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ENGR 3360U Winter 2014

Unit 4.4

Annual Payments

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

UOIT,

Version 2014-I-01

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Unit 4 Interest and Equivalene

!"an#e Re$rd

2014%I%01 Initial !reati$n

&e't re(erenes) !"a*ters 3 and 4

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Unit 4 Interest and Equivalene

4.4 Uniform Series Compound Interest Formulas

Uni($rm series A is de(ined as) An end%$(%*eri$d as" reei*t $r disursement in a

uni($rm series/ $ntinuin# ($r n *eri$ds/ t"e entire

series equivalent t$  P  $r  F  at an interest rate i

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Uni($rm%eries)

Present W$rt" 5at$r PA I( y$u need 7Ayear ($r n years/ 8"at d$ y$u "ave

t$ *ay n$8/ at i9:!a*ital Re$very 5at$r AP I( y$u uy a ma"ine n$8 ($r 7P/ 8"at must it ma;e

t$ *ay $(( t"e det in n years/ at i9:

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Unit 4 Interest and Equivalene

2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%<

PA 5at$r +erivati$n

  etu* t"e ($ll$8in#)

 -ulti*ly y 11+i t$ $tain a se$nd equati$n=

utrat 1 (r$m 2 t$ yield=

1 2 1

1 1 1 1..

1 1 1 1 n n P A

i i i i−

= + + + + + + + +

2 3 1

1 1 1 1..

1 1 1 1 1 n n P 

 Ai i i i i   +

= + + + + + + + + +

(1)

(2)

1

1 1

1 1 2 1 2ni

 P Ai i i

+

−= − + + +

(3)

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Unit 4 Interest and Equivalene

2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%6

PA and AP 5at$r 5$rmulas

  im*li(y 3 t$

yield=

 $lve 4 ($r A t$ #et

AP (at$r 

1 1  0

1

n

n

i P A for i

i i

+ −= ≠

+

(4)

1 1 1

n

ni i A P 

i +=   + −

(P/A,i,n) fa!tor

"#!el$ %P&(i,n,A)

(A/P,i,n) fa!tor"#!el$ %P'(i,n,P)

()

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Unit 4 Interest and Equivalene

E'am*le 4%3

An ener#y%e((iient ma"ine $sts 7

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Unit 4 Interest and Equivalene

ans8er 

2013%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%>

1 2 3 4 <

 P

P @ 9 n @ <

A @ PAP/ >9/

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E'am*le 4%4

An invest$r "$lds a time *ur"ase $ntrat

$n s$me ma"ine t$$ls. &"e $ntrat alls

($r a m$nt"ly *ayment $( 7140 at t"e end $(

ea" m$nt" ($r < years. &"e (irst *ayment isdue in a m$nt". e $((ers t$ sell y$u t"e

$ntrat ($r 76/>00. I( y$u an $t"er8ise

ma;e 19 *er m$nt"/ is it a #$$d deal ($ry$u:

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Unit 4 Interest and Equivalene

ans8er 

P @ APA/ i/ n @ 140PA/ 19 60

  @ 14044.B

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Unit 4 Interest and Equivalene

Part 2

u**$se 8e t$$; t"e deal. W"at is t"e

e((etive interest rate:

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Unit 4 Interest and Equivalene

ans8er 

P @ APA/ i/ 60

6>00 @ 140PA/ i/ 60

PA/ i/ 60 @ 4>.

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Unit 4 Interest and Equivalene

Uni($rm%eries)

in;in# 5und 5at$r A5 I( y$u need 75 in years/ 8"at d$ y$u "ave t$ set

aside annually ($r t"e ne't n years/ at i9:!$m*$und Am$unt 5at$r 5A I( y$u de*$sit 7A annually ($r t"e ne't n years/ 8"at

8ill it e 8$rt" in n years/ at i9:

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2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%14

in;in# 5und 5at$r and Uni($rm eries

!$m*$und Am$unt 5at$r A5 and 5A !as" (l$8 dia#ram ($r A5 (at$r 

 tart 8it" 8"at "as already een devel$*ed

1 1

1 1 1

n

n n

i i A F 

i i

  +=   + + −  

  . . . .

 * 1 2 3 n+2 n+1

nA% per interest period

 i per interest period

F % -ien

Find A, -ien F

1 1ni

 A F i

=   + −

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5A (at$r (r$m A5

 Given)

$lve ($r 5 in terms $( A t$ yield

1 1n

i A F 

i

=   + −

1 1ni F A

i

+ −=  

(A/F,i,n) fa!tor

"#!el$ %P'(i

,n,,F)

(F/A,i,n) fa!tor

"#!el$ %F&(i,n,A)

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E'am*le 4.1

$u de*$sit 7

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E'am*le 4.2

,im 8ants t$ uy a n$te$$; ($r 71/000 ut"e 8ants t$ *ay as". e deides t$ de*$sit

a (i'ed sum m$nt"ly in t"e an; ($r 12

m$nt"s and t"en uy t"e n$te$$;. I( t"e an; *ays 69 $m*$unded m$nt"ly/ "$8

mu" must "e de*$sit ea" m$nt":

A @ 1000A5/ 0.1.10

2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%1?

i 4 d i l

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2014%I%014%1>

Inter*$lati$n in Interest &ales

 W"en usin# taulated interest tales $ne mi#"t e($red t$ a**r$'imate a (at$r t"at is n$t taulated

 !an a**ly linear inter*$lati$n t$ a**r$'imate

ee &ale ($$%($$1

5at$rs are n$nlinear (unti$nsH "ene linear

inter*$lati$n 8ill yield err$rs in t"e 2%49 ran#e

Use a s*reads"eet m$del t$ alulate t"e (at$r

 *reisely

+r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$

U i 4 I d E i l

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&ale 5CC5CC1 109 and 209

n F/P P/F A/F A/P F/A P/A A/ P/ n

1 1.100 .B0B1 1.000 1.100 1.000 0.B0B 0 0 1

2 1.210 .>264 .4?62 . 4

< 1.611 .620B .163> .263> 6.10< 3.?B1 1.>10 6.>62 <

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n F/P P/F A/F A/P F/A P/A A/ P/ n

1 1.200 .>333 1.000 1.200 1.000 0.>33 0 0 1

2 1.440 .6B44 .4

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Unit 4 Interest and Equivalene

u**$se y$u did n$t "ave 1