1 - tvm concepts

8/16/2019 1 - Tvm Concepts

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The Time ValueThe Time Value

of Moneyof Money


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The Time Value of Money The Time Value of Money

Would you prefer toWould you prefer to

have $1 million now orhave $1 million now or

$1 million 10 years$1 million 10 years

from now?from now?

Of course, we wouldOf course, we would

all prefer the moneyall prefer the moneynow!now!

This illustrates that This illustrates that

there is an inherentthere is an inherent


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What is The Time Value ofWhat is The Time Value of

Money?Money?

dollar received today is worth more dollar received today is worth morethan a dollar received tomorrowthan a dollar received tomorrow

This is "ecause a dollar received today This is "ecause a dollar received todaycan "e invested to earn interestcan "e invested to earn interest

The amount of interest earned depends The amount of interest earned dependson the rate of return that can "e earnedon the rate of return that can "e earned

on the investmenton the investment Time value of money #uanties the Time value of money #uanties the

value of a dollar throu%h timevalue of a dollar throu%h time


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&ses of Time Value of&ses of Time Value of

MoneyMoney Time Value of Money, or TVM, is a Time Value of Money, or TVM, is a

concept that is used in all aspects ofconcept that is used in all aspects of

nance includin%'nance includin%' (ond valuation(ond valuation

)toc* valuation)toc* valuation

ccept+reect decisions for proectccept+reect decisions for proect

mana%ementmana%ement -inancial analysis of rms-inancial analysis of rms

nd many others!nd many others!


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-ormulas-ormulas

.ommon formulas that are used in TVM.ommon formulas that are used in TVMcalculations'calculations'// resent value of a lump sum'resent value of a lump sum'

V .-V .-tt + 213r4+ 213r4tt OROR V -V V -Vtt + 213r4+ 213r4tt

-uture value of a lump sum'-uture value of a lump sum'

-V-Vtt .- .-00 / 213r4/ 213r4tt OROR -V -Vtt V / 213r4 V / 213r4tt

resent value of a cash 5ow stream'resent value of a cash 5ow stream'

nn

V V ΣΣ 6.-6.-tt + 213r4+ 213r4tt77 t=0t=0


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-ormulas 2continued4-ormulas 2continued4

-uture value of a cash 5ow stream'-uture value of a cash 5ow stream' nn

-V -V ΣΣ 6.-6.-tt / 213r4/ 213r4n8tn8t77 t=0t=0

resent value of an annuity'resent value of an annuity'

V MT / 9618213r4V MT / 9618213r48t8t7+r:7+r:

-uture value of an annuity'-uture value of an annuity'

-V-Vtt MT / 96213r4 MT / 96213r4tt ;17+r:;17+r:

/


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Varia"lesVaria"les

wherewhere r rate of returnr rate of return t time periodt time period n num"er of time periodsn num"er of time periods MT paymentMT payment .- .ash 5ow 2the su"scripts t and 0 mean at.- .ash 5ow 2the su"scripts t and 0 mean at

time t and at time >ero, respectively4time t and at time >ero, respectively4

V present value 2V present value of anV present value 2V present value of anannuity4annuity4

-V future value 2-V future value of an-V future value 2-V future value of anannuity4annuity4


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Types of TVM .alculations Types of TVM .alculations

There are many types of TVM calculations There are many types of TVM calculations The "asic types will "e covered in this The "asic types will "e covered in this

review module and include'review module and include' resent value of a lump sumresent value of a lump sum -uture value of a lump sum-uture value of a lump sum resent and future value of cash 5ow streamsresent and future value of cash 5ow streams resent and future value of annuitiesresent and future value of annuities

eep in mind that these forms can,eep in mind that these forms can,should, and will "e used in com"ination toshould, and will "e used in com"ination tosolve more comple@ TVM pro"lemssolve more comple@ TVM pro"lems


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(asic Aules(asic Aules

The followin% are simple rules that you should always use no The followin% are simple rules that you should always use nomatter what type of TVM pro"lem you are tryin% to solve'matter what type of TVM pro"lem you are tryin% to solve'

11 )top and thin*' Ma*e sure you understand what the)top and thin*' Ma*e sure you understand what the

pro"lem is as*in% Bou will %et the wron% answer if youpro"lem is as*in% Bou will %et the wron% answer if youare answerin% the wron% #uestionare answerin% the wron% #uestion

CC Draw a representative timeline and la"el the cash 5owsDraw a representative timeline and la"el the cash 5owsand time periods appropriatelyand time periods appropriately

EE Write out the complete formula usin% sym"ols rst andWrite out the complete formula usin% sym"ols rst andthen su"stitute the actual num"ers to solvethen su"stitute the actual num"ers to solve

FF .hec* your answers usin% a calculator.hec* your answers usin% a calculator While these may seem li*e trivial and time consumin% tas*s,While these may seem li*e trivial and time consumin% tas*s,

they will si%nicantly increase your understandin% of thethey will si%nicantly increase your understandin% of thematerial and your accuracy ratematerial and your accuracy rate


