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Chapter 4. Stock And Bond Valuation. Professor John Zietlow MBA 621. Spring 2006. Valuation Fundamentals. Value of any financial asset is the PV of future cash flows Bonds: PV of promised interest & principal payments Stocks: PV of all future dividends
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Chapter 4 Stock And Bond Valuation Professor John ZietlowMBA 621 Spring 2006
Valuation Fundamentals • Value of any financial asset is the PV of future cash flows • Bonds: PV of promised interest & principal payments • Stocks: PV of all future dividends • Patents, trademarks: PV of future royalties • Valuation is the process linking risk & return • Output of process is asset’s expected market price • A key input is the required [expected] return on an asset • Defined as the return an arms-length investor would require for an asset of equivalent risk • Debt securities: risk-free rate plus risk premium(s) • Required return for stocks found using CAPM or other asset pricing model • Beta determines risk premium: higher beta, higher reqd return
The Basic Valuation Model • Can express price of any asset at time 0, P0, mathematically as Equation 4.1: • Where: • P0 = Price of asset at time 0 (today) • CFt = cash flow expected at time t • r = discount rate, reflecting asset’s risk • n = number of discounting periods (usually years)
Illustration Of Simple Asset Valuation • Assume you are offered a security that promises to make four $2,000 payments at the end of years 1-4. • If the appropriate discount rate for securities of this risk is 2%, what price should you pay for this security (same as asking what is its present value)? • Security would be worth $7,615.53 ($7,615.46 calculator) each.
Illustration Of Bond Valuation Using U.S. Treasury Securities • The simplest debt instruments to value are U.S. Treasury securities since there is no default risk. • Instead of r, the discount rate to use, rf, is the pure cost of borrowing. • Assume you are asked to value two Treasury securities, when rf is 1.75 percent (r = 0.0175): • A (pure discount) Treasury bill with a $1,000 face value that matures in three months, and • A 1.75% coupon rate Treasury note, also with a $1,000 face value, that matures in three years. • For the T-Bill, three months is one-quarter year (n=0.25) • For 3-year bond, n = 3
Illustration Of Bond Valuation Using U.S. Treasury Securities (Continued) • The 3-month T-Bill pays no interest; return comes from difference between purchase price and maturity value. • 3-year T-Note makes two end-of-year $17.5 coupon payments (CF1=CF2=$17.5), plus end-of-year 3 payment of interest plus principal (CF3 = $1,017.5) • Can value both with variation of Equation 4.1:
Bond Valuation Fundamentals • Most U.S. corporate bonds: • Pay interest at a fixed coupon interest rate • Have an initial maturity of 10 to 30 years, and • Have a par value (also called face or principal value) of $1,000 that must be repaid at maturity. • The Sun Company, on January 3, 2004, issues a 5 percent coupon interest rate, 10‑year bond with a $1,000 par value • Assume annual interest payments for simplicity • Will value later assuming semi-annual coupon payments • Investors in Sun Company’s bond thus receive the contractual right to: • $50 coupon interest (C) paid at the end of each year and • The $1,000 par value (Par) at the end of the tenth year.
Bond Valuation Fundamentals (Continued) • Assume required return, r, also equal to 5% • The price of Sun Company’s bond, P0, making ten (n=10) annual coupon interest payments (C = $50), plus returning $1,000 principal (Par) at end of year 10, determined as: • So this bond would be selling at par value of $1,000
Bond Valuation Fundamentals (Continued) • Bond’s value has two separable parts: (1) PV of stream of annual interest payments, t=1 to t=10 (2) PV of principal repayment at end of year 10. • Can thus also value bond as the PV of an annuity plus the PV of a single cash flow using PVFA and PVF from tables. P0 = C x (PVFA5%,10yr) + Par x (PVF5%,10yr) = $50 (7.7220) + $1,000 (0.6139) = $1,000.00 • Bonds with a few cash flows can be valued with Eq 4.1; for bonds with many cash flows, use PVFA/PVF factors, calculator or Excel.
