LECTURE 3 :LECTURE 3 :

VALUATION MODELS : VALUATION MODELS : EQUITIES AND BONDSEQUITIES AND BONDS

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

Market price and fair value price Market price and fair value price – Gordon growth model, widely used Gordon growth model, widely used

simplification of the rational valuation simplification of the rational valuation model (RVF) model (RVF)

Are earnings data better than dividend Are earnings data better than dividend information ? information ?

Stock market bubblesStock 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 Duration and modified duration

Discounted Present Discounted Present ValueValue

Rational Valuation Rational Valuation Formula Formula EEttRRt+1t+1 = [E = [EttVVt+1t+1 – V – Vtt + E + EttDDt+1t+1] / V] / Vtt (1.)(1.)

where where

VVtt = value of stock at end of time t = value of stock at end of time t

DDt+1t+1 = dividends paid between t and t+1 = dividends paid between t and t+1

EEtt = expectations operator based on information = expectations operator based on information tt at time t or earlier E(Dat time t or earlier E(Dt+1t+1 | |tt) ) E EttDDt+1t+1

Assume investors expect to earn constant return (= Assume investors expect to earn constant return (= k)k)

EEttRRt+1t+1 = k = k k > 0 k > 0 (2.) (2.)

Rational Valuation Rational Valuation Formula (Cont.)Formula (Cont.) Excess return are ‘fair game’ : Excess return are ‘fair game’ :

EEtt(R(Rt+1t+1 – k | – k |tt) = 0 ) = 0 (3.)(3.)

Using (1.) and (2.) : Using (1.) and (2.) : VVtt = = EEtt(V(Vt+1t+1 + D + Dt+1t+1) ) (4.) (4.)

where where = 1/(1+k) and 0 < = 1/(1+k) and 0 < < 1 < 1

Leading (4.) one period Leading (4.) one period VVt+1t+1 = = EEt+1t+1(V(Vt+2t+2 + D + Dt+2t+2) ) (5.) (5.) EEttVVt+1t+1 = = EEtt(V(Vt+2t+2 + D + Dt+2t+2) ) (6.) (6.)

Rational Valuation Rational Valuation Formula (Cont.) Formula (Cont.) Equation (6.) holds for all periods : Equation (6.) holds for all periods :

EEttVVt+2t+2 = = EEtt(V(Vt+3t+3 + D + Dt+3t+3) ) etc. etc.

Substituting (6.) into (4.) and all other Substituting (6.) into (4.) and all other time periodstime periodsVVtt = E = Ett[[DDt+1t+1 + + 22DDt+2t+2 + + 33DDt+3t+3 + … + + … + nn(D(Dt+nt+n + +

VVt+nt+n)])]

VVtt = E = Et t iiDDt+it+i

Rational Valuation Rational Valuation Formula (Cont.)Formula (Cont.) Assume : Assume :

– Investors at the margin have Investors at the margin have homogeneous expectations homogeneous expectations

(their subjective probability (their subjective probability distribution of fundamental value distribution of fundamental value reflects the ‘true’ underlying reflects the ‘true’ underlying probability). probability).

– Risky arbitrage is instantaneousRisky arbitrage is instantaneous

Special Case of RVF (1) : Special Case of RVF (1) : Expected Div. are Expected Div. are ConstantConstantDDt+1t+1 = D = Dtt + w + wt+1t+1

RE : ERE : EttDDt+jt+j = D = Dtt

PPtt = = (1 + (1 + + + 22 + … )D + … )Dtt = = (1-(1-))-1-1DDtt = (1/k)D = (1/k)Dtt

or Por Ptt/D/Dtt = 1/k = 1/k

or Dor Dtt/P/Ptt = k = k

Prediction : Prediction :

Dividend-price ratio (dividend yield) is constantDividend-price ratio (dividend yield) is constant

Real Dividends : USA, Real Dividends : USA, Annual Data, 1871 - Annual Data, 1871 - 20022002

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Special Case of RVF (2) : Special Case of RVF (2) : Exp. Div. Grow at Constant Exp. Div. Grow at Constant RateRate Also known as the Gordon growth Also known as the Gordon growth

modelmodelDDt+1t+1 = (1+g)D = (1+g)Dtt + w + wt+1t+1

(E(EttDDt+1t+1 – D – Dtt)/D)/Dtt = g = g

EEttDDt+jt+j = (1+g) = (1+g)j j DDtt

PPtt = = ii(1+g)(1+g)i i DDtt

PPtt = [(1+g)D = [(1+g)Dtt]/(k–g) ]/(k–g) with (k - g) > 0 with (k - g) > 0

or Por Ptt = D = Dt+1t+1/(k-g) /(k-g)

Gordon Growth ModelGordon Growth Model

Constant growth dividend discount model is Constant growth dividend discount model is widely used by stock market analysts. widely used by stock market analysts.

