1 The Time Value of Money Learning Module. 2 The Time Value of Money Would you prefer to have $1...

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The Time Value The Time Value of Moneyof Money

Learning ModuleLearning Module

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The Time Value of MoneyThe Time Value of Money

Would you prefer to Would 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 would Of course, we would all prefer the money all prefer the money now!now!

This illustrates that This illustrates that there is an inherent there is an inherent monetary value monetary value attached to time.attached to time.

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What is The Time Value of What is The Time Value of Money?Money?

A dollar received today is worth more A dollar received today is worth more than a dollar received tomorrowthan a dollar received tomorrow This is because a dollar received today This is because a dollar received today

can be invested to earn interestcan be invested to earn interest The amount of interest earned depends The amount of interest earned depends

on the rate of return that can be earned on the rate of return that can be earned on the investmenton the investment

Time value of money quantifies the Time value of money quantifies the value of a dollar through timevalue of a dollar through time

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Uses of Time Value of Uses 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 of concept that is used in all aspects of finance including:finance including: Bond valuationBond valuation Stock valuationStock valuation Accept/reject decisions for project Accept/reject decisions for project

managementmanagement 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 Common formulas that are used in TVM

calculations:calculations:**

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

PV = CFPV = CFtt / (1+r) / (1+r)tt OROR PV = FV PV = FVt t / (1+r)/ (1+r)tt

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

FVFVtt = CF = CF00 * (1+r) * (1+r)tt OROR FV FVtt = PV * (1+r) = PV * (1+r)tt

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

PV = PV = [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 = [CF[CFtt * (1+r) * (1+r)n-tn-t]] t=0t=0

Present value of an annuity:Present value of an annuity:PVA = PMT * {[1-(1+r)PVA = PMT * {[1-(1+r)-t-t]/r}]/r}

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

FVAFVAtt = PMT * {[(1+r) = PMT * {[(1+r)tt –1]/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 CF = Cash flow (the subscripts t and 0 mean

at time t and at time zero, respectively)at time t and at time zero, respectively) PV = present value (PVA = present value of PV = present value (PVA = present value of

an annuity)an annuity) FV = future value (FVA = future value of an FV = future value (FVA = future value of an

annuity)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 The basic types will be covered in this

review module and include:review 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, Keep in mind that these forms can, should, and will be used in combination should, and will be used in combination to solve more complex TVM problemsto solve more complex TVM problems

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Basic RulesBasic Rules The following are simple rules that you should always use no The 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 the Stop and think: Make sure you understand what the

problem is asking. You will get the wrong answer if you problem is asking. You will get the wrong answer if you are answering the wrong question.are answering the wrong question.

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

3.3. Write out the complete formula using symbols first and Write out the complete formula using symbols first and then 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 the they 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 Present Value of a Lump SumSum

Present value calculations determine Present value calculations determine what the value of a cash flow received in what 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 is The process of finding a present value is called “discounting” (called “discounting” (hint: it gets hint: it gets smallersmaller))

The interest rate used to discount cash The interest rate used to discount cash flows is generally called the discount flows is generally called the discount raterate

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Example of PV of a Lump Example of PV of a Lump SumSum

How much would $100 received five years from now How much would $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 thenumbers 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

$100?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 = $62.09PV = $62.095.5. Check using a financial calculator:Check using a financial calculator:FV = $100FV = $100n = 5n = 5PMT = 0 PMT = 0 i = 10%i = 10%PV = ?PV = ?

Example of PV of a Lump Example of PV of a Lump SumSum

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Future Value of a Lump Future Value of a Lump SumSum

You can think of future value as You can think of future value as the opposite of present valuethe opposite of present value

Future value determines the Future value determines the amount that a sum of money amount that a sum of money invested today will grow to in a invested today will grow to in a given period of time given period of time

The process of finding a future The process of finding a future value is called “compounding” value is called “compounding” ((hint: it gets largerhint: it gets larger))

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Example of FV of a Lump Example of FV of a Lump SumSum

How much money will you have in 5 years if you How much money will you have in 5 years if you invest $100 today at a 10% rate of return?invest $100 today at a 10% rate of return?



