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

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Lecture 3 valuation models equities and bonds

LECTURE 3 :VALUATION MODELS : EQUITIES AND BONDS

(Asset Pricing and Portfolio Theory)


Contents
Contents

  • 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 ?

  • Stock market bubbles

  • How well does the RVF work ?

  • Pricing bonds – DPV again !

    • Duration and modified duration



Rational valuation formula
Rational Valuation Formula

EtRt+1 = [EtVt+1 – Vt + EtDt+1] / Vt (1.)

where

Vt = value of stock at end of time t

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

Et = expectations operator based on information Wt at time t or earlier E(Dt+1 |Wt)  EtDt+1

Assume investors expect to earn constant return (= k)

EtRt+1 = k k > 0 (2.)


Rational valuation formula cont
Rational Valuation Formula (Cont.)

  • Excess return are ‘fair game’ :

    Et(Rt+1 – k |Wt) = 0 (3.)

  • Using (1.) and (2.) :

    Vt = dEt(Vt+1 + Dt+1) (4.)

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

  • Leading (4.) one period

    Vt+1 = dEt+1(Vt+2 + Dt+2) (5.)

    EtVt+1 = dEt(Vt+2 + Dt+2) (6.)


Rational valuation formula cont1
Rational Valuation Formula (Cont.)

  • Equation (6.) holds for all periods :

    EtVt+2 = dEt(Vt+3 + Dt+3)

    etc.

  • Substituting (6.) into (4.) and all other time periods

    Vt = Et[dDt+1 + d2Dt+2 + d3Dt+3 + … + dn(Dt+n + Vt+n)]

    Vt = Et S diDt+i


Rational valuation formula cont2
Rational Valuation Formula (Cont.)

  • 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
Special Case of RVF (1) : Expected Div. are Constant

Dt+1 = Dt + wt+1

RE : EtDt+j = Dt

Pt = d(1 + d + d2 + … )Dt = d(1-d)-1Dt = (1/k)Dt

or Pt/Dt = 1/k

or Dt/Pt = k

Prediction :

Dividend-price ratio (dividend yield) is constant



Special case of rvf 2 exp div grow at constant rate
Special Case of RVF (2) : Exp. Div. Grow at Constant Rate

  • Also known as the Gordon growth model

    Dt+1 = (1+g)Dt + wt+1

    (EtDt+1 – Dt)/Dt = g

    EtDt+j = (1+g)j Dt

    Pt = Sdi(1+g)i Dt

    Pt = [(1+g)Dt]/(k–g) with (k - g) > 0

    or Pt = Dt+1/(k-g)


Gordon growth model
Gordon Growth Model

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

  • 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
More Sophisticated Models : 3 Periods

High Dividend growth period

Low Dividend growth

period

Dividend growth rate

Time


Time varying expected returns
Time-Varying Expected Returns

  • Suppose investors require different expected return in each future period.

  • EtRt+1 = kt+1

  • Pt = Et [dt+1Dt+1 + dt+1dt+2Dt+2 + …

    + … dt+N-1dt+N(Dt+N + Pt+N)]

    where dt+i = 1/(1+kt+i)



Price earnings ratio
Price Earnings Ratio

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

  • E = RE + D

    with D = pE and RE = (1-p)E

    p = proportion of earnings paid out as div.

    P = V = pE1 / (R – g)

    or

    P / E1 = p / (R - g)

    (base on the Gordon growth model.)

    Note : R, return on equity replaced k (earlier).


Price earnings ratio cont
Price Earnings Ratio (Cont.)

  • Important ratio for security valuation is the P/E ratio.

  • Problems :

    • forecasting earnings

    • forecasting price earnings ratio

      Riskier stocks will have a lower P/E ratio.




Ff 2002 the equity premium
FF (2002) : The Equity Premium

  • All variables are in real terms.

    A(Rt) = A(Dt/Pt-1) + A(GPt)

  • Two alternative ways to measure returns

    A(RDt) = A(Dt/Pt-1) + A(GDt)

    A(RYt) = A(Dt/Pt-1) + A(GYt)

    where ‘A’ stands for average

    GPt = growth in prices (=pt/pt-1)*(Lt-1/Lt) – 1)

    GDt = dividend growth (= dt/dt-1)*(Lt-1/Lt) -1)

    GYt = earning growth (= yt/yt-1)*(Lt-1/Lt) -1)

    L is the aggregate price index (e.g. CPI)




Ff 2002 the equity premium cont1
FF (2002) : The Equity Premium (Cont.)

