Discounting Intro/Refresher

H. Scott Matthews12-706/73-359Lecture 11a - Oct. 8, 2004

Project Financing

Recall - will only be skimming this chapter (6) in lecture - it is straightforward and mechanical Especially with excel, calculators, etc. Should know theory regardless Should look at problems in Chapter 6

and ensure you can do them all on your own

Common Monetary Units

Often face problems where benefits and costs occur at different times

Need to adjust values to common units to compare them

Photo sensor - look at values over several years...

Ex: Compounding (Future Value)

Buy painting for $10,000 Will be worth $11,000 in one year (sure) Need to consider ‘opportunity cost’ Make table or diagram of streams of benefits

and costs over time

Have several analysis options Put $10,000 in savings, would earn simple

interest (8%): so $10,000(1.08)=$10,800 So should you buy the painting?

Decision Rules

As always, should choose option that maximizes net benefits Now we are using that same rule with

values adjusted for time value of money Still choose option that gives us the

highest value In this case it is ‘buying the painting’ Called ‘future value’ when you

compound current value to future

Alternative - Present Value

Do the problem in reverse Time - line representation How much money you would need to

invest in savings to get $11,000 in 1 year FV=$10,000(1+i) : $10,000 was ‘present’ PV=FV/(1+i); PV=$11,000/(1.08)=$10,185 Since greater, should buy painting

Has lower investment cost of $10k Last option - convert all to present value

Net Present Value Method

Current investment of $10,000 for painting represented as -$10,000

Receipt of $11,000 in a year as +$11,000/(1.08)

So NPV= -$10,000 + $10,185 = $185Since NPV positive, should buy

painting (it has positive net benefits) Relevance to Kaldor-Hicks, BCA rule?

General Terms

Three methods: PV, FV, NPVFV = $X (1+i)n

X : present value, i:interest rate and n is number of periods (eg years) of interest

Rule of 72PV = $X / (1+i)n

NPV=NPV(B) - NPV(C) (over time)Real vs. Nominal values

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

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: r=(i-m)/(1+m)

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

Rates to Use for Analysis

In example, investments vs. savings We assumed an actual option for rate

But can use any rate to discount FV! Called a discount rate- may be set for us

MARR: opportunity cost of fundsAssume all values ‘real’ unless

stated otherwise

Minimum Attractive Rate of Return

MARR usually resolved by top management in view of numerous considerations. Among these are: Amount of money available for investment, and

the source and cost of these funds (i.e., equity or borrowed funds).

Number of projects available for investment and purpose (i.e., whether they sustain present operations and are essential, or expand present operations)

MARR part 2

The amount of perceived risk associated with investment opportunities available to the firm and the estimated cost of administering projects over short planning horizons versus long planning horizons.

The type of organization involved (i.e., government, public utility, or competitive industry)

In the end, we are usually given MARR

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

Sensitivity Analysis

How do NPV results change with i?Back to our garbage trucks example

Vary the real discount rate from 4-10% NPV declines as rate i increases Future benefits ‘discounted more’

See updated RealNominal.xls

Other Issues

Inflation hard to predict Tend to use historical trends/estimates

Terminal or residual values Value of equipment at end of

investmentTiming - typically assume beginning

of period values, not end of period

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 .014hr = $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 1993 was 1 cent

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

Another Analysis Tool

Assume 2 projects (power plants) Equal capacities, but different lifetimes

35 years vs. 70 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

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

Equivalent Annual Benefit

EANB=NPV/Annuity Factor Annuity factor (i=5%,n=70) = 19.343 Ann. Factor (i=5%,n=35) = 16.374

EANB(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

Internal Rate of Return

Defined as discount rate where NPV=0Graphically it is between 8-9%But we could solve otherwise

E.g. 0=-100k/(1+i) + 150k /(1+i)2 100k/(1+i) = 150k /(1+i)2

100k = 150k /(1+i) <=> 1+i = 1.5, i=50% -100k/1.5 + 150k /(1.5)2 <=> -

66.67+66.67

Decision Making

Choose project if discount rate < IRRReject if discount rate > IRROnly works if unique IRRCan get quadratic, other NPV eqns

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?