Implications : Implications : The stock value will be greater : The stock value will be greater :

… … the larger its expected dividend per sharethe larger its expected dividend per share… … the lower the discount rate (e.g. interest rate)the lower the discount rate (e.g. interest rate)… … the higher the expected growth rate of the higher the expected growth rate of dividendsdividends

Also implies that stock price grows at the same Also implies that stock price grows at the same rate as dividends. rate as dividends.

More Sophisticated More Sophisticated Models : 3 PeriodsModels : 3 Periods

Div

iden

d g

row

th r

ate

Time

High Dividend growth period

Low Dividend growth period

Time-Varying Expected Time-Varying Expected ReturnsReturns Suppose investors require different Suppose investors require different

expected return in each future period. expected return in each future period. EEttRRt+1t+1 = k = kt+1t+1

PPtt = E = Ett [ [t+1t+1DDt+1t+1 + + t+1t+1t+2t+2DDt+2t+2 + … + …

+ … + … t+N-1t+N-1t+Nt+N(D(Dt+Nt+N + + PPt+Nt+N)])]

where where t+it+i = 1/(1+k = 1/(1+kt+it+i))

Using Earnings (Instead Using Earnings (Instead of Dividends)of Dividends)

Price Earnings Ratio Price Earnings Ratio

Total Earnings (per share) = retained earnings Total Earnings (per share) = retained earnings + dividend payments+ dividend payments

E = RE + DE = RE + Dwith with D = pE and RE = (1-p)ED = pE and RE = (1-p)E

p = proportion of earnings paid out as p = proportion of earnings paid out as div. div.

P = V = pEP = V = pE11 / (R – g) / (R – g) or or

P / EP / E11 = p / (R - g) = p / (R - g) (base on the Gordon growth model.) (base on the Gordon growth model.) Note : R, return on equity replaced k (earlier).Note : R, return on equity replaced k (earlier).

Price Earnings Ratio Price Earnings Ratio (Cont.)(Cont.) Important ratio for security valuation is Important ratio for security valuation is

the P/E ratio. the P/E ratio.

Problems : Problems : – forecasting earningsforecasting earnings– forecasting price earnings ratioforecasting price earnings ratio

Riskier stocks will have a lower P/E ratio. Riskier stocks will have a lower P/E ratio.

Industrial P/E Ratios Industrial P/E Ratios Based on EPS Based on EPS ForecastsForecasts

23.1

17.618.8

11.7

22.5

35.6

24.2

15.5

31.9

23.7

33.1

12.5

19.3

15.8

23.5

10.3

19.9

23.7

17.5

12.7

18.5

9.5

13.4

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2000

The Equity Premium The Equity Premium Puzzle (Fama and French, Puzzle (Fama and French, 2002)2002)

FF (2002) : The Equity FF (2002) : The Equity PremiumPremium All variables are in real terms. All variables are in real terms.

A(RA(Rtt) = A(D) = A(Dtt/P/Pt-1t-1) + A(GP) + A(GPtt) )

Two alternative ways to measure returns Two alternative ways to measure returns A(RDA(RDtt) = A(D) = A(Dtt/P/Pt-1t-1) + A(GD) + A(GDtt) ) A(RYA(RYtt) = A(D) = A(Dtt/P/Pt-1t-1) + A(GY) + A(GYtt) )

where where ‘‘A’ stands for averageA’ stands for averageGPGPtt = growth in prices (=p = growth in prices (=ptt/p/pt-1t-1)*(L)*(Lt-1t-1/L/Ltt) – 1)) – 1)GDGDtt = dividend growth (= d = dividend growth (= dtt/d/dt-1t-1)*(L)*(Lt-1t-1/L/Ltt) -1)) -1)GYGYtt = earning growth (= y = earning growth (= ytt/y/yt-1t-1)*(L)*(Lt-1t-1/L/Ltt) -1)) -1)L is the aggregate price index (e.g. CPI)L is the aggregate price index (e.g. CPI)

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

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perc

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FF (2002) : The Equity FF (2002) : The Equity Premium (Cont.)Premium (Cont.)

