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ENGR 3360U Winter 2014

Unit 4.5-6

Arithmetic and Geometric Gradients

Dr. J. Michael Bennett, P. Eng., PMP,

UOIT,

Version 2014-I-01

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Unit 4 Interest and Eqi!a"ence

#han$e Record

2014-I-01 Initia" #reation

%e&t re'erences( #ha)ters 3 and 4

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4.5 Arithmetic Gradient

))ose that or A increases at a 'i&ed amont

e!er ear.

A A10 A20.

We ca"" this the Arithmetic Gradient

%he A is the same.

%he Gradient is ca""ed Gnote con'sion ith $eometric $radient ne&t

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2014-I-014-4

Arithmetic Gradient 7actors/8G and A8G

#ash '"o #7 )ro'i"e

0 1 2 3 n-1n

A1+G

A1+2G

A1+(n-2)G

A1

+(n-1)G

Find P, given gradient cash flow G

CFn = A1 ± (n-1)G

Base amont

= A1

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2014-I-014-5

Gradient E&am)"e

0 1 2 3 4 5 67

$100$200

$300

$400

$500

$600

$700

Gradients have two com!onents"

1# $he %ase amont and the gradient

$he %ase amont (a%ove) = '1time !eriod


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2014-I-014-6

Gradient #om)onents

..

1 & * n-& n-1n

ase amont

)er )eriod

G

1G&G

(n-*)G(n-&)G

(n-1)G

Present worth !oint is 1 !eriod to the left of the G cash flow

For !resent worth of the %ase amont, se the PA factor (alread+ nown)

For !resent worth of the gradient series, se the PG factor (to %e derived)

7ind / o' $radient series


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2014-I-014-9

Gradient *ecom)osition

As e :no arithmetic $radients arecom)rised o' to com)onents

1. Gradient com)onent

2. ase amont

When or:in$ ith a cash '"ocontainin$ a $radient the /8G 'actor ison" 'or the $radient com)onent

A))" the /8A 'actor to or: on the ;ase amont com)onent

/ < /W$radient /W;ase amont


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2014-I-01 *r. +.,. ennett /.En$. /,/ ENGR 3360U En$ Eco4-=

*eri!ation mmar 'or /8G

tart ith(

,"ti)" 1 ; 1i1 to create a second eqation

;tract 1 'rom the second eqation and sim)"i'

>ie"ds

8 26 2 8 36 3 8 46 ...

?n-26G@/87in-16?n-16G@/87in6

P G P F i G P F i G P F i= + + +1)

2

G 1 6 1 1 6 1/<

i 1 6 1 6 1 6

n n

n n n

i n i in

i i i i i

+ − + − −− = + + +

(PG,i,n) factor

o ./cel relation e/ists

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2014-I-014-

Use o' the A8G 7actor

G

2G

(n-2)G

(n-1)G

Find A, given gradient cash flow G

CFn = (n-1)G

.0ivalent A

of gradient

series

A A A # # # AA

A = G(AG,i,n)


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E&am)"e

#N Rai" is considerin$ the de)osit o' B500000 in an accont 'orthe re)air o' o"d and sa'et-qestiona;"e ;rid$es in ritish

#o"m;ia. 7rther the estimate that the de)osits i"" increase

; B100000 )er ear 'or on" ears therea'ter and then cease.

*etermine a )resent orth and ; anna" series amonts that

o"d $i!e the same res"t i' the 'nds earn interest at 5C )er

ear.


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2014-I-014-11

o"tion

%o cash '"os mst ;e made and added( the 'irst 'or the )resentorth o' the ;ase amont P A and a second 'or the )resent orth

o' the $radient P G

%here'ore P T

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4.6 Geometric Gradient eries 7actor

Geometric Gradient

#ash '"o series that starts ith a ;ase amont A1

Increases or decreases 'rom )eriod to )eriod ; a

constant )ercenta$e amont

%his ni'orm rate o' chan$e de'ines A GED,E%RI# GRA*IEN%

Notation($ < the constant rate o' chan$e in decima" 'orm ; hich 'tre

amonts increase or decrease 'rom one time )eriod to the ne&t


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2014-I-014-14

asic *eri!ation( Geometric Gradient2 1

1 1 1 1

1 2 3

1 6 1 6 1 6...

1 6 1 6 1 6 1 6

n

g n

A A g A g A g P

i i i i

−+ + += + + + +

+ + + +

1 2 1

1 2 3

1 1 6 1 6 1 6...

1 6 1 6 1 6 1 6

n

g n

g g g P A

i i i i

− + + += + + + + + + + +

(1)

3%tract .0# (& ) from .0# (* ) to +ield

1 1

12$ 41 6 1112i 41 6 1

n

g n

g P A i i

+

+ − = − ÷ + +

Solve for Pg and simlif! "o

!ield#

3tart with"

Factor ot A1 ot and re-write

(&)

1 2 1

1 2 3

41$6 41$6 1 41 6 41 6 41 6...

41i6 41i6 41 6 41 6 41 6 41 6

n

g n

g g g P A

i i i i

− + + += + + + + + + + + (*)

1

11

1 $ i

n

g

g

i P A

i g

+ − + = ≠ −

4lti!l+ %+ (1g)(1i) to o%tain .0# (* )

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%o 7orms to #onsider

1

1 6 g

nA P i

= +1

1

1 1 $ i

n

g

g

i P A

i g

+ − + = ≠

−

%ase& g ' i %ase& g i

A1 is the starting cash flow

$here is 5 %ase amont associated with a geometric gradient

$he remaining cash flows are generated from the A1 starting vale

o ta%les availa%le to ta%late this factor6too man+ com%inations of i

2 and g2 to s!!ort ta%les

%o se the /8A$CiCn 'actor


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E&am)"e 4-13

%he 'irst ear maintenance cost o' a ne caris B100. It increases 10C )er ear

therea'ter.

Usin$ a rate o' =C hat is the )resent!a"e o' 5 ears o' maintenance

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o"tion 1( ; hand

2014-I-01 *r. +.,. ennett /.En$. /,/ ENGR 3360U En$ Eco4-19

>ear ,ain #ost /87 =C n /W

1 100.00 .25 2.5

2 110.00 .=593 4.30

3 121.00 .93= 6.05

4 133.10 .9350 9.=3

5 146.41 .6=0= .65

4=0.42

i 4 d i "

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o"tion 2( ; 'orm"a

2014-I-01 *r + , ennett / En$ /,/ ENGR 3360U En$ Eco0 1-1=

/ < A1 F 1$8 1in8i-$

/ < 1001-1.181.0=58-0.02 < 4=0.42

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3360 Unit 04.3 2014-I-01