time value of money illustrations updated
TRANSCRIPT
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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.
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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
" # ,
$& # 5
'( # 0C'( )* 'V # 42.66
ne has to invest 24.66 today in o!de! to ea!n an ann#ity o% s. 200 %o! 5 yea!s beginning at
end o% to yea!s %!om no.
PV % an ann#ity d#e
. 7iven a discount rate of ,0%/ hat is the present value of a 1)year annuity that ma8es a series o
+eginning of each of the net three years/ starting today 9
:ns. First/ let;s solve this pro+lem using the calculator;s 2nd? >
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1)year annuity. (hen multiply this 'V +y =,$&. (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
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. 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
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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!
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, 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 -
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(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
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"#25
$,2.5
FV#66/612.66
C'()*'(# ) 120.0
(hus/ the investor must deposit Rs. 120.0 per year at the end of each of the net 25 years in orde
accumulate Rs. 66/612.66. Nith this amount she ill +e a+le to ithdra Rs. 5/000 per year for th
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unded
sted.
for net 20 year/
net 20 years.
ars/ +eginning 2 years
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he
Rs.200 payments at the
modeompute 'V.
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ompute 'V.
lans to follo this
ch share of
scri+ed a+ove
flo/ then summing the
4/1-.
e the 'V of an ordinary
C'()*FV#FV,#),/6,0.5,
C'()*FV#FV2#)-12.05
C'()*FV#FV1#0.00
C'()*FV#FV#/40.00
C'()*FV#FV5#1/450.00
C'()*FV#FV6#2/000.00
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rate of return.
pVs of the individual cash
P/-,,.,
gure/ e have omitted the
articular cash flo amount
stated annual interest rate
Present alue
P1.0
2.562.,4
,.05
,.-6
ding freIuencies!
compounding
C'()*'V#'V,#)0.0
C'()*'V#'V2
#),1.22
C'()*'V#'V1#0
C'()*'V#'V#2/-12.05
C'()*'V#'V5#2/,-1.22
C'()*'V#'V6#,/,24.5
d the net present value
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r compounding period/and
mpounding periods per year/
rate of ,0% and reIuires
nt that the company must .
FV#0 in this computationD
ear payments for a ,0%/
e +y entering the
Principal Ending
Component "%$ !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
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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
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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
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r to
e folloing 10 years.
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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
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$RR criteria recommends that 'roQect : should +e accepted. Gn the other hand/ the "'V criteria i
proQect
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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
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indicates acceptance of
,P at #)-
Rs. 55.5Rs. ,2/-2-.2-
< increases shareholder
ect < should +e selected
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e!ientia& ea!ning>>> ?asic e&ements in %inancia& mode&ing
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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
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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!
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nd P100 three years from today.
losest to!
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e! %o! ce& %#nction &iCe PV
utflo
+ac8 to you
to outflo
cel as +uilt
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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
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oney does he need to set aside today if the appropriateJ
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,
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
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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 $-year, ero Coupon bon
'ssume a preferred stoc( ' has a)"** par value and a dividend
1hat is the value of a stoc( that last year paid a )" dividend. Yo
'ssumin a holdin period of three years with the followin esti
1hat is the value of the stoc( that paid )# dividend last year if d
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4
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* Coupon
Eiscount rate U Coupon rate
Eiscount rate # Coupon rate
Eiscount rate * Coupon rate
ethod 1
time CF 'V=CF
, P ,00 2.525
2 P ,00 45.-1144
1 P ,00 -.14122
P ,00 -1.502
5 P ,00 64.05412
6 P ,00 61.0,66
- P ,00 54.10
4 P ,00 5.0264 P ,00 50.02
,0 P ,/,00 50.5,24
(otal ,,1.202
upon bond. Calculate the value of the bond with a Y
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of return is ,5%.
. Calculate the value of the bond with a Y! of &%.
of )& a year. +ecause of ination, uncertainties and ta advantae,
thin( net year2s dividend will be "*% hiher and the stoc( will be
ated dividend payments Year " 4 )"."*5 Year # 4 ".#$ and Year 3
ividends are epected to row at $% forever he re/uired rate of re
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of "#%.
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he re/uired rate of return is 0%.
ellin for )#$ at year-end.
".$*.
urn is "#%
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