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LECTURE 3 : VALUATION MODELS : EQUITIES AND BONDS

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LECTURE 3 :VALUATION MODELS : EQUITIES AND BONDS

(Asset Pricing and Portfolio Theory)

- 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

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.)

- 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.)

- 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

- 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

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

- 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)

- 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.

High Dividend growth period

Low Dividend growth

period

Dividend growth rate

Time

- 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)

- 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).

- 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.

The Equity Premium Puzzle (Fama and French, 2002)

- 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)

- 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)

- 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 :

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%

Stock Market Bubbles

- 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

- 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) !

- 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 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

- 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

- 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)

Actual (real) stock price

Perfect foresight price

(discount rate = real interest rate)

Perfect foresight price

(constant discount rate)

Valuation : Bonds

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

Price ($)

Time to maturity

- 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 price at different interest rates

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

Price

Interest Rate

Price

Convex function

P +

P

P -

y -

y

y +

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’.

- 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
- 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

- 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

Price

P +

$ 897.26

YTM = 9%

P

P -

y -

y

y +

(9.1%)

Yield to Maturity

- 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

- 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

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

END OF LECTURE