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

00 11 22 33

$100$100 ??i = 10%i = 10%

44 55

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Example of FV of a Lump Example of FV of a Lump SumSum

3.3. Substitute the numbers into the formula:Substitute the numbers into the formula:

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

4.4. Solve for the future value:Solve for the future value:

FV = $161.05FV = $161.05

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

i = 10%i = 10%

n = 5n = 5

PV = $100PV = $100

PMT = $0PMT = $0

FV = ?FV = ?

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

to perform the opposite operation on the to perform the opposite operation on the answers (i.e., find the future value of $62.09 or answers (i.e., find the future value of $62.09 or the present value of $161.05) you will end up the present value of $161.05) you will end up with your original investment of $100. with your original investment of $100.

This illustrates how present value and future This illustrates how present value and future value concepts are intertwined. In fact, they value concepts are intertwined. In fact, they are the same equation . . .are the same equation . . . Take PV = FVTake PV = FVt t / (1+r)/ (1+r)tt and solve for FV and solve for FVtt. You will get . You will get

FVFVtt = PV * (1+r) = PV * (1+r)tt.. As you get more comfortable with the formulas As you get more comfortable with the formulas

and calculations, you may be able to do the and calculations, you may be able to do the calculations on your calculator alone. Be sure calculations on your calculator alone. Be sure you understand WHAT you are entering into you understand WHAT you are entering into each register and WHY.each register and WHY.

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Present Value of a Cash Present Value of a Cash Flow StreamFlow Stream

A cash flow stream is a finite set of A cash flow stream is a finite set of payments that an investor will receive or payments that an investor will receive or invest over time.invest over time.

The PV of the cash flow stream is equal The PV of the cash flow stream is equal to the sum of the present value of each to the sum of the present value of each of the individual cash flows in the stream.of the individual cash flows in the stream.

The PV of a cash flow stream can also be The PV of a cash flow stream can also be found by taking the FV of the cash flow found by taking the FV of the cash flow stream and discounting the lump sum at stream and discounting the lump sum at the appropriate discount rate for the the appropriate discount rate for the appropriate number of periods. appropriate number of periods.

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Example of PV of a Cash Example of PV of a Cash Flow StreamFlow Stream

Joe made an investment that will pay $100 the first Joe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is ten $1000 the fourth year. If the interest rate is ten percent, what is the present value of this cash flow percent, what is the present value of this cash flow stream?stream?

1.1. 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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2.2. Write out the formula using symbols:Write out the formula using symbols: nn

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

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4.4. Solve for the present value:Solve for the present value:

PV = $90.91 + $247.93 + $375.66 + $683.01PV = $90.91 + $247.93 + $375.66 + $683.01

PV = $1397.51PV = $1397.51

5.5. Check using a calculator:Check using a calculator: Make sure to use the appropriate rate of return, number Make sure to use the appropriate rate of return, number

of periods, and future value for each of the calculations. of periods, and future value for each of the calculations. To illustrate, for the first cash flow, you should enter To illustrate, for the first cash flow, you should enter FV=100, n=1, i=10, PMT=0, PV=?. Note that you will FV=100, n=1, i=10, PMT=0, PV=?. Note that you will have to do four separate calculations.have to do four separate calculations.

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Future Value of a Cash Future Value of a Cash Flow StreamFlow Stream

The future value of a cash flow stream The future value of a cash flow stream is equal to the sum of the future values is equal to the sum of the future values of the individual cash flows.of the individual cash flows.

The FV of a cash flow stream can also The FV of a cash flow stream can also be found by taking the PV of that same be found by taking the PV of that same stream and finding the FV of that lump stream and finding the FV of that lump sum using the appropriate rate of return sum using the appropriate rate of return for the appropriate number of periods.for the appropriate number of periods.

