Chapter 6: Time Value of Money

1
Chapter 6 Time Value of Money

• Purpose Provide students with the math skills
  needed to make long-term decisions.
• Future Value of a Single Sum
• Present Value of a Single Sum
• Future Value of an Annuity
• Present Value of an Annuity
• Annuity Due
• Perpetuities
• Non-annual Periods
• Effective Annual Rates

2
Calculators

• Students are strongly encouraged to use a
  financial calculator, the most efficient approach
  to solving discounted cash flow problems.
  Throughout the lecture materials, setting up the
  problem and tabular solutions have been
  emphasized. In the text, financial calculator and
  regular calculator solutions are also presented.

3
More on Calculators

• Note Read the instructions accompanying your
  calculator. Procedures vary at times among
  calculators (e.g., some require outflows to be
  entered as negative numbers, and some do not).
• The Sharp EL-733A is required. I will be using
  this calculator in class to illustrate problems.

4
The Time Value of Money

• Most financial decisions involve benefits and
  costs that are spread out over time- the time
  value of money establishes a relationship between
  cash flows received and/or paid at different
  points in time.

5
The Role of Time Value in Finance

• Future versus Present Value
• Time lines can be used to depict cash flows
• Compounding is used to find future value
• Discounting is used to find present value
• Computational aids
• Business/Financial calculators

6
The Concept of Future Values

• Terminology Six Functions of a Dollar
• 1. Future Value of a Single Sum
• 2. Future Value of a Series
• 3. Sinking Fund Factor

t
7
Future Value of Single Sum
8
Future Value of an Annuity
9
Sinking Fund
10
Simple Interest

• Justin Jones has borrowed 1000 from his
  grandmother for two years at 8 simple interest.
  How much will Justin pay Granny in two years?

11
Simple Interest
Justin Jones has borrowed 1000 from his
grandmother for two years at 8 simple interest.
How much will Justin pay Granny in two
years? 1,000
? 0
1
2 1,000 X .08 X 2 160
Original Principal 1,000 Amount owed to
Granny 1,160
12
Compounded Interest

• Granny just found Justins finance book and
  discovered the concept of compounded interest. At
  8 annual compounded interest, how much will
  Justin owe Granny at the end of two years?
• Solution -1,000 PV 8 i 2 n
• compute FV
• Answer1,166.40

13
Compounded Interest
Year 1 1,000 X .08 X 1 year 80
Interest This now becomes part of the principal
for year 2 1,000 80 1,080 Year 2 1,080
X .08 X 1 year 86.40 Interest Total Interest
80 86.40 166.40 Original Principal
1,000.00 Amount owed to Granny
1,166.40
14
Future Value Formula
FVn PV (1i)n
FVn Future value at the end of the year n PV
Present value, or original principal amount i
Annual rate of interest paid n
Number of periods (usually years) separating the
present value and the future value, or number of
years the money is left on deposit
15
Compounded Interest Example
If Rita Brown deposits 800 today in an account
paying 9 annual interest, how much will she
have at the end of three years? Solution -800
PV 9 i 3 n compute FV Answer
1,036.02
16
Compounded Interest Example
If Rita Brown deposits 800 today in an account
paying 9 annual interest, how much will she
have at the end of three years? PV 800

FV3? 0 1
2 3 PV -800 i
9 n 3 compute FV
Answer 1036.02
17
A Graphic View of Future Value
Future Value of One Dollar ()
1.00
Periods
18
You Try One!

• What is the future value of 1,000 compounded at
  7 annually for nine years?
• Solution -1,000 PV 9 n 7 i
• compute FV
• Answer 1,838.46
  

19
Important notes regarding FVIF tables

• We will not be using nor be allowed to use tables
  for exam purposes. Therefore disregard the
  tables and rely totally on your calculator -- the
  Sharp EL-733A or EL-735.

