cffm6 ch 05 slides

7/24/2019 CFFM6 Ch 05 Slides

1/44

Time Value of Money

Chapter 5

Future Value

Present Value

Annuities Rates of Return

Amortization

5-1


2/44

Time Lines

Show the timing of cash ows.

Tick marks occur at the end of periods soTime ! is toda"# Time $ is the end of the %rstperiod &"ear month etc.' or the (eginning ofthe second period.

)F! )F$ )F*)F+

! $ + *,-

5-2


3/44

rawing Time Lines

5-3

$!! $!!$!!

! $ + *,-

* "ear /$!! ordinar"

annuit"

$!!

! $ +,-

/$!! lump sum due in + "ears


4/44

rawing Time Lines

5-4

$!! 0!10

! $ + *,-

20!

3ne4en cash ow stream


5/44

5hat is the future 4alue &FV' of an initial/$!! after * "ears if ,67R 8 $!-9

Finding the FV of a cash ow or series ofcash ows is called compounding.

FV can (e sol4ed (" using the step2("2

step %nancial calculator andspreadsheet methods.

FV 8 9

! $ + *

$!-

$!!

5-5


6/44

Sol4ing for FV:The Step2("2Step and Formula ;ethods

After $ "ear:

FV$ 8 PV&$ < ,' 8 /$!!&$.$!' 8 /$$!.!!

After + "ears:

FV+ 8 PV&$ < ,'+ 8 /$!!&$.$!'+8 /$+$.!!

After * "ears:

FV* 8 PV&$ < ,'* 8 /$!!&$.$!'*8 /$**.$!

After = "ears &general case':FV= 8 PV&$ < ,'=

5-6


7/44

Sol4ing for FV:The )alculator ;ethod

Sol4es the general FV e>uation.

Re>uires ? inputs into calculator and willsol4e for the %fth. &Set to P67R 8 $ and

@= mode.'

5-7

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1& &

133'1&

-1&&


8/44

5hat is the present 4alue &PV' of /$!!due in * "ears if ,67R 8 $!-9

Finding the PV of a cash ow or series ofcash ows is called discounting &there4erse of compounding'.

The PV shows the 4alue of cash ows interms of toda"s purchasing power.

PV 8 9 $!!

! $ + *

$!-

5-(


9/44

Sol4ing for PV:The Formula ;ethod

Sol4e the general FV e>uation for PV:

PV 8 FV=6&$ < ,'=

PV 8 FV*6&$ < ,'*

8 /$!!6&$.$!'*

8 /10.$*

5-)


10/44

Sol4ing for PV:The )alculator ;ethod

Sol4es the general FV e>uation for PV.

@Bactl" like sol4ing for FV eBcept weha4e diCerent input information and are

sol4ing for a diCerent 4aria(le.

5-1&

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1& & 1&&

-75'13


11/44

Sol4ing for ,: 5hat interest rate wouldcause /$!! to grow to /$+0.D1 in *"ears9

Sol4es the general FV e>uation for ,.

Eard to sol4e without a %nancialcalculator or spreadsheet.

5-11

INPUT

!UTPUT

N I"#$ PMTPV %V

3

(

& 125')7-1&&


12/44

Sol4ing for =: ,f sales grow at +!- per"ear how long (efore sales dou(le9

Sol4es the general FV e>uation for =.

Eard to sol4e without a %nancialcalculator or spreadsheet.

5-12

INPUT

!UTPUT

N I"#$ PMTPV %V

3'(

2& & 2-1


13/44

5hat is the diCerence (etween anordinar" annuit" and an annuit" due9

5-13

rdinar" Annuit"

P;T P;TP;T

! $ + *,-

P;T P;T

! $ + *,-

P;T

Annuit" ue


14/44

Sol4ing for FV:*27ear rdinar" Annuit" of /$!! at $!-

/$!! pa"ments occur at the end ofeach period (ut there is no PV.

5-14

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1& -1&&

331

&


15/44

Sol4ing for PV:*2"ear rdinar" Annuit" of /$!! at $!-

/$!! pa"ments still occur at the end ofeach period (ut now there is no FV.

5-15

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1& 1&& &

-24('6)


16/44

Sol4ing for FV:*27ear Annuit" ue of /$!! at $!-

=ow /$!! pa"ments occur at the (eginningof each period.

FVAdue8 FVAord&$ < ,' 8 /**$&$.$!' 8 /*G?.$!

Alternati4el" set calculator to HI@J,=Kmode and sol4e for the FV of the annuit":

5-16

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1& -1&&

364'1&

&

*+,IN


17/44

Sol4ing for PV:*27ear Annuit" ue of /$!! at $!-

Again /$!! pa"ments occur at the (eginningof each period.

