Cost and Time Value of $$

Prof. Eric Suuberg

ENGINEERING 90

Cost and Time Value Lecture

What is our goal?» To gain an understanding

of what is and what is not a good project to undertake from a financial point of view.

What are our tools?» Material presented by

Prof. Crawford» Discounting / Time Value

of Money» Tax Savings through

Depreciation

So, what are we starting today?

Go through some of the “fun” math for Present Value Calculations

Do a teaching example of purchasing a machine for a manufacturing plant

Talk about costs – both the obvious kind as well as the non-obvious types

Time value of money calculations

Cost Comparisons Depreciation Put it all together – inc.

continuous discounting and after-tax cost comparisons

Have I Got a Deal for You Would you be interested in investing in

a company that has $1 million in annual sales?

What More Would You Like to Know?

Annual operating expenses (salaries, raw materials, etc.)

Suppose these were $900,000/yr Are you interested? (Come on - I’ve got

to know now. There are a lot of people interested)

ProfitProfit = Sales (revenues) - expenses (costs) Basis for taxation - What goes into the

calculation is of great interest to Uncle Sam

In Our Example Profit = $1,000,000/yr - $900,000/yr

= $100,000/yr Is this a good business?

What Would You be Willing to Pay Me for this

Business? $1 million?

$2 million? How do you decide? This is one of the questions that we will

answer in this part of the course.

Present Value Calculations

Essential element of evaluating a business opportunity

Different variants» Simple discounting» Replacement and abandonment» Venture Worth, Present Value, Discounted

Cash Flow Rate of Return

What information do we need?

Investment (Capital assets, working capital)

Lifetime and Salvage Values Operating Costs

» Fixed» Variable

Interest Rate Tax Rate Depreciation Method Revenues

Information

Required

Capital Investment -

Facility Purchased Process Equipment Field Constructed Equipment Wiring, Piping, Instrumentation Construction, Installation Costs Site Preparation, Buildings Storage Areas Utilities Services (Cafeterias, Parking lots, etc.) Contingency

Capital Investment- Manufacturing

Costs of process equipment may represent only 25% of actual investment!

Costs of process equipment scale according to the “six-tenths rule”» C2/C1 = (Q2/Q1)0.6

See, for example:» “Cost and Optimization Engineering”

by F.C. Jelen and J.H. Black, McGraw-Hill, 1983.

Other Items Working Capital

» Raw materials and supplies inventory» Finished goods in stock and Work in Progress» Accounts Receivable, Taxes payable

Operating Costs» Labor and Raw Materials» Utilities and Maintenance» Royalties

Fixed Costs» Insurance, rent, debt service, some taxes

Time Value of Money $1 today is more valuable than

the promise of $1 tomorrow» Has nothing to do with inflation

“Discounting” is the term used to describe the process of correcting for the reduced value of future payments

Discount rate is the return that can be earned on capital invested today

Future Worth of an Investment

P = Principal i = Annual Interest Rate S = Future value of investment

Compound Interest Law S1 = P (1+i) at the end of one year

S2 = S1(1+i) = P(1+i)2 at end of year 2

Sn = P (1+i)n at end of year n

Present Value of a Future Amount

P = Sn / (1 + i)n

= Sn (1 + i)-n

(1 + i)-n = Present Value Factor or Discount Factor

The promise of $1 million at a time 50 years in the future @ i = 15%/yr

P = $1,000,000(1+0.15)-50 = $923

Simple Example What is the PV of $10.00

today if I promise to give it to you in fifteen years, given a discount rate of 20%?

PV = 10(1.20)-15

= $.65 Not enough to buy a soda

these days

Take Home Message Not all dollars of profit are the same Those that come earlier are “worth”

more

Start with Simple Example from Everyday Life

Do you buy the better made equipment with the higher price tag? or the low first cost equipment that has high maintenance?

Cost ComparisonsWhat are we doing here? Comparing one project to another Deciding to buy the expensive computer

that has free maintenance versus the cheap one that makes you pay for service

vs.

