Time Value of Money Illustrations Updated

8/18/2019 Time Value of Money Illustrations Updated

1/51

Calculate the Future Value of Rs. 500 investment at the end of 20 years if it earns an anually compo

rate of return 6%

Soln. Compute FV using the calculator and folloing values !

" # 20$& # 6

'V # 500

C'( )* FV = -1603.57, -ve sign indicates money has to be invested.

Calculate the 'V of Rs. 500 that ould +e received in ,0 years ith a discout rate of -%

Soln. Calculate using the calculator !

" # ,0FV # 500

$& # -

C'()* PV = -254.175 (he negative shos money has to +e inv

!dina!y "nn#ity $

Calculate the future value of an ordinary annuity that pays Rs. 200 per year at the end of each year

given a rate of $nterest of 5%

Soln.

" # 20

'( # ) 200/ "egative sign indicates the amount that has to +e invested at the end of every year for

'V # 0$& # 5

C'( )* FV # 6/6,1.,

PV o% an o!dina!y ann#ity &ate! the time = 1

Calculate the 'resent value of an annuity that pays Rs. 200 per year at the end of each year for 5 y

F#t#!e Vae o% a sing&e s#m $ FV = PV'1()*+

P!esent Vae o% a sing&e s#m $ PV = FV* '1()*+

200 200 200 200

, 2 1 200

6/6,1.,

3..

33.


2/51

from no given a rate of return of 5%

(his pro+lem can +e done in to steps !

a. a&c#&ate the PV %o! the o!dina!y ann#ity $

" # 5

'( # 200

$& # 5

FV # 0

C'( )* 'V # ) 465.5 / )ve sign indicates money has to +e invested.

b. /hen disco#nting this vae %o! 1 yea! $

FV # 465.5

" # ,

$& # 5

'( # 0C'( )* 'V # 42.66

ne has to invest 24.66 today in o!de! to ea!n an ann#ity o% s. 200 %o! 5 yea!s beginning at

end o% to yea!s %!om no.

PV % an ann#ity d#e

. 7iven a discount rate of ,0%/ hat is the present value of a 1)year annuity that ma8es a series o

+eginning of each of the net three years/ starting today 9

:ns. First/ let;s solve this pro+lem using the calculator;s 2nd? >


3/51

1)year annuity. (hen multiply this 'V +y =,$&. (o use this approach/ enter the relevant inputs and

"#1D $,0D '(#),00D C'()* 'V:G#Rs. 24.6

P!esent Vae o% a e!et#ity

. :ssume the preffered stoc8 of Hhaitan Corporation pays Rs. 6 per year in annual dividends and p

dividend policy forever. 7iven an ,0% rate of return/ hat is the value of Hodon;s preffered stoc89

:ns. 7iven that the value of the stoc8 is the 'V of all future dividends/ e have !

(hus/ if an investor reIuires an ,0% rate of return/ the investor should +e illing to pay Rs. 60 for ea

HhaitanJs preffered stoc8.

om#ting the FV o% an #neven cash %&o se!ies

. Ksing a rate of return of ,0%/ compute the fututre value of the 6)year uneven cash flo stream d

at the end of the sith year.

:ns. (he FV for the cash flo stream is determined +y first computing the FV of each individual cas

FVs of the individual cash flos. "ote that e need to preerve the signs of the cash flos.

'V#L,/000D $,0D "#5D

'V#L500D $,0D "#D

'V#0D $,0D "#1D

'V#000D $,0D "#2D

'V#1/500D $,0D "#,D

'V#2/000D $,0D "#0D

FV of cash flo stream

om#ting PV o% an #neven cash %&o se!ies

"&te!native&y/ this pro+lem can +e solved +y leaving your calculator in the mode. First/ comput

'V:E # 'V:

G =,$& #Rs. 611.- ,.,0 # Rs. 6-.1-

'Vperpetuity

#60.,#Rs. 60

FV,!

FV2!

FV1!

FV!

FV5!

FV6!

Sum of FVindividual

,

200

1

200

'V # 6-.1-

200


4/51

. Compute the present value of this 6)year uneven cash flo stream descri+ed a+ove using a ,0%

:ns. (his pro+lem is solved +y first computing the 'V of each individual cash flo/ then summing th

flos/ hich yield the 'V of the cash flo stream. :gain the signs of te cash flos are preserved.