20
Example

• How long will it take Dan Smits to double his
  100 if he deposits it in an account earning 10
  compounded annually?
• Solution -100 PV 10 i 200 FV
• compute n
• Solution 7.27 years

21
Example

• How long will it take Dan Smits to double his
  100 if he deposits it in an account earning 7
  compounded annually?
• Solution -100 PV 7 i 200 FV
• compute n
• Answer 10.24 years

22
Example

• Pete Brown received 1,000 on his tenth birthday.
  He plans on buying a new motorcycle on his
  sixteenth birthday, but figures he'll need at
  least 1,500 for the purchase. What is the
  minimum annual compound rate of interest Pete's
  deposit will need to earn?
• Solution -1,000 PV 1,500 FV 6 n
• compute i
• Answer 6.99

23
Compounding More Frequently Than Annually

• New variable m number of compounding
  periods per year
• Divide i by m
• Multiply m times n
• Example Suppose you have 10,000 on deposit
  earning 6, semi-annually. How much will you
  have in 5 years?
• Solution -10,000 PV 10 n 3 i
  compute FV
• Answer 13,439.16

24
Example

• What is the future value of 1,000 deposited in
  an account paying 12 interest compounded
  quarterly after four years?
• PV -1,000 i 12, m 4, n
  4
• Therefore -1,000 PV 3 i 16
  n
• compute FV
• Answer 1,604.71

25
Nominal And Effective Interest Rates

• Nominal, or stated, rates
• Effective, or stated, rates
• annual percentage rate (or APR)
• The effective rate (keff)

26
Equation for Effective Rate of Interest, keff

• keff (1 k/m)m - 1
• Assume you can earn 6 compounded monthly. What
  is the effective rate of interest?
• Solution
• -1 PV .5 i 12 n compute FV
• Answer 1.0617 or 6.17

27
Example

• Nominal Rate, k, 12
• Compounding Effective
• Period m keff (1k/m)m - 1 Rate
• Annual 1 (1 .12/1) -1 1 .12 -1 .12
  12.00
• Semiannual 2 (1 .12/2)2 -1 1.1236 -1
  .1236 12.36
• Quarterly 4 (1 .12/4)4 -1 1.1255 -1
  .1255 12.55
• Monthly 12 (1 .12/12)12 -1 1.1268 -1
  .1268 12.68
• Daily 360 (1 .12/360)360 -1 1.1275-1
  .1275 12.75

28
Future value of an Annuity
Types of Annuities 1. Ordinary Annuities The
ordinary annuity occurs when the level payment is
received at the end of the payment period. 2.
Annuities Due The annuity due occurs when the
level payment is received at the beginning of the
payment period.
29
Example

• What is the future value of an ordinary, 3-year
  annuity of 1,000 at 10 annual compounding?
• 0 1,000 1,000
  1,000
• Solution -1,000 pmt 10 i 3 n
• compute FV
• Answer 3,310.00

30
Example

• What is the future value of an ordinary,
  eight-year annuity of 100 compounded at 7
  annually?
• 
  
• 0 100 100 100 100 100
  100 100 100
• Solution -100 pmt 8 n
  7 i
• Compute FV
• Answer 1,025.98

31
Finding The Future Value Of An Annuity Due

• Since each payment of an annuity due is received
  one period sooner, each is compounded for one
  year longer than for an ordinary annuity.
• Solution to the previous problem if it were an
  annuity due
• BGN -100 pmt 7 i 8 n compute FV
• Answer 1,097.80
• Notice that the payments are all compounded for
  one more time period. Or, we can take the
  previous answer of 1,025.98 times (1.07) to get
  1,097.80.

