Chapter 8, 9 and Bond Valuation

Konan Chan Corporate Finance, 2018

Bond Valuation Bond Valuation

 Bond pricing model  Annual vs. semi-annual coupon bonds  -to-maturity (YTM)  Credit ratings

Corporate Finance Konan Chan 3

Bond Cash Flows  Annual coupons coupon coupon+par 0 1 2 ……. T

 Value today = PV of expected cash flows

Corporate Finance Konan Chan 4 Bond Characteristics

 Five basic variables  FV : par value (or face value) - usually $1000 to be paid at maturity  PMT : annual coupon = par value*coupon rate (paid periodically to bondholder)  T : years to maturity  r : required rate of return (discount rate)  PV : PV of future cash flows (value today)

Corporate Finance Konan Chan 5

Bond Pricing

Example What is the price of a 6.5 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 3.9%.

Corporate Finance Konan Chan 6 Bond Price and Interest Rate

 There is a negative relationship between bond price and interest rate (discount rate)  If discount rate is higher (lower) than coupon rate, bond prices should be less (more) than par value

Corporate Finance Konan Chan 7

Bond Price Over Time

1,080 Price path for 1,060 Premium Bond

1,040 1,020 Both premium and discount bonds 1,000 approach face value as their

980 maturity date approaches

Bond Price 960

940

920 Price path for Today Maturity 900 Discount Bond

880 0 5 10 15 20 25 30 Time to Maturity

Corporate Finance Konan Chan 8 Bond Cash Flows  Annual coupons coupon coupon+par 0 1 2 ……. T  Semi-annual coupons coupon/2 coupon/2+par 0 1 2 ……. 2T  Value today = PV of expected cash flows

Corporate Finance Konan Chan 9

Bond Pricing

Example (continued) What is the price of the bond if the required rate of return is 3.9% and the coupons are paid semi- annually?

Corporate Finance Konan Chan 10 Bond Yields

 Current Yield

 annual coupon payments divided by bond price.  Yield To Maturity (YTM)

 interest rate for which the present value of bond equals the market price

 total annual if you buy the bond today and hold to the maturity date

Corporate Finance Konan Chan 11

Corporate Bond

 Corporate bond quotation (on Sep 2005) Company Coupon Maturity Last price Last yield Wal-mart 7.55 Sep 30, 2031 125.314 5.675  How to compute yield to maturity?

 Bond price = 1253.14

 Annual coupon = 7.55%*1000 = 75.5

 N = 2*26 = 52, PMT = 75.5/2 = 37.75, FV = 1000

 The YTM to meet the current price is 5.676%

Corporate Finance Konan Chan 12 Yield To Maturity

 In Excel, RATE(52,37.75,-1253.14,1000,0)

Corporate Finance Konan Chan 13

Clean versus Dirty Prices

 Clean price: quoted price  Dirty price: price actually paid = quoted price plus accrued interest  Example: Buy a T-bond with annual coupon 8%  Ask quote is 132.24 (i.e (132+24/32)% of face value)  Number of days since last coupon = 61  Number of days in the coupon period = 184  Accrued interest = (61/184)(8%*1,000/2) = 13.26  Prices:  Clean price = 1,327.50  Dirty price = 1,327.50 + 13.26 = 1,340.76  So, you would actually pay $1,340.76 for the bond.

Corporate Finance Konan Chan 17 Interest Rate Risk

 Measures bond price sensitivity to changes in interest rates  All things equal, -term bonds have more interest rate risk than -term bonds.  All things equal, low coupon bonds also have more interest rate risk than high coupon bonds

Corporate Finance Konan Chan 15

Interest Rate Risk Example

 Let’s compare two bonds with everything the same except the time-to-maturity (1 vs. 30 years)

 PVs of 10% annual coupons with r at 5%, 10%, 15%, 20%.

