Chapter 8, 9 Stock and Bond Valuation Bond Valuation
Konan Chan Corporate Finance, 2018
Bond Valuation Bond Cash Flows Annual coupons Bond pricing model coupon coupon+par Annual vs. semi-annual coupon bonds 0 1 2 ……. T Yield-to-maturity (YTM) Credit ratings Value today = PV of expected cash flows
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Bond Characteristics Bond Pricing
Five basic variables Example FV : par value (or face value) - usually $1000 to What is the price of a 6.5 % annual coupon bond, be paid at maturity with a $1,000 face value, which matures in 3 years? Assume a required return of 3.9%. 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 Corporate Finance Konan Chan 6 Bond Price and Interest Rate Bond Price Over Time
1,080 Price path for There is a negative relationship between bond 1,060 Premium Bond price and interest rate (discount rate) 1,040 If discount rate is higher (lower) than coupon 1,020 Both premium and discount bonds 1,000 approach face value as their
rate, bond prices should be less (more) than 980 maturity date approaches
par value Price Bond 960 940
920 Price path for Today Maturity 900 Discount Bond
880 0 5 10 15 20 25 30 Time to Maturity
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Bond Cash Flows Bond Pricing
Annual coupons Example (continued) coupon coupon+par What is the price of the bond if the required rate of 0 1 2 ……. T return is 3.9% and the coupons are paid semi- Semi-annual coupons annually? coupon/2 coupon/2+par 0 1 2 ……. 2T Value today = PV of expected cash flows
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Bond Yields Corporate Bond
Current Yield Corporate bond quotation (on Sep 2005) annual coupon payments divided by bond price. Company Coupon Maturity Last price Last yield Yield To Maturity (YTM) Wal-mart 7.55 Sep 30, 2031 125.314 5.675 interest rate for which the present value of bond equals the market price How to compute yield to maturity? Bond price = 1253.14 total annual expected return if you buy the bond today and hold to the maturity date 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 11 Corporate Finance Konan Chan 12 Yield To Maturity 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% In Excel, RATE(52,37.75,-1253.14,1000,0) 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.
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Interest Rate Risk Interest Rate Risk Example
Measures bond price sensitivity to changes in Let’s compare two bonds with everything the interest rates same except the time-to-maturity (1 vs. 30 years) All things equal, long-term bonds have more interest rate risk than short-term bonds. PVs of 10% annual coupons with r at 5%, 10%, 15%, 20%. All things equal, low coupon bonds also have more interest rate risk than high coupon bonds
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Bond Price Sensitivity Credit Rating (default risk)
Credit ratings proxy for default risk, the risk that bond
When the interest rate equals issuer may default on its obligations the 10% coupon rate, both Default premium: difference between corporate bond bonds sell at face value 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
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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
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Yield Spread 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
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Inflation Fisher Effect (Inflation)
Inflation
Rate at which prices as a whole are increasing. Nominal Interest Rate
Rate at which money invested grows. Real Interest Rate Approximation formula
Rate at which the purchasing power of an investment increases.
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Real interest Inflation Interest rate risk Stock Valuation 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
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Stock Valuation Stock Valuation
Dividend discount model Dividend discount model (DDM) Constant growth discount future dividends back to present where T is time horizon for your investment Zero growth Non-constant growth Expected stock return Multiple valuation
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Stock Valuation 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. We will assume stocks fall into 3 categories D0: dividend just paid (the most recent dividend) Constant growth rate in dividends g: constant growth rate of dividends
Zero growth rate in dividends r: required rate of return for stock “Supernormal” (non-constant) growth rate in dividends
Corporate Finance Konan Chan 29 Corporate Finance Konan Chan 30 Constant Growth DDM Constant Growth DDM - example
D1 = D0 (1 + g) What is the value of a stock that expects to pay a 2 $3.00 dividend next year, and then increase the D2 = D1 (1 + g) = D0 (1 + g) dividend at a rate of 8% per year, indefinitely? Assume a 12% expected return
Using geometric series formula
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Same example Zero Growth DDM
If the same stock is selling for $100 in the stock If we forecast no growth for the stock (i.e., dividends market, what might the market assume about the keep constant forever), the stock will become a growth in dividends? perpetuity
This is exactly the valuation for preferred stocks The market assumes the dividend will grow at 9% per year, indefinitely.
