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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?


Perpetuities2

Perpetuities


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


Growing perpetuities1

Growing Perpetuities


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 annuity

Ordinary Annuity


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


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