Corporate Finance Konan Chan 16 Bond Price Sensitivity

When the interest rate equals the 10% coupon rate, both bonds sell at face value

Corporate Finance Konan Chan 17

Credit Rating (default risk)

 Credit ratings proxy for default risk, the risk that bond issuer may default on its obligations  Default premium: difference between corporate bond yield and T-bond yield (assume same coupon, maturity)  Bonds are generally classified into two groups

grade bonds: BBB and above

 Junk (speculative grade) bonds: below BBB  Investment grade bonds are generally legal for purchase by banks; junk bonds are not

Corporate Finance Konan Chan 18 Credit Rating

Corporate Finance Konan Chan 19

Credit Rating and Yield, 2011

Price, % of Yield to Issuer Coupon Maturity S&P Rating Face Value Maturity Johnson & Johnson 5.15% 2017 AAA 122.88% 1.27% Walmart 5.38 2017 AA 117.99 1.74 Walt Disney 5.88 2017 A 121.00 2.07 Suntrust Banks 7.13 2017 BBB 109.76 4.04 U.S. Steel 6.05 2017 BB 97.80 6.54 American Stores 7.90 2017 B 97.50 8.49 Caesars Entertainment 5.75 2017 CCC 41.95 25.70

Corporate Finance Konan Chan 20 Yield Spread

Corporate Finance Konan Chan 21

Government Bonds

 Treasury Securities

 Issued by federal government

 Examples: T-bills, T-notes, T-bonds

 No default risk  Municipal Securities (munis)

 Issued by state or local governments

 Varying degrees of default risk, rated similar to corporate debt

 Coupons are tax-exempt at the federal level

Corporate Finance Konan Chan 22 Inflation

 Inflation

 Rate at which prices as a whole are increasing.  Nominal Interest Rate

 Rate at which money invested grows.  Real Interest Rate

 Rate at which the purchasing power of an investment increases.

Corporate Finance Konan Chan 23

Fisher Effect (Inflation)

Approximation formula

Corporate Finance Konan Chan 24 Corporate Bond Yield Factors

 Real interest  Inflation  Interest rate risk  Default risk premium – bond ratings  Taxability premium – municipal versus taxable  Liquidity premium – bonds with more trading have lower yield  Anything else that affects the risk of the cash flows to the bondholders, will affect the bond yield

Corporate Finance Konan Chan 25

Stock Valuation

discount model  Constant growth  Zero growth  Non-constant growth  Expected stock return  Multiple valuation

Corporate Finance Konan Chan 27

Stock Valuation

(DDM)

 discount future back to present where T is time horizon for your investment

Corporate Finance Konan Chan 28 Stock Valuation

 We will assume fall into 3 categories

 Constant growth rate in dividends

 Zero growth rate in dividends

 “Supernormal” (non-constant) growth rate in dividends

Corporate Finance Konan Chan 29

Constant Growth DDM

 A dividend discount model where dividends are assume to grow at a constant rate forever  Given any combination of variables in the equation, you can solve for the unknown variable.

 D0: dividend just paid (the most recent dividend)  g: constant growth rate of dividends

 r: required rate of return for stock

Corporate Finance Konan Chan 30 Constant Growth DDM

 D1 = D0 (1 + g) 2  D2 = D1 (1 + g) = D0 (1 + g) 

 Using geometric series formula

Corporate Finance Konan Chan 31

Constant Growth DDM - example

 What is the value of a stock that expects to pay a $3.00 dividend next year, and then increase the dividend at a rate of 8% per year, indefinitely? Assume a 12% expected return

Corporate Finance Konan Chan 32 Same example

 If the same stock is selling for $100 in the stock market, what might the market assume about the growth in dividends? 

 The market assumes the dividend will grow at 9% per year, indefinitely.

Corporate Finance Konan Chan 33

Zero Growth DDM

 If we forecast no growth for the stock (i.e., dividends keep constant forever), the stock will become a perpetuity

 This is exactly the valuation for preferred stocks

Corporate Finance Konan Chan 34

 Stated dividend that must be paid before dividends can be paid to common stockholders  Dividends are not a liability of the firm and preferred dividends can be deferred indefinitely  Most preferred dividends are cumulative – any missed preferred dividends have to be paid before common dividends can be paid  Preferred stock generally does not carry voting rights

Corporate Finance Konan Chan 35

What if CGDDM Doesn’t Apply? 

