Bond and Stock Valuations

1
Bond and Stock Valuations

• 1. Definition and Example of a Bond
• 2. How to Value Bonds
• 3. Bond Concepts
• 4. The Present Value of Common Stocks
• 5. Estimates of Parameters in the
  Dividend-Discount Model
• 6. Growth Opportunities
• 7. Price Earnings Ratio
• 8. Stock Market Reporting
• 9. Summary and Conclusions

2
Valuation of Bonds and Stock

• First Principles
• Value of financial securities PV of expected
  future cash flows
• To value bonds and stocks we need to
• Estimate future cash flows
• Size (how much) and
• Timing (when)
• Discount future cash flows at an appropriate
  rate
• The rate should be appropriate to the risk
  presented by the security.

3
Financial Asset Valuation
4
Definition and Example of a Bond

• A bond is a legally binding agreement between a
  borrower and a lender
• Specifies the principal amount of the loan.
• Specifies the size and timing of the cash flows
• In dollar terms (fixed-rate borrowing)
• As a formula (adjustable-rate borrowing)

5
Definition and Example of a Bond

• Consider a U.S. government bond listed as
• 6 3/8 of December 2009.
• The Par Value of the bond is 1,000.
• Coupon payments are made semi-annually (June 30
  and December 31 for this particular bond).
• Since the coupon rate is 6 3/8 the payment is
  31.875.
• On January 1, 2002 the size and timing of cash
  flows are

6
How to Value Bonds

• Identify the size and timing of cash flows.
• Discount at the correct discount rate.
• If you know the price of a bond and the size and
  timing of cash flows, the yield to maturity is
  the discount rate.

7
Pure Discount Bonds

• Information needed for valuing pure discount
  bonds
• Time to maturity (T) Maturity date
• Face value (F)
• Discount rate (R)

Present value of a pure discount bond at time 0
8
Pure Discount Bonds Example

• Find the value of a 30-year zero-coupon bond with
  a 1,000 par value and a YTM of 6.

9
Level-Coupon Bonds

• Information needed to value level-coupon bonds
• Coupon payment dates and time to maturity (T)
• Coupon payment (C) per period and Face value (F)
• Discount rate

Value of a Level-coupon bond PV of coupon
payment annuity PV of face value
10
Level-Coupon Bonds Example

• Find the present value (as of January 1, 2002),
  of a 6-3/8 coupon T-bond with semi-annual
  payments, and a maturity date of December 2009 if
  the YTM is 5-percent.
• On January 1, 2002 the size and timing of cash
  flows are

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The bond consists of a 10-year, 10 annuity of
100/year plus a 1,000 lump sum at t 10
12
What would happen if expected inflation rose by
3, causing R 13?
When R rises, above the coupon rate, the bonds
value falls below par, so it sells at a discount.
13
What would happen if inflation fell, and R
declined to 7?
If coupon rate gt R, price rises above par, and
bond sells at a premium.
14
Example of Bond valuation for different Yields

• Suppose the bond was issued 20 years ago and now
  has 10 years to maturity. What would happen to
  its value over time if the required rate of
  return remained at 10, or at 13, or at 7?

15
Bond Value () vs Years remaining to Maturity
R 7.
1,372
1,211
R 10.
M
1,000
837
R 13.
775
30 25 20 15 10 5 0
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Bond Concepts

• Bond prices and market interest rates move in
  opposite directions.
• 2. When coupon rate YTM, price par value.
• When coupon rate gt YTM, price gt par value
  (premium bond)
• When coupon rate lt YTM, price lt par value
  (discount bond)
• A bond with longer maturity has higher relative
  () price change than one with shorter maturity
  when interest rate (YTM) changes. All other
  features are identical.
• 4. A lower coupon bond has a higher relative
  price change than a higher coupon bond when YTM
  changes. All other features are identical.

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Rate of Return on a Bond
Annual coupon pmt Current price
Current yield Capital gains yield
YTM
Change in price Beginning price
Exp total return
Exp Curr yld
Exp cap gains yld
18
YTM and Bond Value
1400
When the YTM lt coupon, the bond trades at a
premium.
1300
Bond Value
1200
When the YTM coupon, the bond trades at par.
1100
1000
800
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Discount Rate
When the YTM gt coupon, the bond trades at a
discount.
19
Maturity and Bond Price Volatility
Consider two otherwise identical bonds. The
long-maturity bond will have much more volatility
with respect to changes in the discount rate
20
Coupon Rate and Bond Price Volatility
Consider two otherwise identical bonds. The
low-coupon bond will have much more volatility
with respect to changes in the discount rate
21
Semiannual Bonds
1. Multiply years by 2 to get periods
2n. 2. Divide nominal rate by 2 to get periodic
rate R/2. 3. Divide annual INT by 2
to get PMT INT/2.
22
Value of 10-year, 10 coupon, semiannual bond if
rd 13.
23
Callable Bonds and Yield to Call

• A 10-year, 10 semiannual coupon,1,000 par
  value bond is selling for1,135.90 with an 8
  yield to maturity.It can be called after 5 years
  at 1,050.

