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LECTURE 3 : VALUATION MODELS : EQUITIES AND BONDSPowerPoint Presentation

LECTURE 3 : VALUATION MODELS : EQUITIES AND BONDS

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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 ?
- Stock market bubbles
- How well does the RVF work ?
- Pricing bonds – DPV again !
- Duration and modified duration

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 (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 (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

- Investors at the margin have homogeneous expectations

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

- 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

- 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

High Dividend growth period

Low Dividend growth

period

Dividend growth rate

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)

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.

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 (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, f = 0.9

s(Etht+1) = 1% s(pt – dt) = 6.9%

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 with probability p
- Bt+1 = 0 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.

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

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

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

Price ($)

Time to maturity

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.

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

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