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Chapter 2

Time Value of Money

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Time Value Topics Future value

Present value

Rates of return

Amortization

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Time lines show timing of

cash ows!

CF"

CF1

CF#

CF2

" 1 2 #$%

Tic& mar&s at en's of perio's( so Time " is to'ay) Time 1 is the en' of

Perio' 1) or the *eginning of Perio'2!

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Time line for a +1"" lump

sum 'ue at the en' of ,ear 2!

1""

" 1 2 ,ear$%

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Time line for an or'inary

annuity of +1"" for #years

1"" 1""1""

" 1 2 #$%

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Time line for uneven CFs

1"" -".-

" 1 2 #$%

/-"

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FV of an initial +1"" after

# years 0$ 1"%

FV 3

" 1 2 #

1"%

Fin'ing FVs 0moving to the righton a time line is calle' compoun'ing!

1""

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After 2 years

FV2 FV1014$ PV01 4 $014$

PV014$2

+1""01!1"2

+121!""!

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After # years

FV# FV2014$PV01 4 $2014$

PV014$#

+1""01!1"#

+1##!1"

$n general(FV5 PV01 4 $

5!

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Three 6ays to Fin' FVs

7olve the e8uation with a regularcalculator!

9se a :nancial calculator!

9se a sprea'sheet!

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Financial calculator;

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Financial calculators solve thise8uation;

FV5 4 PV 014$5 "!

There are D varia*les! $f # are&nown( the calculator will solvefor the Dth!

Financial Calculator

7olution

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3 10 -100 0

N I/YR PV PMT FV 133.10

Clearing automatically setseverything to "( *ut for safety enterPMT "!

7et; P,R 1( B5@!

INPUTS

OUTPUT

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6hatEs the PV of +1""

'ue in # years if i 1"%3

1"%

Fin'ing PVs is 'iscounting( an' itEsthe reverse of compoun'ing!

1""

" 1 2 #

PV 3

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1!1"

7olve FV5 PV01 4 $ 5 for

PV

PV

FV5

014$5 FV

5

1

1 4$

5

PV

+1"" 1

+1""0"!.-1# +.-!1#

3

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3 10 0 100N I/YR PV PMT FV

-75.13

Bither PV or FV must *e negative!

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7prea'sheet 7olution

9se the PV function; seesprea'sheet in CF3 Ch02 Mini

Case.xls

PV0$( 5( PMT( FV

PV0"!1"( #( "( 1"" /.-!1#

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2"%

2

" 1 2 3

/1

FV PV01 4 $

5

Continue' on net sli'e

Fin'ing the Time to @ou*le

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Time to @ou*le0Continue'

+2 +101 4 "!2"5 01!25 +2+1 25 G501!2 G502

5 G502G501!2 5 "!HI#"!1J2 #!J!

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20 -1 0 2

N I/YR PV PMT FV

3.8

INPUTS

OUTPUT

Financial Calculator7olution

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9se the 5PBR function; seesprea'sheet in CF3 Ch02 Mini

Case.xls

5PBR0$( PMT( PV( FV

5PBR0"!1"( "( /1( 2 #!J

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3%

2

" 1 2 #

/1 FV PV01 4 $5

+2 +101 4 $#

0201# 01 4 $ 1!2-II 01 4 $ $ "!2-II 2-!II%!

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3 -1 0 2

N I/YR PV PMT FV

25.99

INPUTS

OUTPUT

Financial Calculator

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9se the RATB function;

RATB05( PMT( PV( FV

RATB0#( "( /1( 2 "!2-II

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?r'inary Annuity

PMT PMTPMT

" 1 2 #$%

PMT PMT

" 1 2 #$%

PMT

Annuity @ue

?r'inary Annuity vs! Annuity@ue

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FV Annuity Formula

The future value of an annuity with 5perio's an' an interest rate of $ can*e foun' with the following formula;

PMT

014$5/1

$

1""

014"!1"#/1

"!1"##1

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Financial CalculatorFormulafor Annuities

Financial calculators solve this e8uation;

FV5 4 PV014$5 4

PMT

014$5/1

$ "!

