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Financial
Calculators andExcel
Week 2 Lecture 2
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Three Ways to Solve Time
Value of Money ProlemsUse Equations
Use Financial Calculator
Use Electronic Spreadsheet
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Ty!es of Prolems
Future Value
Present Value
Uneven Cash Flow StreamsTime or Funds to !ccumulate
Future Value o an !nnuit"
Present Value o an !nnuit"E!#
!morti$ation
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After 1 year:FV1 = PV + Interest1 = PV + PV (r)
= PV(1 + r)= $100 (1.10)= $110.00.
After 2 years:FV
2
= PV(1 + r)2
= $100 (1.10)2
= $121.00.
After 3 years:FV
3
= PV(1 + r)3
= 100 (1.10)3
= $133.10.
In general, FVn = PV (1 + r)n
Future Value
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Solve t!s e"#at!on y %l#gg!ng !n tea%%ro%r!ate val#es:
"umerical #E$uation%
Solution
= + ttFV PV(1 r)
PV = $100, r = 10&, an' t =3
$133.10)$100(1.331
$100(1.10)FV 3n
==
=
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ere are var!ales. If 3 are
*non, te al#lator !ll solve forte t.
Financial Calculator
Solution
= +t
tFV PV(1 r)
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INPUTS
CPT
3 10 -100 0 ? N I/YR PV PMT FV
FV =
133.10
-eres te set#% to /n' FV:
lear!ng a#toat!ally sets everyt!ng to 0, #tfor safety enter P = 0.
P FV
Financial Calculator
Solution
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S!readsheet Solution
4eferene ells:
4ate = !nterest
rate, r5%er = n#erof %er!o's!nterest !s
earne'
Pt = %er!o'!%ayent
PV = %resentval#e of te
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Present Value Present value is the value toda" o a uture cash
low or series o cash lows%
Discounting is the process o indin& the presentvalue o a uture cash low or series o uturecash lows' it is the reverse o compoundin&%
Example
What is the PV o ()** due in three "ears i r +)*,-
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100
0 1 2 310%
PV = ?
What is the PV of &'(( due
in three years if ) * '(+,
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Solve FVt = PV (1 + r )t for PV:
( ) ÷
t
t tt
FV 1PV = = FV 1+r1+r
( ) $67.13=0.6713$100=
1.101$100=PV
3
What is the PV of &'(( due
in three years if r * '(+,
!s !s te n#er!al sol#t!on to solve for PV.
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INPUTS
CPT
3 10 ? 0
100N I/YR PV PMT FV
PV = -75.13
Financial Calculator
Solution
8!ter PV or FV #st e negat!ve. -erePV = 967.13. Invest $67.13 to'ay, ta*eo#t $100 after 3 years.
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S!readsheet Solution
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Time for Funds to
-ccumulate. sales &row at 2*, per"ear/how lon& 0eore sales
dou0le-
Solve or n
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Solve for !FV = 1"1 # I$
2 = 1"1.20$
Time for Funds to
-ccumulate
e n#er!al sol#t!on !s soeat'!#lt.
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Time for Funds to
-ccumulate 2 + )1)%2*n
Ln 2 + n ln )%2
%345) + n %)6252
7 + 5%6
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INPUTS
CPT
? 20 -1 0 2
N I/YR PV PMT FV
=3.
;ra%!al
Ill#strat!on:
0 1 2 3 &
1
2
FV
3.
Ye'r
Financial Calculator
Solution
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S!readsheet Solution
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Fractional Time Periods
0 0.25 0.50 0.75
10%
- 100
1.00
FV = ?
8<a%le: $100 'e%os!te' !n a an* at8A4 = 10& for 0.67 of te year
INPUTS
CPT
0.75 10 -100 0?
FV
N I/YR PV PMT FV
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S!readsheet Solution
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Future Value of an -nnuity
Annuity: ! series o pa"ments o equalamounts at i8ed intervals or a speciiednum0er o periods%
Ordinary (deferred) Annuity: !n annuit"whose pa"ments occur at the end o eachperiod%
Annuity Due: !n annuit" whose pa"ments
occur at the 0e&innin& o each period%
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PMT PMTPMT
0 1 2 3r%
PMT PMT
0 1 2 3
r%
PMT
.rdinary -nnuity Versus -nnuity /uer'!nary Ann#!ty
Ann#!ty >#e
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100 100100
0 1 2 310%
110
121
FV = 331
What0s the FV of a 12year
.rdinary -nnuity of &'((at '(+,
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Financial Calculator
Solution
INPUTS
CPT
3 10 0 -100 ?
FV
=331.00
N I/YR PV PMT FV
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S!readsheet Solution
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Present Value of an -nnuity
PV!n + the present value o an annuit"
with n pa"ments%
Each pa"ment is discounted/ and the sumo the discounted pa"ments is the presentvalue o the annuit"%
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2&.() = PV
100 100100
0 1 2 310%
)0.)1
2.(&
75.13
What is the PV of this
.rdinary -nnuity,
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"umerical Solution
+
=
= =
+
∑n
1n(1 I)
n t
t 1
191PVA P P
(1 I) I
$2?.@?7)$100(2.?@
0.10
91$100PVA
3(1.10)
1
3
==
=
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Be *no te %ayents #t no l#% s# Fso enter 0 for f#t#re val#e.
