# LECTURE 3 : VALUATION MODELS : EQUITIES AND BONDS (Asset Pricing and Portfolio Theory)

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LECTURE 3 : VALUATION MODELS : EQUITIES AND BONDS (Asset Pricing and Portfolio Theory)

Contents Market price and fair value price Market price and fair value price –Gordon growth model, widely used simplification of the rational valuation model (RVF) Are earnings data better than dividend information ? Are earnings data better than dividend information ? Stock market bubbles Stock market bubbles How well does the RVF work ? How well does the RVF work ? Pricing bonds – DPV again ! Pricing bonds – DPV again ! –Duration and modified duration

Discounted Present Value

Rational Valuation Formula E t R t+1 = [E t V t+1 – V t + E t D t+1 ] / V t (1.) where V t = value of stock at end of time t D t+1 = dividends paid between t and t+1 E t = expectations operator based on information  t at time t or earlier E(D t+1 |  t )  E t D t+1 Assume investors expect to earn constant return (= k) E t R t+1 = k k > 0 (2.)

Rational Valuation Formula (Cont.) Excess return are ‘fair game’ : Excess return are ‘fair game’ : E t (R t+1 – k |  t ) = 0 (3.) Using (1.) and (2.) : Using (1.) and (2.) : V t =  E t (V t+1 + D t+1 ) (4.) where  = 1/(1+k) and 0 <  < 1 Leading (4.) one period Leading (4.) one period V t+1 =  E t+1 (V t+2 + D t+2 ) (5.) E t V t+1 =  E t (V t+2 + D t+2 ) (6.)

Rational Valuation Formula (Cont.) Equation (6.) holds for all periods : Equation (6.) holds for all periods : E t V t+2 =  E t (V t+3 + D t+3 ) etc. Substituting (6.) into (4.) and all other time periods Substituting (6.) into (4.) and all other time periods V t = E t [  D t+1 +  2 D t+2 +  3 D t+3 + … +  n (D t+n + V t+n )] V t = E t  i D t+i

Rational Valuation Formula (Cont.) Assume : Assume : –Investors at the margin have homogeneous expectations (their subjective probability distribution of fundamental value reflects the ‘true’ underlying probability). –Risky arbitrage is instantaneous

Special Case of RVF (1) : Expected Div. are Constant D t+1 = D t + w t+1 RE : E t D t+j = D t P t =  (1 +  +  2 + … )D t =  (1-  ) -1 D t = (1/k)D t or P t /D t = 1/k or D t /P t = k Prediction : Dividend-price ratio (dividend yield) is constant

Real Dividends : USA, Annual Data, 1871 - 2002

Special Case of RVF (2) : Exp. Div. Grow at Constant Rate Also known as the Gordon growth model Also known as the Gordon growth model D t+1 = (1+g)D t + w t+1 (E t D t+1 – D t )/D t = g E t D t+j = (1+g) j D t P t =   i (1+g) i D t P t = [(1+g)D t ]/(k–g) with (k - g) > 0 or P t = D t+1 /(k-g)

Gordon Growth Model Constant growth dividend discount model is widely used by stock market analysts. Constant growth dividend discount model is widely used by stock market analysts. Implications : Implications : The stock value will be greater : … the larger its expected dividend per share … the lower the discount rate (e.g. interest rate) … the higher the expected growth rate of dividends Also implies that stock price grows at the same rate as dividends.

More Sophisticated Models : 3 Periods Dividend growth rate Time High Dividend growth period Low Dividend growth period

Time-Varying Expected Returns Suppose investors require different expected return in each future period. Suppose investors require different expected return in each future period. E t R t+1 = k t+1 E t R t+1 = k t+1 P t = E t [  t+1 D t+1 +  t+1  t+2 D t+2 + … P t = E t [  t+1 D t+1 +  t+1  t+2 D t+2 + … + …  t+N-1  t+N (D t+N + P t+N )] + …  t+N-1  t+N (D t+N + P t+N )] where  t+i = 1/(1+k t+i )

Price Earnings Ratio Total Earnings (per share) = retained earnings + dividend payments Total Earnings (per share) = retained earnings + dividend payments E = RE + D E = RE + D with D = pE and RE = (1-p)E p = proportion of earnings paid out as div. P = V = pE 1 / (R – g) or P / E 1 = p / (R - g) (base on the Gordon growth model.) Note : R, return on equity replaced k (earlier).

Price Earnings Ratio (Cont.) Important ratio for security valuation is the P/E ratio. Important ratio for security valuation is the P/E ratio. Problems : Problems : –forecasting earnings –forecasting price earnings ratio Riskier stocks will have a lower P/E ratio.

