Brooks PP Ch 4 RTP

8/18/2019 Brooks PP Ch 4 RTP

1/41

The Time Value of Money(Part Two)


2/41

4-2

1. Explain and illustrate an annuity.2. Determine the future value of an annuity.

3. Determine the present value of an annuity.4. Adjust the annuity equation for present valueand future value for an annuity due.5. Distinguish et!een the di"erent types ofloan repayments.#. $uild and analy%e amorti%ation s&hedules.'. (al&ulate !aiting time and interest rates foran annuity.

LEARNING OBJECTIVES


3/41


4/41

Fig. 4.1 The time-line of anest egg

4-4


5/41

4.2 Future Value of an nnuityStream

4-5


6/41

4.2 Future Value of an nnuityStream

!"ample# Future Value of an $r%inarynnuity Stream

-ill has een faithfully depositing 2+/// at theend of ea&h year over the past 1/ years intoan a&&ount that pays a guaranteed 0 peryear. o! mu&h money has she have

a&&umulated in the a&&ount

4-6


7/41

Example Ans!er via the long !ay

6uture 7alue of 8ayment 9ne : 2+/// x 1./0 ; : 3+;;0./16uture 7alue of 8ayment


8/41

4.2 Future Value of annnuity Stream

4-11


9/41

4.2 Future Value of an nnuityStream>,?=@ A= E (EB ,8CEAD, EE<

Enter :67 0 + 1/+ 2///+ /+ / F 9utput :Enter :67 0 + 1/+ 2///+ /+ / F 9utput : 20+;'3.1320+;'3.13 Cate : /./0+ =per : 1/+ 8mt : 2+///+ 87 :/+

and ,?=@ 67?6A


10/41

F&' ! 4.* &nterest an% prin+ipalgrowth with %i,erent interest ratesfor 1 annual payments .

4-10


11/41

4.* Present Value of annnuity

4-11

( )r

r PMT PV

n +−×= 1

11


12/41

F&' ! 4.4 Time line of present

/alue of annuity stream.

4-12


13/41

4.* Present Value of an

nnuityExample pro lem for the four solution methods Iou are no! holding the !inning lottery ti&Jet that!ill pay the holder of the ti&Jet 1/+/// per year forthe next 2/ years. A friend has o"ered to uy the!inning ti&Jet from you. )hat should you sell theti&Jet for assuming you have a dis&ount rate of #on future dollars this is your opportunity rate for thefuture &ash *o!

6our !ays to solve6ormula(al&ulator,preadsheet


14/41

4.* Present Value of annnuity6ormula

?nputs = : 2/+ r : /./#+ 8K< : 1/+/// and

(ompute 87+87 : 1/+/// x L1 M 1N 1./# 2/ O N /./# :114+#;;.21114+#;;.21

(al&ulator

?nputs = : 2/+ ?NI : #./+ 8K< : 1/+///+ 67 : /(ompute 8787 : 114+#;;.21114+#;;.21

4-#


15/41

4.* Present Value of annnuity,preadsheet+ use 87 fun&tion

?nputs Cate : /./#+ =per : 2/+ 8mt : 1/+///+6v : /87 : 114+#;;.21114+#;;.21


16/41

4.4 nnuity 0ue an%Perpetuity

4-16

A &ash *o! stream su&h as rent+ lease+ andinsuran&e payments+ !hi&h involves equalperiodi& &ash *o!s that egin right a!ay or atthe eginning of ea&h time interval is Jno!n

as an annuity due.


17/41

4.4 nnuity 0ue an%

Perpetuity6ormula Adjustment87 annuity due : 87 ordinary annuity x 1Pr67 annuity due : 67 ordinary annuity x 1Pr87 annuity due Q 87 ordinary annuity67 annuity due Q 67 ordinary annuity

an you see why (!"+el Sprea%sheet !"ample)

6inan&ial &al&ulatorKode set to $@= for annuity due

Kode set to E=D for an ordinary annuity,preadsheet


18/41

4.4 nnuity 0ue an%

Perpetuity!"ample# nnuity 0ue /ersus $r%inary

nnuityBetRs say that you are saving up for retirementand de&ide to deposit 3+/// ea&h year for thenext 2/ years into an a&&ount !hi&h pays arate of interest of 0 per year. $y ho! mu&h!ill your a&&umulated nest egg vary if youmaJe ea&h of the 2/ deposits at the eginningof the year+ starting right a!ay+ rather than atthe end of ea&h of the next t!enty years

4-18


19/41

4.4 nnuity 0ue an%

PerpetuityExample Ans!er@iven informationH 8K< : 3+///F n:2/F i:

0 F 87:/F

4-19

67 ordinary annuity : 3+/// x L 1./0 2/ 1 N./0O

: 3+/// x 45.'#1;#3+/// x 45.'#1;# : $137,285.89$137,285.89

67 of annuity due : 67 of ordinary annuity x

1Pr67 of annuity due : 13'+205.0; x 1./0 :13'+205.0; x 1./0 :

140+2#0.'#140+2#0.'#

Di"eren&e is 1/+;02.0'Di"eren&e is 1/+;02.0'


20/41

4.4 nnuity 0ue an%Perpetuity

PerpetuityA 8erpetuity is an equal periodi& &ash *o!

stream that !ill never &ease.


