Lecture 2 Time Value of Money0

8/12/2019 Lecture 2 Time Value of Money0

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Cash Flows

Inflows and outflows of money.

Estimate or observed values. Future = estimate

Accuracy of estimates Quality of economic analysis andconclusions.

Inflows = Cash Receipts (+) Revenue, Income

Outflows = Cash Disbursements (-)

Expenses, Cost

Net Cash Flow = Receipts Disbursements

= Cash inflows Cash outflows


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Examples of Cash Flows

Inflows Revenues (incremental when

comparing alternatives)

Operating cost reductions (from analternative)

Asset salvage value Receipt of loan principal

Income tax savings

Receipts from stock and bond sales

Construction and facility cost savings

Saving or return of corporate capitalfunds

Outflows First cost of assets

Engineering design costs

Operating Costs (annual andincremental)

Periodic maintenance and rebuildcosts.

Loan interest and principal payments

Major expected/unexpectedupgrade costs

Income taxes Expenditure of corporate capital

funds


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End-of-period convention

All cash flows are assumed to occur at the end of an interestperiod.

When several receipts and disbursements occur within agiven interest period, the netcash flow is assumed to occurat the endof the interest period.


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Cash Flow Diagram

Helps to visualise complex cash flows. Graphical representation.

Vertical axis = Cash Flows ($)

Known, estimated

and needed Horizontal axis = Time

scale (Interest periods)

Most often,

Interest period = Year Draw to scale


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Cash Flow Diagram: Example 1


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Present Value Future Value

Find the present (or past) or future value of a single cash flow.

F = P(1+i)n = Future Value

F/P factor = (1+i)n

Single-payment compound amount factor (SPCAF)

P = F / (1+i)n = Present Value

P/F factor = 1 / (1+i)n

Single-payment present worth factor (SPPWF)


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Annual Value Present Value

What is the equivalent present value, in year 0, for a uniformseries A of end-of-period cash flows beginning at the end ofyear 1, and extending for n periods.


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Consider each amount individually as a future value, calculatetheir present value, and sum the results.

P = A/(1+i) + A/(1+i)2 + A/(1+i)3 + + A/(1+i)n

Show that we therefore have:

P/A Factor =

Uniform-series present worth factor (USPWF)

( )( )

01

11

+

+= iii

iAP

n

n

( )

( )0

1

11

+

+i

ii

i

n

n


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Divide by 1 + i

Subtract line 2 fromline 3

Rearrange

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

( )( )

( )

( )( )

01

11

1

11

11

1

1

1

1

1

1

1

1...

1

1

1

1

1

1...

1

1

1

1

1

1

1...

1

1

1

1

1

1

1

1

1...

1

1

1

1

1

1

1...

111

1

2132

1432

32

32

+

+=

+

+=

+=

+

+=

+

+++

++

+

+++

++

+=

+

+++

++

++

+=

+

+++

++

++

+=

+++

++

++

+=

+

+

+

iii

iAP

i

i

Ai

AiP

iA

iAP

i

i

iiiA

iiiAP

i

P

iiiiA

i

P

iiiiAP

i

A

i

A

i

Ai

AP

n

n

n

n

n

n

nn

n

n

n


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Capital Recovery Factor (CRF)

Equivalent uniform annual worth A over n years for a given Pin year 0, where interest rate is i.

CRF = A/P Factor = ( )( ) 11

1

+

+n

n

i

ii


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Annual Value Future Value

Homework: Use the equations and concepts from theprevious slides to calculate the following.

1) Sinking Fund Factor = A/F Factor

Eqn 2.8 and Eqn 2.9 (page 60 Blank 6th Ed)

2) Uniform-series compound amount factor (USCAF) = F/A Factor


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Arithmetic Gradient Factors

Cash Flow series thatchanges by a constant

amount each period.

Arithmetic Gradient, G= Constant arithmetic change in themagnitude of receipts or disbursements from one time period tothe next. (Eg. $50)

Base amount, A1, is paid at the end of period 1. ($1500) At the end of period 2: A1+G

Cash Flow at the end of period n = CFn = A1 + (n-1)G


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P/G Factor

Multiply by (1+i)

Subtract line 1

from line 2

First term on RHS

the same as P/A

( ) ( )

( )

( )

( )( ) ( )

( )( )

( )( ) ( ) ( )

( )( )

( ) ( )( )

( )( )( )

( ) ( ) ( )( )( )

( ) ( ) ( ) ( ) ( )

( )( ) ( )

( )( )

+

+=

+

+

+=

+

++

+++

++

+=

+

+++

++

+=

+

+

+

++

++

+

+

+

++

++

+=+

+

++

++

+=+

+

++

+

+

+

=

n

n

nn

n

G

nnnG

nnG

nn

nGG

nG

nG

ii

niiG

ii

nG

ii

i

i

GP

i

nGiiii

GiP

i

n

iiiGiP

i

n

i

n

iiG

i

n

iiiGPiP

i

n

iiGiP

i

n

ii

GP

1

11

11

11

11

1

1

1...1

1

1

1

1

1

1

1...

