McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
CHAPTER
5 How to Value Bonds and Stocks
Slide 2
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Chapter Outline5.1 Definitions and Example of a Bond5.2 How to Value Bonds5.3 Bond Concepts5.4 The Present Value of Common Stocks5.5 Estimates of Parameters in the Dividend-
Discount Model5.6 Growth Opportunities5.7 The Dividend Growth Model and the
NPVGO Model5.8 Price-Earnings Ratio
Slide 3
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.1 Definition of a Bond
• A bond is a legally binding agreement between a borrower and a lender that specifies the:– Par (face) value– Coupon rate– Coupon payment– Maturity Date
• The yield to maturity (YTM) is the required market interest rate on the bond.
Slide 4
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.2 How to Value Bonds
• Primary Principle:– Value of financial securities = PV of expected
future cash flows
• Bond value is, therefore, determined by the present value of the coupon payments and par value.
• Interest rates are inversely related to present (i.e., bond) values.
Slide 5
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
The Bond Pricing Equation
T
T
)(1
FV
RR)(1
1-1
C Value BondR
Slide 6
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Pure Discount Bonds
• Make no periodic interest payments (coupon rate = 0%)
• The entire yield to maturity comes from the difference between the purchase price and the par value.
• Cannot sell for more than par value• Sometimes called zeroes, deep discount bonds,
or original issue discount bonds (OIDs)
Slide 7
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Pure Discount BondsInformation needed for valuing pure discount bonds:
– Time to maturity (T) = Maturity date - today’s date– Face value (F)– Discount rate (r)
TR
FVPV
)1(
Present value of a pure discount bond at time 0:
0
0$
1
0$
2
0$
1T
F$
T
Slide 8
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Pure Discount Bond: Example
Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.
11.174$)06.1(
000,1$
)1( 30
TR
FVPV
0
0$
1
0$
2
0$
29
000,1$
30
0
0$
1
0$
2
0$
29
000,1$
30
Slide 9
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Level Coupon Bonds
• Make periodic coupon payments in addition to the maturity value
• The payments are equal each period. Therefore, the bond is just a combination of an annuity and a terminal (maturity) value.
• Coupon payments are typically semiannual.• Effective annual rate (EAR) =
(1 + R/m)m – 1
Slide 10
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Level Coupon Bond: Example
Slide 11
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Consols
• Not all bonds have a final maturity.• British consols pay a set amount (i.e., coupon)
every period forever.• These are examples of a perpetuity.
R
CPV
Slide 12
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.3 Bond Concepts
Bond prices and market interest rates move in opposite directions.
When coupon rate = YTM, price = par value
When coupon rate > YTM, price > par value (premium bond)
When coupon rate < YTM, price < par value (discount bond)
Slide 13
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
YTM and Bond Value
800
1000
1100
1200
1300
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Discount Rate
Bon
d V
alu
e
6 3/8
When the YTM < coupon, the bond trades at a premium.
When the YTM = coupon, the bond trades at par.
When the YTM > coupon, the bond trades at a discount.
Slide 14
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Bond Example Revisited (p.133)
Slide 15
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Computing Yield to Maturity (p.134)
• Yield to maturity is the rate implied by the current bond price.
Slide 16
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.4 The Present Value of Common Stocks
• The value of any asset is the present value of its expected future cash flows.
• Stock ownership produces cash flows from: – Dividends – Capital Gains
• Valuation of Different Types of Stocks– Zero Growth– Constant Growth– Differential Growth
Slide 17
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Case 1: Zero Growth
• Assume that dividends will remain at the same level forever
RP
RRRP
Div
)1(
Div
)1(
Div
)1(
Div
0
33
22
11
0
321 DivDivDiv Since future cash flows are constant, the value of a zero
growth stock is the present value of a perpetuity:
Slide 18
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Case 2: Constant Growth
)1(DivDiv 01 g
Since future cash flows grow at a constant rate forever, the value of a constant growth stock is the present value of a growing perpetuity:
gRP
1
0
Div
Assume that dividends will grow at a constant rate, g, forever, i.e.,
2012 )1(Div)1(DivDiv gg
3023 )1(Div)1(DivDiv gg
.
..
Slide 19
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Case 3: Differential Growth• Assume that dividends will grow at different
rates in the foreseeable future and then will grow at a constant rate thereafter.
• To value a Differential Growth Stock, we need to:– Estimate future dividends in the foreseeable future.– Estimate the future stock price when the stock
becomes a Constant Growth Stock (case 2).– Compute the total present value of the estimated
future dividends and future stock price at the appropriate discount rate.