FV#),/000D $,0D "#,D

FV#)500D $,0D "#2DFV#0D $,0D "#1D

FV#/000D $,0D "#D

FV#1/500D $,0D "#5D

FV#2/000D $,0D "#6D

'V of cash flo stream

= "'V function on your calculator. (his procedure is illustrated in the ta+les in Figure +elo. $n the fi

FG,/ F02/ etc. values +ecause they are all eIual to ,. (he Fn varia+le indicates ho many times a p

is repeated.

Fig.! "'V Calculator Heystro8es)($ ? > ? 500 >)? >A"(AR? 'eriod 2 Cash Flo C02 # ) 500.00000

> ? > ? 0 >A"(AR? 'eriod 1 Cash Flo C01 # 0.00000

> ? > ? /000 >A"(AR? 'eriod Cash Flo C0 # /000.00000

> ? > ? 1/500 >A"(AR? 'eriod 5 Cash Flo C05 # 1/500.00000

> ? > ? 2/000 >A"(AR? 'eriod 6 Cash flo C06 # 2/000.00000

>"'V? ,0 >A"(AR? ,0% Eiscount Rate $ # ,0.00000> ? >C'(? Calculate "'V "'V # /-,,.,226

"ote that the


5/51

2. Heep the calculator in the annual compounding mode =', and enter $& as the interest rate pe

" as the num+er of compounding period in the investment horiOon. Metting m eIual the num+er of co

the +asic formulas for the calculator input data are determined as !

$& # the annual interest rate m /

"# the num+er of years m

(he computations for the FV and 'V amounts in the given fig. are!

FV#),/000D $,#6D "#,,#,!

FV#),/000D $2#1D "#,2#2!

FV#),/000D $#,.5D "#,#!

FV#),/000D $,2#0.5D "#,,2#,2!

FV#),/000D $165#0.0,614D "#,165#165!

'V#),/000D $,#6D "#,,#,!

'V#),/000D $2#1D "#,2#2!

'V#),/000D $#,.5D "#,#!

'V#),/000D $,2#0.5D "#,,2#,2!

'V#),/000D $165#0.0,614D "#,165#165!

oan ayment ca&c#&ation$ #a!te!&y ayments

. : company plans to +orro Rs.40/000 for five years. (he company;s +an8 ill lend the money at

that the loan +e paid off in Iuarterly payments for net five years. Calculate the amount of the paym

ma8e in order to fuly amortiOe this loan in five years

:ns. (o determine the annual loan payment/ input the relevant data and compute '(.

"#20 = 5

$,0 #2.5

'V#)40/000

C'()* '(#Rs. 5/,1,.--

(hus/ the loan can +e paid off in (enty eIual Iuarterly payments of Rs. 5/,1,.--. 'lease note that

(he loan ill +e fully paid off =amortiOed after the (enty payents have +een made.

onst!#cting an amo!ti8ation sched#&e

. Constuct an amortiOation schedule to sho the interest and principal componenets of the end)of)

5)year/ Rs. 20/000 loan.

:ns. (he first step in soving this pro+lem is to compute the amount of the loan payments. (his is do

relevant data and computing '(!

"#5

$,0%

'V#)Rs. 20/000

C'()*'(# Rs. 52-5.5 52-5.5

Period !eginning !alance Payment Interest

Component "#$

'V :!

'VS!

'V!

'V!

'VE!

FV :!

FVS!

FV!

FV!FV

E!


6/51

, Rs. 20000 52-5.5 2000

2 Rs. ,6/-2.05 52-5.5 ,6-2.,

1 ,1/,20.5, 52-5.5 ,1,2.05

/,56.6, 52-5.5 ,5.66

5 /-6.12 52-5.5 -.61

,. $nterest component # +eginning +alance periodic interest rate. $n period 1/ the interest componeRs. ,1/,20.5, 0.,0#Rs. ,1,2.05,

2. 'rincipal component # payment)interest. For eample/ the perod principal component is 52-5.

of the payment/ here the +eginning +alance for period t is the ending +alance from period t),. For e

the period 2 ending +alance eIuals Rs. 4/162.01)Rs. ,/40,.--#Rs. 6/560.26/hich +ecomes the per

+alance.

