LCBF_Time Value of Money

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THE TIME VALUE OF MONEY

LCBF_BA(Hons)_L-5Accounting and Finance for Managers


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The Time Value of Money

Would you prefer to

have $1 million now or

$1 million 10 yearsfrom now?

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Of course, we would all

prefer the money now!

This illustrates that thereis an inherent monetary

value attached to time.


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Uses of Time Value of Money

Time Value of Money, or TVM, is a concept that isused in all aspects of finance including:

Bond valuation

Stock valuation

Accept/reject decisions for project management

Financial analysis of firms

And many others!

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Formulas (continued)

Future value of a cash flow stream:n

FV = S [CFt

* (1+r)n-t]t=0

Present value of an annuity:

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

Future value of an annuity:

FVAt = PMT * {[(1+r)t1]/r}

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* List adapted from the Prentice Hall Website


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Variables

where r = rate of return

t = time period

n = number of time periods PMT = payment

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

PV = present value (PVA = present value of an annuity)

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

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Types of TVM Calculations

There are many types of TVM calculations

The basic types will be covered in this reviewmodule and include: Present value of a lump sum Future value of a lump sum

Present and future value of cash flow streams

Present and future value of annuities

Keep in mind that these forms can, should, andwill be used in combination to solve morecomplex TVM problems

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Basic Rules

The following are simple rules that you should always use nomatter what type of TVM problem you are trying to solve:

1. Stop and think: Make sure you understand what the problemis asking. You will get the wrong answer if you are answeringthe wrong question.

2. Draw a representative timeline and label the cash flows andtime periods appropriately.

3. Write out the complete formula using symbols first and thensubstitute the actual numbers to solve.

4. Check your answers using a calculator.

While these may seem like trivial and time consuming tasks, theywill significantly increase your understanding of the material andyour accuracy rate.

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Present Value of a Lump Sum

Present value calculations determine what thevalue of a cash flow received in the future

would be worth today (time 0) The process of finding a present value is called

discounting (hint: it gets smaller)

The interest rate used to discount cash flows isgenerally called the discount rate

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Example of PV of a Lump Sum

How much would $100 received five years from now beworth today if the current interest rate is 10%?

1. Draw a timeline

The arrow represents the flow of money and thenumbers under the timeline represent the time period.

Note that time period zero is today.

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0 1 2 3 4 5

$100?i = 10%


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Example of PV of a Lump Sum

2. Write out the formula using symbols:

PV = CFt / (1+r)t

3. Insert the appropriate numbers:

PV = 100 / (1 + .1)5

4. Solve the formula:

PV = $62.09

5. Check using a financial calculator:

FV = $100

n = 5

PMT = 0

i = 10%

PV = ?

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Future Value of a Lump Sum

You can think of future value as the oppositeof present value

Future value determines the amount that asum of money invested today will grow to in agiven period of time

The process of finding a future value is called

compounding (hint: it gets larger)

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Example of FV of a Lump Sum

How much money will you have in 5 years if you invest $100

today at a 10% rate of return?

1. Draw a timeline


FVt = CF0 * (1+r)t

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0 1 2 3

$100 ?i = 10%

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Example of FV of a Lump Sum

3. Substitute the numbers into the formula:

FV = $100 * (1+.1)5

4. Solve for the future value:

FV = $161.05

5. Check answer using a financial calculator:

i = 10%

n = 5

PV = $100

PMT = $0

FV = ?

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Some Things to Note

In both of the examples, note that if you were to performthe opposite operation on the answers (i.e., find thefuture value of $62.09 or the present value of $161.05)you will end up with your original investment of $100.

This illustrates how present value and future valueconcepts are intertwined. In fact, they are the sameequation . . . Take PV = FVt / (1+r)

t and solve for FVt. You will get FVt = PV *(1+r)t.

As you get more comfortable with the formulas andcalculations, you may be able to do the calculations onyour calculator alone. Be sure you understand WHAT youare entering into each register and WHY.