There are - varia*les! $f D are&nown( the calculator will solvefor the -th!

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3 10 0 -100

331.00

N I/YR PV PMT FV

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9se the FV function; seesprea'sheet!

FV0$( 5( PMT( PV

FV0"!1"( #( /1""( " ##1!""

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6hatEs the PV of this

or'inary annuity3

1"" 1""1""

" 1 2 #

1"%

I"!I1

J2!HD

.-!1#

2DJ!HI PV

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PV Annuity Formula

The present value of an annuity with5 perio's an' an interest rate of $ can

*e foun' with the following formula;

PMT

1

$

1K

$014$5

1"" 1

"!1

1K

"!1014"!1#

2DJ!HI

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9se the PV function; seesprea'sheet!

PV0$( 5( PMT( FV

PV0"!1"( #( 1""( " /2DJ!HI

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Fin' the FV an' PV if theannuity were an annuity'ue!

1"" 1""

" 1 2 #1"%

1""

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PV an' FV of Annuity @uevs! ?r'inary Annuity

PV of annuity 'ue;

0PV of or'inary annuity 014$

02DJ!HI 014 "!1" 2.#!-H

FV of annuity 'ue; 0FV of or'inary annuity 014$

0##1!"" 014 "!1" #HD!1

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3 10 100 0

-273.55

N I/YR PV PMT FV

INPUTS

OUTPUT

PV of Annuity @ue; 7witchfrom LBn' to L=egin

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3 10 0 100

-364.1

N I/YR PV PMT FV

INPUTS

OUTPUT

FV of Annuity @ue; 7witchfrom LBn' to L=egin

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Bcel Function forAnnuities @ue

Change the formula to; PV01"%(#(/1""("(1

The fourth term( "( tells the functionthere are no other cash ows! The :fthterm tells the function that it is anannuity 'ue! A similar function givesthe future value of an annuity 'ue;

FV01"%(#(/1""("(1

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6hat is the PV of thisuneven cash ow stream3

"

1""

1

#""

2

#""

#1"%

/-"

D

I"!I1

2D.!I#22-!#I

/#D!1-

-#"!"J PV

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Financial calculator;

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Financial calculator; 1 0to see

CF 1! To see CF 1"/1D( hit RCG CF> !0perio' 1 0to see CF 11!

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$nput in LCFG? register; CF" "

CF1 1"" CF2 #""

CF# #""

CFD /-"

Bnter $ 1"%( then press 5PV *uttonto get 5PV -#"!"I! 0

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Bcel Formula in cell A#;5PV01"%(=2;B2

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Perio'ic rate 0$PBR

$PBR $5?MM( where M is num*er of

compoun'ing perio's per year! M D for

8uarterly( 12 for monthly( an' #H" or #H- for'aily compoun'ing!

9se' in calculations( shown on time lines!

Bamples; J% 8uarterly; $PBR J%D 2%!

J% 'aily 0#H-; $PBR J%#H- "!"21I1J%!

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The $mpact ofCompoun'ing

6ill the FV of a lump sum *e largeror smaller if we compoun' more

often( hol'ing the state' $%constant3

6hy3

h f

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The $mpact ofCompoun'ing 0Answer

GARBR

$f compoun'ing is more fre8uentthan once a year//for eample(semiannually( 8uarterly( or 'aily//

interest is earne' on interest moreoften!

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FV Formula with @iQerentCompoun'ing Perio's

$5?MFV5 PV 1

4M

M 5

+ i l

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+1"" at a 12% nominal ratewith semiannual compoun'ing

for - years

+1""01!"H1" +1.I!"J

$5?MFV5 PV 1

4 M

M 5

"!12FV-7 +1"" 1

42

2-

FV f +1"" t 12% i l

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FV of +1"" at a 12% nominalrate for - years with 'iQerent

compoun'ing

FV0Ann! +1""01!12-

+1.H!2#

FV07emi!

+1""01!"H1" +1.I!"

J

FV0Ouar!

+1""01!"#2" +1J"!H

1

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BQective Annual Rate0BAR BFF%

The BAR is the annual rate whichcauses PV to grow to the same FV

as un'er multi/perio'compoun'ing!

