LCBF_Time Value of Money
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Transcript of 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
2. Write out the formula using symbols:
FVt = CF0 * (1+r)t
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0 1 2 3
$100 ?i = 10%
4 5
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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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Example of PV of a Cash Flow
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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Example of PV of a Cash Flow
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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Example of FV of a Cash Flow
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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Example of FV of a Cash Flow
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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Example of PV of an Annuity
2. Write out the formula using symbols:
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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Example of PV of an Annuity
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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Example of FV of an Annuity
2. Write out the formula using symbols:
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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Example of FV of an Annuity
5. Check using calculator:
Make sure that the calculator is set to one period per year
PMT = $100
n = 20i = 15%
FV = ?