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Present Value of a Cash Flow

Stream

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

The PV of the cash flow stream is equal to the

sum of the present value of each of the individualcash flows in the stream.

The PV of a cash flow stream can also be foundby taking the FV of the cash flow stream anddiscounting the lump sum at the appropriate

discount rate for the appropriate number ofperiods.

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Example of PV of a Cash Flow

Stream Joe made an investment that will pay $100 the first year, $300 the

second year, $500 the third year and $1000 the fourth year. If theinterest rate is ten percent, what is the present value of this cashflow stream?

1. Draw a timeline:

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0 1 2 3 4

?

$100 $300 $500 $1000

?

?

?

i = 10%


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Stream2. Write out the formula using symbols:

n

PV =S [CFt / (1+r)t]

t=0

ORPV = [CF1/(1+r)

1]+[CF2/(1+r)2]+[CF3/(1+r)

3]+[CF4/(1+r)4]

3. Substitute the appropriate numbers:

PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]

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Stream4. Solve for the present value:

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

PV = $1397.51

5. Check using a calculator:

Make sure to use the appropriate rate of return, number of periods,and future value for each of the calculations. To illustrate, for the firstcash flow, you should enter FV=100, n=1, i=10, PMT=0, PV=?. Note

that you will have to do four separate calculations.

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Future Value of a Cash Flow

Stream

The future value of a cash flow stream is equal tothe sum of the future values of the individual

cash flows.

The FV of a cash flow stream can also be foundby taking the PV of that same stream and finding

the FV of that lump sum using the appropriate

rate of return for the appropriate number ofperiods.

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Example of FV of a Cash Flow

Stream Assume Joe has the same cash flow stream from his investment

but wants to know what it will be worth at the end of the fourthyear

1. Draw a timeline:

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0 1 2 3 4

$100 $300 $500 $1000

i = 10%

$1000

?

?

?


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Stream

2. Write out the formula using symbols

n

FV =S [CFt * (1+r)n-t]

t=0

OR

FV = [CF1*(1+r)n-1]+[CF2*(1+r)

n-2]+[CF3*(1+r)n-3]+[CF4*(1+r)

n-4]

3. Substitute the appropriate numbers:

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

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Stream4. Solve for the Future Value:FV = $133.10 + $363.00 + $550.00 + $1000

FV = $2046.10

5. Check using the calculator:

Make sure to use the appropriate interest rate, time period andpresent value for each of the four cash flows. To illustrate, for the firstcash flow, you should enter PV=100, n=3, i=10, PMT=0, FV=?. Notethat you will have to do four separate calculations.

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Annuities

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

intervals.

Note that annuities can be a fixed amount, anamount that grows at a constant rate over time,

or an amount that grows at various rates of

growth over time. We will focus on fixedamounts.

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Example of PV of an Annuity

Assume that Sally owns an investment that will pay her$100 each year for 20 years. The current interest rate is15%. What is the PV of this annuity?

1. Draw a timeline

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0 1 2 3.

19 20

$100 $100 $100 $100 $100

?

i = 15%


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

3. Substitute appropriate numbers:PVA = $100 * {[1-(1+.15)-20]/.15}

4. Solve for the PV

PVA = $100 * 6.2593

PVA = $625.93

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5. Check answer using a calculator

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

PMT = $100

n= 20i = 15%

PV = ?

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

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Example of FV of an Annuity

Assume that Sally owns an investment that will pay her$100 each year for 20 years. The current interest rate is15%. What is the FV of this annuity?

1. Draw a timeline

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0 1 2 3 . 19 20

$100 $100 $100$100 $100

i = 15%

?


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FVAt = PMT * {[(1+r)t1]/r}

3. Substitute the appropriate numbers:FVA20 = $100 * {[(1+.15)

201]/.15

4. Solve for the FV:

FVA20 = $100 * 102.4436FVA20 = $10,244.36

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5. Check using calculator:

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

PMT = $100

n = 20i = 15%

FV = ?

LCBF_Time Value of Money

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