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FINE 3010-01 Financial Management. Instructor: Rogério Mazali Lecture 05: 09/23/2011. FINE 3010-01 Instructor: Rogério Mazali. Chapter 5 : The Time Value of Money. Fundamentals of Corporate Finance Sixth Edition Richard A. Brealey Stewart C. Myers Alan J. Marcus McGraw Hill/Irwin.

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fine 3010 01 financial management

FINE 3010-01Financial Management

Instructor: RogérioMazali

Lecture 05: 09/23/2011

fine 3010 01 instructor rog rio mazali
FINE 3010-01Instructor: RogérioMazali

Chapter 5:

The Time Value of Money

Fundamentals of Corporate Finance

Sixth Edition

Richard A. Brealey

Stewart C. Myers

Alan J. Marcus

McGraw Hill/Irwin

agenda
Agenda
  • Perpetuities
  • Annnuities
    • Ordinary Annuities
    • Delayed Annuities
    • Annuities-Due
  • Effective Annual Interest Rates and Inflation
    • Real vs. Nominal Cash Flows
    • Inflation and Interest Rates
    • Valuing Real Cash Payments
    • Real or Nominal?
perpetuities and annuities
Perpetuities and Annuities
  • Streams of equal cash flows:
    • Home mortgage
    • Car loans
    • Student loans
    • Coupon paying Government Bonds
    • Coupon paying Corporate Bonds
  • Annuity: any sequence of equally spaced, level cash flows
    • Example: fixed-rate mortgage
  • Perpetuity: any sequence of equally spaced, level, everlasting cash flows
    • Example: Consols (British Government Bonds that pay a yearly coupon forever
perpetuities
Perpetuities
  • A perpetuity will pay a constant cash flow CFt = C forever

C

C

C

C

C

C

C

0

1

2

3

4

5

6

7

perpetuities1
Perpetuities
  • How to evaluate the PV of a perpetuity?
perpetuities3
Perpetuities
  • Perpetuity Formula:
  • Example: British consols that promise to pay £100 as interest yearly (Take r = 10% yearly):

PV0 = C/r

delayed perpetuities
Delayed Perpetuities
  • Consider that you work for a company who has just sold your business in the UK to a British company
  • It will take two years to finish the deal
  • You will be paid in British Consol bonds that will pay a total of £3 million in coupons (regular payments).
  • What is the value of the deal today?

£ 3M

£ 3M

£ 3M

£ 3M

£ 3M

0

1

2

3

4

5

6

7

delayed perpetuities1
Delayed Perpetuities
  • We know how to find the value of our bonds when we receive them:
  • Once we have that, we can find the consols value at today:
growing perpetuities
Growing Perpetuities
  • Annual payments grow at a constant rate g
example
Example
  • What is PV if C = $100, r = 10%, and g = 2%?
example1
Example
  • An investment in a growing perpetuity costs $5000, it is expected to pay $200 next year.
  • If the interest is 10%, what is the growth rate of the annual payment?
  • A: we have C = $200, r = 10%, and PV = $5,000; g = ?
  • Note: this formula only works if g < r
annuities
Annuities
  • An annuity is a series of equal payments made at fixed intervals for a specific length of time
    • Ordinary Annuity: payments occur at the end of each period
    • Annuity Due: payments occur at the beginning of each period

C

C

C

C

C

0

1

2

3

4

5

6

7

annuities1
Annuities
  • How to find the PV of an annuity?
  • Consider, for example, a 3-year annuity

C

C

C

0

1

2

3

4

5

6

7

ordinary annuities
Ordinary Annuities
  • Now consider the following strategy:
  • Buy today perpetuity paying C starting at t=1;
  • Issue perpetuity at t = 3 promising to pay C starting at t = 4;
  • Payoffs are:

C

C

C

C

C

C

C

0

1

2

3

4

5

6

7

C

C

C

C

ordinary annuity1
Ordinary Annuity
  • PV of an ordinary annuity paying C dollars every year, for t years:
example 1
Example 1
  • Compute the present value of a 3-year ordinary annuity with payments of $100 at r=10%
example 2
Example 2
  • You agree to lease a car for 4 years at $300 per month, payable at the end of the month. If the discount rate is 0.5% per month, what is the cost of the lease?
delayed annuity
Delayed Annuity
  • The Problem: No payment for 5 years…
  • Then pay 4-year annuity of Example 1

$100

$100

$100

0

1

2

3

4

5

6

7

8

delayed annuity1
Delayed Annuity
  • Step 1: Calculate the PV at time 5 using the following formula
  • Step 2: Determine the PV at time zero:
example 3
Example 3
  • What is the value today of a 10-year annuity that pays $300 a year (at yearend) if the annuity’s first cash flow starts at the end of year 6 and the interest rate is 15% for years 1 through 5 and 10% thereafter?
  • Steps:
    • Get value of annuity at t= 5 (year end)
    • Bring value in step 1 to t=0
annuities due
Annuities Due
  • Annuity and Perpetuity formulas: payment at the end of period
  • What if payments are made in the beginning of the period?
  • Often, cash payments start immediately
  • A level stream of payments starting immediately (beginning of period) is known as annuity due.
annuity due
Annuity Due
  • Annuity-Due PV formula:
future value of an annuity
Future Value of an Annuity
  • Example: if you save $3,000 a year, at 8% interest rate, how much you would have at the end of 4 years?
future value of an annuity1
Future Value of an Annuity
  • With many cash flows, calculation can be hard
  • However, cash flows are the same as annuities’.
future value of an annuity2
Future Value of an Annuity
  • Future Value of an Annuity paying C dollars for t years:
  • Future Value of an Annuity-Due paying C dollars for t years:
inflation and the time value of money
Inflation and the Time Value of Money
  • Inflati0n erodes the purchase power of money
  • So far we have computed PVs and FVs disregarding this issue
  • Inflation: GENERAL increase in prices, effect of money’s loss of value
  • Measure of Inflation: Consumer Price Index (CPI)
inflation and the time of money
Inflation and the Time of Money
  • Nominal vs. Real Values
    • Nominal Values: actual numbers of dollars of the day
    • Real Values: amount of purchasing power; stated in number of dollars of reference period
  • Example: 6% interest rate and 6% inflation rate => you gain NOTHING!
  • Approximation commonly used:
inflation and the time value of money1
Inflation and the Time Value of Money
  • Discounting Cash Flows: $100 to be received 1 year from today when annual interest rate is 10%:
  • Discounting $100 to be received 1 year from today when real interest rate is 2.8% and inflation is expected to be 7%.
  • Note:
    • NOMINAL cash flows discounted using NOMINAL interest rates
    • REAL cash flows must be discounted using REAL interest rates