Download Discounting, Real/Nominal Values

Discounting, Real, Nominal
Values
Costa Samaras
12-706 / 19-702
Agenda
Net Present Value
Discounting and decision making
Real and nominal interest rates
Why are we learning this?
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“The most powerful force in the universe is compound
interest” - Albert Einstein
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“You will use these methods on the EPP Part B exam
(and probably throughout your life)” - EPP faculty
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 sample problems 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.
n
F
P  (1i)n  F(1 i)
Present Value:
NPV=NPV(B) - NPV(C) (over time)
Assume flows at end of period unless stated

Notes on Notation
P
F
(1i)n
n
 F(1 i)
P
F

1
(1i )n
 (1  i) n
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
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.

$100k
1i

$150k
0  $100k

1i
(1i)2
$150k
 (1i)
2
$100k  $150k
1i
1+i = 1.5, i=50%
 $100k  $150k 2
10.5
(10.5)
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
6.5M
6.5M
NPV1  $100M  6.5M


...
 $25.73M
1.05
1.052
1.0570
4.2 M
4.2M
NPV2  $50M  4.2M


...

 $18.77M
1.05
1.052
1.0535
NPV2R  $18.77M  18.77M
 $22.17M
1.0535
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)!
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%
A
A
A
A
P  1i
 (1i)


..

2
(1i)3
(1i)n
A
A
A
P * (1  i)  A  (1i)
 (1i)

..

2
(1i)n1
----- Subtract above 2 equations.. ------A
P * (1  i)  P  A  (1i)
n
P * (i)  A(1  (1i1 )n )  A(1  (1  i) n )
A(1(1i) n )
(1(1i) n )
P
;P / A 
i
i
a.k.a “annuity factor”; usually listed as (P|A,i,n)
Equivalent Annual Benefit “Annualizing” cash flows
EANB 
NPV
annuity_ factor
recall : annuity _ factor 
Annuity factor (i=5%,n=70) = 19.343
Ann. Factor (i=5%,n=35) = 16.374

$25.73M
EANB1  19.343  $1.33M
$18.77M
EANB2  16.374  $1.15M
Of course, still higher for option 1
Note we assumed end of period pays
(1(1i) n )
i
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), $3800 (PIT)
This is a more “useful” metric for decision
making because it is easier to compare this
project with other yearly costs (e.g. electricity)
Benefit-Cost Ratio
BCR = NPVB/NPVC
Look out - gives odd results. Only very
useful if constraints on B, C exist.
Example
3 projects being considered R, F, W
Recreational, forest preserve, wilderness
Which should be selected?
Alternative
R
R w/ Road
F
F w/ Road
W
W w/ Road
Road only
Benefits
($)
10
18
13
18
5
4
2
Costs
($)
8
12
10
14
1
5
4
B/C
Ratio
1.25
1.5
1.3
1.29
5
0.8
0.5
Net
Benefits ($)
2
6
3
4
4
-1
-2
Example
Base Case Net Benefits ($)
Road only
Project
“R with Road”
has highest NB
W w / Road
W
F w / Road
F
R w / Road
R
-4
-2
0
2
4
6
8
Beyond Annual Discounting
We generally use annual compounding of
interest and rates (i.e., i is “5% per year”)
i kn
Generally, F  P(1 )
k
Where i is periodic rate, k is frequency of
compounding, n is number of years
 k=1/year, i=annual rate: F=P*(1+i)n
For
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, 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)
$50,000
$45,000
$40,000
$35,000
$30,000
Nominal
$25,000
Real (2005)
$20,000
$15,000
$10,000
$5,000
$0
1967
1972
1977
1982
1987
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
Real and Nominal Price of Gasoline (2007 cents/gallon)
US Gasoline Prices (1970-2008)
450
400
350
300
Real Prices
250
200
150
100
Nominal Prices
50
Time
0
70
19
72
19
74
19
76
19
78
19
80
19
82
19
84
19
86
19
88
19
90
19
92
19
94
19
96
19
98
19
00
20
02
20
04
20
06
20
08
20
Income in current and 2007 CPI-U-RS adjusted dollars
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 thencurrent 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: r  (im)
1m
Example: If i=10%, m=4%
Simple: r=6%, Precise: r=5.77%

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 values
See “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