Simple Cost Comparisons

Strategy» Reduce costs (and/or revenues) to a

common instant, usually the present time» Work on full year periods» approximate costs or revenues which occur

over the year as single year-end amounts Basic Rule: All comparisons must be

performed on an equal time period basis

Unequal Lifetime Cost Comparisons

Repeatability Assumption

(to get to same time basis)

Annuity Comparison

Co-termination assumption

First Some Useful Mathematical Machinery

Uniform periodic annual payments (annuities)

Projects frequently generate recurring income or cost streams on an annual basis

1 65432 (m-1) m0

x x x x x x x x

x = annuity

Discounting a Series of Payments

m

j 2 mj 1

1 1 1 1P X X ...(1 i) (1 i) (1 i) (1 i)

1(1 i) a

2 3 mP X a a a ... a 2 3 4 m 1Pa X a a a ... a

m 1P X(1 a) x(a a )

Discounting a Series of Payments con’t

m 1 ma a 1 aP X Xa1 a 1 a

m

m

m

11 1 i 11 1 iX X11 i i(1 i)1

1 i

m 1 m

m

(1 i) 1 (1 i)P X Xi(1 i) i

Capital Recovery Factor

m

i captial recovery factor1 (1 i)

m

iX P1 (1 i)

Future Equivalents of Annuities

mm

m

(1 i) 1P S(1 i) Xi(1 i)

m(1 i) 1S Xi

m

iX S(1 i) 1

Link to summary of useful formulae

Examples What future payment N years from now shall I

accept in return for an investment of $P now, given I could instead invest my money elsewhere (e.g. a bank) and earn i %/yr?

What set of annual revenues for N years will entice me to invest $P, given the same alternative as above?

NS P(1 i)

N

iX P1 (1 i)

What price should I pay for an investment which returns $X/yr for N years, if i %/yr is available to me in a bank?

What annual interest rate (bank, etc.) would be required to make an investment returning $S in N years on a present investment of $P?

Examples

N1 (1 i)P Xi

NSi 1P

A Simple Replacement Problem

Process to be operated for 4 years and then junked

Do you buy a new low-maintenance machine now or not???

DATA (neglect tax effects)

Options Stick w/old Buy newPurchase Price ($) 0 4000Operating Cost ($/yr) 2000 500Lifetime (yrs) 4 4

Cash Flow Time Lines1 4320

$2000 $2000 $2000$2000

OLD

1 4320

$500 $500 $500$500

NEW

4

old1 (1 i)P $2000

i

4

new1 (1 i)P $4000 500

i

$4000

The Key Role of Interest Rates

4

old1 (1 i)P $2000

i

4

new1 (1 i)P $4000 500

i

If management demands i = 10 %/yrPold=$6340, Pnew=$5585 new is better choice

If management demands i = 20 %/yrPold=$5180, Pnew=$5295 old is better choice

Note In a replacement problem like this you

could have added revenues to the analysis, but no need to do so if they are the same for both options.

Financial Comparisons with Unequal Lifetimes

Simple Example: Choose between 2 pieces of equipment, one of which is better built and has a longer lifetime

N is not the same for both

Not a fair comparison with N=2 unless process is to be shut down and both options have no residual value

Well Built

Poorly Built

20 year life

2 year life

What to Do? Option 1 - Repeatability

Well Built 20 year life

Poorly Built 2 year life

(Buy 1)

(Buy 10)

Option 2 - Annualized Costs

Convert the investment and maintenance for both options into a single annual payment

Alternative 1 Alternative 2Purchase Price ($)

Annual Op. Cost ($/yr)

Salvage Value ($)Service Life (yrs)

10,000 20,000

1500 1000

500 1000

2 3

i = 0.15 / yr

Annualized Cost of Alternative 1

50 500

$1500$10,000 $15001 2

0

$7418 $74181 2=

2

1500 1500 500P 10,000 $12,0601 .15 (1 .15)

Now 2

iX P1 (1 i)

2 2

i .15X 12060 12060 $74181 (1 i) 1 (1.15)

0 500

150010,000 15001 2

Annualized Cost of Alternative 2

01000

100020,000 1000

1

1000

0

9472 9472

1

9472

32=

In this case, choose alternative 1 because yearly cost is lower.

2 3