ñòð. 10 |

+ + hkbik

E(Ri,) = ho hlbil ***

Comparing (3.88) with regressions with unrestricted constant terms a, (a

ho = r):

+ +

(Ri, - r l ) = ai CbijFj, &it

Comparing (3.88) and (3.89) we see that there are non-linear restrictions,

in (3.89), if the APT is the correct model. In the combined time series cr

regressions we can apply non-linear least squares to (3.88) and test these restri

hs and bijs are jointly estimated in (3.88) and results on US portfolios (Mc

(1985)) suggest that the restrictions do not hold.

The key issues in testing and finding an acceptable empirical APT model a

how to measure the â€˜newsâ€™ factors, F j f . Should one use first differences o

0

ables, or residuals from single equation regressions or VAR models, and

allow the coefficients in such models to vary, in order to mimic learning

(e.g. recursive least squares, time varying parameter models)?

are the set of factors Fjt and the resulting values of hj constant over differ

0

periods and across different portfolios (e.g. by size or by industry)? If

different in different sample periods then the price of risk for factor j is tim

(contrary to the theory) and the returns equation for R; is non-unique

Although there has been considerable progress in estimating and testing th

empirical evidence on the above issues is far from definitive.

36 SUMMARY

.

This chapter has extended our analysis of the CAPM and analysed an alterna

of equilibrium returns, the APT. It explained how the concepts behind the C

be used to construct alternative performance indicators and briefly looked at

empirical tests of the CAPM and tests of the APT. In addition, it showed how

variance approach can be used to derive an individualâ€™s asset demand function

compared this approach to the CAPM.

4

I

Valuation Models

In this chapter we look at models that seek to determine how investors in

decide what is the correct, fundamental or fair value V, for a particular sto

decided on what is the fair value for the stock, market participants (in an efficie

should set the actual price P , equal to the fundamental value. If P , # V, then u

profit opportunities exist in the market. For example, if P , < V, then risk neutra

would buy the share since they anticipate they will make a capital gain as P , ris

its â€˜correct valueâ€™ in the future. As investors purchase the share with Pi < V

would tend to lead to a rise in the current price as demand increases, so that

moves tawards its fundamental value. The above, of course, assumes that inve

margin have homogeneous expectations or more precisely that their subjective

probability distribution of the fundamental value reflects the â€˜trueâ€™ underlying d

A key proposition running through this chapter is that stock returns and s

are closely linked. Indeed, alternative models of expected returns give rise t

expressions for the determination of fundamental yalue and hence stock prices

themes covered in the chapter are as follows:

We begin with a simple model where equilibrium expected stock returns a

to be constant and this implies that stock prices equal the discounted pre

(DPV) of future dividends, with a constant discount rate. This express

further simplified if one also assumes either a constant level of dividends or

dividend growth rate.

We explore the implications for the determination of stock prices when e

expected returns are assumed to vary over time and then show how the ris

of the CAPM helps to determine a (possibly time varying) discount rate i

formula for stock prices.

We discuss a somewhat more general CAPM than that discussed in Chap

3 which is known as the consumption CAPM (or C-CAPM). This model i

poral, in that investors are assumed to maximise expected utility of current

consumption. Financial assets allow the consumer to smooth his consumpt

over time, selling assets to finance consumption in â€˜badâ€™ times and saving

times. Assets whose returns have a high negative covariance with consumpt

willingly held even though they have low expected returns. This is becaus

be â€˜cashed inâ€™ at a time when they are most needed, namely when consump

4.1 THE RATIONAL VALUATION FORMULA (RV

Expected Returns are Constant

One of the simplest assumptions one can make is that expected returns are co

expected return is defined as

where V, is the value of the stock at the end of time t and Dr+l are divi

+

between t and t 1. E, is the expectations operator based on information a

earlier, S2,. The superscript â€˜eâ€™ is equivalent to E,; it helps to simplify the not

used when no ambiguity is likely to arise, that is:

Assume investors are willing to hold the stock as long as it is expected to earn

return (= k). We can think of this â€˜required returnâ€™ k as that rate of return

sufficient to compensate investors for the inherent riskiness of the stock:

The form of (4.2) is known as the fair game property of excess returns (see

The stochastic behaviour of R,+l - k is such that no abnormal returns are

average: the expected (conditional) excess return on the stock is zero:

Using (4.1) and (4.2) we obtain a differential (Euler) equation which dete

movement in â€˜valueâ€™ over time:

+ k) with 0 < 6 < 1. Leading (4.4) one per

where 6 = discount factor = 1/(1

Now take expectations of (4.5) assuming information is only available up to

In deriving (4.6) we have used the law of iterated expectations:

The expectation formed today (t) of what oneâ€™s expectation will be tomorr

+

of the value at t 2 is the LHS of (4.7). This simply equals oneâ€™s expectatio

Vf+2 (i.e. the RHS of (4.7)) since one cannot know how one will alter oneâ€™s e

The next part of the solution requires substitution of (4.8) in (4.6):

By successive substitution

Now let N + 00 and hence aN -+ 0. If the expected growth in D is not explo

D;+N is finite and if V;+N is also finite then:

Equation (4.10) is known as a terminal condition or transversality condition

out rational speculative bubbles (see Chapter 7). Equation (4.9) then becomes

Vl = 6'D;+,

i=l

+ k ) we have derived (4.11) under the assumptions:

Where 6 = 1/(1

expected returns are constant

0

the law of iterated expectations (i.e. RE) holds for all investors

0

dividend growth is not explosive and the terminal condition holds

0

all investors have the same view (model) of the determinants of return

0

homogeneous expectations

Hence the correct or fundamental value V , of a share is the DPV of expe

dividends. If we add the assumption that

investors instantaneously set the actual market price P, equal to fundament

0

then we obtain the rational valuation formula (RVF) for stock prices with

If investors ensure (4.12) holds at all times then the Euler equation (4.6) an

formula can be expressed in terms of P I , the actual price (rather than V t ) ,

simplifies the notation, we use this whenever possible.