Q i l

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BQective Annual RateBample

Bample; $nvest +1 for one year at 12%(semiannual;

FV PV01 4 $5?MMM

FV +1 01!"H2 1!12#H!

BFF% 12!#H%( *ecause +1 investe' forone year at 12% semiannualcompoun'ing woul' grow to the samevalue as +1 investe' for one year at12!#H% annual compoun'ing!

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Comparing Rates

An investment with monthlypayments is 'iQerent from one with

8uarterly payments! Must put onBFF% *asis to compare rates ofreturn! 9se BFF% only for

comparisons! =an&s say Linterest pai' 'aily!

7ame as compoun'e' 'aily!

BFF% f i l t f

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BFF% 1 4 K

1

$5?M

M

M

BFF% for a nominal rate of12%( compoun'e'

semiannually

1 4 K1

"!12

2

2

01!"H2 / 1!" "!12#H

12!#H%!

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Fin'ing BFF with

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BAR 0or BFF% for a5ominal Rate of of 12%

BARAnnual 12%!

BARO 01 4 "!12DD / 1 12!--%!

BARM 01 4 "!121212 / 1 12!HJ%!

BAR@0#H- 01 4 "!12#H-#H- / 1

12!.-%!

C th Q ti t

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Can the eQective rateever *e e8ual to thenominal rate3 ,es( *ut only if annual

compoun'ing is use'( i!e!( if M 1!

$f M 1( BFF% will always *egreater than the nominal rate!

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6hen is each rate use'3

$5?M; 6ritten into contracts(

8uote' *y *an&s an'*ro&ers! 5ot use' incalculations or shownon time lines!

6h i h t '3

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$PBR; 9se' in calculations( shown

on time lines!

$f $5?M has annual

compoun'ing(then $PBR $5?M1 $5?M!

6hen is each rate use'30Continue'

6h i h t '3

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6hen is each rate use'30Continue'

BAR 0or BFF%; 9se' to comparereturns on investments with

'iQerent payments per year! 9se' for calculations if an' only if

'ealing with annuities where

payments 'onEt match interestcompoun'ing perio's!

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Amortization

Construct an amortizationsche'ule for a +1("""( 1"% annual

rate loan with # e8ual payments!

7t 1 Fi ' th i '

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7tep 1; Fin' the re8uire'payments!

PMT PMTPMT

" 1 2 #1"%

/1("""

# 1" /1""" "

$5P9T7

?9TP9T

5 $,R PV FVPMT

D"2!11

7t 2 Fi ' i t t

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7tep 2; Fin' interestcharge for ,ear 1!

$5Tt =eg *alt 0$

$5T1 +1("""0"!1" +1""!

7t # Fi ' t f

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Repmt PMT / $5T +D"2!11 / +1"" +#"2!11!

7tep #; Fin' repayment ofprincipal in ,ear 1!

7tep D Fin' en'ing

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7tep D; Fin' en'ing*alance after ,ear 1!

Bn' *al =eg *al / Repmt

+1(""" / +#"2!11 +HI.!JI.

Repeat these steps for ,ears 2 an' #to complete the amortization ta*le!

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Amortization Ta*le

,BAR=B=AG PMT $5T

PR$5PMT

B5@=AG

1 +1(""

"

+D"2 +1"" +#"2 +HIJ

2 HIJ D"2 ." ##2 #HH

# #HH D"2 #. #HH "

T?T 1(2"H!#D 2"H!#D 1("""

$nterest 'eclines *ecause

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$nterest 'eclines *ecauseoutstan'ing *alance'eclines!

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Amortization ta*les are wi'ely use'//for home mortgages( auto loans(*usiness loans( retirement plans( an'more! They are very important

Financial calculators 0an'sprea'sheets are great for setting

up amortization ta*les!

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Fractional Time Perio's

?n Sanuary 1 you 'eposit +1"" inan account that pays a nominal

interest rate of 11!##DH#%( with'aily compoun'ing 0#H- 'ays!

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$PBR 11!##DH#%#H- "!"#1"-D% per 'ay!

