module time pp t

7/27/2019 Module Time Pp t

1/31

1

The Time ValueThe Time Value

of Moneyof MoneyLearning ModuleLearning Module


2/31

2

The Time Value of MoneyThe Time Value of Money

Would you prefer toWould you prefer to

have $1 million nowhave $1 million now

oror$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 thatThis illustrates that

there is an inherentthere is an inherent


3/31

3

What is The Time Value ofWhat is The Time Value of

Money?Money?

A dollar received today is worthA dollar received today is worthmore than a dollar receivedmore than a dollar received

tomorrowtomorrowThis is because a dollar receivedThis is because a dollar received

today can be invested to earn interesttoday can be invested to earn interestThe amount of interest earnedThe amount of interest earned

depends on the rate of return thatdepends on the rate of return thatcan be earned on the investmentcan be earned on the investment

Time value of money quantifies theTime value of money quantifies the

value of a dollar through timevalue of a dollar through time


4/31

4

Uses of Time Value ofUses of Time Value of

MoneyMoney

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

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

finance including:finance including: Bond valuationBond valuation

Stock valuationStock valuation

Accept/reject decisions for projectAccept/reject decisions for project

managementmanagement Financial analysis of firmsFinancial analysis of firms

And many others!And many others!


5/31

5

FormulasFormulas

Common formulas that are used in TVMCommon formulas that are used in TVMcalculations:calculations:** Present value of a lump sum:Present value of a lump sum:

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

tt/ (1+r)/ (1+r)tt

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

FVFVtt= CF= CF

00* (1+r)* (1+r)ttOROR FVFV

tt= 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[CF

tt / (1+r)/ (1+r)tt

]]

t=0t=0


6/31

6

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


7/317

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 meanCF = 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 ofPV = present value (PVA = present value of

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

annuity)annuity)


8/31

8

Types of TVM CalculationsTypes of TVM Calculations

There are many types of TVMThere are many types of TVMcalculationscalculations

The basic types will be covered in thisThe basic types will be covered in thisreview 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 flowPresent and future value of cash flowstreamsstreams 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 combinationshould, and will be used in combinationto solve more complex TVM problemsto solve more complex TVM problems


9/31

9

Basic RulesBasic Rules

The following are simple rules that you should alwaysThe following are simple rules that you should alwaysuse no matter what type of TVM problem you areuse no matter what type of TVM problem you aretrying to solve:trying to solve:

1.1. Stop and think: Make sure you understand whatStop and think: Make sure you understand whatthe problem is asking. You will get the wrongthe problem is asking. You will get the wronganswer if you are answering the wrong question.answer if you are answering the wrong question.

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

3.3. Write out the complete formula using symbols firstWrite out the complete formula using symbols firstand then substitute the actual numbers to solve.and 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 timeWhile these may seem like trivial and time

consuming tasks, they will significantly increase yourconsuming tasks, they will significantly increase yourunderstanding of the material and your accuracyunderstanding of the material and your accuracy


10/31

10

Present Value of a LumpPresent Value of a Lump

SumSum

Present value calculationsPresent value calculations

determine what the value of a cashdetermine what the value of a cash

flow received in the future would beflow received in the future would beworth today (time 0)worth today (time 0)

The process of finding a presentThe process of finding a present

value is called discounting (value is called discounting (hint: ithint: itgets smallergets smaller))

The interest rate used to discountThe interest rate used to discount

cash flows is generally called thecash flows is generally called the


11/31

11

Example of PV of a LumpExample of PV of a Lump

SumSum

How much would $100 received five years fromHow much would $100 received five years fromnow be worth today if the current interest rate isnow be worth today if the current interest rate is10%?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 timenumbers under the timeline represent the timeperiod.period.

0 1 2 3 4 5

$100?i = 10%


12/31

12

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 = $62.09

5.5. Check using a financial calculator:Check using a financial calculator:FV = $100FV = $100

n = 5n = 5

PMT = 0PMT = 0

i = 10%i = 10%

PV = ?PV = ?

Example of PV of a LumpExample of PV of a Lump

SumSum


13/31

13

Future Value of a LumpFuture Value of a Lump

SumSum

You can think of future value asYou can think of future value asthe opposite of present valuethe opposite of present value

Future value determines theFuture value determines theamount that a sum of moneyamount that a sum of moneyinvested today will grow to in ainvested today will grow to in agiven period of timegiven period of time

The process of finding a futureThe process of finding a futurevalue is called compoundingvalue is called compounding((hint: it gets largerhint: it gets larger))


14/31

14

Example of FV of a LumpExample of FV of a Lump

SumSum

How much money will you have in 5 years if youHow 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= CF

00* (1+r)* (1+r)tt

00 11 22 33

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

44 55


15/31

15

Example of FV of a LumpExample 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 = ?


16/31

16

Some Things to NoteSome Things to Note

In both of the examples, note that if you wereIn both of the examples, note that if you wereto perform the opposite operation on theto perform the opposite operation on theanswers (i.e., find the future value of $62.09 oranswers (i.e., find the future value of $62.09 orthe present value of $161.05) you will end upthe present value of $161.05) you will end upwith your original investment of $100.with your original investment of $100.

