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

€

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!

€

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%

€

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