The Time Value of Money

1
The Time Value of Money

• Learning Module

2
The Time Value of Money

• Would you prefer to
• have 1 million now or
• 1 million 10 years
• from now?

Of course, we would all prefer the money
now! This illustrates that there is an inherent
monetary value attached to time.
3
What is The Time Value of Money?

• A dollar received today is worth more than a
  dollar received tomorrow
• This is because a dollar received today can be
  invested to earn interest
• The amount of interest earned depends on the rate
  of return that can be earned on the investment
• Time value of money quantifies the value of a
  dollar through time

4
Uses of Time Value of Money

• Time Value of Money, or TVM, is a concept that is
  used in all aspects of finance including
• Bond valuation
• Stock valuation
• Accept/reject decisions for project management
• Financial analysis of firms
• And many others!

5
Formulas

• Common formulas that are used in TVM
  calculations
• Present value of a lump sum
• PV CFt / (1r)t OR PV FVt / (1r)t
• Future value of a lump sum
• FVt CF0 (1r)t OR FVt PV (1r)t
• Present value of a cash flow stream
• n
• PV S CFt / (1r)t
• t0

6
Formulas (continued)

• Future value of a cash flow stream
• n
• FV S CFt (1r)n-t
• t0
• Present value of an annuity
• PVA PMT 1-(1r)-t/r
• Future value of an annuity
• FVAt PMT (1r)t 1/r

List adapted from the Prentice Hall Website
7
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 t and at time zero, respectively)
• PV present value (PVA present value of an
  annuity)
• FV future value (FVA future value of an
  annuity)

8
Types of TVM Calculations

• There are many types of TVM calculations
• The basic types will be covered in this review
  module 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, and
  will be used in combination to solve more complex
  TVM problems

9
Basic Rules

• The following are simple rules that you should
  always use no matter what type of TVM problem you
  are trying to solve
• Stop and think Make sure you understand what the
  problem is asking. You will get the wrong answer
  if you are answering the wrong question.
• Draw a representative timeline and label the cash
  flows and time periods appropriately.
• Write out the complete formula using symbols
  first and then substitute the actual numbers to
  solve.
• Check your answers using a calculator.
• While these may seem like trivial and time
  consuming tasks, they will significantly increase
  your understanding of the material and your
  accuracy rate.

10
Present Value of a Lump Sum

• Present value calculations determine what the
  value 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 is
  generally called the discount rate

11
Example of PV of a Lump Sum

• How much would 100 received five years from now
  be worth today if the current interest rate is
  10?
• Draw a timeline
• The arrow represents the flow of money and the
• numbers under the timeline represent the time
  period.
• Note that time period zero is today.

i 10
100
?
0
1
2
3
4
5
12
Example of PV of a Lump Sum

• Write out the formula using symbols
• PV CFt / (1r)t
• Insert the appropriate numbers
• PV 100 / (1 .1)5
• Solve the formula
• PV 62.09
• Check using a financial calculator
• FV 100
• n 5
• PMT 0
• i 10
• PV ?

13
Future Value of a Lump Sum

• You can think of future value as the opposite of
  present value
• Future value determines the amount that a sum of
  money invested today will grow to in a given
  period of time
• The process of finding a future value is called
  compounding (hint it gets larger)

14
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?
• Draw a timeline
• Write out the formula using symbols
• FVt CF0 (1r)t

i 10
100
?
0
1
2
3
4
5
15
Example of FV of a Lump Sum

• Substitute the numbers into the formula
• FV 100 (1.1)5
• Solve for the future value
• FV 161.05
• Check answer using a financial calculator
• i 10
• n 5
• PV 100
• PMT 0
• FV ?

16
Some Things to Note

• In both of the examples, note that if you were to
  perform the opposite operation on the answers
  (i.e., find the future 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
  value concepts are intertwined. In fact, they
  are the same equation . . .
• Take PV FVt / (1r)t and solve for FVt. You
  will get FVt PV (1r)t.
• As you get more comfortable with the formulas and
  calculations, you may be able to do the
  calculations on your calculator alone. Be sure
  you understand WHAT you are entering into each
  register and WHY.

