal Rate of Return
Defined as discount rate where NPV=0 Literally, solving for “breakeven” discount rate
If we graphed IRR, it might be 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 < IRR
Reject 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 lifetimes70 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 ($3.5 M)!
€
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), $3800 (PIT)This is a more “useful” metric for decision making because it is easier to compare this project with other yearly costs (e.g. electricity)
Benefit-Cost Ratio
BCR = NPVB/NPVCLook out - gives odd results. Only very useful if constraints on B, C exist.
Example
3 projects being considered R, F, W Recreational, forest preserve, wilderness Which should be selected?
Alternative Benefits($)
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
Example
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, P*(1 + i/k)kn --> P*ein
$1,083.29 continuing 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 $5k per year in a retirement account
Start doing this ASAP after you get a job.
See ‘IRA worksheet’ in RealNominal
US Household Income (1967-90)
$0
$5,000
$10,000
$15,000
$20,000
$25,000
$30,000
$35,000
$40,000
$45,000
$50,000
1967 1972 1977 1982 1987
Nominal
Real (2005)
Income in current and 2005 CPI-U-RS adjusted dollars
Real and Nominal
Nominal: ‘current’ or historical data
Real: ‘constant’ or adjusted data Use inflation deflator or price index for real
US Gasoline Prices (1970-2008)
Income in current and 2007 CPI-U-RS adjusted dollars
0
50
100
150
200
250
300
350
400
450
1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008
Real and Nominal Price of Gasoline (2007 cents/gallon)
Real Prices
Nominal Prices
Time
Adjusting to Real Values
Price Index (CPI, PPI) - need base year Market baskets of goods, tracks price changes
E.g., http://www.minneapolisfed.org/research/data/us/calc/
CPI-U-RS1990=198.0; CPI2005=286.7 So $30,7571990$* (286.7/198.0) = $44,536 2005$
Price Deflators (GDP Deflator, etc.) Work in similar ways but based on output of economy not prices
Other Real and Nominal Values
Example: real vs. nominal GDP If GDP is $990B in $2000.. (this is nominal) and GDP is $1,730B in $2001 (also nominal) Then nominal GDP growth = 75% If 2001 GDP equal to $1450B “in $2000”, then that is a real value and real growth = 46%
Then we call 2000 a “base year” Use this “GDP deflator” to adjust nominal to real
GDP deflator = 100 * Nominal GDP / Real GDP =100*(1730/1450) = 119.3 (changed by 19.3%)
Nominal Discount Rates
Market interest rates are nominal They ideally reflect inflation to ensure value
Buy $100 certificate of deposit (CD) paying 6% after 1 year (get $106 at the end). Thus the bond pays an interest rate of 6%. This is nominal. Whenever people speak of the “interest rate” they're
talking about the nominal interest rate, unless they state otherwise.
Real Discount Rates
Suppose inflation rate is 3% for that year i.e., if we can buy a “basket of goods” today for $100, then we can
buy that basket next year and it will cost $103.
If buy the $100 CD at 6% nominal interest rate.. Sell it after a year and get $106, buy the basket of goods at then-
current cost of $103, we will have $3 left over. So after factoring in inflation, our $100 bond will earn us $3 in net
income; a real interest rate of 3%.
Real / Discount Rates
Market interest rates are nominal They reflect inflation to ensure value
Real rate r, nominal i, inflation m “Real rates take inflation into account”
Simple method: r ~ i-m <-> r+m~i More precise: Example: If i=10%, m=4% Simple: r=6%, Precise: r=5.77%
€
r = (i−m )1+m
Discount Rates - Similar
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
Unless told otherwise, assume we are using (or are given!) real rates.
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
Summary and Take Home Messages
Three methods for getting common units PV, FV, NPV
Projects with unequal lifetimes require “annualizing” flows of costs and benefits
Keep nominal with nominal and real with real