PVAdue8 PVAord&$ < ,' 8 /+?.GD&$.$!' 8 /+1*.00

Alternati4el" set calculator to HI@J,=K modeand sol4e for the PV of the annuit":

5-17

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1& 1&& &

-273'55

*+,IN


18/44

5hat is the present 4alue of a 02"ear/$!! ordinar" annuit" at $!-9

Ie sure "our %nancial calculator is set(ack to @= mode and sol4e for PV:

= 8 0 ,67R 8 $! P;T 8 2$!! FV 8 !.

PV 8 /*1D.!.

5-1(


19/44

5hat if it were a $!2"ear annuit"9 A+02"ear annuit"9 A perpetuit"9

$!2"ear annuit"

= 8 $! ,67R 8 $! P;T 8 2$!! FV 8 !#sol4e for PV 8 /G$?.?G.

+02"ear annuit"

= 8 +0 ,67R 8 $! P;T 8 2$!! FV 8 !#sol4e for PV 8 /D!1.1!.

Perpetuit" PV 8 P;T6, 8 /$!!6!.$ 8 /$!!!.

5-1)


20/44

The Power of )ompound,nterest

A +!2"ear2old student wants to sa4e /* ada" for her retirement. @4er" da" sheplaces /* in a drawer. At the end of the

"ear she in4ests the accumulated sa4ings&/$!D0' in a (rokerage account with aneBpected annual return of $+-.

Eow much mone" will she ha4e when sheis G0 "ears old9

5-2&


21/44

Sol4ing for FV: ,f she (egins sa4ingtoda" how much will she ha4e when sheis G09

,f she sticks to her plan she will ha4e/$?1+G$.D when she is G0.

5-21

INPUT

!UTPUT

N I"#$ PMTPV %V

45 12 -1&)5

14(7262

&


22/44

Sol4ing for FV: ,f "ou dont start sa4inguntil "ou are ?! "ears old how much will"ou ha4e at G09

,f a ?!2"ear2old in4estor (egins sa4ingtoda" and sticks to the plan he or shewill ha4e /$?G!!!.0D at age G0. This is/$.* million less than if starting at age +!.

Lesson: ,t pa"s to start sa4ing earl".

5-22

INPUT

!UTPUT

N I"#$ PMTPV %V

25 12 -1&)5

146&&1

&


23/44

Sol4ing for P;T: Eow much must the?!2"ear old deposit annuall" to catchthe +!2"ear old9 To %nd the re>uired annual contri(ution

enter the num(er of "ears untilretirement and the %nal goal of

/$?1+G$.D and sol4e for P;T.

5-23

INPUT

!UTPUT

N I"#$ PMTPV %V

25 12

-11154'42

14(7262&


24/44

5hat is the PV of this une4en cashow stream9

5-24

!

$!!

$

*!!

+

*!!

*$!-

20!

?

D!.D$

+?1.D*

++0.*D2*?.$0

0*!.! 8 PV


25/44

Sol4ing for PV:3ne4en )ash Flow Stream

,nput cash ows in the calculatorsH)FLK register:

)F!8 !

)F$8 $!!

)F+8 *!!

)F*8 *!!

)F?8 20!

@nter ,67R 8 $! press =PV (utton to get=PV 8 /0*!.!1. &Eere =PV 8 PV.'

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26/44

5ill the FV of a lump sum (e larger orsmaller if compounded more often holdingthe stated ,- constant9

LARJ@R as the more fre>uentl"compounding occurs interest is earned oninterest more often.

5-26

Annuall": FV*8 /$!!&$.$!'*8 /$**.$!

! $ + *

$!-

$!! $**.$!

Semiannuall": FVG8 /$!!&$.!0'G8 /$*?.!$

! $ + *0-

? 0 G

$*?.!$

$ + *!

$!!


27/44

)lassi%cations of ,nterestRates

=ominal rate &,=;' M also called the >uotedor stated rate. An annual rate that ignorescompounding eCects.

,=;is stated in contracts. Periods must also (e

gi4en e.g. - >uarterl" or - dail" interest.

Periodic rate &,P@R' M amount of interestcharged each period e.g. monthl" or>uarterl".

,P@R8 ,=;6; where ; is the num(er ofcompounding periods per "ear. ; 8 ? for>uarterl" and ; 8 $+ for monthl" compounding.

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28/44

)lassi%cations of ,nterestRates

@Cecti4e &or e>ui4alent' annual rate &@AR8 @FF-' M the annual rate of interestactuall" (eing earned accounting forcompounding.

@FF- for $!- semiannual in4estment

@FF- 8 & $ < ,=;6; ';M $

8 & $ < !.$!6+ '+M $ 8 $!.+0-

Should (e indiCerent (etween recei4ing$!.+0- annual interest and recei4ing $!-interest compounded semiannuall".

