lec sept 18 ch3 lec cash flow analysis ii

8/10/2019 Lec Sept 18 Ch3 Lec Cash Flow Analysis II

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Engineering Economics

Chapter 3

Cash Flow Analysis - II

3 - 1

Sept. 18, 2014


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Example: Uneven Multiple Cash Flows, but we separate(decompose) these into single cash f lows!

How much do you need to deposit today (P) to withdraw $25,000at n=1, $3,000 at n= 2, and $5,000 at n=4, if your account earns10% annual interest?

0

1 2 3 4

$25,000

$3,000 $5,000

P


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3 - 3

Decomposit ion of uneven multiple cash flows

Notice that in this example, using the factor seems to beeasier than using the equation!


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Example: Future Value of an Uneven CashFlows with Varying Interest Rates

Given: Deposit series as given over 5 years

Find: Balance at the end of year 5=?

Complicated


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Example: Future Value of an Uneven Serieswith Varying Interest Rates

Given: Deposit series as given

over 5 years

Find: Balance at the end of year 5


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So far, for single cash flow problems, we can

use one of the following approaches:1. Use the equations

2. Use the Compound Interest Factors (the tables)

3. In addition, we can use the Excel Spreadsheet

Functions3 - 6

F = P(1+ i)N

P = F/(1+ i)N

Compound Amount Factor : (F/P,i,N)

Present Worth Factor: (P/F, i, N)


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3 - 7

Compound Interest Factors and Excel Function(Textbook p.50)


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3 - 8

Single Cash Flow - Finding the Future Value F=?


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In Excel: FVfunction (Future Value function).

Syntax FV rate, nper, [pmt], -pv, [type])

Rate Required. The interest rate per period. (e.g. 10%/12, or 0.83%, or 0.0083)Nper Required. The number of periods.Pmt Optional. The payment made each period (annuity).

Pv Required. The present value (required if Pmt is omitted).Type Optional. The number 0 or 1 and indicates when payments are due. If type isomitted, it is assumed to be 0.


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In Excel: PVfunction (Present Value function).

SyntaxPV rate, nper, 0, - fv, [type])

Rate Required. The interest rate per period. (e.g. 10%/12, or 0.83%, or 0.0083)Nper Required. The total number of periods.Pmt Optional. The payment made each period (annuity).Fv Required. The future value, (required if Pmt is omitted)

Type Optional. The number 0 or 1 and indicates when payments are due.


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Single Cash Flow - Finding the interest rate i=?


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Excel: RATEfunction

RATE(nper, pmt, -pv, [fv], [type], [guess])Syntax

The RATE function syntax has the following arguments:

Nper Required. The total number of periods.Pmt Required. The payment made each period (annuity). If pmt is omitted, you must

include the fv argument.Pv Required. The present value.Fv Optional. The future value. If fv is omitted, it is assumed to be 0.

Type Optional. The number 0 or 1 and indicates when payments are due.


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Single Cash Flow - Finding the compoundingperiods N=?


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Excel: NPERfunction

NPER(rate, pmt, -pv, [fv], [type])Syntax

The NPER function syntax has the following arguments:

Rate Required. The interest rate per period.Pmt Required if no Fv value specified. If no annuity, Pmt=0.Pv Required. The present value.Fv Optional. The future value. If fv is omitted, it is assumed to be 0.Type Optional. The number 0 or 1 and indicates when payments are due.


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3 - 15

Annuity:series of Nreceipts or disbursements that beginat end of period 1and continue to end of period N.Notice that if there is any cash flow occurred at period0, it can not be considered as part of the series cashflows, even though the amount is the same as A!

loan payments are classical examples of annuities.

3.5 Compound Interest Factors forAnnuities

0 1 2 3 4 5 6

A

.. NN-1

..

A A A AA A A


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3 - 16

TheSinking Fund Factor: (A/F, I, N)

To find a series of Nreceipts/disbursements(A) thatshould be set aside each period in order to meet a major

financial need in the future (F)

3.5 Compound Interest Factors forAnnuities

1)1(),,,(

+

=N

i

iNiFA


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3 - 17

TheCapital Recovery Factor: (A/P, i, N)

Is used to compute how much to be set aside eachperiod to repay a present use of money.

easily derived from the sinking fund factor and theuniform series compound amount factor:

3.5 Compound Interest Factors for Annuities(contd)

1)1(

)1(

),,/)(,,/(),,/(

+

+=

=

N

N

i

ii

NiPFNiFANiPA


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How did we get these equations?Example: Find the Equation for F, given A, i, N

0 1 2 N 0 1 2 N

A A A

F

A(1+i)N-1

A(1+i)N-2

A

1 2 (1 ) 1(1 ) (1 )

N

N N iF A i A i A A

i

+ = + + + + + =


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How did we get these equations?Example: Find the Equation for F, given A, i, N

1 2 (1 ) 1(1 ) (1 )N

N N iF A i A i A Ai

+ = + + + + + =

Based on the above equation, we can also derive theEquation for finding P,givenA, i, N as

1 2 (1 ) 1(1 ) (1 )

N

N N iF A i A i A A

i

+ = + + + + + =

+

+=

N

N

ii

iAP

)1(

1)1(

To see how the formula derivation, check appendix 3A (p.82)


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Equations- Find P given F

- Find F given P

- Find A given F

- Find A given P

- Find F given A

- Find P given A

ni

FP

)1( +=

niPF )1( +=

+=

i

iAF

n 1)1(

+=

1)1( ni

iFA

+

+=

n

n

ii

iAP

)1(

1)1(

+

+=

1)1(

)1(n

n

i

iiPA


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3 - 21

Practice Problem 3.5a

A Ford Mustang costs $17 000. It can be financed at5.9% for 48 months, with monthly compounding. Howmuch will the monthly payments be?

Answeri= 0.059/12 = 0.00492 per month

A = P(A/P, i, N)= $17 000(A/P, 0.00492, 48) =$398.50

Cannot find values from the Tables!

+

+=

1)1(

)1(n

n

i

iiPA


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Practice Problem 3.5b

What is the present worth of a series of 15 annualpayments of $1000 each, when the first payment isnowand the interest rate is 5%, compoundedmonthly?

AnswerAn effective annual interest rate must be calculated first:

P = 1000 + 1000(P/A, 5.116%, 14)= 1000 + 1000 (9.82563) = $10 826

0.05116=1)12/05.01( 12

+=ei

lec sept 18 ch3 lec cash flow analysis ii

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