The Time Value of Money 2

8/6/2019 The Time Value of Money 2

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The Time ValueThe Time Value

of Moneyof Money Presented By:Presented By:Anshul KothariAnshul Kothari

Ravi SomaraRavi Somara

Kapil SharmaKapil Sharma

Ajay MeenaAjay Meena

Ayush JainAyush Jain

Jyoti RaniJyoti Rani

MamtaMamta


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The Time Value of MoneyThe Time Value of MoneyWould you prefer toWould you prefer to

have Rs.1 million nowhave Rs.1 million now

ororRs. 1 million 10 yearsRs. 1 million 10 years

from now?from now?

Of course, we would allOf course, we would allprefer the money now!prefer the money now!

This illustrates that thereThis illustrates that thereis an inherent monetaryis an inherent monetaryvalue attached to time.value attached to time.


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What is The Time Value ofWhat is The Time Value of

Money?Money?A rupee received today is worth moreA rupee received today is worth more

than a rupee received tomorrowthan a rupee received tomorrow This is because a rupee received today canThis is because a rupee received today can

be invested to earn interestbe invested to earn interest

The amount of interest earned depends onThe amount of interest earned depends onthe rate of return that can be earned onthe rate of return that can be earned on

the investmentthe investment Time value of money quantifies theTime value of money quantifies the

value of a rupee through timevalue of a rupee through time


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Uses of Time Value of MoneyUses of Time Value of Money Time Value of Money, or TVM, is a conceptTime Value of Money, or TVM, is a concept

that is used in all aspects of financethat is used in all aspects of finance

including:including: Bond valuationBond valuation

Stock valuationStock valuation

Accept/reject decisions for project managementAccept/reject decisions for project management

Financial analysis of firmsFinancial analysis of firms

And many others!And many others!


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FormulasFormulas Common formulas that are used in TVM calculations:Common formulas that are used in TVM calculations:**

Present value of a lump sum:Present value of a lump sum:

PV = CFPV = CFtt/ (1+r)/ (1+r)tt

OROR PV = FVPV = FVtt/ (1+r)/ (1+r)tt

Future value of a lump sum:Future value of a lump sum:

FVFVtt= CF= CF00 * (1+r)* (1+r)ttOROR FVFVtt= PV * (1+r)= PV * (1+r)

tt

Present value of a cash flow stream:Present value of a cash flow stream:

nn

PV =PV = 77[CF[CFtt/ (1+r)/ (1+r)tt]]

t=0t=0


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Formulas (continued)Formulas (continued) Future value of a cash flow stream:Future value of a cash flow stream:

nn

FV =FV = 77[CF[CFtt* (1+r)* (1+r)nn--tt

]]t=0t=0

Present value of an annuity:Present value of an annuity:

PVA = PMT * {[1PVA = PMT * {[1--(1+r)(1+r)--tt]/r}]/r}

Future value of an annuity:Future value of an annuity:

FVAFVAtt= PMT * {[(1+r)= PMT * {[(1+r)tt1]/r}1]/r}

* List adapted from the Prentice Hall Website


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VariablesVariables wherewhere

r = rate of returnr = rate of return

t = time periodt = time period n = number of time periodsn = number of time periods

PMT = paymentPMT = payment

CF = Cash flow (the subscripts t and 0 mean atCF = Cash flow (the subscripts t and 0 mean attime t and at time zero, respectively)time t and at time zero, respectively)

PV = present value (PVA = present value of anPV = present value (PVA = present value of anannuity)annuity)

FV = future value (FVA = future value of anFV = future value (FVA = future value of anannuity)annuity)


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Types of TVM CalculationsTypes of TVM Calculations There are many types of TVM calculationsThere are many types of TVM calculations

The basic types will be covered in this reviewThe basic types will be covered in this review

module and include:module and include: Present value of a lump sumPresent value of a lump sum

Future value of a lump sumFuture value of a lump sum

Present and future value of cash flow streamsPresent and future value of cash flow streams

Present and future value of annuitiesPresent and future value of annuities Keep in mind that these forms can, should,Keep in mind that these forms can, should,

and will be used in combination to solve moreand will be used in combination to solve morecomplex TVM problemscomplex TVM problems


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Basic RulesBasic Rules The following are simple rules that you should always use noThe following are simple rules that you should always use no

matter what type of TVM problem you are trying to solve:matter what type of TVM problem you are trying to solve:

1.1. Stop and think: Make sure you understand what theStop and think: Make sure you understand what theproblem is asking. You will get the wrong answer if youproblem is asking. You will get the wrong answer if youare answering the wrong question.are answering the wrong question.

2.2. Draw a representative timeline and label the cash flowsDraw a representative timeline and label the cash flowsand time periods appropriately.and time periods appropriately.

