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)!

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NPV1 = −$100M + 6.5M1.05 + 6.5M

1.052 + ...+ 6.5M1.0570 = $25.73M

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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

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EANB = NPVannuity _ factor

€

recall : annuity _ factor = (1−(1+i)−n )i

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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

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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%

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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