Bond Values If Required Return Is Not Equal To The Coupon Rate • Whenever the required return on a bond (r) differs from its coupon interest rate, the bond's value will differ from its par, or face, value. • Will only sell at par ifr= coupon rate • When r is greater than the coupon interest rate, P0 will be less than par value, and the bond will sell at a discount. • For Sun, ifr>5%, P0 will be less than $1,000 • When r is below the coupon interest rate, P0 will be greater than par, and the bond will sell at a premium. • For Sun, ifr<5%, P0 will be greater than $1,000 • Exercise: Value Sun Company, 10-year, 5% coupon rate bond if required return, r =6% and again if r = 4%.
Bond Values If Required Return Is Not Equal To The Coupon Rate (Continued) • Value Sun Company bond if r = 6% P0 = $50 x (PVFA6%,10yr) + Par x (PVF6%,10yr) = $50 (7.3601) + $1,000 (0.5584) = $926.405 approx • Bond sells at a discount of $1,000 - $926.405 = $73.595 • Value Sun Company bond if r = 4% P0 = C x (PVFA4%,10yr) + Par x (PVF4%,10yr) = $50 (8.1109) + $1,000 (0.6756) = $1081.145 approx • Bond sells at a premium of $1,081.45 - $1,000 = $81.45 • Premiums & discounts change systematically as r changes
Bond Value & Required Return, Sun Company’s 5 % Coupon Rate, 10-year, $1,000 Par, January 1, 2004 Issue Paying Annual Interest 1,200 1,100 1,081 Premium Market Value of Bond P0 ($) Par 1,000 Discount 926 900 800 0 1 2 3 4 5 6 7 8 Required Return, r (%)
The Dynamics Of Bond Valuation Changes For Different Times To Maturity • Whenever r is different from the coupon interest rate, the time to maturity affects bond value • even if the required return remains constant until maturity. • The shorter is n, the less responsive is P0 to changes in r. Assume r falls from 5% to 4% • For n=8 years,P0 rises from $1,000 to $1,067.33, or 6.73% • For n=3 years,P0 rises from $1,000 to $1,027.75, or 2.775% • Same relationship if r rises from 5% to 6%, though percentage declines in price less than increases (maximum decline is 100%, increase unlimited) • For n=8 years,P0 falls from $1,000 to $937.89, or 6.21% • For n=3 years,P0 falls from $1,000 to $973.25, or 2.675% • Even if r doesn’t change, premiums and discounts will decline towards par as bond nears maturity.
Relation Between Time to Maturity, Required Return & Bond Value, Sun Company’s 5%, 10-year, $1,000 Par Issue Paying Annual Interest 1,100 Premium Bond, Required Return, r = 4% 1,081 1,067.3 1,050 1,027.75 Par-Value Bond, Required Return, r = 5% 1,000 M Market Value of Bond P0 ($) 950 Discount Bond, Required Return, r = 6% 926 900 10 9 8 7 6 5 4 3 2 1 0 Time to maturity (years)
Relationship Between Bond Prices & Yields, Bonds Of Differing Current Maturities But Same 6.5% Coupon Rates
Semi-Annual Bond Interest Payments • Most bonds pay interest semi-annually rather than annually • Can easily modify basic valuation formula; divide both coupon payment (C) and discount rate (r) by 2, as in Eq 4.3: • In Eq 4.3, C is the annual coupon payment, so C/2 is the semi-annual payment. • r is the annual required return, so r/2 is the semi-annual discount rate. • n is the number of years, so there are 2n semi-annual payments.
Valuing A Bond With Semi-Annual Bond Interest Payments • Value a T-Bond with a par value of $1,000 that matures in exactly 2 years and pays a 4% coupon if r = 4.4% per year. • Insert known variables into Equation 4.3: C = $40, so C/2 = $20, r = 0.044, so r/2 = 0.022, n = 2, so 2n = 4:
The Importance And Calculation Of Yield To Maturity • Yield to Maturity (YTM) is the rate of return investors earn if they buy the bond at P0 and hold it until maturity. • The YTM on a bond selling at par (P0 = Par) will always equal the coupon interest rate. • When P0Par, the YTM will differ from the coupon rate. • YTM is the discount rate that equates the PV of a bond’s cash flows with its price. If P0, CFs, n known, can find YTM • Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43 • The YTM can be found by trial and error, calculator or with spreadsheet program (Excel).
