The Time Value of Money

1
The Time Value of Money

• Time Value of Money Concept
• Future and Present Values of single payments
• Future and Present values of periodic payments
  (Annuities)
• Present value of perpetuity
• Future and Present values of annuity due
• Annual Percentage Yield (APY)

2
The Time Value of Money Concept

• We know that receiving 1 today is worth more
  than 1 in the future. This is due to
  opportunity costs
• The opportunity cost of receiving 1 in the
  future is the interest we could have earned if we
  had received the 1 sooner

3
The Future Value

• Future Value equation

4
Future Value Single Sums

• If you deposit 100 in an account earning 6, how
  much would you have in the account after 1 year?
• Mathematical Solution
• FVn 1 x (1 i)n
• FVn 100 x (1 0.06)1
• FVn 106

5
Future Value Single Sums (Continued)
Calculator Solution (TI BA II PLUS) Calculator Solution (TI BA II PLUS)
N Number of periods
I/Y Interest per Year
P/Y Payment per Year
C/Y Compounding per Year
PV Present Value
PMT PayMenT (Periodic and Fixed)
FV Future Value
MODE END for ending and BGN for beginning
N I/Y P/Y PV PMT FV MODE
1 6 1 -100 0 106
6
Future Value Single Sums (Continued)

• If you deposit 100 in an account earning 6, how
  much would you have in the account after 5 years?
• Mathematical Solution
• FVn 1 x (1 i)n
• FVn 100 x (1 0.06)5
• FVn 133.82

N I/Y P/Y PV PMT FV MODE
5 6 1 -100 0 133.82
7
Future Value Single Sums (Continued)

• If you deposit 100 in an account earning 6 with
  quarterly compounding, how much would you have in
  the account after 5 year?
• Mathematical Solution
• FVn 1 x (1 i)n
• FVn 100 x (1 0.06/4)5x4
• FVn 134.69

N I/Y P/Y PV PMT FV MODE
20 6 4 -100 0 134.69
8
Future Value Single Sums (Continued)

• If you deposit 100 in an account earning 6 with
  monthly compounding, how much would you have in
  the account after 5 year?
• Mathematical Solution
• FVn 1 x (1 i)n
• FVn 100 x (1 0.06/12)5x12
• FVn 134.89

N I/Y P/Y PV PMT FV MODE
60 6 12 -100 0 134.89
9
Future Value Single Sums (Continued)

• If you deposit 1,000 in an account earning 8
  with daily compounding, how much would you have
  in the account after 100 year?
• Mathematical Solution
• FVn 1 x (1 i)n
• FVn 1,000 x (1 0.08/365)100x365
• FVn 2,978,346.07

N I/Y P/Y PV PMT FV MODE
36,500 8 365 -1000 0 2,978,346.07
10
The Present Value

• Present Value equation

11
Present Value Single Sums (Continued)

• If you receive 100 one year from now, what is
  the PV of that 100 if your opportunity cost is
  6?
• Mathematical Solution
• PV0 1 / (1 i)n
• PV0 100 / (1 0.06)1
• PV0 -94.34

N I/Y P/Y PV PMT FV MODE
1 6 1 -94.37 0 100
12
Present Value Single Sums (Continued)

• If you receive 100 five year from now, what is
  the PV of that 100 if your opportunity cost is
  6?
• Mathematical Solution
• PV0 1 / (1 i)n
• PV0 100 / (1 0.06)5
• PV0 -74.73

N I/Y P/Y PV PMT FV MODE
5 6 1 -74.73 0 100
13
Present Value Single Sums (Continued)

• If you sold land for 11,933 that you bought 5
  years ago for 5,000, what is your annual rate of
  return?
• Mathematical Solution

N I/Y P/Y PV PMT FV MODE
5 19 1 -5,000 0 11,933
14
Present Value Single Sums (Continued)

• Suppose you placed 100 in an account that pays
  9.6 interest, compounded monthly. How long will
  it take for your account to grow to 500?
• Mathematical Solution

N I/Y P/Y PV PMT FV MODE
202 9.6 12 -100 0 500
15
Hint for Single Sum Problems

• In every single sum future value and present
  value problem, there are 4 variables
• FV, PV, i, and n
• When doing problems, you will be given 3 of these
  variables and asked to solve for the 4th variable
• Keeping this in mind makes time value problems
  much easier!

16
Compounding and Discounting Cash Flow Streams

• Annuity a sequence of equal cash flows,
  occurring at the end of each period
• If you buy a bond, you will receive equal
  semi-annual coupon interest payments over the
  life of the bond
• If you borrow money to buy a house or a car, you
  will pay a stream of equal payments

17
Future Value Annuity

• If you invest 1,000 each year at 8, how much
  would you have after 3 years?
• Mathematical Solution

N I/Y P/Y PV PMT FV MODE
3 8 1 0 -1000 3,246.40
18
Present Value Annuity

• What is the PV of 1,000 at the end of each of
  the next 3 years, if the opportunity cost is 8?
• Mathematical Solution

N I/Y P/Y PV PMT FV MODE
3 8 1 2,577.10 -1000 0
19
Perpetuities

• Suppose you will receive a fixed payment every
  period (month, year, etc.) forever. This is an
  example of a perpetuity
• You can think of a perpetuity as an annuity that
  goes on

20
Perpetuities (Continued)
21
Perpetuities (Continued)

• What should you be willing to pay in order to
  receive 10,000 annually forever, if you require
  8 per year on the investment?
• PV 10,000 / 0.08 125,000

22
Future Value Annuity Due Annuity Due The cash
flows occur at the beginning of each year, rather
than at the end of each year

• If you invest 1,000 at the beginning of each of
  the next 3 years at 8, how much would you have
  at the end of year 3?
• Mathematical Solution

N I/Y P/Y PV PMT FV MODE
3 8 1 0 -1000 3,506.11 BEGIN
23
Present Value Annuity Due Annuity Due The cash
flows occur at the beginning of each year, rather
than at the end of each year

• What is the PV of 1,000 at the beginning of each
  of the next 3 years, if your opportunity cost is
  8?
• Mathematical Solution

N I/Y P/Y PV PMT FV MODE
3 8 1 2,783.26 -1000 0 BEGIN
24
Uneven Cash FlowsHow do we find the PV of a cash
flow stream when all of the cash flows are
different? (Use a 10 discount rate)
Period CF PVCF
0 -10,000 -10,000.00
1 2,000 1,818.15
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
Total Total 4,412.95
25
Uneven Cash Flows
26
Annual Percentage Yield (APY) or Effective
Annual Rate (EAR)

• Which is the better loan
• 8.00 compounded annually, or
• 7.85 compounded quarterly?
• We cant compare these nominal (quoted) interest
  rates, because they dont include the same number
  of compounding periods per year!
• We need to calculate the APY

27
Annual Percentage Yield (APY) or Effective
Annual Rate (EAR) (Continued)

• Find the APY for the quarterly (m 4) loan
• The quarterly loan is more expensive than the 8
  loan with annual compounding!

28
Annual Percentage Yield (APY) or Effective
Annual Rate (EAR) (Continued)

• 2nd ICONV
• NOM 7.85 ENTER (EFF)
• C/Y 4 ENTER (EFF)
• CPT 8.08