Discounting, Real, Nominal Values

H. Scott Matthews12-706 / 19-702

Admin IssuesPipeline case study writeup - due MondayFormat expectations:

Framing of problem (see p. 7!), Answer/justify with preliminary calculations Don’t just estimate the answer! Do not need to submit an excel printout, but

feel free to paste a table into a documentLength: Less than 2 pages.

AnnouncementsHW 1 Returned

Solutions / “best answers” posted this afternoon

Pipeline Case (for next Monday) posted

Project FinancingGoal - 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 DefinitionsThree 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

€

P = F(1+i)n

= F(1+ i)−n PF =

1(1+i)n

=(1+ i)−n

Timing of Future ValuesNormally assume ‘end of period’

valuesWhat 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

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 MakingChoose 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 ToolAssume 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 ExampleYou 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 RatioBCR = NPVB/NPVCLook out - gives odd results. Only

very useful if constraints on B, C exist.

Example from Boardman3 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 2.4Base Case Net Benefits ($)

-4 -2 0 2 4 6 8

R

R w/ Road

F

F w/ Road

W

W w/ Road

Road onlyProject“R with Road”has highest NB

Beyond Annual DiscountingWe 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 exampleWhile thinking about careers ..Government allows you to invest $5k

per year in a retirement account and deduct from your income tax Investment values will rise past $5k soon

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

Real and NominalNominal: ‘current’ or historical data

Real: ‘constant’ or adjusted data Use inflation deflator or price index for

real

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

NominalReal (2005)

Income in current and 2005 CPI-U-RS adjusted dollars

Adjusting to Real ValuesPrice Index (CPI, PPI) - need base year

Market baskets of goods, tracks price changes E.g., http://www.minneapolisfed.org/research/data/us/calc/ With ‘Dec 1977=100’ (for income graphs) CPI-U-RS1990=198.0; CPI2005=286.7 So $30,7571990$* (286.7/198.0) = $44,536 2005$ These are values graphed

Price Deflators (GDP Deflator, etc.) Work in similar ways but based on output of

economy not prices

Other Real and Nominal ValuesExample: 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 2000 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 RatesMarket interest rates are nominal

They ideally reflect inflation to ensure valueBuy $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 RatesMarket interest rates are nominal

They reflect inflation to ensure valueReal 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 - SimilarFor 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 ExampleCity: 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

Similar Idea : Exchange Rates / PPPBig Mac handoutCommon Definition of inputsShould be able to compare cost

across countriesInteresting results? Why?What are limitations?

Hyperbolic Discounting