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What if CGDDM Doesn’t Preferred Stock Apply? 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 Any restriction on constant growth DDM? Most preferred dividends are cumulative – any What does it mean? How to deal with it if this missed preferred dividends have to be paid before restriction exists? common dividends can be paid Two-stage or multiple stage of growth Preferred stock generally does not carry voting rights
Corporate Finance Konan Chan 35 Corporate Finance Konan Chan 36 Non-constant Growth Model Non-constant Growth - Example
The growth for firm A.net is expected to be 20% for next two Two stages of growth years, and 6% thereafter. The current dividend is $1.60, and the assume stock has a period of non-constant growth firm’s required rate of return is 10%. What’s stock worth today? in dividend, and then eventually settles into a g1 = 20% g1 = 20% g = 6% normal constant growth pattern Step 1 D1=$1.6(1.2)=$1.92 D2=$1.92(1.2)=$2.304 Generally, high growth in the first stage, then low growth stage in the second stage Step 2 Young, start-up firms, or technology firms with new product will have high growth rates Step 3 Multiple stages if necessary
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Sustainable Growth Rate Estimate Expected Return
Payout ratio Given constant growth dividend discount model, we Fraction of earnings paid out as dividends can estimate stock return Plowback (retention) ratio = 1 - payout ratio
Fraction of earnings retained by the firm g = return on equity (ROE) * retention ratio Expected return = expected dividend yield + growth rate Steady rate at which a firm can grow This estimation of growth rate applies to stable Previous example: firms only r = $3/$75 + 8% = 12%
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Components of Expected Growth in Constant Growth DDM 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 P1 / P0 = 1 + g (i.e., the firm will grow constantly)
Corporate Finance Konan Chan 41 Corporate Finance Konan Chan 42 Chapter 10, 11 Risk and Return Risk and Return Return measures Expected return and risk? Portfolio risk and diversification Konan Chan CAPM (Capital Asset Pricing Model) Corporate Finance, 2018 Beta
Calculating Return - Single period Calculating Return - Multi periods
Holding period return (HPR) Arithmetic average Arithmetic mean of returns
Good measure for future performance Geometric average This assumes we only have one investment Geometric mean of returns period. What about multiple periods? The return measure that gives the same cumulative performance as actual returns (buy- and-hold)
Required for mutual fund performance
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Returns - Example 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) Arithmetic Average: (0.14 -0.1455 + 0.10) / 3 = 3.15% How to estimate the expected return and risk? Geometric Average: (1 + R )3 = (1 + 0.14)*(1 - 0.1455)*(1 + 0.10) G based on probability distribution R = [(1 + 0.14)*(1 - 0.1455)*(1 + 0.10)]1/3 – 1=2.33% G based on historical data RG RA , RG is a better measure for past performance
Corporate Finance Konan Chan 47 Corporate Finance Konan Chan 48 The Normal Distribution 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
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- Expected Return & Risk Return & Risk - Historical data Example Initial investment : $100 Treat each historical outcome equally and assign a probability of 1/n ( n is number of observations) Return
Use sample average
• Expected return = 0.3(0.5)+0.5(0.2)+0.2(-0.4)=17% Risk 2 2 • Variance = 0.3(0.5-0.17) +0.5(0.2-0.17) Use sample variance +0.2(-0.4-0.17)2 = 0.0981 • Standard deviation = (0.0981)0.5 = 0.313
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Return & Risk -Historical data Return and Risk – Two Assets (example)
Using Excel functions State of Probability Stock Bond economy Boom 25% 80% 5%
Normal 60% 30% 10%
Recession 15% -30% 15%
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rS = 0.25*0.8+0.6*0.3+0.15*(-0.3) = 0.335 What is the expected return of a portfolio consisting of 60% stock and 40% bond? rB = 0.25*0.05+0.6*0.1+0.15*0.15 = 0.095 Standard deviation 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
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Portfolio Risk - Example Portfolio Risk
State of Prob. Portfolio (60% S+40% B) We need to account for covariance economy Variance for a two-asset portfolio : Boom 25% 50% Normal 60% 22%
Recession 15% -12% Expected return=23.9% Standard deviation=19.1%
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Covariance and Correlation Covariance and Coefficient (Example) What is covariance?