24
Nominal Yield to Call (YTC)
25
If you bought bonds, would you be more likely to
earn YTM or YTC?

• Coupon rate 10 vs. YTC Rd 7.53. Could
  raise money by selling new bonds which pay 7.53.
• Could thus replace bonds which pay 100/year with
  bonds that pay only 75.30/year.
• Investors should expect a call, hence YTC 7.5,
  not YTM 8.

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Relationship Between YTM, Coupon Rate, and YTC

• In general, if a bond sells at a premium, then
  (1) coupon gt R, so (2) a call is likely.
• So, expect to earn
• YTC on premium bonds.
• YTM on par discount bonds.

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R r IP DRP LP MRP.

• Here
• R Required rate of return on a debt
  security.
• r Real risk-free rate.
• IP Inflation premium.
• DRP Default risk premium.
• LP Liquidity premium.
• MRP Maturity risk premium.

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What is the nominal risk-free rate?

• RRF (1r)(1IP)-1
• r IP (rxIP)
• r IP. (Because rxIP is small)
• RRF Rate on Treasury securities.

29
Bond Spreads, the DRP, and the LP

• A bond spread is often calculated as the
  difference between a corporate bonds yield and a
  Treasury securitys yield of the same maturity.
  Therefore
• Spread DRP LP.
• Bonds of large, strong companies often have very
  small LPs. Bonds of small companies often have
  LPs as high as 2.

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Bond Ratings Provide One Measure of Default Risk
Investment Grade Investment Grade Investment Grade Investment Grade Junk Bonds Junk Bonds Junk Bonds Junk Bonds
Moodys Aaa Aa A Baa Ba B Caa C
SP AAA AA A BBB BB B CCC D
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Bond Ratings and Bond Spreads (YahooFinance, 2006)
Long-term Bonds Yield Spread
U.S. Treasury 5.25
AAA 6.26 1.01
AA 6.42 1.17
A 6.54 1.29
BBB 6.60 1.35
BB 7.80 2.55
B 8.42 3.17
CCC 10.53 5.28
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What factors affect default risk and bond
ratings?

• Financial performance
• Debt ratio
• Coverage ratios, such as interest coverage ratio
  or EBITDA coverage ratio
• Current ratios

(More)
33
What factors affect default risk and bond
ratings?

• Other factors
• Earnings stability
• Regulatory environment
• Potential product liability
• Accounting policies

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What is reinvestment rate risk?

• The risk that CFs will have to be reinvested in
  the future at lower rates, reducing income.
• Illustration Suppose you just won 500,000
  playing the lottery. Youll invest the money and
  live off the interest. You buy a 1-year bond
  with a YTM of 10.

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The Maturity Risk Premium

• Long-term bonds High interest rate risk, low
  reinvestment rate risk.
• Short-term bonds Low interest rate risk, high
  reinvestment rate risk.
• Nothing is riskless!
• Yields on longer term bonds usually are greater
  than on shorter term bonds, so the MRP is more
  affected by interest rate risk than by
  reinvestment rate risk.

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Term Structure Yield Curve

• Term structure of interest rates the
  relationship between interest rates (or yields)
  and maturities.
• A graph of the term structure is called the yield
  curve.

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Hypothetical Treasury Yield Curve
38
Stock Valuations

• Features of common stock
• Determining common stock values
• Efficient markets
• Preferred stock

39
Common Stock Owners, Directors, and Managers

• Represents ownership.
• Ownership implies control.
• Stockholders elect directors.
• Directors hire management.
• Since managers are agents of shareholders,
  their goal should be Maximize stock price.

40
Different Approaches for Valuing Common Stock

• Dividend growth model
• Using the multiples of comparable firms
• Free cash flow method

41
The Present Value of Common Stocks

• Dividends versus Capital Gains
• Valuation of Different Types of Stocks
• Zero Growth
• Constant Growth
• Differential Growth

42
Case 1 Zero Growth

• Assume that dividends will remain at the same
  level forever

• Since future cash flows are constant, the value
  of a zero growth stock is the present value of a
  perpetuity

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If g 0, the dividend stream is a perpetuity.
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Stock Value PV of Dividends
Case 2 What is a constant growth stock?
One whose dividends are expected to grow forever
at a constant rate, g.
45
Case 2For a constant growth stock
D1 D0(1g)1 D2 D0(1g)2 Dt D0(1g)t
If g is constant and less than R, then
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Intrinsic Stock Value D0 2.00, R 13, g
6.
Constant growth model
47
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.

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Case 3 Differential Growth

• Assume that dividends will grow at rate g1 for N
  years and grow at rate g2 thereafter

.
.
.
.
.
.
49
Case 3 Differential Growth

• Dividends will grow at rate g1 for N years and
  grow at rate g2 thereafter

0 1 2


N N1
50
Case 3 Differential Growth
We can value this as the sum of an N-year
annuity growing at rate g1
plus the discounted value of a perpetuity growing
at rate g2 that starts in year N1
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Case 3 Differential Growth
To value a Differential Growth Stock, we can use

• Or we can cash flow it out.

52
A Differential Growth Example

• A common stock just paid a dividend of 2. The
  dividend is expected to grow at 8 for 3 years,
  then it will grow at 4 in perpetuity.
• What is the stock worth?