Financial Calculator
Solution
INPUTS
*UTPUT
3 10 ? 100 0
-2&.()
N I/YR PV PMT FV
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S!readsheet Solution
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100 100
0 1 2 310%
100
Find the FV and PV if the
-nnuity 4ere an -nnuity/ue5
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"umerical Solution
+
=
= = × + +
∑t
1t91(1 r)
n tt 0
191PVA P P (1 r)
(1 r) r
[ ]
$263.773)$100(2.637
1.10(2.?@?7)$100
(1.10)
0.10
91$100PVA
3(1.10)
1
3
==
×=
×
=
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S!t fro C8n'D to CE;5D 9 2n' ten E;5 (Pt) en2n' S8. Ee s#re to ange !t a* to 85> to ret#rn to r'!naryAnn#!t!es.
en enter var!ales to /n' PVA3 = $263.77
o /n' FV, enter PV = 0 an' %ress FV to/n'
FV = [email protected].
Financial CalculatorSolution
INPUTS
CPT
3 10 ? 100 0
-273.55
N I/YR PV PMT FV
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S!readsheet Solution
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250 250
0 1 2 3r = ?
- &(.0
&
250 250
o# %ay $?@.?0 for an !nvestent tat%ro!ses to %ay yo# $270 %er year for te
ne<t fo#r years, !t %ayents a'e at teen' of ea year. Bat !nterest rate !llyo# earn on t!s !nvestentG
Solving for )nterest 6ates
4ith -nnuities
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Hse tr!al9an'9error y s#st!t#t!ng '!erentval#es of r !nto te follo!ng e"#at!on #nt!l
te r!gt s!'e e"#als $?@.?0.
"umerical Solution
+
=
1
(1 r)19
$?@.?0 $270 r
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Financial Calculator
Solution
INPUTS
CPT
& ? -&(.0 250
0
7.0
N I/YR PV PMT FV
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S!readsheet Solution
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+100 "1 # r $3 = +125.)7.
What interest rate 4ould cause&'(( to gro4 to &'7859: in 1
years,
INPUTS
*UTPUT
N I/YR PV PMT FV
3 ? -100 0
125.)7
%
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S!readsheet Solution
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EFF vs -P6 To clear 2nd #eset+
.C97V :ispla" will 0e 79;
)2 E7TE#
:ispla" will 0e C<=
)2 E7TE#
:ispla" will 0e EFF
CPT :ispla" will 0e )2%36
Choose F> 1conversion
7+)2 i,+)2
E + )2%36,
!P#+)2,
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-morti;ed <oans !morti$ed Loan? ! loan that is repaid in equal pa"ments
over its lie%
!morti$ation ta0les are widel" used or home mort&a&es/auto loans/ retirement plans/ and so orth to determine howmuch o each pa"ment represents principal repa"ment andhow much represents interest% The" are ver" important/ especiall" to homeowners@
Financial calculators 1and spreadsheets are &reat orsettin& up amorti$ation ta0les%
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PMT PMTPMT
0 1 2 310%
-1000
Ste! '3 /etermine there$uired !ayments
( )
3
11
1 .1$1000.1
Pmt
−
+ = ×
$402.11 Pmt =
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PMT PMTPMT
0 1 2 310%
-1000
INPUTS
*UTPUT
3 10 -1000 ?
0
&02.11
N I/YR PV PMT FV
Ste! '3 /etermine the
re$uired !ayments
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Ste! 73 Find interest
charge for >ear 'I5t = Eeg!nn!ng alanet (I)
I51 = 1,000(0.10) = $100.00
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4e%ayent = P 9 I5
= $02.11 9 $100.00= $302.11.
Ste! 13 Find re!ayment
of !rinci!al in >ear '
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8n'!ng al. = Eeg!nn!ng al. 9
4e%ayent= $1,000 9 $302.11 =
$@6.?.
4e%eat tese ste%s for te rea!n'erof te %ayents (ears 2 an' 3 !n t!sase)to o%lete te aort!Jat!on tale.
Ste! ?3 Find ending
alance after >ear '
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Calculator 7ot worth the trou0le
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S!readsheet Solution
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Interest 'el!nes, ! as ta< !%l!at!ons.
<oan -morti;ation Tale
'( Percent )nterest 6ate YR Beg Bal PMT INT Prin PMT End Bal
1 $1000.00 $402.11 $100.00 $302.11 $697.89
2 697.89 402.11 69.79 332.32 365.57
3 365.57 402.11 36.56 365.55 0.02
Toal 1!206.33 206.35 999.98 "
K 4o#n'!ng '!erene
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Uneven Cash Flo4 Streams ! series o cash lows in which the amount varies
rom one period to the ne8t?
Pa"ment 1P;T desi&nates constant cash lowsAthat is/ an annuit" stream%
Cash low 1CF desi&nates cash lows in&eneral/ 0oth constant cash lows and unevencash lows%
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0
100
1
300
2
300
310%
-50
&
)0.)1
2&7.)3
225.3)
-3&.15
530.0 = PV
What is the PV of this
Uneven Cash Flo4Stream,
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"umerical Solution
t
= + + +
+ + + 1 21 2 t
1 1 1PV F F ... F
(1 r) (1 r) (1 r)
−+
+
+
=
321 (1.10)
170)(
(1.10)
1300
(1.10)
1300
(1.10)
1100PV
$730.0
01)$70)(0.@?3(31)$300(0.6717)$300(0.?2@0)$100(0.0
=
−+++=
Fi i l C l l
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Financial CalculatorSolution CL# TV; B 0ut this doesnt clear ever"thin&
7eed to #ESET to clear ever"thin& CF
C** + *
C*) + )** Enter F*)+) C*2 + 5** Enter F*2+) C*5 + 5** Enter F*5+) C*D + >* Enter B F*D+)
CPT 7PV . + )* Enter
7PV CPT 7PV + >5*%*6
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S!readsheet Solution
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Three Ways to Solve Time
Value of Money ProlemsUse Equations
Use Financial Calculator
Use Electronic Spreadsheet