The Time Value of Money

1
The Time Value of Money

• Single Cash Payments

2
Future Value

• You have 100 to invest in a bank account. Banks
  are paying an interest rate of 6 per cent per
  year on deposits. After a year, your account will
  earn interest of 6
• Interest interest rate x initial
  investment
• .06 x 100 6.

3
Future Value

• Value of investment after 1 year
• 1006
• 106.
• Value after 1 year 100 (100 x .06)
• 100 x (1.06)
• initial investment x (1r)
• r interest rate

4
Future Value

• What if you leave your money in the bank for a
  second year?
• Interest in year 2 .06 x 106 6.36
• Value after year 2 106 6.36 112.36
• 100 x (1.06) x (1.06)
• 100 x (1.06)2 112.36

5
Simple Vs. Compound Interest

• Compound Interest
• Interest earned on interest
• Simple Interest
• Interest earned only on the original investment
  no interest is earned on interest

6
Compound Interest
7
Future Value

• Amount to which an investment will grow after
  earning interest
• Future value at the end of n periods Initial
  investment x (1r)n
• The expression (1 r)n is the future value
  interest factor or compounding factor.

8
Future Value of 1 for different periods and rates
9
Example Manhattan Island

• Peter Minuit bought the Manhattan island for 24
  in 1626. Based on new York real estate prices
  today, it seems that Minuit got a deal. But
  consider the future value of that 24 if it had
  been invested for 377 years (2003 minus 1626) at
  an interest rate of 8 per year
• 24 x (1.08)377 95,712,000,000,000
• 95.712 trillion

10
Example Manhattan Island

• Is assuming that the interest rate was 8
  throughout these 375 years realistic? What if it
  was 3.5?
• 24 x (1.035)377 10,297,294

11
Example

• Deposit 5,000 today in an account paying 12
  interest. How much will you have in 6 years? How
  much is simple interest? How much is compound
  interest?

12
Present Value

• Value today of a future cash flow
• Discount rate
• Interest rate used to compute present values of
  future cash flows

13
Present Value of 1 for Different Periods and
Rates
14
Present Value and Discounting
In general, the present value of 1 to be
received in n years when the interest rate is r
is
Present Value Interest Factor Present Value
Factor Discount Factor
15
Present Value

• Suppose you need 20,000 in three years to pay
  for your college tuition. You can earn 8 per
  year on your money in the bank. How much money
  should you set aside now?

16
Present Value Vs. Future Value

• Present value interest factor is just the
  reciprocal of the future value interest factor.
• FVn PV x (1 r)n
• PV FVn / (1 r)n
• With any three of PV, FV, r, or n known, you can
  solve for the fourth one.

17
Finding the Value of Free Credit

• Kangaroo Autos is offering free credit on a
  10,000 car. You pay 4,000 down and then the
  balance at the end of 2 years. Turtle Motors next
  door does not offer free credit but will give you
  500 off the list price. If the interest rate is
  10, which company is offering the better deal?

18
Finding r

• Suppose you deposit 5000 today in an account
  paying r percent per year. If you will get
  10,000 in 10 years, what rate of return are you
  being offered?

19
Finding r

• Assume the total cost of a college education will
  be 75,000 when your child enters college in 18
  years. You have 7,000 to invest. What rate of
  interest must you earn on your investment to
  cover the cost of your childs education?
• Present value 7,000
• Future value 75,000
• n 18 r ?