17
Present Value of a Cash Flow Stream

• A cash flow stream is a finite set of payments
  that 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
  individual cash flows in the stream.
• The PV of a cash flow stream can also be found by
  taking the FV of the cash flow stream and
  discounting the lump sum at the appropriate
  discount rate for the appropriate number of
  periods.

18
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 the interest
  rate is ten percent, what is the present value of
  this cash flow stream?
• Draw a timeline

100
300
500
1000
0
1
2
3
4
?
?
i 10
?
?
19
Example of PV of a Cash Flow Stream

• Write out the formula using symbols
• n
• PV S CFt / (1r)t
• t0
• OR
• PV CF1/(1r)1CF2/(1r)2CF3/(1r)3CF4/(
  1r)4
• Substitute the appropriate numbers
• PV 100/(1.1)1300/(1.1)2500/(1.1)31
  000/(1.1)4

20
Example of PV of a Cash Flow Stream

• Solve for the present value
• PV 90.91 247.93 375.66 683.01
• PV 1397.51
• 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 first
  cash flow, you should enter FV100, n1, i10,
  PMT0, PV?. Note that you will have to do four
  separate calculations.

21
Future Value of a Cash Flow Stream

• The future value of a cash flow stream is equal
  to the sum of the future values of the individual
  cash flows.
• The FV of a cash flow stream can also be found by
  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 of periods.

22
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 fourth year
• Draw a timeline

100
300
500
1000
0
1
2
3
4
1000
i 10
?
?
?
23
Example of FV of a Cash Flow Stream

• Write out the formula using symbols
• n
• FV S CFt (1r)n-t
• t0
• OR
• FV CF1(1r)n-1CF2(1r)n-2CF3(1r)n-3
  CF4(1r)n-4
• Substitute the appropriate numbers
• FV 100(1.1)4-1300(1.1)4-2500(1.1
  )4-3 1000(1.1)4-4

24
Example of FV of a Cash Flow Stream

• Solve for the Future Value
• FV 133.10 363.00 550.00 1000
• FV 2046.10
• Check using the calculator
• Make sure to use the appropriate interest rate,
  time period and present value for each of the
  four cash flows. To illustrate, for the first
  cash flow, you should enter PV100, n3, i10,
  PMT0, FV?. Note that you will have to do four
  separate calculations.

25
Annuities

• An annuity is a cash flow stream in which the
  cash flows are all equal and occur at regular
  intervals.
• Note that annuities can be a fixed amount, an
  amount that grows at a constant rate over time,
  or an amount that grows at various rates of
  growth over time. We will focus on fixed amounts.

26
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 is 15. What is the PV of this
  annuity?
• Draw a timeline

100
100
100
100
100
0
1
2
3
.
19
20
?
i 15
27
Example of PV of an Annuity

• Write out the formula using symbols
• PVA PMT 1-(1r)-t/r
• Substitute appropriate numbers
• PVA 100 1-(1.15)-20/.15
• Solve for the PV
• PVA 100 6.2593
• PVA 625.93

28
Example of PV of an Annuity

• Check answer using a calculator
• Make sure that the calculator is set to one
  period per year
• PMT 100
• n 20
• i 15
• PV ?
• Note that you do not need to enter anything for
  future value (or FV0)

29
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 is 15. What is the FV of this
  annuity?
• Draw a timeline

100
100
100
100
100
0
1
2
3
.
19
20
?
i 15
30
Example of FV of an Annuity

• Write out the formula using symbols
• FVAt PMT (1r)t 1/r
• Substitute the appropriate numbers
• FVA20 100 (1.15)20 1/.15
  
• Solve for the FV
• FVA20 100 102.4436
• FVA20 10,244.36

31
Example of FV of an Annuity

• Check using calculator
• Make sure that the calculator is set to one
  period per year
• PMT 100
• n 20
• i 15
• FV ?