  • Ft = risk free rate

  • Rt = return on equity

  • RXDt = equity premium, calculated using dividend growth

  • RXYt = equity premium, calculated using earnings growth

  • RXt = actual equity premium (= Rt – Ft)


Linearisation of rvf
Linearisation of RVF

  • ht+1 ln(1+Ht+1) = ln[(Pt+1 + Dt+1)/Pt]

  • ht+1 ≈ rpt+1 – pt + (1-r)dt+1 + k

  • where pt = ln(Pt)

  • and r = Mean(P) / [Mean(P) + Mean(D)]

  • dt = dt – pt

  • ht+1 = dt – rdt+1 + Ddt+1 + k

    Dynamic version of the Gordon Growth model :

    pt – dt = const. + Et [Srj-1(Ddt+j – ht+j)] + lim rj(pt+j-dt+j)


Expected returns and price volatility
Expected Returns and Price Volatility

Expected returns :

ht+1 = fht + et+1

Etht+2 = fEtht+1(Expected return is persistent)

Etht+j = fjht

  • (pt – dt) = [-1/(1 – rf)] ht

  • Example :

    r = 0.95, f = 0.9

    s(Etht+1) = 1% s(pt – dt) = 6.9%



Bubbles examples
Bubbles : Examples

  • South Sea share price bubble 1720s

  • Tulipmania in the 17th century

  • Stock market : 1920s and collapse in 1929

  • Stock market rise of 1994-2000 and subsequent crash 2000-2003


Rational bubbles
Rational Bubbles

  • RVF : Pt = Sdi EtDt+i + Bt = Ptf + Bt (1)

    Bt is a rational bubble

    d = 1/(1+k) is the discount factor

    EtPt+1 = Et[dEt+1Dt+2 + d2Et+1Dt+3 + … + Bt+1]

    = (dEtDt+2 + d2EtDt+3 + … + EtBt+1)

    d[EtDt+1 + EtPt+1] = dEtDt+1

    + [d2EtDt+2 + d3EtDt+3 +…+ dEtBt+1]

    = Ptf + dEtBt+1 (2)

    Contraction between (1) and (2) !


Rational bubbles cont
Rational Bubbles (Cont.)

  • Only if EtBt+1 = Bt/d = (1+k)Bt are the two expression the same.

  • Hence EtBt+m = Bt/dm

  • Bt+1 = Bt(dp)-1 with probability p

  • Bt+1 = 0 with probability 1-p


Rational bubbles cont1
Rational Bubbles (Cont.)

  • Rational bubbles cannot be negative : Bt ≥ 0

    • Bubble part falls faster than share price

    • Negative bubble ends in zero price

    • If bubbles = 0, it cannot start again Bt+1–EtBt+1 = 0

    • If bubble can start again, its innovation could not be mean zero.

  • Positive rational bubbles (no upper limit on P)

    • Bubble element becomes increasing part of actual stock price


Rational bubble cont
Rational Bubble (Cont.)

  • Suppose individual thinks bubble bursts in 2030.

  • 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

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



Shiller volatility tests
Shiller Volatility Tests

  • RVF under constant (real) returns

    Pt = Sdi EtDt+i + dn EtPt+n

    Pt* = Sdi Dt+i + dn Pt+n

    Pt* = Pt + ht

    Var(Pt*) = Var(Pt) + Var(ht) + 2Cov(ht, Pt)

    Info. efficiency (orthogonality condition) implies Cov(ht, Pt) = 0

    Hence : Var(Pt*) = Var(Pt) + Var(ht)

    Since : Var(ht) ≥ 0

    Var(Pt*) ≥ Var(Pt)


Us actual and perfect foresight stock price
US Actual and Perfect Foresight Stock Price

Actual (real) stock price

Perfect foresight price

(discount rate = real interest rate)

Perfect foresight price

(constant discount rate)




Price of a 30 year zero coupon bond over time
Price of a 30 Year Zero-Coupon Bond Over Time

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

Price ($)

Time to maturity


Bond pricing
Bond Pricing

  • Fair value of bond

    = present value of coupons

    + present value of par value

  • Bond value = S[C/(1+r)t] + Par Value /(1+r)T

    (see DPV formula)

  • Example :

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


Bond prices and interest rates
Bond Prices and Interest Rates

Bond price at different interest rates

for 8% coupon paying bond, coupons paid semi-annually.



Inverse relationship between bond price and yields
Inverse Relationship between Bond Price and Yields

Price

Convex function

P +

P

P -

y -

y

y +

Yield to Maturity


Yield to maturity
Yield to Maturity

  • 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

    $1,276.76 = S [($40)/(1+r)t] + $1,000/(1+r)60

    Solve equation above for ‘r’.


Interest rate risk
Interest Rate Risk

  • Changes in interest rates affect bond prices

  • 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 :

    D = S t wt where wt = [CFt/(1+y)t] / Bond price

  • Properties of duration :

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

    • Duration falls as yields rise


Modified duration
Modified Duration

  • 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

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

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

    Hence : DP/P = -D* Dy


Duration approximation to price changes
Duration Approximation to Price Changes

Price

P +

$ 897.26

YTM = 9%

P

P -

y -

y

y +

(9.1%)

Yield to Maturity


Summary
Summary

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

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

  • Formula can easily amended to accommodate/explain bubbles

  • Empirical evidence : excess volatility

  • Earnings data is better in explaining the large equity premium


References
References

  • 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


References1
References

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



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