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I@ample of V of a ero is today

0 1 2 3 4 5

$100?i = 10%


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-uture Value of a


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I@ample of -V of a


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I@ample of -V of a


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)ome Thin%s to Kote)ome Thin%s to Kote

Ln "oth of the e@amples, note that if you were toLn "oth of the e@amples, note that if you were toperform the opposite operation on the answersperform the opposite operation on the answers2ie, nd the future value of $NC0 or the2ie, nd the future value of $NC0 or the

present value of $1N104 you will end up withpresent value of $1N104 you will end up withyour ori%inal investment of $100your ori%inal investment of $100

This illustrates how present value and future This illustrates how present value and futurevalue concepts are intertwined Ln fact, they arevalue concepts are intertwined Ln fact, they arethe same e#uation the same e#uation Ta*e V -V Ta*e V -Vtt + 213r4+ 213r4tt and solve for -Vand solve for -Vtt Bou will %et -V Bou will %et -Vtt

V / 213r4 V / 213r4tt s you %et more comforta"le with the formulass you %et more comforta"le with the formulas

and calculations, you may "e a"le to do theand calculations, you may "e a"le to do thecalculations on your calculator alone (e surecalculations on your calculator alone (e sureyou understand W=T you are enterin% intoyou understand W=T you are enterin% into

each re%ister and W=Beach re%ister and W=B


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resent Value of a .ashresent Value of a .ash

-low )tream-low )tream

cash 5ow stream is a nite set of cash 5ow stream is a nite set ofpayments that an investor will receive orpayments that an investor will receive orinvest over timeinvest over time

The V of the cash 5ow stream is e#ual The V of the cash 5ow stream is e#ualto the sum of the present value of eachto the sum of the present value of eachof the individual cash 5ows in the streamof the individual cash 5ows in the stream

The V of a cash 5ow stream can also "e The V of a cash 5ow stream can also "e

found "y ta*in% the -V of the cash 5owfound "y ta*in% the -V of the cash 5owstream and discountin% the lump sum atstream and discountin% the lump sum atthe appropriate discount rate for thethe appropriate discount rate for theappropriate num"er of periodsappropriate num"er of periods


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I@ample of V of a .ashI@ample of V of a .ash


Poe made an investment that will pay $100 the rst Poe made an investment that will pay $100 the rst

year, $E00 the second year, $00 the third year andyear, $E00 the second year, $00 the third year and

$1000 the fourth year Lf the interest rate is ten$1000 the fourth year Lf the interest rate is ten

percent, what is the present value of this cash 5owpercent, what is the present value of this cash 5owstream?stream?

11 Draw a timeline'Draw a timeline'

00 11 22 33 44

??

$100$100 $300$300 $500$500 $1000$1000

??

??

??

i = 10%i = 10%


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CC Write out the formula usin% sym"ols'Write out the formula usin% sym"ols' nn

V V ΣΣ 6.-6.-tt + 213r4+ 213r4tt77

t=0t=0

OAOA

V 6.-V 6.-11+213r4+213r411736.-736.-CC+213r4+213r4CC736.-736.-EE+213r4+213r4EE736.-736.-FF+213r4+213r4FF77

EE )u"stitute the appropriate num"ers')u"stitute the appropriate num"ers'

V 6100+21314V 6100+2131411736$E00+21314736$E00+21314CC73600+2131473600+21314EE7361000+7361000+

21142114FF77


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FF )olve for the present value')olve for the present value'

V $01 3 $CFQE 3 $EQNN 3 $NRE01V $01 3 $CFQE 3 $EQNN 3 $NRE01

V $1EQ1V $1EQ1

.hec* usin% a calculator'.hec* usin% a calculator' Ma*e sure to use the appropriate rate of return, num"erMa*e sure to use the appropriate rate of return, num"er

of periods, and future value for each of the calculationsof periods, and future value for each of the calculations

To illustrate, for the rst cash 5ow, you should enter To illustrate, for the rst cash 5ow, you should enter

-V100, n1, i10, MT0, V? Kote that you will-V100, n1, i10, MT0, V? Kote that you willhave to do four separate calculationshave to do four separate calculations


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-uture Value of a .ash-uture Value of a .ash


The future value of a cash 5ow stream is The future value of a cash 5ow stream is

e#ual to the sum of the future values ofe#ual to the sum of the future values of

the individual cash 5owsthe individual cash 5ows The -V of a cash 5ow stream can also The -V of a cash 5ow stream can also

"e found "y ta*in% the V of that same"e found "y ta*in% the V of that same

stream and ndin% the -V of that lumpstream and ndin% the -V of that lump

sum usin% the appropriate rate of returnsum usin% the appropriate rate of returnfor the appropriate num"er of periodsfor the appropriate num"er of periods


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I@ample of -V of a .ashI@ample of -V of a .ash


ssume Poe has the same cash 5ow stream from hisssume Poe has the same cash 5ow stream from his

investment "ut wants to *now what it will "e worth atinvestment "ut wants to *now what it will "e worth at

the end of the fourth yearthe end of the fourth year

11 Draw a timeline'Draw a timeline'

00 11 22 33 44

$100$100 $300$300 $500$500 $1000$1000

i = 10%i = 10%

$1000$1000

??