53
With the Formula
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A Differential Growth Example (continued)

0 1 2 3 4
The constant growth phase beginning in year 4 can
be valued as a growing perpetuity at time 3.
0 1 2 3
55
Supernormal Growth Stock

• Supernormal growth of 30 for 3 years, and then
  long-run constant g 6.
• Can no longer use constant growth model.
• However, growth becomes constant after 3 years.

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Nonconstant growth followed by constant growth
(D0 2)
R13
0
1
2
3
4
g 30
g 30
g 30
g 6
2.60 3.38 4.394 4.6576
2.3009
2.6470
3.0453

4.6576
46.1135
P3
66.5371
0.13 0.06

54.1067 P0
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Estimates of Parameters in the Dividend-Discount
Model

• The value of a firm depends upon its growth rate,
  g, and its discount rate, R.
• Where does g come from?
• Where does R come from?

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Formula for Firms Growth Rate

• g Retention ratio Return on retained earnings

59
Where does R come from?

• The discount rate can be broken into two parts.
• The dividend yield
• The growth rate (in dividends)
• In practice, there is a great deal of estimation
  error involved in estimating R.

60
Price Earnings Ratio

• Many analysts frequently relate earnings per
  share to price.
• The price earnings ratio is a.k.a the multiple
• Calculated as current stock price divided by
  annual EPS
• The Wall Street Journal uses last 4 quarters
  earnings
• Firms whose shares are in fashion sell at high
  multiples. Growth stocks for example.
• Firms whose shares are out of favor sell at low
  multiples. Value stocks for example.

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Other Price Ratio Analysis

• Many analysts frequently relate earnings per
  share to variables other than price, e.g.
• Price/Cash Flow Ratio
• cash flow net income depreciation cash flow
  from operations or operating cash flow
• Price/Sales
• current stock price divided by annual sales per
  share
• Price/Book (a.k.a Market to Book Ratio)
• price divided by book value of equity, which is
  measured as assets - liabilities

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Preferred Stock

• Hybrid security.
• Similar to bonds in that preferred stockholders
  receive a fixed dividend which must be paid
  before dividends can be paid on common stock.
• However, unlike bonds, preferred stock dividends
  can be omitted without fear of pushing the firm
  into bankruptcy.

63
Expected return, given Vps 50 and annual
dividend 5
64
Are volatile stock prices consistent with
rational pricing?

• Small changes in expected g and R cause large
  changes in stock prices.
• As new information arrives, investors continually
  update their estimates of g and R.
• If stock prices arent volatile, then this means
  there isnt a good flow of information.

65
What is market equilibrium?

• In equilibrium, stock prices are stable. There is
  no general tendency for people to buy versus to
  sell.
• The expected price, P, must equal the actual
  price, P. In other words, the fundamental value
  must be the same as the price.

(More)
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Whats the Efficient MarketHypothesis (EMH)?

• Securities are normally in equilibrium and are
  fairly priced. One cannot beat the market
  except through good luck or inside information.

(More)
67
Weak-form EMH

• Cant profit by looking at past trends. A recent
  decline is no reason to think stocks will go up
  (or down) in the future. Evidence supports
  weak-form EMH, but technical analysis is still
  used.

68
Semistrong-form EMH

• All publicly available information is reflected
  in stock prices, so it doesnt pay to pore over
  annual reports looking for undervalued stocks.
  Largely true.

69
Strong-form EMH

• All information, even inside information, is
  embedded in stock prices. Not true--insiders can
  gain by trading on the basis of insider
  information, but thats illegal.

70
Markets are generally efficient because

• 100,000 or so trained analysts--MBAs, CFAs, and
  PhDs--work for firms like Fidelity, Merrill,
  Morgan, and Prudential.
• These analysts have similar access to data and
  megabucks to invest.
• Thus, news is reflected in P0 almost
  instantaneously.

71
Stock Market Reporting
Gap ended trading at 19.25, down 1.75 from
yesterdays close
72
Stock Market Reporting
Gap Incorporated is having a tough year, trading
near their 52-week low. Imagine how you would
feel if within the past year you had paid 52.75
for a share of Gap and now had a share worth
19.25! That 9-cent dividend wouldnt go very far
in making amends. Yesterday, Gap had another
rough day in a rough year. Gap opened the day
down beginning trading at 20.50, which was down
from the previous close of 21.00 19.25
1.75 Looks like cargo pants arent the only
things on sale at Gap.
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Summary and Conclusions

• In this section, we used the time value of money
  formulae from previous discussion to value bonds
  and stocks.
• The value of a zero-coupon bond is
• The value of a perpetuity is

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Summary and Conclusions (continued)

1. The value of a coupon bond is the sum of the PV
   of the annuity of coupon payments plus the PV of
   the par value at maturity.
2. The yield to maturity (YTM) of a bond is that
   single rate that discounts the payments on the
   bond to the purchase price.

75
Summary and Conclusions (continued)

• A stock can be valued by discounting its
  dividends. There are three cases
• Zero growth in dividends
• Constant growth in dividends
• Differential growth in dividends