This illustrates how present value and futureThis illustrates how present value and futurevalue concepts are intertwined. In fact, theyvalue concepts are intertwined. In fact, theyare the same equation . . .are the same equation . . . Take PV = FVTake PV = FV

tt/ (1+r)/ (1+r)tt and solve for FVand solve for FV

tt. You will get. You will get

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

and calculations, you may be able to do theand calculations, you may be able to do thecalculations on your calculator alone. Be surecalculations on your calculator alone. Be sureyou understand WHAT you are entering intoyou understand WHAT you are entering into

each register and WHY.each register and WHY.


17/31

17

Present Value of a CashPresent Value of a Cash

Flow StreamFlow Stream

A cash flow stream is a finite set ofA cash flow stream is a finite set ofpayments that an investor will receivepayments that an investor will receiveor invest over time.or invest over time.

The PV of the cash flow stream is equalThe PV of the cash flow stream is equalto the sum of the present value of eachto the sum of the present value of eachof the individual cash flows in theof the individual cash flows in thestream.stream.

The PV of a cash flow stream can alsoThe PV of a cash flow stream can alsobe found by taking the FV of the cashbe found by taking the FV of the cashflow stream and discounting the lumpflow stream and discounting the lumpsum at the appropriate discount rate forsum at the appropriate discount rate forthe appropriate number of periods.the appropriate number of periods.


18/31

18

Example of PV of a CashExample of PV of a Cash


Joe made an investment that will pay $100 the firstJoe made an investment that will pay $100 the first

year, $300 the second year, $500 the third year andyear, $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 flowpercent, what is the present value of this cash flowstream?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%


19/31

19

Example of PV of a CashExample of PV of a Cash



nn

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

t=0t=0

OROR

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

22/(1+r)/(1+r)22]+[CF]+[CF

33/(1+r)/(1+r)33]+[CF]+[CF

44/(1+r)/(1+r)44]]

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

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

(1.1)(1.1)44]]


20/31


21/31

21

Future Value of a CashFuture Value of a Cash


The future value of a cash flow streamThe future value of a cash flow stream

is equal to the sum of the future valuesis 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 alsoThe FV of a cash flow stream can also

be found by taking the PV of that samebe found by taking the PV of that same

stream and finding the FV of that lumpstream and finding the FV of that lump

sum using the appropriate rate of returnsum using the appropriate rate of returnfor the appropriate number of periods.for the appropriate number of periods.


22/31

22

Example of FV of a CashExample of FV of a Cash


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 atinvestment 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

??

??

??


23/31

23



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]+[CF

22*(1+r)*(1+r)n-2n-2]+[CF]+[CF

33*(1+r)*(1+r)n-3n-3]+[CF]+[CF

44*(1+r)*(1+r)n-4n-4]]

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


24/31

24



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, timeMake 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 enterTo illustrate, for the first cash flow, you should enter

PV=100, n=3, i=10, PMT=0, FV=?. Note that you willPV=100, n=3, i=10, PMT=0, FV=?. Note that you willhave to do four separate calculations.have to do four separate calculations.


25/31

25

AnnuitiesAnnuities

An annuity is a cash flow stream inAn annuity is a cash flow stream in

which the cash flows are all equal andwhich the cash flows are all equal and

occur at regular intervals.occur at regular intervals. Note that annuities can be a fixedNote that annuities can be a fixed

amount, an amount that grows at aamount, an amount that grows at a

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

that grows at various rates of growththat grows at various rates of growthover time. We will focus on fixedover time. We will focus on fixed

amounts.amounts.


26/31

26

Example of PV of anExample of PV of an

AnnuityAnnuity

Assume that Sally owns an investment thatAssume that Sally owns an investment that

will pay her $100 each year for 20 years. Thewill pay her $100 each year for 20 years. The

current interest rate is 15%. What is the PVcurrent 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%


27/31

27


AnnuityAnnuity


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


28/31

28


AnnuityAnnuity

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

per yearper year

PMT = $100PMT = $100n= 20n= 20

i = 15%i = 15%

PV = ?PV = ?

Note that you do not need to enter anything forNote that you do not need to enter anything forfuture value (or FV=0)future value (or FV=0)


29/31

29

Example of FV of anExample of FV of an

AnnuityAnnuity

Assume that Sally owns an investment thatAssume that Sally owns an investment that

will pay her $100 each year for 20 years. Thewill pay her $100 each year for 20 years. The

current interest rate is 15%. What is the FVcurrent 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%

??


30/31

30


AnnuityAnnuity

2.2. Write out the formula usingWrite out the formula using

symbols:symbols:

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

3.3. Substitute the appropriateSubstitute the appropriate

numbers:numbers:

FVAFVA2020

= $100 * {[(1+.15)= $100 * {[(1+.15)2020 1]/.151]/.15

4.4.

Solve for the FV:Solve for the FV:

*


31/31

31


AnnuityAnnuity

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

per yearper year

PMT = $100PMT = $100n = 20n = 20

i = 15%i = 15%

FV = ?FV = ?

module time pp t

Documents