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FINE 3010-01 Financial Management

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FINE 3010-01Financial Management

Instructor: RogérioMazali

Lecture 05: 09/23/2011

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

- 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?

- 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

- A perpetuity will pay a constant cash flow CFt = C forever

C

C

C

C

C

C

C

…

0

1

2

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4

5

6

7

- How to evaluate the PV of a perpetuity?

- Perpetuity Formula:
- Example: British consols that promise to pay £100 as interest yearly (Take r = 10% yearly):

PV0 = C/r

- 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

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7

- 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:

- Annual payments grow at a constant rate g

- What is PV if C = $100, r = 10%, and g = 2%?

- 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

- 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

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7

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

C

C

C

0

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3

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6

7

- 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

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5

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7

C

C

C

C

- PV of an ordinary annuity paying C dollars every year, for t years:

- Compute the present value of a 3-year ordinary annuity with payments of $100 at r=10%

- 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?

- The Problem: No payment for 5 years…
- Then pay 4-year annuity of Example 1

$100

$100

$100

0

1

2

3

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5

6

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8

- Step 1: Calculate the PV at time 5 using the following formula
- Step 2: Determine the PV at time zero:

- 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

- 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 PV formula:

- Example: if you save $3,000 a year, at 8% interest rate, how much you would have at the end of 4 years?

- With many cash flows, calculation can be hard
- However, cash flows are the same as annuities’.

- Future Value of an Annuity paying C dollars for t years:
- Future Value of an Annuity-Due paying C dollars for t years:

- 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)

- 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:

- 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