21/41

4.4 nnuity 0ue an%

Perpetuity!"ample# PV of a perpetuity

?f you are &onsidering the pur&hase of a &onsol thatpays #/ per year forever+ and the rate of interestyou !ant to earn is 1/ per year+ ho! mu&h moneyshould you pay for the &onsol

nswer# r:1/ + 8K< : #/F and 87 : #/N/.1/ :r:1/ + 8K< : #/F and 87 : #/N/.1/ : #//#//

#// is the most you should pay for the &onsol.

Iou &an thinJ of it this !ay+ if you put #// in theanJ earning 1/ you &an !ithdra! the annual

interest of #/ every year forever !ithouttou&hing the prin&ipal.

4-21


22/41

4. Three PaymentMetho%s

Boan payments &an e stru&tured in one of 3!aysH

1 Dis&ount loan• 8rin&ipal and interest is paid in lump sum at

end2 ?nterest only loan

• 8eriodi& interest only payments+ prin&ipaldue at end.

3 Amorti%ed loan• Equal periodi& payments of prin&ipal and

interest4-22


23/41

4. Three PaymentMetho%s!"ample# 0is+ount /ersus &nterest-only /ersus morti5e%

loans

Coseanne !ants to orro! 4/+/// for a period of 5 years.


24/41

4. Three Payment Metho%sMetho% 1# 0is+ount 6oan.

,in&e all the interest and the prin&ipal is paid at theend of 5 years !e &an use the 67 of a lump sum

equation to &al&ulate the payment required+ i.e.6767 : 87 x 1 P r: 87 x 1 P r nn

6767 55 : 4/+/// x 1P/.1/: 4/+/// x 1P/.1/ 55 : 4/+/// x 1.#1/51: 4/+/// x 1.#1/51

:: #4+ 42/.4/#4+ 42/.4/?nterest paid :


25/41

4. Three Payment Metho%sMetho% 2# &nterest-$nly 6oan.

Annual ?nterest 8ayment Iears 1 4Annual ?nterest 8ayment Iears 1 4

: 4/+/// x /.1/ : 4+/// ea&h year 1#+/// Iear 5 payment Iear 5 payment

: Annual interest payment P 8rin&ipalpayment: 4+/// P 4/+/// : 44+///


26/41

4. Three Payment Metho%s

Metho% *# morti5e% 6oan.n : 5F ?NI : 1/./F 87: 4/+///F 67 : /Fn : 5F ?NI : 1/./F 87: 4/+///F 67 : /F

(8< 8K


27/41

4. Three Payment Metho%s6oan Type Total Payment &nterest Pai%

Dis&ount Boan #4+42/.4/ 24+42/.4/?nterest only Boan #/+///.// 2/+///.//Amorti%ed Boan 52+'5;.31 12+'5;.3112+'5;.31

)hy does the equal annual payments of prin&ipal andinterest ea&h period have the lo!est total interest

4-#


28/41

4.7 morti5ation S+he%ules


29/41

4.7 morti5ation S+he%ules!"ample# 6oan amorti5ation s+he%ule.

8repare a loan amorti%ation s&hedule for theamorti%ed loan option given in the previous

Example !ith the Gve annual payments for the4/+/// at 1/ annual interest rate. )hat is theloan payo" amount at the end of 2 years

,tep 9ne+ determine the annual paymentH

87 : 4/+///F n:5F ?NI:1/./F 67:/F87 : 4/+///F n:5F ?NI:1/./F 67:/F

(8< 8K< :(8< 8K< : 1/+551.;/ rounded to nearest1/+551.;/ rounded to nearest!hole &ent!hole &ent

4-29


30/41


31/41

4.;


32/41

4.;


33/41

4.; sing a Gnan&ial &al&ulator?=8>< 0./ / 0///1/////

sing an Ex&el spreadsheet>sing the T=8ECU fun&tion !e enter the follo!ingH

Cate : 0 F 8mt : 0///F 87 : /F 67 : 1/////F


34/41

4.;


35/41

4.= Fin%ing the interestrateSol/ing a 6ottery Pro:lem

?n the &ase of lottery !innings+ 2 &hoi&es

1 Annual lottery payment for Gxednum er of years+ 9C2 Bump sum payout.

o! do !e maJe an informed judgment=eed to Ggure out the implied rate of return of

oth options using


36/41

4.= Fin%ing theinterest rate!"ample# al+ulating an implie% rate ofreturn gi/en an annuity

BetRs say that you have just !on the statelottery.


37/41

4.= Fin%ing the interestrate>sing the


38/41

4.= Fin%ing the interestrate)e &ould use the ,preadsheet fun&tions Cate

fun&tion to Gnd the 3.;0 .)e &ould use the


39/41

4.> Ten &mportant Points a:out

the TVM !?uation1. Amounts of money &an e added or su tra&ted

only if they are at the same point in time.2.


40/41

4.> Ten &mportant Points a:out

the TVM !?uation#. An annuity is a series of equal &ash payments at

regular intervals a&ross time.'. sing the sing Gnan&ial fun&tions from a spreadsheet.

4-40


41/41

4.> Ten &mportant Points a:outthe TVM !?uation;.

Brooks PP Ch 4 RTP

Documents