FFtt RRtt RXDRXDtt RXYRXYtt RXRXtt

Mean of annual values of variablesMean of annual values of variables1872-1872-20002000

3.243.24 8.818.81 3.543.54 NANA 5.575.57

1872-1872-19501950

3.903.90 8.308.30 4.174.17 NANA 4.404.40

1951-1951-20002000

2.192.19 9.629.62 2.552.55 4.324.32 7.437.43

Standard deviation of annual values of Standard deviation of annual values of variablesvariables

1872-1872-20002000

8.488.48 18.0318.03 13.0013.00 NANA 18.5118.51

1872-1872-19501950

10.6310.63 18.7218.72 16.0216.02 NANA 19.5719.57

1951-1951-20002000

2.462.46 17.0317.03 5.625.62 14.0214.02 16.7316.73

FF (2002) : The Equity FF (2002) : The Equity Premium (Cont.)Premium (Cont.) FFtt = risk free rate = risk free rate

RRtt = return on equity = return on equity

RXDRXDtt = equity premium, calculated = equity premium, calculated using dividend growth using dividend growth

RXYRXYtt = equity premium, calculated = equity premium, calculated using earnings growthusing earnings growth

RXRXtt = actual equity premium (= R = actual equity premium (= Rtt – – FFtt) )

Linearisation of RVF Linearisation of RVF

hht+1t+1 ln(1+H ln(1+Ht+1t+1) = ln[(P) = ln[(Pt+1t+1 + D + Dt+1t+1)/P)/Ptt]]

hht+1t+1 ≈ ≈ ppt+1t+1 – p – ptt + (1- + (1-)d)dt+1t+1 + k + k

where pwhere ptt = ln(P = ln(Ptt) ) and and = Mean(P) / [Mean(P) + Mean(D)] = Mean(P) / [Mean(P) + Mean(D)] tt = d = dtt – p – ptt

hht+1t+1 = = tt – – t+1t+1 + + ddt+1t+1 + k + k

Dynamic version of the Gordon Growth model : Dynamic version of the Gordon Growth model :

pptt – d – dtt = const. + E = const. + Et t [[j-1j-1((ddt+jt+j – h – ht+jt+j)] + lim )] + lim jj(p(pt+jt+j-d-dt+jt+j))

Expected Returns and Expected Returns and Price VolatilityPrice Volatility

Expected returns : Expected returns :

hht+1t+1 = = hhtt + + t+1t+1

EEtthht+2t+2 = = EEtthht+1t+1 (Expected return is persistent)(Expected return is persistent)

EEtthht+jt+j = = jjhhtt

(p(ptt – d – dtt) = [-1/(1 – ) = [-1/(1 – )] h)] htt

Example : Example : = 0.95, = 0.95, = 0.9 = 0.9

(E(Etthht+1t+1) = 1%) = 1% (p(ptt – d – dtt) = 6.9%) = 6.9%

Stock Market BubblesStock Market Bubbles

Bubbles : ExamplesBubbles : Examples

South Sea share price bubble South Sea share price bubble 1720s1720s

Tulipmania in the 17Tulipmania in the 17thth century century Stock market : 1920s and Stock market : 1920s and

collapse in 1929collapse in 1929 Stock market rise of 1994-2000 Stock market rise of 1994-2000

and subsequent crash 2000-2003and subsequent crash 2000-2003

Rational Bubbles Rational Bubbles

RVF : PRVF : Ptt = = i i EEttDDt+it+i + B + Btt = P = Pttff + B + Btt (1) (1)

BBtt is a rational bubble is a rational bubble = 1/(1+k) is the discount factor= 1/(1+k) is the discount factor

EEttPPt+1t+1 = E = Ett[[EEt+1t+1DDt+2t+2 + + 22EEt+1t+1DDt+3t+3 + … + B + … + Bt+1t+1] ] = (= (EEttDDt+2t+2 + + 22EEttDDt+3t+3 + … + E + … + EttBBt+1t+1) )

[E[EttDDt+1 t+1 + E+ EttPPt+1t+1] = ] = EEttDDt+1t+1 + [+ [22EEttDDt+2t+2 + + 33EEttDDt+3t+3 +…+ +…+

EEttBBt+1t+1]] = P= Ptt

ff + + EEttBBt+1t+1 (2) (2)

Contraction between (1) and (2) !Contraction between (1) and (2) !

