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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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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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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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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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E'am*le 4%3
An ener#y%e((iient ma"ine $sts 7
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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:
2013%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$0.1%B
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Unit 4 Interest and Equivalene
ans8er
P @ APA/ i/ n @ 140PA/ 19 60
@ 14044.B
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Part 2
u**$se 8e t$$; t"e deal. W"at is t"e
e((etive interest rate:
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ans8er
P @ APA/ i/ 60
6>00 @ 140PA/ i/ 60
PA/ i/ 60 @ 4>.
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Unit 4 Interest and Equivalene
Uni($rm%eries)
in;in# 5und 5at$r A5 I( y$u need 75 in years/ 8"at d$ y$u "ave t$ set
aside annually ($r t"e ne't n years/ at i9:!$m*$und Am$unt 5at$r 5A I( y$u de*$sit 7A annually ($r t"e ne't n years/ 8"at
8ill it e 8$rt" in n years/ at i9:
2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%13
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2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%14
in;in# 5und 5at$r and Uni($rm eries
!$m*$und Am$unt 5at$r A5 and 5A !as" (l$8 dia#ram ($r A5 (at$r
tart 8it" 8"at "as already een devel$*ed
1 1
1 1 1
n
n n
i i A F
i i
+= + + −
. . . .
* 1 2 3 n+2 n+1
nA% per interest period
i per interest period
F % -ien
Find A, -ien F
1 1ni
A F i
= + −
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2014%I%01 +r. ,.-. ennett/ P.En#./ P-P ENGR 3360U En# E$4%1<
5A (at$r (r$m A5
Given)
$lve ($r 5 in terms $( A t$ yield
1 1n
i A F
i
= + −
1 1ni F A
i
+ −=
(A/F,i,n) fa!tor
"#!el$ %P'(i
,n,,F)
(F/A,i,n) fa!tor
"#!el$ %F&(i,n,A)
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E'am*le 4.1
$u de*$sit 7
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E'am*le 4.2
,im 8ants t$ uy a n$te$$; ($r 71/000 ut"e 8ants t$ *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