??

??


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CC Write out the formula usin% sym"olsWrite out the formula usin% sym"ols nn

-V -V ΣΣ 6.-6.-tt / 213r4/ 213r4n8tn8t77 t=0t=0

OAOA

-V 6.--V 6.-11/213r4/213r4n81n81736.-736.-CC/213r4/213r4n8Cn8C736.-736.-EE/213r4/213r4n8En8E736.-736.-FF/213r4/213r4n8n8

FF77

EE )u"stitute the appropriate num"ers')u"stitute the appropriate num"ers'

-V 6$100/21314-V 6$100/21314F81F81736$E00/21314736$E00/21314F8CF8C736$00/21314736$00/21314F8EF8E7 37 3

6$1000/213146$1000/21314F8FF8F77


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FF )olve for the -uture Value')olve for the -uture Value'

-V $1EE10 3 $ENE00 3 $000 3 $1000-V $1EE10 3 $ENE00 3 $000 3 $1000

-V $C0FN10-V $C0FN10

.hec* usin% the calculator'.hec* usin% the calculator' Ma*e sure to use the appropriate interest rate, timeMa*e sure to use the appropriate interest rate, time

period and present value for each of the four cash 5owsperiod and present value for each of the four cash 5ows

To illustrate, for the rst cash 5ow, you should enter To illustrate, for the rst cash 5ow, you should enter

V100, nE, i10, MT0, -V? Kote that you willV100, nE, i10, MT0, -V? Kote that you willhave to do four separate calculationshave to do four separate calculations


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nnuitiesnnuities

n annuity is a cash 5ow stream inn annuity is a cash 5ow stream in

which the cash 5ows are all e#ual andwhich the cash 5ows are all e#ual and

occur at re%ular intervalsoccur at re%ular intervals Kote that annuities can "e a @edKote that annuities can "e a @ed

amount, an amount that %rows at aamount, an amount that %rows at a

constant rate over time, or an amountconstant rate over time, or an amount

that %rows at various rates of %rowththat %rows at various rates of %rowthover time We will focus on @edover time We will focus on @ed

amountsamounts


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I@ample of V of anI@ample of V of an

nnuitynnuity ssume that )ally owns an investment thatssume that )ally owns an investment that

will pay her $100 each year for C0 years Thewill pay her $100 each year for C0 years The

current interest rate is 1J What is the Vcurrent interest rate is 1J What is the V

of this annuity?of this annuity?

11 Draw a timelineDraw a timeline

00 11 22 33 …………………………………………………….. 1919 2020

$100$100 $100$100 $100$100 $100$100 $100$100

??

i = 15%i = 15%


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nnuitynnuityCC Write out the formula usin% sym"ols'Write out the formula usin% sym"ols'

V MT / 9618213r4V MT / 9618213r48t8t7+r:7+r:

EE )u"stitute appropriate num"ers')u"stitute appropriate num"ers'

V $100 / 961821314V $100 / 9618213148C08C07+1:7+1:

FF )olve for the V)olve for the V

V $100 / NCEV $100 / NCE

V $NCEV $NCE


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nnuitynnuity .hec* answer usin% a calculator.hec* answer usin% a calculator

Ma*e sure that the calculator is set to one periodMa*e sure that the calculator is set to one period

per yearper year

MT $100MT $100

n C0n C0

i 1Ji 1J

V ?V ?

Kote that you do not need to enter anythin% forKote that you do not need to enter anythin% forfuture value 2or -V04future value 2or -V04


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I@ample of -V of anI@ample of -V of an

nnuitynnuity ssume that )ally owns an investment thatssume that )ally owns an investment that

will pay her $100 each year for C0 years Thewill pay her $100 each year for C0 years The

current interest rate is 1J What is the -Vcurrent interest rate is 1J What is the -V

of this annuity?of this annuity?

11 Draw a timelineDraw a timeline

00 11 22 33 …………………………………………………….. 1919 2020

$100$100 $100$100 $100$100$100$100 $100$100

i = 15%i = 15%

??


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nnuitynnuityCC Write out the formula usin% sym"ols'Write out the formula usin% sym"ols'

-V-Vtt MT / 96213r4 MT / 96213r4tt ;17+r:;17+r:

EE )u"stitute the appropriate num"ers')u"stitute the appropriate num"ers'-V-VC0C0 $100 / 9621314 $100 / 9621314C0C0 ;17+1;17+1

FF )olve for the -V')olve for the -V'-V-VC0C0 $100 / 10CFFEN $100 / 10CFFEN

-V-VC0C0 $10,CFFEN $10,CFFEN


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nnuitynnuity .hec* usin% calculator'.hec* usin% calculator'

Ma*e sure that the calculator is set to one periodMa*e sure that the calculator is set to one period

per yearper year

MT $100MT $100

n C0n C0

i 1Ji 1J

-V ?-V ?

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