Rational Bubbles Rational Bubbles (Cont.) (Cont.) Only if EOnly if EttBBt+1t+1 = B = Btt// = (1+k)B = (1+k)Btt are are

the two expression the same. the two expression the same. Hence EHence EttBBt+mt+m = B = Btt//mm

BBt+1t+1 = B = Btt(())-1-1 with probability with probability

BBt+1t+1 = 0 = 0 with probability 1-with probability 1-

Rational Bubbles Rational Bubbles (Cont.)(Cont.)

Rational bubbles cannot be negative : BRational bubbles cannot be negative : Btt ≥ 0 ≥ 0– Bubble part falls faster than share price Bubble part falls faster than share price – Negative bubble ends in zero priceNegative bubble ends in zero price

– If bubbles = 0, it cannot start again BIf bubbles = 0, it cannot start again Bt+1t+1–E–EttBBt+1t+1 = 0 = 0

– If bubble can start again, its innovation could not be If bubble can start again, its innovation could not be mean zero. 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 Bubble element becomes increasing part of actual

stock price stock price

Rational Bubble Rational Bubble (Cont.)(Cont.) Suppose individual thinks bubble bursts in Suppose individual thinks bubble bursts in

2030. 2030. Then in 2029 stock price should only reflect Then in 2029 stock price should only reflect

fundamental value (and also in all earlier fundamental value (and also in all earlier periods). periods).

Bubbles can only exist if individuals horizon is Bubbles can only exist if individuals horizon is less than when bubbles is expected to burstless than when bubbles is expected to burst

Stock price is above fundamental value Stock price is above fundamental value because individual thinks (s)he can sell at a because individual thinks (s)he can sell at a price higher than paid for. price higher than paid for.

Stock Price VolatilityStock Price Volatility

Shiller Volatility Tests Shiller Volatility Tests

RVF under constant (real) returnsRVF under constant (real) returnsPPtt = = i i EEttDDt+it+i + + n n EEttPPt+nt+n

PPtt** = = i i DDt+it+i + + n n PPt+nt+n

PPtt** = P = Ptt + + tt

Var(PVar(Ptt**) = Var(P) = Var(Ptt) + Var() + Var(tt) + 2Cov() + 2Cov(tt, P, Ptt))

Info. efficiency (orthogonality condition) implies Cov(Info. efficiency (orthogonality condition) implies Cov(tt, P, Ptt) = 0) = 0

Hence : Hence : Var(PVar(Ptt*) = Var(P*) = Var(Ptt) + Var() + Var(tt))Since : Since : Var(Var(tt) ≥ 0 ) ≥ 0

Var(PVar(Ptt**) ≥ Var(P) ≥ Var(Ptt) )

US Actual and Perfect US Actual and Perfect Foresight Stock Price Foresight Stock Price

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Perfect foresight price Perfect foresight price (discount rate = real (discount rate = real interest rate) rate)

Actual (real) stock price

Perfect foresight price (constant discount rate)

Variance Bounds TestsVariance Bounds Tests

(P(Ptt**)) (P(Ptt

**)) (P(Ptt)) VR VR (MCS)(MCS)

DividendsDividendsConst. Const. disc. disc. FactorFactor

0.1330.133 4.7034.703 0.620.62 6.036.03 1.281.28

Time vary. Time vary. disc. factor disc. factor

0.060.06 7.7797.779 0.470.47 6.036.03 1.291.29

EarningEarningConst. Const. disc. disc. FactorFactor

0.2960.296 1.6111.611 0.470.47 6.7066.706 3.773.77

Time vary. Time vary. disc. factor disc. factor

0.0480.048 4.654.65 0.220.22 6.7066.706 1.441.44

Valuation : BondsValuation : Bonds

Price of a 30 Year Price of a 30 Year Zero-Coupon Bond Zero-Coupon Bond Over Time Over Time

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Time to maturity

Face value = $1,000, Maturity date = 30 years, i. r. = 10%

Pri

ce (

$)

Bond PricingBond Pricing

Fair value of bondFair value of bond

= present value of coupons = present value of coupons

+ present value of par value+ present value of par value Bond value = Bond value = [C/(1+r)[C/(1+r)tt] + Par Value ] + Par Value

/(1+r)/(1+r)TT

(see DPV formula)(see DPV formula) Example : Example :

8%, 30 year coupon paying bond with a par 8%, 30 year coupon paying bond with a par value of $1,000 paying semi annual coupons. value of $1,000 paying semi annual coupons.