32
The Present Value Functions

• So far we have discussed the first three
  functions. The next three functions are all PV
  functions.
• 4. PV of a Single Sum
• 5. PV of a Series of Payments
• 6. Determining the Amortization Factor

33
Present Value of a Single Sum

• The Concept of Present Value
• Present Value concerns the current dollar value
  (today's value) of a future amount of money

34
A Graphic View of Present Value
0
35
Present Value Single Payment
36
Present Value of the Annuity
37
Amortized loans
38
Example

• What is the present value of 1,000 due to be
  received four years from today assuming that if
  you had the dollar equivalent today, you could
  invest it at 8?
• 
  1,000
• 0 1 2 3 4
• Value
• Today?

39
Solution

• What we know
• 4 n 1,000 FV 8 i
• Therefore compute PV
• Solution 735.03

40
You Try One

• What is the value today of 1,000 to be received
  in five years for a person with an opportunity
  cost of 11.5?
• Solution 1,000 FV 5 n
• 11.5 i
• Compute PV
• Answer 580.26

41
Present Value of a Series of Payments (Annuity)

• An annuity is a series of equal pay-ments over a
  given time period.
• Example Suppose you will receive a pay-ment of
  5,000 starting immediately for four years to
  complete your college education. What is the
  total value of those payments today given your
  opportunity cost for money is 6?

42
Example

• What do we know?
• 1. 5,000 is the payment (pmt)
• 2. 4 is the value for (n)
• 3. 6 is the interest rate (i)
• Therefore, we are computing the present value of
  the series (PV)
• Answer 18,365.06

43
Determining the Payment

• Suppose you are considering purchasing a new car
  at a price of 22,000. If the dealer offers a
  low 6 financing rate for a five year loan,
  monthly payments, what will your monthly payments
  be?
• What we know 22,000 PV 6/12 i
• 60 n Compute pmt
• Answer 425.32

44
Another Example

• At what rate of interest would you be indifferent
  between 6500 today and 1,000 per year for the
  next eight years?
• Solution 6,500 PV -1,000 pmt 8 n
  compute i
• Answer 4.86

45
Cash Flow Example

• Suppose you had a project cost of 10,000 today
  that yield a series of payments for the next
  three years of 4,000 in year one, 3,500 in year
  two, and 4,200 in year three. If you can earn
  8 on your money, which of the two alternatives
  would you take?
• Solution 8 i -10,000 CFi 4,000 CFi
• 3,500 CFi 4,200 CFi press NPV
• Answer 38.48
• Select the cash flow because it is worth
  10,038.48

46
Example

• Margaret Taylor wants to have 4,000 at the
  beginning of each of her four years of college -
  starting one year from today. If she can earn 8
  annually, how much does she need to deposit in
  the account today?
• 0 1
  2 3 4
• 4000 4000
  4000 4000
• Solution -4,000 pmt 4 n
  8 i comp PV
• Answer 13,248.51

47
What about the Annuity Due?

• Note that the PV of an annuity due will be
  greater than that of an ordinary annuity of the
  same size and duration. Because the payments
  occur sooner, they are worth more. Therefore,
  press the BGN button on your calculator before
  beginning the problem. Then, your inputs are the
  same. Therefore, the answer would be
• 14,308.39

48
ExamplePresent Value of a Perpetuity

• I
• R V
• Where I is equal to a level income stream
• R is equal to the overall discount rate
• V is the value of the asset

49
Example
If you wanted to endow a Finance scholarship of
1,000 per year forever, given that the
University will always earn 5 on such deposits,
how large a deposit would you have to
make? Solution 1,000/.05 20,000 Therefore
you would have to donate a total of 20,000.
50
Special Applications of Time Value

• Deposits to Accumulate a Future Sum (Sinking
  Fund)
• A common application of time value is to
  determine the annual deposit required to
  accumulate a particular sum of money at a
  particular time in the future.

51
Example

• How much would you have to deposit annually at
  the end of each of the next five years into an
  account paying 8 annual interest in order to
  accumulate 20,000 needed for a down payment on a
  new home?
• Solution 5 n 8 i 20,000 FV
• compute pmt
• Answer3,409.13

52
Loan Amortization

• Loan Amortization
• Definition The determination of the equal annual
  (periodic) loan payments necessary to provide a
  lender with a specified interest return and repay
  the loan principal over a specified period.

