E02 CashFlow

Centre for Management of Technology and Entrepreneurship University of Toronto

Cash Flow Analysis Part A

Copyright: Joseph C. Paradi 1996-2004 Centre for Management of Technology and Entrepreneurship

Course: CHE349 File: CHE349/CashFlowA4

Categories of Cash Flows  First cost - expense to build or to buy and install  Operations and maintenance (O&M) - annual expense, can include ; electricity, labour, repairs, etc.  Salvage value(s) - receipt at project termination for disposal of the equipment (can be a salvage cost)  Revenues - annual receipts due to sale of products or services  Overhauls - major capital expenditure that occurs part way through the life of the asset  Prepaid expenses - annual expenses, such as leases and insurance payments, that must be paid in advance Centre for Management of Technology and

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Economic Equivalence  How do we measure and compare economic worth of various cash flow profiles? We need to know:  Magnitude of cash flows  Their direction (receipt or disbursement)  Timing (when does transaction occur)  Applicable interest rate(s) during time period under consideration

 We should be economically indifferent to choosing between two alternatives that are economically equivalent and could therefore be traded for one another in the financial marketplace.  Any cash flow can be converted to an equivalent cash flow at any point in time. Centre for Management of Technology and

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An Example of Equivalence  Suppose you are offered the alternative of receiving either $3,000 at the end of 5 yrs (guaranteed) or P dollars today. You would deposit the P dollars into an account that pays 8% interest. What value of P would make you indifferent to your choice between P dollars today and the $3,000 at the end of 5 yrs?  Determine the present amount that is economically equivalent to $3,000 in 5 yrs given the investment potential of 8% per year. F = $3,000, N = 5 years, i = 8% per year Find P P = F/(1+i)N = $2,042 Centre for Management of Technology and

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Economic Equivalence: General Principles  Equivalence calculations to compare alternatives require a common time basis  Equivalence depends on the interest rate  May require conversion of multiple payment cash flow to a single cash flow  Equivalence is maintained regardless of the individual's point of view (borrower or lender)

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Equivalence - A Factor Approach  Now we can look at organizing the approach to Engineering Economy by defining its language and notations.  Engineering Economy Factors apply compound interest to calculate equivalent cash flow values. The tabulated values at the end of the book (or you can use the functions in spreadsheets) convert from one cash flow quantity (P, F, A, or G) to another.  Assumptions: 1. Interest is compounded once per period 2. Cash flow occurs at the end of the period 3. Time 0 is period 0 or the start of period 1 4. All periods are the same length Centre for Management of Technology and

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Definitions  The following are the definitions for the variables used: i interest rate per period, expressed as a decimal N Number of periods (also called the study horizon) P Present cash flow, or value equivalent to a cash flow series F Future cash flow at the end of period N, or future worth at the end of period N equivalent to a cash flow series A Uniform periodic cash flow (annuity) at the end of every period from 1 to N. Also a uniform constant amount equivalent to a cash flow series. G Gradient or constant period-by-period change in cash flows from period 1 to N (arithmetic series) Centre for Management of Technology and

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Factor Notation  These have their roots in the pre-computer age when prepared tables were used by engineers for many design needs. However, they still serve to state the problem and can be used to solve it too.  The format of engineering economy factors is:  (X/Y, i%, N) where X and Y are chosen from the cash flow symbols P,F,A and G.  So, if you have Y multiplied by a factor, you get the equivalent value of X: P = A(P/A, i ,N) e.g. convert from a cash flow (F) in year 10 to an equivalent present value (P), the factor is:  (P/F, i N) - see text pages after 559 for the tables Centre for Management of Technology and

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Names of the EE Factors see pp 83 in text  So now we know how all this works together, but these factors have names as follows: (P/F,i,N) Present worth factor (F/P,i,N) Compound amount factor (P/A,i,N) Series present worth factor (A/P,i,N) Capital recovery factor, how much an investment has to return to recover its cost - no salvage value (A/F,i,N) Sinking fund factor - where a savings account is used to accumulate funds for future investment (F/A,i,N) Series compound amount factor  Note that the first letter is the variable you are seeking and the second the one you have P/A want First Cost (investment), have annuity payment

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Compound Interest Factors for Discrete Compounding  The four discrete cash flow patterns are:  a single disbursement or receipt  a set of equal amounts in/out over a sequence of periods annuity  a set of equal amounts in/out that change by a constant amount from one period to the next in a sequence of periods arithmetic gradient series  a set of equal amounts in/out that change by a constant proportion from one period to the next in a sequence of periods - geometric gradient series

 The following assumptions apply:  compounding periods are equal  cash flows at the end of the period (consider payment at 0 to be at period -1  annuities and gradients occur at period ends Centre for Management of Technology and

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The Basic Single Payment Factors  P and F are the basic single payment quantities and they are related as follows: F = P(1 + i)N can be developed as: Fn= P(1+ in) - where: i is the annual interest rate simple interest - but this is not very useful, so we develop the compound interest model: Fn = Fn-1 (1+ i)  So we can see this in the EE notation as: (F/P,i,N) = (1 + i)N and for the reverse (P/F,i,N) = (1 + i)-N  The limits of P/F and F/P factors are:  1 when i and N approach 0  P/F approaches 0; F/P infinity when i and N approach infinity. Centre for Management of Technology and

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The Relationship between P and F F If you had $2,000 F occurs N periods after P 1

0

2

N-1

N

now and invested it at 10%, how much would it be worth in 8 years?

