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 ?
Stock market bubbles
How well does the RVF work ?
Pricing bonds – DPV again !
– Duration and modified duration
Discounted Present Value
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.)

Excess return are ‘fair game’ :
Et(Rt+1 – k |Wt) = 0

Using (1.) and (2.) :
Vt = dEt(Vt+1 + Dt+1)
where d = 1/(1+k) and 0 < d < 1

Leading (4.) one period
Vt+1 = dEt+1(Vt+2 + Dt+2)
EtVt+1 = dEt(Vt+2 + Dt+2)
(3.)
(4.)
(5.)
(6.)
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 (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
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
Real Dividends : USA,
Annual Data, 1871 - 2002
20
18
16
14
12
10
8
6
4
2
0
1860
1880
1900
1920
1940
1960
1980
2000
2020
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 = S di(1+g)i Dt
Pt = [(1+g)Dt]/(k–g)
or
with (k - g) > 0
Pt = Dt+1/(k-g)
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.
Dividend growth rate
More Sophisticated
Models : 3 Periods
High Dividend growth period
Low Dividend growth
period
Time
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)
Using Earnings (Instead of
Dividends)
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.)


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.
I/B
/E
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/S
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Ba
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Industrial P/E Ratios
Based on EPS Forecasts
40
35
35.6
31.9
25
20
15
10
33.1
30
27
23.1
22.5
24.2
18.5 17.6
18.8
15.8
12.7
11.7
23.7 23.7
19.3
13.4
23.5
19.9
17.5
15.5
1999
2000
9.5
12.5
5
0
10.3
The Equity Premium Puzzle
(Fama and French, 2002)
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)
US Data (1872-2002) :
Div/P and Earning/P ratios
18
16
14
percent
12
10
8
6
4
2
0
1865
1885
1905
1925
1945
1965
1985
2005
FF (2002) : The Equity
Premium (Cont.)
Ft
Rt
RXDt
RXYt
RXt
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
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
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






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 :
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,
s(Etht+1) = 1%
f = 0.9
s(pt – dt) = 6.9%
Stock Market Bubbles
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

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.)




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
Bt+1 = 0
with probability p
with probability 1-p
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.)




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.
Stock Price Volatility
Shiller Volatility Tests

RVF under constant (real) returns
Pt = S di EtDt+i + dn EtPt+n
Pt* = S di 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 :
Since :
Var(Pt*) = Var(Pt) + Var(ht)
Var(ht) ≥ 0
Var(Pt*) ≥ Var(Pt)
US Actual and Perfect
Foresight Stock Price
700
Actual (real) stock price
600
500
Perfect foresight price
(discount rate = real interest rate)
400
300
200
Perfect foresight price
(constant discount rate)
100
0
1860
1880
1900
1920
1940
1960
1980
2000
Variance Bounds Tests
r
s(Pt*) rs(Pt*)
s(Pt)
VR
(MCS)
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
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
Valuation : Bonds
Price of a 30 Year ZeroCoupon Bond Over Time
Face value = $1,000, Maturity date = 30 years, i. r. = 10%
1000
900
Price ($)
800
700
600
500
400
300
200
100
0
0
5
10
15
20
Time to maturity
25
30
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 price at different interest rates
for 8% coupon paying bond, coupons paid semi-annually.
Bond Price at given market interest rate
Time to
maturity
10%
12%
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
1 year
4%
6%
8%
Bond Price and Int. Rate :
8% semi ann. 30 year bond
3000
2500
Price
2000
1500
1000
500
0
0
0.02
0.04
0.06
0.08
Interest Rate
0.1
0.12
0.14
Inverse Relationship between
Bond Price and Yields
Price
Convex function
P+
P
P-
y- y
y+
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


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
– 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


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
Price
P+
$ 897.26
YTM = 9%
P
P-
y- y
y+
(9.1%)
Yield to Maturity
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


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
References

Fama, E.F. and French, K.R. (2002)
‘The Equity Premium’, Journal of
Finance, Vol. LVII, No. 2, pp. 637-659
END OF LECTURE