LECTURE 3 : VALUATION MODELS : EQUITIES AND BONDS

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LECTURE 3 VALUATION MODELS EQUITIES AND
BONDS

• (Asset Pricing and Portfolio Theory)

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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

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Discounted Present Value
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Rational Valuation Formula

• EtRt1 EtVt1 Vt EtDt1 / Vt (1.)
• where
• Vt value of stock at end of time t
• Dt1 dividends paid between t and t1
• Et expectations operator based on
  information Wt at time t or earlier E(Dt1 Wt) ?
  EtDt1
• Assume investors expect to earn constant return
  ( k)
• EtRt1 k k gt 0 (2.)

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Rational Valuation Formula (Cont.)

• Excess return are fair game
• Et(Rt1 k Wt) 0 (3.)
• Using (1.) and (2.)
• Vt dEt(Vt1 Dt1) (4.)
• where d 1/(1k) and 0 lt d lt 1
• Leading (4.) one period
• Vt1 dEt1(Vt2 Dt2) (5.)
• EtVt1 dEt(Vt2 Dt2) (6.)

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Rational Valuation Formula (Cont.)

• Equation (6.) holds for all periods
• EtVt2 dEt(Vt3 Dt3)
• etc.
• Substituting (6.) into (4.) and all other time
  periods
• Vt EtdDt1 d2Dt2 d3Dt3 dn(Dtn
  Vtn)
• Vt Et S diDti

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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

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

• Dt1 Dt wt1
• RE EtDtj 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

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Real Dividends USA, Annual Data, 1871 - 2002
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Special Case of RVF (2) Exp. Div. Grow at
Constant Rate

• Also known as the Gordon growth model
• Dt1 (1g)Dt wt1
• (EtDt1 Dt)/Dt g
• EtDtj (1g)j Dt
• Pt S di(1g)i Dt
• Pt (1g)Dt/(kg) with (k - g) gt 0
• or Pt Dt1/(k-g)

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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.

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More Sophisticated Models 3 Periods
High Dividend growth period
Low Dividend growth period
Dividend growth rate
Time
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Time-Varying Expected Returns

• Suppose investors require different expected
  return in each future period.
• EtRt1 kt1
• Pt Et dt1Dt1 dt1dt2Dt2
• dtN-1dtN(DtN PtN)
• where dti 1/(1kti)

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Using Earnings (Instead of Dividends)
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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).

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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.

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Industrial P/E Ratios Based on EPS Forecasts
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The Equity Premium Puzzle (Fama and French, 2002)
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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)

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US Data (1872-2002) Div/P and Earning/P ratios
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FF (2002) The Equity Premium (Cont.)
Ft Rt RXDt RXYt RXt
Mean of annual values of variables Mean of annual values of variables Mean of annual values of variables Mean of annual values of variables Mean of annual values of variables Mean of annual values of variables
1872-2000 3.24 8.81 3.54 NA 5.57
1872-1950 3.90 8.30 4.17 NA 4.40
1951-2000 2.19 9.62 2.55 4.32 7.43
Standard deviation of annual values of variables Standard deviation of annual values of variables Standard deviation of annual values of variables Standard deviation of annual values of variables Standard deviation of annual values of variables Standard deviation of annual values of variables
1872-2000 8.48 18.03 13.00 NA 18.51
1872-1950 10.63 18.72 16.02 NA 19.57
1951-2000 2.46 17.03 5.62 14.02 16.73
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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)

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Linearisation of RVF

• ht1 ? ln(1Ht1) ln(Pt1 Dt1)/Pt
• ht1 rpt1 pt (1-r)dt1 k
• where pt ln(Pt)
• and r Mean(P) / Mean(P) Mean(D)
• dt dt pt
• ht1 dt rdt1 Ddt1 k
• Dynamic version of the Gordon Growth model
• pt dt const. Et Srj-1(Ddtj htj) lim
  rj(ptj-dtj)

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Expected Returns and Price Volatility

• Expected returns
• ht1 fht et1
• Etht2 fEtht1 (Expected return is
  persistent)
• Ethtj fjht
• (pt dt) -1/(1 rf) ht
• Example
• r 0.95, f 0.9
• s(Etht1) 1 s(pt dt) 6.9

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Stock Market Bubbles
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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

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Rational Bubbles

• RVF Pt Sdi EtDti Bt Ptf Bt (1)
• Bt is a rational bubble
• d 1/(1k) is the discount factor
• EtPt1 EtdEt1Dt2 d2Et1Dt3 Bt1
• (dEtDt2 d2EtDt3 EtBt1)
• dEtDt1 EtPt1 dEtDt1
• d2EtDt2 d3EtDt3 dEtBt1
• Ptf dEtBt1 (2)
• Contraction between (1) and (2) !

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Rational Bubbles (Cont.)

• Only if EtBt1 Bt/d (1k)Bt are the two
  expression the same.
• Hence EtBtm Bt/dm
• Bt1 Bt(dp)-1 with probability p
• Bt1 0 with probability 1-p

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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 Bt1EtBt1
  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

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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.

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Stock Price Volatility
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Shiller Volatility Tests

• RVF under constant (real) returns
• Pt S di EtDti dn EtPtn
• Pt S di Dti dn Ptn
• 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)

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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)
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Variance Bounds Tests
r s(Pt) rs(Pt) s(Pt) VR (MCS)
Dividends Dividends Dividends Dividends Dividends Dividends
Const. disc. Factor 0.133 4.703 0.62 6.03 1.28
Time vary. disc. factor 0.06 7.779 0.47 6.03 1.29
Earning Earning Earning Earning Earning Earning
Const. disc. Factor 0.296 1.611 0.47 6.706 3.77
Time vary. disc. factor 0.048 4.65 0.22 6.706 1.44
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Valuation Bonds
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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
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Bond Pricing

• Fair value of bond
• present value of coupons
• present value of par value
• Bond value SC/(1r)t Par Value /(1r)T
• (see DPV formula)
• Example
• 8, 30 year coupon paying bond with a par value
  of 1,000 paying semi annual coupons.

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Bond Prices and Interest Rates
Bond price at different interest rates for 8
coupon paying bond, coupons paid semi-annually.
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Bond Price and Int. Rate 8 semi ann. 30 year
bond
Price
Interest Rate
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Inverse Relationship between Bond Price and Yields
Price
Convex function
P
P
P -
y
y -
y
Yield to Maturity
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Yield to Maturity

• YTM is defined as the discount rate which makes
  the present value of the bonds 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)/(1r)t 1,000/(1r)60
• Solve equation above for r.

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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
  bonds 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

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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/(1y)t / Bond
  price
• Properties of duration
• Duration of portfolio equals duration of
  individual assets weighted by the proportions
  invested.
• Duration falls as yields rise

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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(1y)) / (1y)
• Modified duration D D/(1y)
• Hence DP/P -D Dy

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Duration Approximation to Price Changes
Price
P
897.26 YTM 9
P
P -
Yield to Maturity
y
y -
y (9.1)
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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

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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

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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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END OF LECTURE