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Time Value of Money. TVM - Compounding $ Today Future $ Discounting. Future Value (FV). Definition -. FV n = PV(1 + i) n. 1. 2. 0. N. FV = ?. PV=x. Future Value Calculations.

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time value of money
Time Value of Money
  • TVM -

Compounding

$ Today Future $

Discounting

future value fv
Future Value (FV)
  • Definition -

FVn = PV(1 + i)n

1

2

0

N

FV = ?

PV=x

future value calculations
Future Value Calculations
  • Suppose you have $10 million and decide to invest it in a security offering an interest rate of 9.2% per annum for six years. At the end of the six years, what is the value of your investment?
  • What if the (interest) payments were made semi-annually?
  • Why does semi-annual compounding lead to higher returns?
future value of an annuity fva
Future Value of an Annuity (FVA)
  • Definition -

0

1

2

N

A

A

A

FVA = ?

ordinary annuity vs annuity due
Ordinary Annuity vs. Annuity Due

Ordinary Annuity

0

1

2

N

i%

A

A

A

Annuity Due

0

1

2

N

i%

A

A

A

future value of an annuity examples
Future Value of an Annuity Examples
  • Suppose you were to invest $5,000 per year each year for 10 years, at an annual interest rate of 8.5%. After 10 years, how much money would you have?
  • What if this were an annuity due?
  • What if you made payments of $2,500 every six-months instead?
present value pv
Present Value (PV)
  • Definition -

PV = P0 = FV / (1 + i)n

1

2

0

N

FV = x

PV= ?

present value calculations
Present Value Calculations
  • How much would you pay today for an investment that returns $5 million, seven years from today, with no interim cashflows, assuming the yield on the highest yielding alternative project is 10% per annum?
  • What if the opportunity cost was 10% compounded semi-annually?
  • Why does semi-annual compounding lead to lower present values?
present value of an annuity pva
Present Value of an Annuity (PVA)
  • Definition -

0

1

2

N

A

A

A

PVA = ?

present value of an annuity examples
Present Value of an Annuity Examples
  • How much would you spend for an 8 year, $1,000, annual annuity, assuming the discount rate is 9%?
  • What if this were an annuity due?
  • What if you were to receive payments of $500 every six-months instead?
tvm properties
TVM Properties
  • Future Values
      • An increase in the discount rate
      • An increase in the length of time until the CF is received, given a set interest rate,
  • Present Values
      • An increase in the discount rate
      • An increase in the length of time until the CF is received, given a set interest rate,
  • Note: For this class, assume nominal interest rates can’t be negative!
perpetuities
Perpetuities
  • Definition -

0

1

2

$

$

$

PVperpetuity = ?

perpetuity examples
Perpetuity Examples
  • What is the value of a $100 annual perpetuity if the interest rate is 7%?
  • What if the interest rate rises to 9%?
  • Principles of Perpetuities:
uneven cash flow streams
Uneven Cash Flow Streams
  • Description -
  • Ex. Given a discount rate of 8%, how much would you be willing to pay today for an investment which provided the following cash flows:
uneven cash flow streams1
Uneven Cash Flow Streams
  • Ex. Given a discount rate of 8%, what is the future value of the following cash flows stream:
nominal vs effective rates
Nominal vs. Effective Rates
  • Nominal Rate -
  • Effective Rate -
  • What’s the difference?
nom vs eff rate examples
Nom. vs. Eff. Rate Examples
  • Ex. #1: A bond pays 7% interest semi-annually, what is the effective yield on the bond?
  • A credit card charges 1.65% per month (APR=19.8%), what rate of interest are they effectively charging?
  • What nominal rate would produce an effective rate of 9.25% if the security pays interest quarterly?
amortization
Amortization
  • Amortized Loan -
  • Ex. Suppose you borrow $10,000 to start up a small business. The loan offers a contract interest rate of 8.5%, and must be repaid in equal, annual installments over the next 4 years. How much is your annual payment?
  • What percentage of your payments go toward the repayment of principal in each year?
amortization schedules
Amortization Schedules

Year #1, Principal % =

Year #2, Principal % =

Year #3, Principal % =

Year #4, Principal % =

continuous compounding
Continuous Compounding
  • Definition/Description -
does compounding matter
Does Compounding Matter?
  • What is the present value of $200 to be received 2 years from today, if the discount rate is 9% compounded continuously?
      • How much more would the cash flow be worth if the discount rate were 9% compounded annually?
  • What is the future value, in 10 years, of a $5,000 investment today, if the interest rate is 8.75% compounded continuously?
      • How much lower would the future value be if the interest rate were 8.75% compounded annually?
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