Discounting Overview

H. Scott Matthews12-706 / 19-702 / 73-359Lecture 3

Book Progress Update

They’ve been shipped - not sure when they’ll arrive. Paperback price.

First 20 get them, I think I have 18 so far.

Project Financing

Goal - common monetary unitsRecall - will only be skimming this

chapter 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, NPVFV = $PV (1+i)n

PV: present value, i:interest rate and n is number of periods (e.g., years) of interest

Rule of 72 i is discount rate, MARR, opportunity cost, etc.PV = $FV / (1+i)n

NPV=NPV(B) - NPV(C) (over time)Other methods: IRR (rate i at which NPV=0)All methods give same qualitative answer.Assume flows at end of period unless stated

Notes on Notation

PV = $FV / (1+i)n = $FV * [1 / (1+i)n ] But [1 / (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

Ex: The Value of Money (pt 1)When did it stop becoming worth it for the

avg American to pick up a penny?Two parts: time to pick up money?

Assume 5 seconds to do this - what fraction of an hour is this? 1/12 of min = .0014 hr

And value of penny over time? Assume avg American makes $30,000 / yr About $14.4 per hour, so .0014hr = $0.02 Thus ‘opportunity cost’ of picking up a

penny is 2 cents in today’s terms

Ex: The Value of Money (pt 2)

If ‘time value’ of 5 seconds is $0.02 now Assuming 5% long-term inflation, we can

work problem in reverse to determine when 5 seconds of work ‘cost’ less than a penny

Using Excel (penny.xls file): Adjusting per year back by factor 1.05 Value of 5 seconds in 1984 was 1 cent

Better method would use ‘actual’ CPI for each year..

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 + 1000/1.05 + 1000/1.052 +

1000/1.053 + 1000/1.054

NPV1 =1000 + 952 + 907 + 864 + 823 = $4,545

NPV2 = 1000/1.05 + 1000/1.052 + 1000/1.053 + 1000/1.054 + 1000/1.055

NPV2 = 952 + 907 + 864 + 823 + 784 = $4,329

NPV1 - NPV2 ~ $216 Note on Notation: use U for uniform $1000 value

(or A for annual) so (P|U,i,n) or (P|A,i,n)

Relative NPV AnalysisIf 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

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 (using Cambpell notation)

Market interest rates are nominal They reflect inflation to ensure value

Real rate r, inflation i, nominal rate m Simple method: r ~ m-i <-> r + i ~ m More precise: r=(m - i)/(1+i)

Example: If m=10%, i=4% Simple: r=6%, Precise: r=5.77%

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 spreadsheet for nominal values

Annuities

Consider the PV of getting the same amount ($1) for many years Lottery pays $P / yr for n yrs at i=5% PV=P/(1+i)+P/(1+i)2+ P/(1+i)3+…+P/(1+i)n

PV(1+i)=P+P/(1+i)1+ P/(1+i)2+…+P/(1+i)n-1

------- PV(1+i)-PV=P- P/(1+i)n

PV(i)=P(1- (1+i)-n) PV=P*[1- (1+i)-n]/i : “annuity factor”

Perpetuity (money forever)

Can we calculate PV of $A received per year forever at i=5%?

PV=A/(1+i)+A/(1+i)2+…PV(1+i)=A+A/(1+i) + …PV(1+i)-PV=APV(i)=A , PV=A/i E.g. PV of $2000/yr at 8% = $25,000When can/should we use this?

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 NPV(1)=-100+(6.5/1.05)+..

+6.5/1.0570=25.73 NPV(2)=-50+(4.2/1.05)+..

+4.2/1.0535=18.77 NPV(2R)=18.77+(18.77/1.0535)=22.17 Makes them comparable - Option 1 is best There is another way - consider

“annualized” net benefits

Equivalent Annual Benefit

EANB=NPV/Annuity Factor Annuity factor (i=5%,n=70) = 19.343

Ann. Factor (i=5%,n=35) = 16.374EANB(1)=$25.73/19.343=$1.330EANB(2)=$18.77/16.374=$1.146

Still higher for option 1Note we assumed end of period pays

Benefit-Cost Ratio

BCR = NPVB/NPVC

Look out - gives odd results. Only very useful if constraints on B, C exist.

Beyond Annual Discounting

We generally use annual compounding of interest and rates (i.e., i is “5% per year”)

Generally, FV = PV (1 + i/k)kn

Where i is periodic rate, k is frequency of compounding, n is number of years

For k=1/year, i=annual rate: FV=PV(1+i)n

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, PV*(1 + i/k)kn -->

PV*ein

$1083.29 using our previous exampleWhat types of problems might find

this equation useful?

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

Examples (from Campbell)