1

1

1

1

1

1

1

2...

1

2

1

1

1

1...

1

3

1

2

1

11

1

1...

1

2

1

11

1

1...

1

2

1

1

2

121

121

132

1321

121

32


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Arithmetic Gradient Factors

P/G Factor = Arithmetic-Gradient Present Worth factor:

Extension: Find the following factors: A/G Factor = Arithmetic-Gradient Uniform-series factor.

F/G Factor = Arithmetic-gradient future worth factor.

In each case the cash flow series can be separated into the base amountseries, A and the gradient G, and the present, future or annual worthcalculated from these separately, and then added together.

( )( )

+

+=

n

n

G

ii

niiGP

1

11

2


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Cash Flow Series

To derive equations: Describe cash flow using a simple equation, usually as a

combination of:

One-off cash flows: F, CF = F

Uniform series: A, CF = A

Arithmetic gradient: G, CFn = (n-1)G Geometric gradient: g, CFn = A1(1+g)

n

Convert each component of the cash flow series to a presentvalue, and then sum present values.

Convert to a future or annual value if need.


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Cash Flow Series Example

P = F / (1+i)n

( )( )

+

+=

n

n

G

ii

niiGP

1

11

2

( )( )

01

11

+

+= i

ii

iAP

n

n


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Calculate the present value of each component


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Calculate the annual value of each component


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Geometric Gradient Series

Extension: Calculate the equivalent present value of a cash flowseries that changes from period to period by a constantpercentage.

g = constant rate of change,

in decimal form, by which

amounts increase or

decrease from one period to

the next. Eg. g = 0.05

Cash Flow Series = A1(1+g)n

P(A1, g, i) = ?


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Combining Cash Flows Series

What if the cash flow series is more complicated?


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Combining Cash Flows

Use the following steps to avoid errors Draw a cash flow diagram.

Identify the component cash flows: Single payments, annual series,arithmetic gradient series or geometric gradient series.

Locate the appropriate reference point at which to calculate the

present value of each component cash flow on the cash flowdiagram.

Renumber the cash flow diagram to determine n for each calculation.

Solve the equations for each present value.

Calculate the value of each present value at the reference time youneed, and add them together.


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Shifting Single Cash Flows

P = F / (1+i)n

P is at the end of period 0, F is at the end of period n.

Use to shift any future single cash flow, back to a previous valueby n years, at a compound interest rate, i.

Use the inverse to shift any single cash flow forward.

Single cash flows can therefore be easily evaluated at anyreference year.


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Shifted Annual Value

Uniform Annual Cash Flows: Start at the end of period 1, and continue consecutively, and

uniformly until the end of period n.

( )

( )0

1

11

+

+= i

ii

iAP

n

n


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Shifted Arithmetic Gradient

Arithmetic Gradients are made up of: A uniform cash flow, starting at the end of period 1, and

continuing until the end of period n.

A gradient, starting at the end of period 2, and continuing untilthe end of period n.

( )( )

01

11

+

+=

+=

iii

iAP

PPP

n

n

A

GAT

( )( )

+

+=

n

n

G

ii

niiGP

1

11

2


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Shifted Geometric Gradient

Extension: How do we represent geometric gradients?

When does each component start and end?

What are the equations for converting each component topresent value?


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Combining Cash Flows Example


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Summary

What did we learn today?


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Summary

Interest Rates (Simple, Compound)

Rates of Return, Minimum Attractive Rate of Return (MARR)

Interest ($) = Final ($) Original ($)

Interest Rate (%) = Interest accrued per time unit x 100%

Original Amount

Rate of Return (%) = Interest accrued per time unit x 100%

Original AmountSimple Interest: Final Amount = Principal (1 +ni)

Compound Interest: Final Amount = Principal (1 + i)n

Equivalence

Time Value of Money

Present Value Future Value Annual Value

Cash Flows, Series

( )

( )0

1

11

+

+

=

+=

iii

iAP

PPP

n

n

A

GAT

( )( )

+

+=

n

n

G

ii

niiGP

1

11

2

( )niF

P+

=1

( )( )

01

11

+

+= i

ii

iAP

n

n

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Lecture 2 Time Value of Money0