3.3. Write out the complete formula using symbols first andWrite out the complete formula using symbols first andthen substitute the actual numbers to solve.then substitute the actual numbers to solve.

4.4. Check your answers using a calculator.Check your answers using a calculator.

While these may seem like trivial and time consuming tasks,While these may seem like trivial and time consuming tasks,they will significantly increase your understanding of thethey will significantly increase your understanding of the

material and your accuracy rate.material and your accuracy rate.


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Present Value of a Lump SumPresent Value of a Lump Sum Present value calculations determinePresent value calculations determine

what the value of a cash flow received inwhat the value of a cash flow received in

the future would be worth today (time 0)the future would be worth today (time 0)

The process of finding a present value isThe process of finding a present value iscalled discounting (called discounting (hint: it gets smallerhint: it gets smaller))

The interest rate used to discount cashThe interest rate used to discount cashflows is generally called the discount rateflows is generally called the discount rate


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Example of PV of a Lump SumExample of PV of a Lump Sum How much would Rs.100 received five years from nowHow much would Rs.100 received five years from now

be worth today if the current interest rate is 10%?be worth today if the current interest rate is 10%?

1.1. Draw a timelineDraw a timeline

The arrow represents the flow of money and theThe arrow represents the flow of money and the

numbers under the timeline represent the time period.numbers under the timeline represent the time period.

Note that time period zero is today.Note that time period zero is today.

0 1 2 3 4 5

Rs.10

0

?i = 10%


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2.2. Write out the formula using symbols:Write out the formula using symbols:

PV = CFPV = CFtt/ (1+r)/ (1+r)tt

3.3. Insert the appropriate numbers:Insert the appropriate numbers:

PV = 100 / (1 + .1)PV = 100 / (1 + .1)55

4.4. Solve the formula:Solve the formula:

PV = Rs.62.09PV = Rs.62.09

5.5. Check using a financial calculator:Check using a financial calculator:

FV = Rs.100FV = Rs.100n = 5n = 5

PMT = 0PMT = 0

i = 10%i = 10%

PV = ?PV = ?

Example of PV of a Lump SumExample of PV of a Lump Sum


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Future Value of a Lump SumFuture Value of a Lump SumYou can think of future value as theYou can think of future value as the

opposite of present valueopposite of present value

Future value determines the amountFuture value determines the amountthat a sum of money invested today willthat a sum of money invested today willgrow to in a given period of timegrow to in a given period of time

The process of finding a future value isThe process of finding a future value iscalled compounding (called compounding (hint: it getshint: it getslargerlarger))


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Example of FV of a Lump SumExample of FV of a Lump Sum How much money will you have in 5 years if you investHow much money will you have in 5 years if you invest

Rs.100 today at a 10% rate of return?Rs.100 today at a 10% rate of return?


2.2. Write out the formula using symbols:Write out the formula using symbols:

FVFVtt= CF= CF00 * (1+r)* (1+r)tt

00 11 22 33

Rs.100Rs.100 ??i = 10%i = 10%

44 55


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Example of FV of a Lump SumExample of FV of a Lump Sum3.3. Substitute the numbers into the formula:Substitute the numbers into the formula:

FV = Rs.100 * (1+.1)FV = Rs.100 * (1+.1)55

4.4. Solve for the future value:Solve for the future value:FV = Rs.161.05FV = Rs.161.05

5.5. Check answer using a financial calculator:Check answer using a financial calculator:

i = 10%i = 10%

n = 5n = 5

PV = Rs.100PV = Rs.100

PMT = Rs.0PMT = Rs.0

FV = ?FV = ?


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Some Things to NoteSome Things to Note In both of the examples, note that if you were toIn both of the examples, note that if you were to

perform the opposite operation on the answers (i.e.,perform the opposite operation on the answers (i.e.,find the future value of Rs.62.09 or the present valuefind the future value of Rs.62.09 or the present value

of Rs.161.05) you will end up with your originalof Rs.161.05) you will end up with your originalinvestment of Rs.100.investment of Rs.100. This illustrates how present value and future valueThis illustrates how present value and future value

concepts are intertwined. In fact, they are the sameconcepts are intertwined. In fact, they are the sameequation . . .equation . . . Take PV = FVTake PV = FVtt/ (1+r)/ (1+r)

ttand solve for FVand solve for FVtt. You will get FV. You will get FVtt==PV * (1+r)PV * (1+r)tt..

As you get more comfortable with the formulas andAs you get more comfortable with the formulas andcalculations, you may be able to do the calculationscalculations, you may be able to do the calculationson your calculator alone. Be sure you understandon your calculator alone. Be sure you understandWHAT you are entering into each register and WHY.WHAT you are entering into each register and WHY.