FV=?

0 1 2 273

0.031054%

-100

Convert interest to 'aily rate

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FV2.# +1""

01!"""#1"-D2.#

+1"" 01!"JJDH

+1"J!J-

Fin' FV

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273 -100 0

108.85

INPUTS

OUTPUT

N I/YR PV FVPMT

$PBR $5?MM 11!##DH##H- "!"#1"-D% per 'ay!

Calculator 7olution

5on matching rates an'

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5on/matching rates an'perio's

6hatEs the value at the en' of ,ear# of the following CF stream if the

8uote' interest rate is 1"%(compoun'e' semiannually3

Time line for non

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Time line for non/matching rates an'perio's

" 1

1""

2 #-%

D - H H/mos! perio's

1"" 1""

5on matching rates an'

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5on/matching rates an'perio's

Payments occur annually( *utcompoun'ing occurs each H

months! 7o we canEt use normal annuity

valuation techni8ues!

1st Metho'; Compoun'

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1st Metho'; Compoun'Bach CF

" 1

1""

2 #-%

D - H

1"" 1""!""11"!2-121!--

##1!J"

FVA# +1""01!"-D 4 +1""01!"-2 4 +1""

+##1!J"!

2n' Metho'; Treat as an

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2n' Metho'; Treat as anannuity( use :nancial

calculator

Fin' the BFF% 0BAR for the 8uote' rate

BFF% 1 4 K 1

1"!2-%

"!1"

2

2

9se BAR 1" 2-% as the

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3 10.25 0 -100

INPUTS

OUTPUT

N I/YR PV FVPMT

331.80

9se BAR 1"!2-% as theannual rate in calculator!

6hatEs the PV of this

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6hat s the PV of thisstream3

"

1""

1

-%

2 #

1"" 1""

I"!."J2!2..D!H2

2D.!-I

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Comparing $nvestments

,ou are oQere' a note which pays+1(""" in 1- months 0or D-H 'aysfor +J-"! ,ou have +J-" in a *an&

which pays a H!.HHDI% nominal rate(with #H- 'aily compoun'ing( which isa 'aily rate of "!"1J-#J% an' anBAR of .!"%! ,ou plan to leave the

money in the *an& if you 'onEt *uythe note! The note is ris&less! 7houl' you *uy it3

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$PBR

"!"1J-#J% per 'ay!

1,000

0 365 456 d!"

-850

@aily time line

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Three solution metho's

1! reatest future wealth; FV

2! reatest wealth to'ay; PV

#!

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1! reatest Future 6ealth

Find FV of $850 left in bank for 15 months and compare with

note’s FV = $1,000.

FVBank = $8501.000185!8"#5

= $%.%' in bank.

B() the note* $1,000 + $%.%'.

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456 -850 0

924.97

INPUTS

OUTPUT

N I/YR PV FVPMT

$PBR $5?MM

H!.HHDI%#H- "!"1J-#J% per 'ay!

Calculator 7olution to FV

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Find V of note, and comparewith its $850 cost*

V = $1,000-1.000185!8"#5

= $%18.%5.

2! reatest Present 6ealth

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456 .018538 0 1000

-918.95

INPUTS

OUTPUT

N I/YR PV FVPMT

6.76649/365 =

PV of note is greater than its +J-"cost( so *uy the note! Raises yourwealth!

Financial Calculator 7olution

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Fin' the BFF% on note an'compare with .!"% *an& pays(which is your opportunity cost ofcapital;

FV5 PV01 4 $5

+1(""" +J-"01 4 $D-H

5ow we must solve for $!

#! Rate of Return

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456 -850 0 1000

0.035646%#$ d!

INPUTS

OUTPUT

N I/YR PV FVPMT

Convert % to 'ecimal;@ecimal "!"#-HDH1"" "!"""#-HDBAR BFF% 01!"""#-HDH#H- / 1 1#!JI%!

Calculator 7olution

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P,R #H-5?M% "!"#-HDH0#H- 1#!"1BFF% 1#!JI

7ince 1#!JI% .!"% opportunity cost(*uy the note!

9sing interest conversion

02 tvm

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