Measures how closely two variables move together
Covariance
Correlation coefficient 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% standardize covariance by dividing standard deviations of individual returns • Coefficient is between +1 and -1. – (rS, rB)=-1.58%/[(34%)* (3.1%)] = -0.997 “+1” means perfect positive correlation and “-1” means perfect negative correlation Corporate Finance Konan Chan 59 Corporate Finance Konan Chan 60 Cov. and Coef. - historical data Investment Opportunity Set
Empirically, we estimate covariance & correlation by using historical time series data
B
Minimum variance portfolio, Z
A
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Mean-Variance Analysis Diversification Effect
B = -1 What will happen if AB 0.3
= -0.3 = 0 = 1
AB = 1.0 : A
AB = -1.0 :
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 63 Corporate Finance Konan Chan 64
Efficient Frontier Optimal Risky Portfolio
Minimum variance portfolio Capital Allocation Line “M” Efficient Frontier Efficient Frontier x • Efficient portfolio is the x portfolio with Return Optimal Risky Portfolio “M” x the highest return for a x x – given amount of risk. Capital Allocation Line “A” Z x – the lowest risk for a A x given amount of return Risk-free Rate Risk Risk, return combination of a portfolio or a single stock
Corporate Finance Konan Chan 65 Corporate Finance Konan Chan 66 Terminology of Return and Optimal Portfolio Selection Risk Optimal Portfolio Selection requires 3 steps: Risk-free rate Construct efficient frontier The rate of return that can be earned with Pick optimal risky portfolio by Capital Allocation Line certainty with risk-free asset
Choose appropriate weights for optimal risky portfolio Risk premium and risk-free asset (depend on risk aversion of investors) Difference between return and risk-free asset Separation property : step 2 and 3 are independent return All rational risk-averse investors will passively Risk aversion
index holdings to an equity fund (portfolio “M”) and The degree to which an investor is unwilling to a money market fund accept risk
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Risk-Free Asset Asset Allocation (continued)
Only the government can issue default-free Capital Allocation Line (CAL)
bonds. varying the weights between a risk-free asset and T-bills viewed as “the” risk-free asset a risky portfolio gives us all portfolio combinations, which fall on a single line Money market funds also considered risk-free in practice The slope of the CAL is the Reward-to- Variability Ratio, or the Sharpe ratio
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Risk Aversion and Allocation Capital Market Line
Assume investors are risk averse, they invest a risky Capital allocation line formed from 1-month security if it provides risk premium. T-bills and a broad index of common stocks Greater (lower) levels of risk aversion lead investors (e.g. the S&P 500). 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 Corporate Finance Konan Chan 72 Historical Evidence on CML Historical Returns, 1926-2011
From 1926 to 2009, the passive risky portfolio Risk-return trade-off Average Standard offered an average risk premium of 7.9% with a Series Annual Return Deviation Distribution standard deviation of 20.8%, resulting in a reward- Large Company Stocks 11.8% 20.3% to-volatility ratio of .38. 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.
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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 75 Corporate Finance Konan Chan 76
Risk and Diversification Risk and Diversification
Market compensates investors for Diversification taking risk Strategy designed to reduce risk by spreading the Only market risks are compensated portfolio across many investments Unique risk should be diversified away 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 77 Corporate Finance Konan Chan 78 Beta Stock Betas
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)
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Market Equilibrium Security Characteristic Line
In equilibrium, all assets and portfolios must have Beta is the slope of the regression line the same reward-to-risk ratio and they all must equal Ri = +β RM , the reward-to-risk ratio for the market regressing a stock’s return (Ri) on the Ri = i + iRM + ei market return (RM) = Slope = Intercept
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CAPM and Valuation CAPM & Valuation - Example
Return = [dividend + capital gain]/price Your stockbroker calls you to buy Fearfree Inc. The stock is currently selling for $15 a share = dividend yield + % capital gain The risk free rate is 5%, and you demand a 17% In equilibrium, the expected return defined return on the market. above should equal CAPM return. Fearfree's current dividend is $4 a share Some analyst has estimated that Fearless's beta is 2.0 Expected return and that the stock's dividend will grow at a constant = expected dividend yield + capital gain yield 8% = CAPM expected return Is recommendation to buy Fearfree a good one? What do you think the stock is worth?
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Single factor model D0=4, g=8%, rm=17%, rf =5%,β=2 Ri = i + iRM + ei From CAPM, r = 5%+2*(17%-5%) = 29% Usually, use market index as the ‘single’ factor
i is factor loading (sensitivity) P0 = $4*(1 + 8%) / (29% - 8%)= $20.57 Multifactor model Intrinsic value $20.57 > market value $15 Ri = i + 1iR1f + 2iR2f + … + ei Price may appreciate by 5.57 later! 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 85 Corporate Finance Konan Chan 86
Expected Return by Factor Model
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