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Present Value of a Cash FlowPresent Value of a Cash Flow

StreamStream A cash flow stream is a finite set of paymentsA cash flow stream is a finite set of payments

that an investor will receive or invest overthat an investor will receive or invest overtime.time.

The PV of the cash flow stream is equal toThe PV of the cash flow stream is equal tothe sum of the present value of each of thethe sum of the present value of each of theindividual cash flows in the stream.individual cash flows in the stream.

The PV of a cash flow stream can also beThe PV of a cash flow stream can also be

found by taking the FV of the cash flowfound by taking the FV of the cash flowstream and discounting the lump sum at thestream and discounting the lump sum at theappropriate discount rate for the appropriateappropriate discount rate for the appropriatenumber of periods.number of periods.


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Example of PV of a Cash FlowExample of PV of a Cash Flow

StreamStream Joe made an investment that will pay Rs.100 the first year,Joe made an investment that will pay Rs.100 the first year,

Rs.300 the second year, Rs.500 the third year and Rs.1000Rs.300 the second year, Rs.500 the third year and Rs.1000the fourth year. If the interest rate is ten percent, what is thethe fourth year. If the interest rate is ten percent, what is the

present value of this cash flow stream?present value of this cash flow stream?1.1. Draw a timeline:Draw a timeline:

00 11 22 33 44

??

Rs.1Rs.1

0000

Rs.30Rs.30

00

Rs.5Rs.5

0000

Rs.10Rs.10

0000

????

??

i = 10%i = 10%


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StreamStream2.2. Write out the formula using symbols:Write out the formula using symbols:

nn

PV =PV = 77[CF[CFtt/ (1+r)/ (1+r)tt]]

t=0t=0

OROR

PV = [CFPV = [CF11/(1+r)/(1+r)11]+[CF]+[CF22/(1+r)/(1+r)

22]+[CF]+[CF33/(1+r)/(1+r)33]+[CF]+[CF44/(1+r)/(1+r)

44]]

3.3. Substitute the appropriate numbers:Substitute the appropriate numbers:

PV =PV =[100/(1+.1)[100/(1+.1)11]+[Rs.300/(1+.1)]+[Rs.300/(1+.1)22]+[500/(1+.1)]+[500/(1+.1)33]+[1000/(1.1)]+[1000/(1.1)44]]


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StreamStream4.4. Solve for the present value:Solve for the present value:

PV = Rs.90.91 + Rs.247.93 + Rs.375.66 + Rs.683.01PV = Rs.90.91 + Rs.247.93 + Rs.375.66 + Rs.683.01

PV = Rs.1397.51PV = Rs.1397.51

5.5. Check using a calculator:Check using a calculator:

Make sure to use the appropriate rate of return, number ofMake sure to use the appropriate rate of return, number ofperiods, and future value for each of the calculations. Toperiods, and future value for each of the calculations. Toillustrate, for the first cash flow, you should enter FV=100, n=1,illustrate, for the first cash flow, you should enter FV=100, n=1,

i=10, PMT=0, PV=?. Note that you will have to do four separatei=10, PMT=0, PV=?. Note that you will have to do four separatecalculations.calculations.


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Future Value of a Cash FlowFuture Value of a Cash Flow

StreamStream The future value of a cash flow stream isThe future value of a cash flow stream is

equal to the sum of the future values of theequal to the sum of the future values of the

individual cash flows.individual cash flows. The FV of a cash flow stream can also beThe FV of a cash flow stream can also be

found by taking the PV of that same streamfound by taking the PV of that same streamand finding the FV of that lump sum usingand finding the FV of that lump sum using

the appropriate rate of return for thethe appropriate rate of return for theappropriate number of periods.appropriate number of periods.


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Example of FV of a Cash FlowExample of FV of a Cash Flow

StreamStream Assume Joe has the same cash flow stream from hisAssume Joe has the same cash flow stream from his

investment but wants to know what it will be worth at theinvestment but wants to know what it will be worth at theend of the fourth yearend of the fourth year

1.1. Draw a timeline:Draw a timeline:

00 1122 33 44

Rs.1Rs.1

0000

Rs.30Rs.30

00

Rs.5Rs.5

0000

Rs.10Rs.10

0000

i = 10%i = 10%

Rs.10Rs.10

0000??

??

??