53
Example

• How much would your annual end-of-year payments
  have to be on a 12,000 loan with a 15 interest
  rate that must be fully repaid in three years?
  12,000 PV 15 i 3 n
• Compute PMT 5,255.72
• Loan Amortization Schedule - A schedule of equal
  payments to repay a loan. It shows the
  allocation of each loan payment to interest and
  principal.

54
Example

• End Loan Beginning Payments
  Payments End-of-Year
• of Payment of Year Interest
  Principal Principal
• Year Principal .15 x
  (2) (1)-(3) (2)-(4)
• (1) (2) (3)
  (4) (5)
• 1 5,256.2412,000.00 1,800.00 3,456.24
  8,543.76
• 2 5,256.24 8,543.76 1,281.56
  3,974.68 4,569.08
• 3 5,254.24 4,569.08 685.36
  4,569.08 0
• 15,766.92 3,766.92
  12,000.00

55
Interest or Growth Rates Example

• What rate of growth in EPS did the firm
  experience between 1989 and 1994 given the
  following data?
• Year EPS
• 1994 5.80 Treat as FV
• 1993 5.50
• 1992 5.00 Ignore Incremental
• 1991 4.75 Changes
• 1990 4.15
• 1989 3.75 Treat as PV

56
Interest or Growth Rates Example

• From the previous example we would use the 3.75
  earning value as our PV. Second, we would use
  the 5.80 as our future value. The value for n
  is 5 years and we are now solving for i.
• Answer 9.11

57
Another Example
Whats the approximate interest rate being paid
on a 7,000 loan requiring equal annual payments
of 1,700 at the end of each year over six
years? Solution 7,000 PV -1,700
PMT 6 n compute i Answer
11.95
58
Study Items to Contemplate

• 1.   Seven years ago today, you deposited 5,000
  in an account that pays 7 compound annual. If
  you left that money in the account and reinvested
  all the interest payments, how much is that
  account worth today?
• 2.   Your mortgage payment is 1,400 per month.
  It is a 30-year mortgage at a annual rate of
  5.5 compounded monthly. How much did you
  borrow?
• 3.   You plan to borrow 12,000 to purchase a car
  at an interest rate of 9 APR. You will repay
  the loan with even monthly payments for 48
  months. What are your monthly payments?
• 4.   You will receive a single payment of 50,000
  fifteen years from today. The appropriate
  discount rate of is 10, compounded annually.
  Find the present value of the payment.
• 5.   An annuity makes annual payments and its
  present value is 50,000. The annual interest
  rate is 6 APR. The number of years in the
  annuity is 12. Calculate the amount of the
  payments.
• 6.  Your mortgage payment is 1,800 per month.
  It is a 30-year mortgage at 12.0 compounded
  monthly. How much did you borrow?
• 7. The principal amount of a bond that is repaid
  at the end of the loan term is called the bond's
  
• 8. The return on investment you earn when the
  price of an asset you own increases is called?

59
Study Items to Contemplate

• 9. The general method of estimating the value a
  stock is to find the present value of all future
  dividends.
• 10. To find the yield-to-maturity, we find the
  interest rate that equates the price of the bond
  to the present values of its future cash flows.
• 11. A bond that has 10 years remaining to
  maturity and has an annual 5 coupon. It has a
  1,000 face value, and its yield-to-maturity is
  9. What is the market price of the bond?
• 12. The market value of a zero-coupon bond is
  600. It matures in 10-years and has a 1,000
  face value? Interest is annual. What is the
  yield-to-maturity on the bond?
• 13. Omega Industries is considering a project
  that requires an initial investment of 12,000
  and will return 2,000 per year for 7 years.
  What is the payback period of the above project?
• 14. Refer to the previous problem. What is the
  NPV of the above project if the discount rate is
  9.