The Fisher Effect And Expected Inflation • The relationship between nominal (observed) and real (inflation-adjusted) interest rates and expected inflation called the Fisher Effect (or Fisher Equation). • Fisher said the nominal rate (r) is approximately equal to the real rate of interest (a) plus a premium for expected inflation (i). • If real rate equals 3% (a = 0.03) and expected inflation equals 2% (i = 0.02): r a + i 0.03 + 0.02 0.05 5% • The true Fisher Effect is multiplicative, rather than additive: (1+r) = (1+a)(1+i) = (1.03)(1.02) = 1.0506; so r = 5.06%
The Term Structure Of Interest Rates • At any point in time, will be a systematic relationship between YTM and maturity for securities of a given risk • Usually, yields on long-term securities higher than short-term • Generally look at risk-free Treasury debt securities • Relationship between yield and maturity called the Term Structure of Interest Rates • Graphical depiction called a Yield Curve • Yield curves normally upwards-sloping (long yields > short) • Can be flat or even inverted during times of financial stress • Won’t cover term structure in depth, but three principal “expectations” theories explain term structure: • Pure expectations hypothesis: YC embodies prediction • Liquidity premium theory: Investors must be paid more to invest L-T • Preferred habitat hypothesis: Investors prefer maturity zones, so different supply and demand characteristics in sub-sectors
Yield Curves for US Treasury Securities 16 May 1981 14 12 10 January 1995 Interest Rate % 8 August 1996 6 October 1993 4 2 1 3 5 10 15 20 30 Years to Maturity
Yield Curve, March 23, 2006From www.cnnfn.com % % Years to maturity
Changes In The Shape And Level Of Treasury Yield Curve During Early October 1998 5.1 October 9 4.9 October 8 4.7 October 2 4.5 Yield % 4.3 4.1 3.9 3.7 1 5 10 30 Maturity in Years
Equity Valuation • As will be discussed in chapter 5, the required return on common stock is based on its beta, derived from the CAPM • Valuing CS is the most difficult, both practically & theoretically • Preferred stock valuation is much easier (the easiest of all)^ • Disequilibrium: Whenever investors feel the expected return, r, is not equal to the required return, r, prices will react: • If exp return declines or reqd return rises, stock price will fall • If exp return rises or reqd return declines, stock price will rise • Asset prices can change for reasons besides their own risk • Changes in asset’s liquidity, tax status can change price • Changes in market risk premium can change all asset values • Most dramatic change in market risk: Russian default Fall 98 • Caused required return on all risky assets to rise, price to fall
Preferred Stock Valuation • PS is an equity security that is expected to pay a fixed annual dividend over its (assumed infinite) life. • Preferred stock’s market price, P0, equals next period’s dividend payment, Dt+1, divided by the discount rate, r, appropriate for securities of its risk class: • A share of PS paying a $2.30 per share annual dividend and with a required return of 11% would thus be worth $20.91: • Formula can be rearranged to compute required return, if price and dividend known:
Common Stock Valuation • Basic formula for valuing a share of stock easy to state; P0 is equal to the present value of the expected stock price at end of period 1, plus dividends received, as in Eq 4.4: • But how to determine P1? This is the PV of expected stock price P2, plus dividends. P2 in turn, the PV of P3 plus dividends, and so on. • Repeating this logic over and over, find that today’s price equals PV of the entire dividend stream the stock will pay in the future, as in Eq 4.5:
The Zero Growth Valuation Model • To value common stock, must make assumption about growth of future dividends. • Simplest approach, the zero growth model, assumes a constant, non-growing dividend stream: D1 = D2= ... = D • Plugging constant value D into Eq 4.5, valuation formula reduces to simple equation for a perpetuity: • Assume the dividend of Disco Company is expected to remain at $1.75/share indefinitely, and the required return on Disco’s stock is 15%. P0 is determined to be $11.67 as:
The Constant Growth Valuation Model • The most widely used simple stock valuation formula, the constant growth model, assumes dividends will grow at a constant rate, g, that is less than the required return (g<r). • If dividends grow at a constant rate forever, can value stock as a growing perpetuity. Denoting next year’s dividend as D1: • This is commonly called the Gordon Growth Model, after Myron Gordon, who popularized model in the 1960s. • The Gordon Company’s dividends have grown by 7% per year, reaching $1.40 per share. This growth is expected to continue, so D1=$1.40 x 1.07=$1.498. If required return is 15%:
Valuing Common Stock Using The Variable Growth Model • Because future growth rates might change, need to consider a variable growth rate model that allows for a change in the dividend growth rate. • Let g1 = the initial, higher growth rate and g2 = the lower, subsequent growth rate, and assume a single shift in growth rates from g1 to g2. • Model can be generalized for two or more changes in growth rates, but keep simple now. • For a single change in growth rates, can use four-step valuation procedure:
Valuing Common Stock Using The Variable Growth Model (Continued) • Step 1: Find the value of the dividends at the end of each year, Dt, during the initial high-growth phase. • Step 2: Find the PV of the dividends during this high-growth phase, and sum the discounted cash flows. • Step 3: Using the Gordon growth model, (a) find the value of the stock at the end of the high-growth phase using the next period’s dividend (after one year’s growth at g2). • (b) Then compute PV of this price by discounting back to time 0. • Step 4: Determine the value of the stock today (P0) by adding the PV of the stock price computed in step 3 to the sum of the discounted dividend payments from step 2.