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StreamStream2.2. Write out the formula using symbolsWrite out the formula using symbols

nn

FV =FV = 77[CF[CFtt* (1+r)* (1+r)nn--tt]]

t=0t=0

OROR

FV = [CFFV = [CF11*(1+r)*(1+r)nn--11]+[CF]+[CF22*(1+r)*(1+r)

nn--22]+[CF]+[CF33*(1+r)*(1+r)nn--33]+[CF]+[CF44*(1+r)*(1+r)

nn--44]]


FV = [Rs.100*(1+.1)FV = [Rs.100*(1+.1)44--11]+[Rs.300*(1+.1)]+[Rs.300*(1+.1)44--22]+[Rs.500*(1+.1)]+[Rs.500*(1+.1)44--33]]+[Rs.1000*(1+.1)+[Rs.1000*(1+.1)44--44]]


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StreamStream4.4. Solve for the Future Value:Solve for the Future Value:

FV = Rs.133.10 + Rs.363.00 + Rs.550.00 + Rs.1000FV = Rs.133.10 + Rs.363.00 + Rs.550.00 + Rs.1000

FV = Rs.2046.10FV = Rs.2046.10

5.5. Check using the calculator:Check using the calculator:

Make sure to use the appropriate interest rate, time period andMake sure to use the appropriate interest rate, time period andpresent value for each of the four cash flows. To illustrate, forpresent value for each of the four cash flows. To illustrate, forthe first cash flow, you should enter PV=100, n=3, i=10,the first cash flow, you should enter PV=100, n=3, i=10,

PMT=0, FV=?. Note that you will have to do four separatePMT=0, FV=?. Note that you will have to do four separatecalculations.calculations.


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AnnuitiesAnnuities


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Example of PV of an AnnuityExample of PV of an Annuity Assume that Sally owns an investment that will payAssume that Sally owns an investment that will pay

her Rs.100 each year for 20 years. The currenther Rs.100 each year for 20 years. The currentinterest rate is 15%. What is the PV of thisinterest rate is 15%. What is the PV of thisannuity?annuity?

1.1. Draw a timelineDraw a timelineRs.100Rs.100Rs.100Rs.100Rs.100Rs.100 Rs.100Rs.100Rs.100Rs.100

00 11 22 33 .. 1919 2020

??

i = 15%i = 15%


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Example of PV of an AnnuityExample of PV of an Annuity2.2. Write out the formula using symbols:Write out the formula using symbols:

PVA = PMT * {[1PVA = PMT * {[1--(1+r)(1+r)--tt]/r}]/r}

3.3. Substitute appropriate numbers:Substitute appropriate numbers:

PVA = Rs.100 * {[1PVA = Rs.100 * {[1--(1+.15)(1+.15)--2020]/.15}]/.15}

4.4. Solve for the PVSolve for the PV

PVA = Rs.100 * 6.2593PVA = Rs.100 * 6.2593

PVA = Rs.625.93PVA = Rs.625.93


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Example of PV of an AnnuityExample of PV of an Annuity5.5. Check answer using a calculatorCheck answer using a calculator

Make sure that the calculator is set to one period per yearMake sure that the calculator is set to one period per year


n= 20n= 20

i = 15%i = 15%

PV = ?PV = ?

Note that you do not need to enter anything for futureNote that you do not need to enter anything for future

value (or FV=0)value (or FV=0)


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Example of FV of an AnnuityExample of FV of an Annuity Assume that Sally owns an investment that will payAssume that Sally owns an investment that will pay

her Rs.100 each year for 20 years. The currenther Rs.100 each year for 20 years. The currentinterest rate is 15%. What is the FV of thisinterest rate is 15%. What is the FV of thisannuity?annuity?


00 11 22 33 .. 1919 2020

Rs.100Rs.100Rs.100Rs.100Rs.100Rs.100 Rs.100Rs.100Rs.100Rs.100

i = 15%i = 15%

??


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Example of FV of an AnnuityExample of FV of an Annuity2.2. Write out the formula using symbols:Write out the formula using symbols:

FVAFVAtt

= PMT * {[(1+r)= PMT * {[(1+r)tt1]/r}1]/r}


FVAFVA2020 = Rs.100 * {[(1+.15)= Rs.100 * {[(1+.15)20201]/.151]/.15

4.4. Solve for the FV:Solve for the FV:FVAFVA2020 = Rs.100 * 102.4436= Rs.100 * 102.4436

FVAFVA2020 = Rs.10,244.36= Rs.10,244.36


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ConclusionConclusion Overall, this presentation would help you toOverall, this presentation would help you to

effectively understand the time value ofeffectively understand the time value of

money.money. Plus, concepts and short formulas for findingPlus, concepts and short formulas for finding

the PV of FV of any amount.the PV of FV of any amount.

This was a sincere attempt towards lighteningThis was a sincere attempt towards lighteningyou on the concept but suggestions andyou on the concept but suggestions andqueries are always welcome.queries are always welcome.


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Example of FV of an AnnuityExample of FV of an Annuity5.5. Check using calculator:Check using calculator:

Make sure that the calculator is set to one period per yearMake sure that the calculator is set to one period per year


n = 20n = 20

i = 15%i = 15%

FV = ?FV = ?

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The Time Value of Money 2