P!incia& and inte!est comonent o% a seci%ic &oan ayment

. Suppose you +orroed Rs. 5/000 at ,0% interest to +e paid semiannually over ,0 years. Calcula

outstanding +alance for the loan after the second payment is made.

:ns. First the amunt of the payment must +e determined +y entering the relevant information and co

'V#)Rs. 5000

$,02#5

"#,0 2#20

C'()*'(#Rs. 0,.22

(he principal and interest component of the second payment can e determined using the folloing p

'ayment ,! $nterest#=Rs.5/000=0.05#Rs. 250

'rincipal#0,.22)250#,5,.22

'ayment 2! $nterest#=5/000),5,.22=0.05#22.

'rincipal#0,.22)22.# ,54.-4

Remaining +alance # 5000 ) ,5,.22 ) ,54.-4 # 60

om#ting the !e#i!ed ayment to %#nd an ann#ity d#e

. Suppose you must ma8e five annual Rs. 2/000 payments/ the first one starting at the +eginning o

(o accumulate the money to ma8e these payments you ant three eIual payments into an investme

made one year from today. :ssuming a ,0% rate of return/ hat is the amount of these three payme

Soln. (he time line for this annuity pro+lem is shon +elo !

1. (he ending +alance in a given period/ t / is the period;s +eginning +alance minus the principal com

20 , 1 5 6 -


7/51

(he first step in this type of pro+lem is to determine the amount of money that must +e availa+le at t

order to satisfy the payment reIuirements.

Meave your calculator in the A"E mode and compute the 'V of a 5)year ordinary annuity and multipl

"#5

$,0

'(#)2/000

C'()*'V#-/54,.5-,.,#'V1#Rs. 411.1

of the three payments necessary to meet this funding reIuirement/ +e sure that your calculator is in t

relevant data/ and compute '(.

"#1

$,0

FV#)411.1

P/-9P:/=s. 2,51;6.61

F#nding " !eti!ement P&an

. :ssume a 15 year)old investor ants to retire in 25 years at the age of 60. She epects to earn ,

prior to her retirement and ,0% thereafter. Bo much must she deposit at the end of each year for t

+e a+le to ithdra Rs. 5/000 per year at the +eginning of each year for the 10 years from age 60 t

:ns. (his is a to)step pro+lem. First determine the amount that must +e on deposit in the retiremen

25 in order to fund the 10)year/ Rs. 5/000 annuity due. Second/ compute the annuity payments that

the reIuired amount

(he reIuired amount is the present value of the Rs. 25/000/ 10)year annuity due at the eginning of y

(his can +e determined +y entering the relevant data/ ith the calculator in the A"E mode/ and com

"#2

$,0

'(#)Rs. 5/000

C'()*'V#Rs. Rs. 2,/612 =for 2 years

"o add the first annuity payment to get Rs. 2,/612 Rs. 5/000# Rs. 66/612.66 (he investor il

at the end of year 25.

entering the relevant data and computing '(.

'V1 +ecomes the FV that you need three years from today from your three eIual end)of)year eposit


8/51

"#25

$,2.5

FV#66/612.66

C'()*'(# ) 120.0

(hus/ the investor must deposit Rs. 120.0 per year at the end of each of the net 25 years in orde

accumulate Rs. 66/612.66. Nith this amount she ill +e a+le to ithdra Rs. 5/000 per year for th


9/51

unded

sted.

for net 20 year/

net 20 years.

ars/ +eginning 2 years


10/51

he

Rs.200 payments at the

modeompute 'V.


11/51

ompute 'V.

lans to follo this

ch share of

scri+ed a+ove

flo/ then summing the

4/1-.

e the 'V of an ordinary

C'()*FV#FV,#),/6,0.5,

C'()*FV#FV2#)-12.05

C'()*FV#FV1#0.00

C'()*FV#FV#/40.00

C'()*FV#FV5#1/450.00

C'()*FV#FV6#2/000.00


12/51

rate of return.

pVs of the individual cash

P/-,,.,

gure/ e have omitted the

articular cash flo amount

stated annual interest rate

Present alue

P1.0

2.562.,4

,.05

,.-6

ding freIuencies!

compounding

C'()*'V#'V,#)0.0

C'()*'V#'V2

#),1.22

C'()*'V#'V1#0

C'()*'V#'V#2/-12.05

C'()*'V#'V5#2/,-1.22

C'()*'V#'V6#,/,24.5

d the net present value


13/51

r compounding period/and

mpounding periods per year/

rate of ,0% and reIuires

nt that the company must .