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29/44

5h" is it important to considereCecti4e rates of return9

,n4estments with diCerent compoundinginter4als pro4ide diCerent eCecti4ereturns.

To compare in4estments with diCerentcompounding inter4als "ou must lookat their eCecti4e returns &@FF- or @AR'.

5-2)


30/44

5-3&

5h" is it important to considereCecti4e rates of return9

See how the eCecti4e return 4aries(etween in4estments with the samenominal rate (ut diCerent

compounding inter4als.@ARA==3AL $!.!!-

@ARN3ART@RL7 $!.*-

@AR;=TEL7 $!.?1-@ARA,L7 &*G0' $!.0+-


31/44

5hen is each rate used9

,=; 5ritten into contracts >uoted ("(anks and (rokers. =ot used incalculations or shown on time lines.

,P@R 3sed in calculations and shown ontime lines. ,f ; 8 $ ,=;8 ,P@R8 @AR.

@AR 3sed to compare returns onin4estments with diCerent pa"ments

per "ear. 3sed in calculations whenannuit" pa"ments dont matchcompounding periods.

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32/44

5hat is the FV of /$!! after * "ears under$!- semiannual compounding9 Nuarterl"compounding9

/$*?.?'/$!!&$.!+0FV

/$*?.!$/$!!&$.!0'FV

+

!.$!$/$!!FV

;

,$PVFV

$+*N

G*S

*+

*S

=;

=;=

==

==

+=

+=

5-32


33/44

)an the eCecti4e rate e4er (ee>ual to the nominal rate9

7es (ut onl" if annual compounding isused i.e. if ; 8 $.

,f ; O $ @FF- will alwa"s (e greater

than the nominal rate.

5-33


34/44


35/44

5-35

;ethod $:)ompound @ach )ash Flow

FV*8 /$!!&$.!0'?< /$!!&$.!0'+< /$!!

FV*8 /**$.!

$$!.+0$+$.00**$.!

! $

$!!

+ *0-

? 0

$!!

G

$!!


36/44

;ethod +:Financial )alculator

Find the @AR and treat as an annuit".

@AR 8 &$ < !.$!6+'+M $ 8 $!.+0-.

5-36

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1&'25 -1&&

331'(&

&


37/44

Find the PV of This *27ear rdinar"Annuit"

)ould sol4e (" discounting each cashow or

3se the @AR and treat as an annuit" to

sol4e for PV.

5-37

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1&'25 1&& &

-247'5)


38/44

Loan Amortization

Amortization ta(les are widel" used forhome mortgages auto loans (usinessloans retirement plans etc.

Financial calculators and spreadsheets aregreat for setting up amortization ta(les.

@QA;PL@: )onstruct an amortization

schedule for a /$!!! $!- annual rateloan with * e>ual pa"ments.

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39/44

Step $:Find the Re>uired Annual Pa"ment

All input information is alread" gi4enust remem(er that the FV 8 ! (ecausethe reason for amortizing the loan and

making pa"ments is to retire the loan.

5-3)

INPUT

!UTPUT

N I"#$ PMTPV %V

3 1&

4&2'11

&-1&&&


40/44

Step +:Find the ,nterest Paid in 7ear $

The (orrower will owe interest upon theinitial (alance at the end of the %rst"ear. ,nterest to (e paid in the %rst "ear

can (e found (" multipl"ing the(eginning (alance (" the interest rate.

,=Tt8 Ieg (alt&,'

,=T$8 /$!!!&!.$!' 8 /$!!

5-4&


41/44

Step *:Find the Principal Repaid in 7ear $

,f a pa"ment of /?!+.$$ was made atthe end of the %rst "ear and /$!! waspaid toward interest the remaining

4alue must represent the amount ofprincipal repaid.

PR,=8 P;T M ,=T

8 /?!+.$$ M /$!! 8 /*!+.$$

5-41


42/44

Step ?:Find the @nding Ialance after 7ear $

To %nd the (alance at the end of theperiod su(tract the amount paid towardprincipal from the (eginning (alance.

@= IAL 8 I@J IAL M PR,=

8 /$!!! M /*!+.$$

8 /GD1.D

5-42


43/44

)onstructing an Amortization Ta(le:Repeat Steps $2? 3ntil @nd of Loan

5-43

,nterest paid declines with each

pa"ment as the (alance declines. 5hatare the taB implications of this9


44/44

,llustrating an Amortized Pa"ment:5here does the mone" go9

)onstant pa"ments

eclining interest pa"ments

eclining (alance

/

! $ + *

?!+.$$,nterest

*!+.$$

Principal Pa"ments

cffm6 ch 05 slides

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