 Any restriction on constant growth DDM?  What does it mean? How to deal with it if this restriction exists?  Two-stage or multiple stage of growth

Corporate Finance Konan Chan 36 Non-constant Growth Model

 Two stages of growth

 assume stock has a period of non-constant growth in dividend, and then eventually settles into a normal constant growth pattern

 Generally, high growth in the first stage, then low growth stage in the second stage

 Young, start-up firms, or technology firms with new product will have high growth rates  Multiple stages if necessary

Corporate Finance Konan Chan 37

Non-constant Growth - Example

 The growth for firm A.net is expected to be 20% for next two years, and 6% thereafter. The current dividend is $1.60, and the firm’s required rate of return is 10%. What’s stock worth today?  g1 = 20% g1 = 20% g = 6%

Step 1 D1=$1.6(1.2)=$1.92 D2=$1.92(1.2)=$2.304

Step 2

Step 3

Corporate Finance Konan Chan 38 Sustainable Growth Rate

 Payout ratio

 Fraction of earnings paid out as dividends  Plowback (retention) ratio = 1 - payout ratio

 Fraction of earnings retained by the firm  g = return on equity (ROE) * retention ratio

 Steady rate at which a firm can grow  This estimation of growth rate applies to stable firms only

Corporate Finance Konan Chan 39

Estimate Expected Return

 Given constant growth dividend discount model, we can estimate stock return 

 Expected return = expected + growth rate  Previous example: r = $3/$75 + 8% = 12%

Corporate Finance Konan Chan 40 Components of Expected Return  Expected Return

 r = total income/ purchase price

 r = [dividend income + capital gain (or loss)]/price

 r = expected dividend yield + capital gain yield

= D1/ P0 + (P1 –P0) / P0

Corporate Finance Konan Chan 41

Growth in Constant Growth DDM

 P1 / P0 = 1 + g (i.e., the firm will grow constantly)

Corporate Finance Konan Chan 42 Chapter 10, 11 Risk and Return

Konan Chan Corporate Finance, 2018

Risk and Return

 Return measures  Expected return and risk?  Portfolio risk and diversification  CAPM (Capital Asset Pricing Model)  Calculating Return - Single period

 Holding period return (HPR)

 This assumes we only have one investment period. What about multiple periods?

Corporate Finance Konan Chan 45

Calculating Return - Multi periods

 Arithmetic average

 Arithmetic mean of returns

 Good measure for future performance  Geometric average

 Geometric mean of returns

 The return measure that gives the same cumulative performance as actual returns (buy- and-hold)

 Required for mutual fund performance

Corporate Finance Konan Chan 46 Returns -Example

 Arithmetic Average: (0.14 -0.1455 + 0.10) / 3 = 3.15% 3  Geometric Average: (1 + RG) = (1 + 0.14)*(1 - 0.1455)*(1 + 0.10) 1/3  RG = [(1 + 0.14)*(1 - 0.1455)*(1 + 0.10)] – 1=2.33%

 RG  RA , RG is a better measure for past performance

Corporate Finance Konan Chan 47

Uncertainty of Investment

 Return and risk tradeoff (every investment has its uncertainty)  At the time when we measure the expected level of returns, we need to quantify the uncertainty (risk)  How to estimate the expected return and risk?

 based on probability distribution

 based on historical data

Corporate Finance Konan Chan 48 The Normal Distribution

Corporate Finance Konan Chan 49

Expected Return & Risk

 Expected Return (Mean)  Find out possible future states  Estimate probability and outcome for each state  Sum of all possible outcomes by multiplying probabilities

 Risk (Variance or Standard deviation)  The degree of various outcomes, or deviation from mean

 Standard deviation is the square root of variance

Corporate Finance Konan Chan 50 Expected Return & Risk - Example  Initial investment : $100

• Expected return = 0.3(0.5)+0.5(0.2)+0.2(-0.4)=17% • Variance = 0.3(0.5-0.17)2+0.5(0.2-0.17) 2 +0.2(-0.4-0.17)2 = 0.0981 • Standard deviation = (0.0981)0.5 = 0.313

Corporate Finance Konan Chan 51

Return & Risk - Historical data

 Treat each historical outcome equally and assign a probability of 1/n ( n is number of observations)  Return

 Use sample average

 Risk

 Use sample variance

Corporate Finance Konan Chan 52 Return & Risk -Historical data (example)  Using Excel functions

Corporate Finance Konan Chan 53

Return and Risk – Two Assets

State of Probability Stock Bond economy Boom 25% 80% 5%

Normal 60% 30% 10%

Recession 15% -30% 15%

Corporate Finance Konan Chan 54 Return and Risk - Example

 rS = 0.25*0.8+0.6*0.3+0.15*(-0.3) = 0.335

 rB = 0.25*0.05+0.6*0.1+0.15*0.15 = 0.095  Standard deviation

Corporate Finance Konan Chan 55

Portfolio Risk and Return

 What is the expected return of a portfolio consisting of 60% stock and 40% bond?