Slide 20
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.5 Estimates of Parameters• The value of a firm depends upon its growth rate,
g, and its discount rate, R. – Where does g come from?
g = Retention ratio × Return on retained earnings (ROE)
Slide 21
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Where does R come from?
• Start with the DGM:
gP
D g
P
g)1(D R
g-R
D
g - R
g)1(DP
0
1
0
0
100
Rearrange and solve for R:
Slide 22
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Where does R come from?
R = Div1/P0 + g
The discount rate can be broken into two parts. – The dividend yield – The growth rate (in dividends) or capital gain
yield
In practice, there is a great deal of estimation error involved in estimating R.
Slide 23
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
• A Healthy Sense of Skepticism – Estimate of g is based on a number of
assumptions:• return on reinvestment• future retention ratio
– Some financial economists suggest calculating the average R for an entire industry.
Slide 24
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Two polar cases
• Case1: A firm paying no dividend, and going from no dividends to a positive number of dividends
• Case2: An analyst whose estimate of g for a particular firm is equal to or above R must have made a mistake.
Slide 25
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.6 Growth Opportunities
• Imagine a company with a level stream of earnings per share in perpetuity
• The company pays off these earnings out to stockholders as dividends. Hence.
EPS = Div (Cash cow)
• It’s value equals
EPS/r = Div/r
Slide 26
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
• Growth opportunities are opportunities to invest in positive NPV projects.
• Suppose the firm retains the entire dividend at date 1 in order to invest in a particular capital budgeting project
• Stock Price after Firm Commits to the New Project: EPS/r + NPVGO
Slide 27
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Example
EPS/r + NPVGO
= 【〔 $1,000,000/1.1 〕 + 〔 $1,000,000/(1.1)2 〕 +…+ 〔 $1,000,000/(1.1)n 〕】 + 【〔 -$1,000,000+($210,000/0.1) 〕 /1.1 】
= 〔 $1,000,000/0.1 〕 + 〔 -$1,000,000+ ($210,000/0.1) 〕 /1.1= $10,000,000+$1,000,000= $11,000,000The price per share:
$11,000,000/100,000 = $110
Slide 28
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
• Two conditions must be met in order to increase value: – Earnings must be retained so that projects
can be funded. – The projects must have positive net present
value.
Slide 29
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Growth in Earnings and Dividends versus Growth Opportunities
• A policy of investing in projects with negative NPVs rather than paying out earnings as dividends will lead to growth in dividends and earnings, but will reduce value.
Slide 30
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Dividends or Earnings: Which to discount?
• Dividends, or would ignore the investment that a firm must make today in order to generate future returns. (only a portion of earnings goes to the stockholders as dividends)
Slide 31
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
The No-Dividend Firm
• A firm with many growth opportunities faces two choices: pays out dividends now, or forgoes dividends now and makes investments.
• The actual application of the dividend discount model is difficult for firms of this type.
Slide 32
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.7 The Dividend Growth Model and the NPVGO Model
• A steady growth in dividends results from a continual investment in growth opportunities, not just in a single opportunity.
Slide 33
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.7 The Dividend Growth Model and the NPVGO Model
• We have two ways to value a stock:– The dividend discount model– The sum of its price as a “cash cow” plus the
per share value of its growth opportunities
Slide 34
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Example
• C has EPS of $10 at the end of the first year, a dividend-payout ratio of 40%, a discount rate of 16%, and a return on its retained earnings of 20%.
Slide 35
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Solution
• The Dividend-Growth Model
P0 = Div1/(R-g)
• The NPVGO Model
P0 = EPS/R + NPVGO (5.10)
Slide 36
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
The Dividend-Growth Model
• Div1/(R-g) = $4/(.16-.12)
=$100
Slide 37
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
Solution• The NPVGO Model
– Value Per Share of a Single Growth Opportunities -$6 + $1.20/0.16 = $1.5
– Value Per Share of All Opportunities NPVGO = $1.50/(0.16-0.12)=$37.50
– Value Per Share if Firm Is a Cash Cow Div/r = $10/0.16 = $62.50
– Summation P0 = EPS/r + NPVGO = $62.5 + $37.5 = $100.0
Slide 38
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.8 Price-Earnings Ratio
• Many analysts frequently relate earnings per share to price.
• The price-earnings ratio is calculated as the current stock price divided by annual EPS.
Slide 39
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw-Hill/Irwin
5.8 Price-Earnings Ratio (P/E ratio)
• P (Price per share) = EPS/r + NPVGO
• P/E = 1/r + NPVGO / EPS
• The market is merely pricing perceptions of the future, not the future itself.
• It implies that P/E ratio is a function of growth opportunities, risk, and the choice of accounting methods.