P P = $2,000, i = 10% per year, N = 8 years F = P(1+i)N =$2,000(1+0.10)8= $4,287.18 F = P (F/P,10%, 8)=$2,000*2.1436 =$4,287.20

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Another P and F Example F F occurs N periods after P 1

0

2

N-1

N

P If $1,000 is to be received in 5 years. At an annual interest rate of 12%, what is the P of this amount? F=$1000, i = 12%, N = 5 years P = F/(1+i)N= $1000/(1+.12)5 = $567.40 P = F(P/F, 12%,5) = $1000(0.5674) = $567.40 Centre for Management of Technology and

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Discrete Compounding, Discrete Cash Flows P. 571

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Combining Factors  We can combine these factors in various ways to create a model for the real problem at hand:  delayed income stream because of startup time  another need may be prepaid expenses - also a way to deal with startup delays or construction delays.

 We can link formulas together to model the actual proposition - in fact deriving one formula’s factor from an other’s  Also, many times the problem has to be defined in more than one part  Then, there are many approaches to a specific problem - one may be longer but the answer must be the same Centre for Management of Technology and

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A Simple Example  We invest $120,000 [P=-120,000]  We pay for 5 years $30,000/year [P=30,000(P/A,12%,5)]  Then pay $35,000 at year 3 [P=-35,000(P/F,12%,3)]  Receive $40,000 at year 4 [P=40,000(P/F,12%,4)]



$40,000

$30,000

$30,000 $35,000

$120,000 Centre for Management of Technology and

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Compound Interest Factors for Annuities: Uniform Series  All cash flows in series are equal (annuity)

P

A

A

1

2

A

A

N-2 N-1

A

N

Note: P occurs 1 period Before 1st A, F would occur At same time as last A.

P = A(1 + i)-1 + A(1 + i)-2 + …. + A(1 + i)-(N-1) + A(1 + i)-N =

n



K1

A(1 + i)-K

 (1  i) N  1 P = A n   i(1  i)  Series Present Worth Factor Centre for Management of Technology and

P = A(P/A i, N)

p.62 Text Course:

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Compound Interest Factors for Annuities: Uniform Series  From before, P = A [(1+i)N-1]/[i(1+i)N]  Then to get (F/A, i, N) we can convert this to a future single payment by multiplying both sides by (1+i)N  (1+i)N * P = A[(1+i)N-1]/i resulting in  F = A[(1+i)N-1]/i

p.60 text

 Thus, we get the (P/A,i,N) and the (F/A,i,N) factors

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Bonds  Financial instrument used by large firms and government to raise funds to finance projects, debt obligations,  A special form of loan, usually with a long term  The creditor (firm or government) promises to pay a stated amount of interest at specified intervals for a defined period and then to repay the principal at a specific date (maturity date)  Canada savings bonds usually represent the lowest interest because they are the safest (set tone for interest rates), then are provincial bonds, next are “Blue-Chip” corporates (banks, large firms)  “Junk” bonds are very high interest and risk Centre for Management of Technology and

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Bond Terminology P = Purchase price of a bond F = Sales price (or redemption value) of a bond V = Par (face) value – the stated value on the bond r = annual interest shown (or coupon rate) n = compounding period (quarterly, six months, etc.) i = the yield rate per annum (usually the market rate) N = number of remaining years to maturity Redemption – when the issuer pays for it in cash Maturity Date – date on which the par value is to be repaid (date bond expires) Market Value – the price one has to pay to buy a bond A = V(r/n) = the value of a single interest payment (a coupon) P = V(r/n) (P/A, i/n, nN) + F (P/F, i, N) Centre for Management of Technology and

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Bond Valuation  Yield to maturity (return on investment) = actual interest earned from a bond over the holding period (equivalent to IRR calculations)  When considering buying or selling bonds, the yield is the fundamental consideration and this depends on a number of issues:    

Current yields for that type of security (economy dependent) Inflation rate (usually included in the yield by market forces) Your tax position Foreign exchange risk if you invest in other currencies

 The actual price of a bond, even at issue, is very unlikely to the face value, except at redemption. Centre for Management of Technology and

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Canadian Bonds

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The Yield Curve

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Example of Arithmetic Gradients  We have two propositions for investment:  New Multi Processor Computer  Natural Gas Pipeline

 Cash Flow Profiles End of Year Computer 0 -$50,000 1 +$20,000 2 +$15,000 3 +$10,000 4 +$5,000

Natural Gas -$50,000 +$5,000 +$10,000 +$15,000 +$20,000

 Which is preferable, the computer or natural gas? Both have revenues of $37,000 over the time period. Centre for Management of Technology and

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CF Diagrams for Computer & Natural Gas Options $20,000 $15,000 (+) $10,000

(+)

$5,000 0 (-)

1

2

Computer Option

$50,000

3

$20,000 $15,000 $10,000 $5,000

4

0 (-)

1

2

3

4

Natural Gas Option

$50,000

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Arithmetic Gradients  An easy definition of an arithmetic gradient is: revenue grows by $10,000 (G) each year. (N-1)G  But the rate is not always constant 3G

2G

Note: No cash flow at end of period 1 Each successive cash flow increases by a fixed amount equal to G 0

G

1

2

3

4

N

 There are four possibilities besides G=0 1. A>0 and G>0 - means positive and increasing 2. A>0 and G