Industrial P/E Ratios Based on EPS Forecasts

The Equity Premium Puzzle (Fama and French, 2002)

FF (2002) : The Equity Premium All variables are in real terms. All variables are in real terms. A(R t ) = A(D t /P t-1 ) + A(GP t ) Two alternative ways to measure returns Two alternative ways to measure returns A(RD t ) = A(D t /P t-1 ) + A(GD t ) A(RY t ) = A(D t /P t-1 ) + A(GY t ) where ‘ A’ stands for average GP t = growth in prices (=p t /p t-1 )*(L t-1 /L t ) – 1) GD t = dividend growth (= d t /d t-1 )*(L t-1 /L t ) -1) GY t = earning growth (= y t /y t-1 )*(L t-1 /L t ) -1) L is the aggregate price index (e.g. CPI)

US Data (1872-2002) : Div/P and Earning/P ratios

FF (2002) : The Equity Premium (Cont.) FtFtFtFt RtRtRtRt RXD t RXY t RX t Mean of annual values of variables 1872-20003.248.813.54NA5.57 1872-19503.908.304.17NA4.40 1951-20002.199.622.554.327.43 Standard deviation of annual values of variables 1872-20008.4818.0313.00NA18.51 1872-195010.6318.7216.02NA19.57 1951-20002.4617.035.6214.0216.73

FF (2002) : The Equity Premium (Cont.) F t = risk free rate F t = risk free rate R t = return on equity R t = return on equity RXD t = equity premium, calculated using dividend growth RXD t = equity premium, calculated using dividend growth RXY t = equity premium, calculated using earnings growth RXY t = equity premium, calculated using earnings growth RX t = actual equity premium (= R t – F t ) RX t = actual equity premium (= R t – F t )

Linearisation of RVF h t+1  ln(1+H t+1 ) = ln[(P t+1 + D t+1 )/P t ] h t+1  ln(1+H t+1 ) = ln[(P t+1 + D t+1 )/P t ] h t+1 ≈  p t+1 – p t + (1-  )d t+1 + k h t+1 ≈  p t+1 – p t + (1-  )d t+1 + k where p t = ln(P t ) where p t = ln(P t ) and  = Mean(P) / [Mean(P) + Mean(D)] and  = Mean(P) / [Mean(P) + Mean(D)]  t = d t – p t  t = d t – p t h t+1 =  t –  t+1 +  d t+1 + k h t+1 =  t –  t+1 +  d t+1 + k Dynamic version of the Gordon Growth model : p t – d t = const. + E t [  j-1 (  d t+j – h t+j )] + lim  j (p t+j -d t+j )

Expected Returns and Price Volatility Expected returns : h t+1 =  h t +  t+1 E t h t+2 =  E t h t+1 (Expected return is persistent) E t h t+j =  j h t (p t – d t ) = [-1/(1 –  )] h t (p t – d t ) = [-1/(1 –  )] h t Example : Example :  = 0.95,  = 0.9  (E t h t+1 ) = 1%  (p t – d t ) = 6.9%

Stock Market Bubbles

Bubbles : Examples South Sea share price bubble 1720s South Sea share price bubble 1720s Tulipmania in the 17 th century Tulipmania in the 17 th century Stock market : 1920s and collapse in 1929 Stock market : 1920s and collapse in 1929 Stock market rise of 1994-2000 and subsequent crash 2000-2003 Stock market rise of 1994-2000 and subsequent crash 2000-2003

Rational Bubbles RVF : P t =  i E t D t+i + B t = P t f + B t (1) RVF : P t =  i E t D t+i + B t = P t f + B t (1) B t is a rational bubble  = 1/(1+k) is the discount factor E t P t+1 = E t [  E t+1 D t+2 +  2 E t+1 D t+3 + … + B t+1 ] = (  E t D t+2 +  2 E t D t+3 + … + E t B t+1 ) = (  E t D t+2 +  2 E t D t+3 + … + E t B t+1 )  [E t D t+1 + E t P t+1 ] =  E t D t+1 + [  2 E t D t+2 +  3 E t D t+3 +…+  E t B t+1 ] = P t f +  E t B t+1 (2) = P t f +  E t B t+1 (2) Contraction between (1) and (2) !

Rational Bubbles (Cont.) Only if E t B t+1 = B t /  = (1+k)B t are the two expression the same. Only if E t B t+1 = B t /  = (1+k)B t are the two expression the same. Hence E t B t+m = B t /  m Hence E t B t+m = B t /  m B t+1 = B t (  ) -1 with probability  B t+1 = B t (  ) -1 with probability  B t+1 = 0 with probability 1-  B t+1 = 0 with probability 1- 

Rational Bubbles (Cont.) Rational bubbles cannot be negative : B t ≥ 0 Rational bubbles cannot be negative : B t ≥ 0 –Bubble part falls faster than share price –Negative bubble ends in zero price –If bubbles = 0, it cannot start again B t+1 –E t B t+1 = 0 –If bubble can start again, its innovation could not be mean zero. Positive rational bubbles (no upper limit on P) Positive rational bubbles (no upper limit on P) –Bubble element becomes increasing part of actual stock price

Rational Bubble (Cont.) Suppose individual thinks bubble bursts in 2030. Suppose individual thinks bubble bursts in 2030. Then in 2029 stock price should only reflect fundamental value (and also in all earlier periods). Then in 2029 stock price should only reflect fundamental value (and also in all earlier periods). Bubbles can only exist if individuals horizon is less than when bubbles is expected to burst Bubbles can only exist if individuals horizon is less than when bubbles is expected to burst Stock price is above fundamental value because individual thinks (s)he can sell at a price higher than paid for. Stock price is above fundamental value because individual thinks (s)he can sell at a price higher than paid for.