 Given rS =33.5% and rB =9.5%

 rP = 0.6*33.5%+0.4*9.5%=23.9%

 How about portfolio risk?

 It’s not a weighted average of standard deviations

Corporate Finance Konan Chan 56 Portfolio Risk - Example

State of Prob. Portfolio (60% S+40% B) economy Boom 25% 50% Normal 60% 22% Recession 15% -12% Expected return=23.9% Standard deviation=19.1%

Corporate Finance Konan Chan 57

Portfolio Risk

 We need to account for covariance  Variance for a two-asset portfolio :

Corporate Finance Konan Chan 58 Covariance and Correlation

 What is covariance?

 Measures how closely two variables move together

 Correlation coefficient

 standardize covariance by dividing standard deviations of individual returns

  is between +1 and -1.

 “+1” means perfect positive correlation and “-1” means perfect negative correlation Corporate Finance Konan Chan 59

Covariance and Coefficient (Example)

 Covariance

 Cov (rS, rB) = 0.25(0.8-0.335) (0.05-0.095)+0.6 (0.3-0.335) (0.1-0.095)+0.15 (-0.3-0.335) (0.15-0.095) = -1.058% • Coefficient

–  (rS, rB)=-1.58%/[(34%)* (3.1%)] = -0.997

Corporate Finance Konan Chan 60 Cov. and Coef. - historical data

 Empirically, we estimate covariance & correlation by using historical time series data

Corporate Finance Konan Chan 61

Investment Opportunity Set

B

Minimum variance portfolio, Z

A

Corporate Finance Konan Chan 62 Mean-Variance Analysis

 What will happen if AB  0.3

  AB = 1.0 :

  AB = -1.0 :

Corporate Finance Konan Chan 63

Diversification Effect

B  = -1

= -0.3   = 0  = 1

A

 As long as  < 1, the standard deviation of a portfolio of two asset is less than the weighted average of the standard deviations of the individual assets Corporate Finance Konan Chan 64

Minimum variance portfolio Efficient Frontier x • Efficient portfolio is the x portfolio with Return x – the highest return for a x x given amount of risk. Z x – the lowest risk for a x given amount of return

Risk Risk, return combination of a portfolio or a single stock

Corporate Finance Konan Chan 65

Optimal Risky Portfolio

Capital Allocation Line “M” Efficient Frontier

Optimal Risky Portfolio “M”

Capital Allocation Line “A” A

Risk-free Rate

Corporate Finance Konan Chan 66 Optimal Portfolio Selection

 Optimal Portfolio Selection requires 3 steps:

 Construct efficient frontier

 Pick optimal risky portfolio by Capital Allocation Line with risk-free asset

 Choose appropriate weights for optimal risky portfolio and risk-free asset (depend on risk aversion of )  Separation property : step 2 and 3 are independent  All rational risk-averse investors will passively index holdings to an equity fund (portfolio “M”) and a money market fund

Corporate Finance Konan Chan 67

Terminology of Return and Risk  Risk-free rate

 The rate of return that can be earned with certainty  Risk premium

 Difference between return and risk-free asset return  Risk aversion

 The degree to which an is unwilling to accept risk

Corporate Finance Konan Chan 68 Risk-Free Asset

 Only the government can issue default-free bonds.  T-bills viewed as “the” risk-free asset  Money market funds also considered risk-free in practice

Corporate Finance Konan Chan 69

Asset Allocation (continued)

 Capital Allocation Line (CAL)

 varying the weights between a risk-free asset and a risky portfolio gives us all portfolio combinations, which fall on a single line  The slope of the CAL is the Reward-to- Variability Ratio, or the

Corporate Finance Konan Chan 70 Risk Aversion and Allocation

 Assume investors are risk averse, they invest a risky security if it provides risk premium.  Greater (lower) levels of risk aversion lead investors to choose larger (smaller) proportions of the risk- free rate  If the reward-to-variability ratio increase, then investors might well decide to take on riskier positions.