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Example of FV of a Cash Example of FV of a Cash Flow StreamFlow Stream

Assume Joe has the same cash flow stream from his Assume Joe has the same cash flow stream from his investment but wants to know what it will be worth at investment but wants to know what it will be worth at the end of the fourth yearthe end of the fourth year

1.1. 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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2.2. Write out the formula using symbolsWrite out the formula using symbols nn

FV = FV = [CF[CFtt * (1+r) * (1+r)n-tn-t]] t=0t=0

OROR

FV = [CFFV = [CF11*(1+r)*(1+r)n-1n-1]+[CF]+[CF22*(1+r)*(1+r)n-2n-2]+[CF]+[CF33*(1+r)*(1+r)n-3n-3]+[CF]+[CF44*(1+r)*(1+r)n-n-

44]]


FV = [$100*(1+.1)FV = [$100*(1+.1)4-14-1]+[$300*(1+.1)]+[$300*(1+.1)4-24-2]+[$500*(1+.1)]+[$500*(1+.1)4-34-3] +] +[$1000*(1+.1)[$1000*(1+.1)4-44-4]]

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4.4. Solve for the Future Value:Solve for the Future Value:

FV = $133.10 + $363.00 + $550.00 + $1000FV = $133.10 + $363.00 + $550.00 + $1000

FV = $2046.10FV = $2046.10

5.5. Check using the calculator:Check using the calculator: Make sure to use the appropriate interest rate, time Make sure to use the appropriate interest rate, time

period and present value for each of the four cash flows. period and present value for each of the four cash flows. To illustrate, for the first cash flow, you should enter To illustrate, for the first cash flow, you should enter PV=100, n=3, i=10, PMT=0, FV=?. Note that you will PV=100, n=3, i=10, PMT=0, FV=?. Note that you will have to do four separate calculations.have to do four separate calculations.

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AnnuitiesAnnuities

An annuity is a cash flow stream in An annuity is a cash flow stream in which the cash flows are all equal and which the cash flows are all equal and occur at regular intervals. occur at regular intervals.

Note that annuities can be a fixed Note that annuities can be a fixed amount, an amount that grows at a amount, an amount that grows at a constant rate over time, or an amount constant rate over time, or an amount that grows at various rates of growth that grows at various rates of growth over time. We will focus on fixed over time. We will focus on fixed amounts.amounts.

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Example of PV of an Example of PV of an AnnuityAnnuity

Assume that Sally owns an investment that Assume that Sally owns an investment that will pay her $100 each year for 20 years. The will pay her $100 each year for 20 years. The current interest rate is 15%. What is the PV current interest rate is 15%. What is the PV of this annuity?of this annuity?


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

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

??i = 15%i = 15%

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

3.3. Substitute appropriate numbers:Substitute appropriate numbers:

PVA = $100 * {[1-(1+.15)PVA = $100 * {[1-(1+.15)-20-20]/.15}]/.15}

4.4. Solve for the PVSolve for the PV

PVA = $100 * 6.2593PVA = $100 * 6.2593

PVA = $625.93PVA = $625.93

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5.5. Check answer using a calculatorCheck answer using a calculator Make sure that the calculator is set to one period Make sure that the calculator is set to one period

per yearper year PMT = $100PMT = $100

n= 20n= 20

i = 15%i = 15%

PV = ?PV = ? Note that you do not need to enter anything for Note that you do not need to enter anything for

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

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Example of FV of an Example of FV of an AnnuityAnnuity

Assume that Sally owns an investment that Assume that Sally owns an investment that will pay her $100 each year for 20 years. The will pay her $100 each year for 20 years. The current interest rate is 15%. What is the FV current interest rate is 15%. What is the FV of this annuity?of this annuity?


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

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

i = 15%i = 15%

??

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


FVAFVA2020 = $100 * {[(1+.15) = $100 * {[(1+.15)2020 –1]/.15 –1]/.15

4.4. Solve for the FV:Solve for the FV:

FVAFVA2020 = $100 * 102.4436 = $100 * 102.4436

FVAFVA2020 = $10,244.36 = $10,244.36

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5.5. Check using calculator:Check using calculator: Make sure that the calculator is set to one period Make sure that the calculator is set to one period

per yearper year PMT = $100PMT = $100

n = 20n = 20

i = 15%i = 15%

FV = ?FV = ?

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