An Example Of Stock Valuation Using The Variable Growth Model • Estimate the current (end‑of‑2003) value of Morris Industries' common stock, P0 = P2003 , using the four‑step procedure presented above, and assuming the following: • The most recent (2003) annual dividend payment of Morris Industries was $4 per share. • The firm's financial manager expects that these dividends will increase at a 8 percent annual rate, g1 , over the next three years (2004, 2005, and 2006). • At the end of the three years (end of 2006) the firm's mature product line is expected to result in a slowing of the dividend growth rate to 5 percent per year forever (noted as g2). • The firm's required return, r, is 12 percent.
An Example Of Stock Valuation Using The Variable Growth Model (Continued) • Step 1: Compute the value of dividends in 2004, 2005, and 2006 as (1+g1)=1.08 times the previous year’s dividend: Div2004= Div2003 x (1+g1) = $4 x 1.08 = $4.32 Div2005= Div2004 x (1+g1) = $4.32 x 1.08 = $4.67 Div2006= Div2005 x (1+g1) = $4.67 x 1.08 = $5.04 • Step 2: Find the PV of these three dividend payments: PV of Div2004= Div2004 (1+r) = $ 4.32 (1.12) = $3.86 PV of Div2005= Div2005 (1+r)2 = $ 4.67 (1.12)2 = $3.72 PV of Div2006= Div2006 (1+r)3 = $ 5.04 (1.12)3 = $3.59 Sum of discounted dividends = $3.86 + $3.72 + $3.59 = $11.17
An Example Of Stock Valuation Using The Variable Growth Model (Continued) • Step 3: Find the value of the stock at the end of the initial growth period (P2006) using constant growth model. • To do this, calculate next period dividend by multiplying D2006 by 1+g2, the lower constant growth rate: D2007 = D2006 x (1+ g2) = $ 5.04 x (1.05) = $5.292 • Then use D2007=$5.292, g =0.05, r =0.12 in Gordon model: • Next, find the PV of this stock price by discounting P2006 by (1+r)3.
An Example Of Stock Valuation Using The Variable Growth Model (Continued) • Step 4: Finally, add the PV of the initial dividend stream (found in Step 2) to the PV of stock price at the end of the initial growth period (P2006): P2003 = $11.17 + $53.81 = $64.98 • The current (end-of-year 2003) stock price is thus $64.98 per share.
Other Approaches To Common Stock Valuation • Book value: simply the net assets per share available to common stockholders after liabilities (and PS) paid in full; equals total common equity on the balance sheet • Assumes assets can be sold at book value, so may over-estimate realizable value • Liquidation value: is the actual net amount per share likely to be realized upon liquidation & payment of liabilities • More realistic than book value, but doesn’t consider firm’s value as a going concern • Price/Earnings (P/E) multiples: reflects the amount investors will pay for each dollar of earnings per share • P/E multiples differ between & within industries • Especially helpful for privately-held firms (think of how many shares will issue as go public, multiply that by P/E for industry to get market value for all new shares)