FV#0 in this computationD

ear payments for a ,0%/

e +y entering the

Principal Ending

Component "%$ !alance "&$

C'()*'V#'V :#1.16

C'()*'V#'VS#2.56

C'()*'V#'V#2.,4

C'()*'V#'V #,.05

C'()*'V#'VE #,.-6

C'()*FV#FV :#,/060.00

C'()*FV#FVS#,/060.0

C'()*FV#FV#,/06,.16

C'()*FV#FV#,/06,.64C'()*FV#FV

E#,/06,.41


14/51

12-5.5 Rs. ,6/-2.05

1/601.55 ,1/,20.5,

1/61.0 /,56.6,

/160.2 /-6.12

/-6.12 0.00

nt of the payment is

),1,2.05#161.0

ample/

iod 1 +eginning

e the amount of the

mpuing he payment.

rocess!

year =end of year 1.

nt account/ the first to +e

nts 9

ponent


15/51

e +eginning of year in

y +y ,.,0.

he A"E mode/ input the

.5% on her investments

e net 25 years in order to

o 0 9

t account at the end of year

must +e made to achieve .

ear 26 =end of year 25.

puting 'V.

l need Rs. 66/612.66

. (o determine the amount

an +e determined +y


16/51

r to

e folloing 10 years.


17/51


18/51


19/51


20/51


21/51


22/51


23/51


24/51


25/51

om#ting PV

. Calculate the "'V of an investment proQect ith an initial cost of Rs. ,0 million and positive cas

at the end of year ,/ Rs. 4 million at the end of year 2/ and Rs. ,0 million at the end of year . Ks

:ns. (he "'V for this proQect is the sum of the 'Vs of the proQect;s individual cash flos and is det"'V # )Rs. ,0.0 Rs. 2,.,0 Rs. 4=,.,02 Rs. ,0=,.,01

# Rs. 5. million

(he procedures for calculating "'V ith a ($ ? ,.6 A"(AR? 'eriod , Cash Flo C0,#2.00> ? > ? 2. >A"(AR? 'eriod 2 Cash Flo C02#4.00

> ? > ? 2.4 >A"(AR? 'eriod 1 Cash Flo C01#,0.00

>"'V? ,2 >A"(AR? ,2% discount rate $#,0.00000

> ? >C'(? Calculate "'V "'V#5.2

om#ting )

. Nhat is the $RR for the investment descri+ed in the preceding eample9

:ns. Su+stituting the investment;s cash flos into the $RR eIuation results in the folloing eIuatio

Solving this eIuation yields an $RR#1.6,%.

$t is possi+le to solve $RR pro+lems throuh a trial and error process. (hat is/ 8eep guessing $RRs u

provides an "'V eIual to Oero. 'ractically spea8ing/ a financial calculator or an elecronic spreads

employed. (he procedures for computing $RR ith the ($ ? > ? 2. >A"(AR? 'eriod 2 Cash Flo C02#4.00> ? > ? 2.4 >A"(AR? 'eriod 1 Cash Flo C01#,0.000

>$RR? >C'(? Calculate $RR $RR#1.6,

on%&icting decisions beteen PV and )

. :ssume "'V and $RR analysis of to mutually eclusive proQects produced the results shon in

0# ),0.0 2=,$RR 4=,$RR2 ,0=,$RR1


26/51

$RR criteria recommends that 'roQect : should +e accepted. Gn the other hand/ the "'V criteria i

proQect


27/51


28/51


29/51


30/51

h flos of Rs. 2 million

,0% as the discount rate.

ermined as follos!

or are presented in Figures

!

til you get the one that

eet can and should +e

Fig. +elo. :s indicated/ the


31/51

indicates acceptance of

,P at #)-

Rs. 55.5Rs. ,2/-2-.2-

< increases shareholder

ect < should +e selected


32/51


33/51


34/51


35/51

e!ientia& ea!ning>>> ?asic e&ements in %inancia& mode&ing


36/51

A 1.:n investment promises to pay P,00 one year from today/ P200 to years from today/ a

$f the reIuired rate of return is , percent/ compounded annually/ the value of this investment today is