Bond Prices and Bond Prices and Interest RatesInterest Rates

Bond Price at given market interest rate

Time to maturity

4% 6% 8% 10% 12%

1 year 1,038.83 1,019.13 1,000 981.41 963.33

10 years 1,327.03 1,148.77 1,000 875.38 770.60

20 years 1.547.11 1,231.15 1,000 828.41 699.07

30 years 1,695.22 1,276.76 1,000 810.71 676.77

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

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

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Pri

ce

Interest Rate

Inverse Relationship Inverse Relationship between Bond Price and between Bond Price and YieldsYieldsPrice

Yield to Maturity

P

y

P +

P -

y - y +

Convex function

Yield to MaturityYield to Maturity

YTM is defined as the ‘discount rate’ which YTM is defined as the ‘discount rate’ which makes the present value of the bond’s makes the present value of the bond’s payments equal to its pricepayments equal to its price

(IRR for investment projects). (IRR for investment projects). Example : Consider the 8%, 30 year coupon Example : Consider the 8%, 30 year coupon

paying bond whose price is $1,276.76 paying bond whose price is $1,276.76 $1,276.76 = $1,276.76 = [($40)/(1+r) [($40)/(1+r)tt] + $1,000/(1+r)] + $1,000/(1+r)6060

Solve equation above for ‘r’. Solve equation above for ‘r’.

Interest Rate RiskInterest 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 Increase in bond YTM results in a smaller price decline than the price gain followed by an equal fall in YTMprice gain followed by an equal fall in YTM

– Prices of long term bonds tend to be more sensitive to interest Prices of long term bonds tend to be more sensitive to interest rate changes than prices of short-term bondsrate changes than prices of short-term bonds

– The sensitivity of bond prices to changes in yields increases at The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases (interest rate risk is a decreasing rate as maturity increases (interest rate risk is less than proportional to bond maturity). less than proportional to bond maturity).

– Interest rate risk is inversely related to the bond’s coupon rate. 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 Sensitivity of a bond price to a change in its yield is inversely

related to YTM at which the bond currently is sellingrelated to YTM at which the bond currently is selling

DurationDuration

Duration Duration – has been developed by Macaulay [1938]has been developed by Macaulay [1938]– is defined as weighted average term to maturityis defined as weighted average term to maturity– measures the sensitivity of the bond price to a change in measures the sensitivity of the bond price to a change in

interest ratesinterest rates– takes account of time value of cash flowstakes account of time value of cash flows

Formula for calculating duration :Formula for calculating duration :

D = D = t wt wtt where w where wtt = [CF = [CFtt/(1+y)/(1+y)tt] / Bond price] / Bond price Properties of duration : Properties of duration :

– Duration of portfolio equals duration of individual assets Duration of portfolio equals duration of individual assets weighted by the proportions invested. weighted by the proportions invested.

– Duration falls as yields riseDuration falls as yields rise

Modified DurationModified Duration

Duration can be used to measure the Duration can be used to measure the interest rate sensitivity of bondsinterest rate sensitivity of bonds

When interest rate change the When interest rate change the percentage change in bond prices is percentage change in bond prices is proportional to its duration proportional to its duration

P/P = -D [(P/P = -D [((1+y)) / (1+y)](1+y)) / (1+y)]

Modified duration : D* = D/(1+y)Modified duration : D* = D/(1+y)

Hence : Hence : P/P = -D* P/P = -D* yy

Duration Duration Approximation to Price Approximation to Price ChangesChanges

Price

Yield to Maturity

P

y

P +

P -

y - y +(9.1%)

$ 897.26YTM = 9%

SummarySummary

RVF is used to calculate the fair price of RVF is used to calculate the fair price of stock and bonds stock and bonds

For stocks, the Gordon growth model For stocks, the Gordon growth model widely used by academics and widely used by academics and practitionerspractitioners

Formula can easily amended to Formula can easily amended to accommodate/explain bubblesaccommodate/explain bubbles

Empirical evidence : excess volatility Empirical evidence : excess volatility Earnings data is better in explaining the Earnings data is better in explaining the

large equity premium large equity premium

References References

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial (2004) ‘Quantitative Financial Economics’, Chapters 10 and 11Economics’, Chapters 10 and 11

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and (2001) ‘Investments : Spot and Derivatives Markets’, Chapters 7, Derivatives Markets’, Chapters 7, 12, 1312, 13

ReferencesReferences

Fama, E.F. and French, K.R. Fama, E.F. and French, K.R. (2002) ‘The Equity Premium’, (2002) ‘The Equity Premium’, Journal of FinanceJournal of Finance, Vol. LVII, No. , Vol. LVII, No. 2, pp. 637-6592, pp. 637-659

END OF LECTUREEND OF LECTURE