Corporate Finance Konan Chan 71

Capital Market Line

 Capital allocation line formed from 1-month T-bills and a broad index of common stocks (e.g. the S&P 500).

Corporate Finance Konan Chan 72 Historical Evidence on CML

 From 1926 to 2009, the passive risky portfolio offered an average risk premium of 7.9% with a standard deviation of 20.8%, resulting in a reward- to- ratio of .38.

Corporate Finance Konan Chan 73

Historical Returns, 1926-2011

 Risk-return trade-off Average Standard Series Annual Return Deviation Distribution

Large Company Stocks 11.8% 20.3%

Small Company Stocks 16.5 32.5

Long-Term Corporate Bonds 6.4 8.4

Long-Term Government Bonds 6.1 9.8

U.S. Treasury Bills 3.6 3.1

Inflation 3.1 4.2

– 90%0% + 90% Source: Global Financial Data (www.globalfinddata.com) copyright 2012.

Corporate Finance Corporate Finance Konan Chan 75

CAPM

Capital Asset Pricing Model (CAPM)

 Theory of relationship between risk and return

 Expected (required) return = risk-free rate + beta * market risk premium

 Market risk premium = rm –rf

 Risk free rate = rf  Beta = (measure of market risk)

ri = rf +  (rm –rf) Corporate Finance Konan Chan 76 Risk and Diversification

 Market compensates investors for taking risk  Only market risks are compensated  Unique risk should be diversified away

Corporate Finance Konan Chan 77

Risk and Diversification

 Diversification

 Strategy designed to reduce risk by spreading the portfolio across many  Unique Risk (diversifiable risk)

 Risk factors affecting only that firm  Market Risk (systematic risk)

 Economy-wide sources of risk that affect the overall stock market

 Measured by Beta Corporate Finance Konan Chan 78 Beta

 Sensitivity of stock’s return to the market return  How stock’s return changes with market return changes  Proxy for market risk

 β = 1.0: same risk as the market (average stock)

 β < 1.0: less risky than market (defensive stock)

 β > 1.0: riskier than market (aggressive stock)

Corporate Finance Konan Chan 79

Stock Betas

Corporate Finance Konan Chan 80 Market Equilibrium

 In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market

Corporate Finance Konan Chan 81

Security Characteristic Line

Beta is the slope of the regression line

Ri =  +β RM , regressing a stock’s

return (Ri) on the Ri = i + iRM + ei market return (RM)  = Slope  = Intercept

Corporate Finance Konan Chan 82 CAPM and Valuation

 Return = [dividend + capital gain]/price = dividend yield + % capital gain  In equilibrium, the expected return defined above should equal CAPM return.  Expected return = expected dividend yield + capital gain yield = CAPM expected return

Corporate Finance Konan Chan 83

CAPM & Valuation - Example

 Your stockbroker calls you to buy Fearfree Inc.  The stock is currently selling for $15 a share  The risk free rate is 5%, and you demand a 17% return on the market.  Fearfree's current dividend is $4 a share  Some analyst has estimated that Fearless's beta is 2.0 and that the stock's dividend will grow at a constant 8%  Is recommendation to buy Fearfree a good one? What do you think the stock is worth?

Corporate Finance Konan Chan 84 CAPM & Valuation - Example

 D0=4, g=8%, rm=17%, rf =5%,β=2  From CAPM, r = 5%+2*(17%-5%) = 29%

 P0 = $4*(1 + 8%) / (29% - 8%)= $20.57  Intrinsic value $20.57 > market value $15  Price may appreciate by 5.57 later!

Corporate Finance Konan Chan 85

Factor Models

 Single factor model

 Ri = i + iRM + ei  Usually, use market index as the ‘single’ factor

 i is factor loading (sensitivity)  Multifactor model

 Ri = i + 1iR1f + 2iR2f + … + ei  Use different factors, such as GNP, inflation, …  Fama-French three-factor model

 Rit = ai + biRMt + siSMBt + hiHMLt + ei  Market, size factor, book-to-market factor  Four factor model (add ) Corporate Finance Konan Chan 86 Expected Return by Factor Model

Corporate Finance Konan Chan 87