Chapter 8, 9 Stock and Bond Valuation
Konan Chan Corporate Finance, 2018
Bond Valuation Bond Valuation
Bond pricing model Annual vs. semi-annual coupon bonds Yield-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 expected return 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, long-term bonds have more interest rate risk than short-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
Investment 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 Stock Valuation
Dividend discount model Constant growth Zero growth Non-constant growth Expected stock return Multiple valuation
Corporate Finance Konan Chan 27
Stock Valuation
Dividend discount model (DDM)
discount future dividends back to present where T is time horizon for your investment
Corporate Finance Konan Chan 28 Stock Valuation
We will assume stocks 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 Preferred Stock
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 dividend yield + 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) Beta 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 Efficient Frontier
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 investors) 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 investor 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 Sharpe ratio
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-volatility 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 investments 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 momentum) Corporate Finance Konan Chan 86 Expected Return by Factor Model
Corporate Finance Konan Chan 87