Eiscount rate

(ime Cash flo


37/51

Financia& a&c#&ato! convention - @hy yo# see -ve sign in an

you have P,00 ith you today and you ant to invest it

let us say e go to $C$C$ +an8 =in "& city and e deposit it

e have to give the money to the +an8/ it is a cash outflo

FV # 'V =, r n 'V e have put P,00D in other ords e are giving ve sign to oFormula '2nd method

after 1 years then hat happens9 $C$C$ +an8 ill return the mone

,,0 that means it is an infloD then calculator says GH you gave sig

,2, therefore it means )ve sign for inflo +ac8 to you

,11.,

,6., this is a convention folloed +y all financial calculator and since e

,6,.05, as a calculator =financial it is also folloing the same convention

,--.,56,

,.4-,-

2,.154

215.-4

25.1-2

uity Eue

ay/ into an account paying percent interest compounded annually.

osest to!


38/51

nd P100 three years from today.

losest to!


39/51

e! %o! ce& %#nction &iCe PV

utflo

+ac8 to you

to outflo

cel as +uilt


40/51


41/51

Suppose an investor ishes to have Rs. , Crores at the time of retirement/ hich is 20 years aay/ ho much

interest rate is ,0% compounded annually

'V ),.46₹

FV ,00

(ime 20 years

$nterest rate ,0%Compounding annual

Simple retirement planning modelTTT


42/51

oney does he need to set aside today if the appropriateJ


43/51

,

Eiscount rate

:ssumptions! , 'remium * ,000

aturity ,0 years 2 'ar ,000

Eiscount rate! 4% 1 Eiscount U ,000

Coupon rate ,0%

Coupon amount P ,00'rincipal P ,/000

ethod , , :nnuity P,00/ ,0 years/ ,0% ethod 2

P 6-,.0, P ,/,1.20

2 lumpsum! P,000/ ,0 years/ ,0%

P 61.,

'rice , 2

P ,/,1.20

2

Eata given :ssumptions!

&ears Semi

aturity 1 6

Coupon rate 6% 1%

&( ,2% 6%

'ar value P ,/000

Coupon P 10 =every 6 months

odel! ,st method

(ime Semiannual CFJs 'V=CFJs

0 0 999

0.5 , P 10 24.10,446-2

, 2 P 10 26.6412

,.5 1 P 10 25.,445-4,

2 P 10 21.-62404-

2.5 5 P 10 22.,--5,46

1 6 P ,/010 -26.,015665

(otal 452.402-022

2nd method you do as homeor8

:nnuity lumpsun

What’s the value of a 10-year, 10% annual coupon bond if k d = 10%?

Suppose that we have a 3-year, 6% semiannual c


44/51

1rd method Acel Functio P 452.4

1

(hus/ the value of this preferred stoc8 : is33

5

(he reIuired return or the discount rate is ,,%.

6

:ssume an estimated sale price of P5-/ three years from no. ReIuired rat

-

Suppose that we have a $-year, ero Coupon bon

'ssume a preferred stoc( ' has a)"** par value and a dividend

1hat is the value of a stoc( that last year paid a )" dividend. Yo

'ssumin a holdin period of three years with the followin esti

1hat is the value of the stoc( that paid )# dividend last year if d


45/51

4


46/51

* Coupon

Eiscount rate U Coupon rate

Eiscount rate # Coupon rate

Eiscount rate * Coupon rate

ethod 1

time CF 'V=CF

, P ,00 2.525

2 P ,00 45.-1144

1 P ,00 -.14122

P ,00 -1.502

5 P ,00 64.05412

6 P ,00 61.0,66

- P ,00 54.10

4 P ,00 5.0264 P ,00 50.02

,0 P ,/,00 50.5,24

(otal ,,1.202

upon bond. Calculate the value of the bond with a Y


47/51

of return is ,5%.

. Calculate the value of the bond with a Y! of &%.

of )& a year. +ecause of ination, uncertainties and ta advantae,

thin( net year2s dividend will be "*% hiher and the stoc( will be

ated dividend payments Year " 4 )"."*5 Year # 4 ".#$ and Year 3

ividends are epected to row at $% forever he re/uired rate of re


48/51


49/51

of "#%.


50/51

he re/uired rate of return is 0%.

ellin for )#$ at year-end.

".$*.

urn is "#%


51/51

Time Value of Money Illustrations Updated

Documents

Transcript of Time Value of Money Illustrations Updated