Stock Price Volatility

Shiller Volatility Tests RVF under constant (real) returns RVF under constant (real) returns P t =   i E t D t+i +  n E t P t+n P t * =   i D t+i +  n P t+n P t * = P t +  t Var(P t * ) = Var(P t ) + Var(  t ) + 2Cov(  t, P t ) Info. efficiency (orthogonality condition) implies Cov(  t, P t ) = 0 Hence : Var(P t *) = Var(P t ) + Var(  t ) Since : Var(  t ) ≥ 0 Var(P t * ) ≥ Var(P t )

US Actual and Perfect Foresight Stock Price Perfect foresight price (discount rate = real rate) (discount rate = real interest rate) Actual (real) stock price Perfect foresight price (constant discount rate)

Variance Bounds Tests   (P t * )  (P t * )  (P t ) VR (MCS) Dividends Const. disc. Factor 0.1334.7030.626.031.28 Time vary. disc. factor 0.067.7790.476.031.29 Earning Const. disc. Factor 0.2961.6110.476.7063.77 Time vary. disc. factor 0.0484.650.226.7061.44

Valuation : Bonds

Price of a 30 Year Zero- Coupon Bond Over Time Time to maturity Face value = \$1,000, Maturity date = 30 years, i. r. = 10% Price (\$)

Bond Pricing Fair value of bond Fair value of bond = present value of coupons + present value of par value Bond value =  [C/(1+r) t ] + Par Value /(1+r) T Bond value =  [C/(1+r) t ] + Par Value /(1+r) T (see DPV formula) Example : Example : 8%, 30 year coupon paying bond with a par value of \$1,000 paying semi annual coupons.

Bond Prices and Interest Rates Bond price at different interest rates for 8% coupon paying bond, coupons paid semi-annually.

Bond Price and Int. Rate : 8% semi ann. 30 year bond Price Interest Rate

Inverse Relationship between Bond Price and Yields Price Yield to Maturity P y P + P - y -y + Convex function

Yield to Maturity YTM is defined as the ‘discount rate’ which makes the present value of the bond’s payments equal to its price YTM is defined as the ‘discount rate’ which makes the present value of the bond’s payments equal to its price (IRR for investment projects). Example : Consider the 8%, 30 year coupon paying bond whose price is \$1,276.76 Example : Consider the 8%, 30 year coupon paying bond whose price is \$1,276.76 \$1,276.76 =  [(\$40)/(1+r) t ] + \$1,000/(1+r) 60 Solve equation above for ‘r’.

Interest Rate Risk Changes in interest rates affect bond prices Changes in interest rates affect bond prices Interest rate sensitivity Interest rate sensitivity –Increase in bond YTM results in a smaller price decline than the price gain followed by an equal fall in YTM –Prices of long term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds –The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases (interest rate risk is less than proportional to bond maturity). –Interest rate risk is inversely related to the bond’s coupon rate. –Sensitivity of a bond price to a change in its yield is inversely related to YTM at which the bond currently is selling

Duration Duration Duration –has been developed by Macaulay [1938] –is defined as weighted average term to maturity –measures the sensitivity of the bond price to a change in interest rates –takes account of time value of cash flows Formula for calculating duration : Formula for calculating duration : D =  t w t where w t = [CF t /(1+y) t ] / Bond price Properties of duration : Properties of duration : –Duration of portfolio equals duration of individual assets weighted by the proportions invested. –Duration falls as yields rise

Modified Duration Duration can be used to measure the interest rate sensitivity of bonds Duration can be used to measure the interest rate sensitivity of bonds When interest rate change the percentage change in bond prices is proportional to its duration When interest rate change the percentage change in bond prices is proportional to its duration  P/P = -D [(  (1+y)) / (1+y)] Modified duration : D* = D/(1+y) Hence :  P/P = -D*  y

Duration Approximation to Price Changes Price Yield to Maturity P y P + P - y -y + (9.1%) \$ 897.26 YTM = 9%

Summary RVF is used to calculate the fair price of stock and bonds RVF is used to calculate the fair price of stock and bonds For stocks, the Gordon growth model widely used by academics and practitioners For stocks, the Gordon growth model widely used by academics and practitioners Formula can easily amended to accommodate/explain bubbles Formula can easily amended to accommodate/explain bubbles Empirical evidence : excess volatility Empirical evidence : excess volatility Earnings data is better in explaining the large equity premium Earnings data is better in explaining the large equity premium

References Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 10 and 11 Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 10 and 11 Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapters 7, 12, 13 Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapters 7, 12, 13

References Fama, E.F. and French, K.R. (2002) ‘The Equity Premium’, Journal of Finance, Vol. LVII, No. 2, pp. 637- 659 Fama, E.F. and French, K.R. (2002) ‘The Equity Premium’, Journal of Finance, Vol. LVII, No. 2, pp. 637- 659

END OF LECTURE

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