In the above analysis we did not distinguish between real and nominal va

indeed the mathematics goes through for either case: hence (4.12) is true w

variables are nominal or are all deflated by an aggregate goods price index. N

intuitive reasoning and causal empiricism suggest that expected real return

likely to be constant than expected nominal returns (not least because of g

If investors have a finite horizon, that is they are concerned with the price they

in the near future, does this alter our view of the determination of fundame

Consider the simple case of an investor with a one-period horizon:

Pf= 6D:+, + S2P;+,

+

The price today depends on the expected price at t 1. But how is this

+

determine the value P:+l at which he can sell at t l? If he is consistent (ra

should determine this in exactly the same way that he does for P,. That is

But by repeated forward induction each investor with a one-period horizon w

that P,+j is determined by the above Euler equation and hence todayâ€™s price

the DPV of dividends in allfutureperiods.

Thus even if some agents have a finite investment horizon they will still c

determine the fundamental value in such a way that it is equal to that of an in

has an infinite horizon. This is the usual counterargument to the view that i

have short horizons then price cannot reflect fundamental value (i.e. short-term

counterargument assumes, of course, that investors are homogeneous, that they a

the underlying equilibrium model of returns given by (4.2) is the true mode

â€˜pushâ€™ price immediately to its equilibrium value. Later in this section the assum

equilibrium returns are constant is relaxed and we find the RVF still holds. I

Chapter 8 allows non-rational agents or noise traders to influence returns and

model price may not equal fundamental value.

Expected Dividends are Constant

Let us simplify the RVF further by assuming a time series model for (real)

which has the property that the best forecast of all future (real) dividends is e

current level of dividends, that is the random walk model:

+ Wf+l

Dr+l = D,

where w,+1 is white noise. Under RE we have E,(w,+jlS2,)= 0 ( j 2 1) and E,

and hence the growth in dividends is expected to be zero. The RVF (4.12) then r

+ S + J2 + * . * ) D = S/(1 - S)D, = ( l / k ) D ,

P, = S(l t

Equation (4.13) predicts that the dividend price (DP) or dividend yield is

ratio

equal to the required (real) return, k. (Note that the ratio D/P is the same w

variables are measured in real or nominal terms.) If the required (real) return

(8 percent per annum) then the zero dividend growth model predicts a constan

price ratio which also equals 8 percent. The model (4.13) also predicts that the

change in stock prices P, equals the percentage change in current dividends. Pric

only occur when new information about dividends becomes available: the mod

dends). The conditional variance of prices therefore depends on the varian

about these fundamentals:

Much later in the book, Chapter 16 examines the more general case where the

prices depends not only on the volatility in dividends but also the volatility in t

rate. In fact, an attempt is made to ascertain whether the volatility in stock price

due to the volatility in dividends, or the discount factor.

Expected Dividend Growth is Constant

A time series model in which (real) dividends grow at a constant rate g is

model:

+

Df+1 = (1 + g)Dr W f + l

where w, is white noise and E(w,+llQ,) = 0. Expected dividend growth fro

easily seen to be equal to g.

Note that if the logarithm of dividends follows a random walk with drift para

then this also gives a constant expected growth rate for dividends (i.e. In D

+

In D, w,). The optimal forecasts of future dividends may be found by lea

one period

+ +

Df+2 = (1 g)Q+1 W f + 2

Hence by repeated substitution:

Substituting the forecast of future dividends from (4.18) in the rationa

formula gives:

00

+ g)â€™D,

P, = 6â€™(1

i=l

which after some simple algebra yields

P, = - â€˜IDf with ( k - g ) > 0

(â€™ +

(k -g)

series properties of dividend payouts is such that it is reasonable (rational) fo

to expect that dividends grow at a constant rate. Equation (4.20) â€˜collapses

when g = 0.

Time Varying Expected Returns

Suppose investors require a different expected return in each future period in

they will willingly hold a particular stock. (Why this may be the case is i

later.) Our model is therefore:

E,R,+l = kf+l

where we have a time subscript on k to indicate it is time varying, Repeating th

steps, involving forward substitution, gives:

+ &+1&+2D;+2 + b+1&+2&+3D;+3+ + & + N E t ( D r + N + P z

Pr = &+1DF+, ***

which can be written in more compact form (assuming the transversality condit

+

where 8,+; = 1/(1 k,+i). The current stock price therefore depends on all fu

tations of the discount rate &+, and all future expected dividends. Note that 0

for all periods and hence expected dividends $-for-$ have less influence on

stock price the further they accrue in the future. However, it is possible (b

unlikely) that an event announced today (e.g. a merger with another company

expected to have a substantial $ impact on dividends starting in say five y

In this case the announcement could have a large effect on the current stock

though it is relatively heavily discounted. Note that in a well-informed (â€˜efficie

one expects the stock price to respond immediately and completely to the ann

even though no dividends will actually be paid for five years. In contrast, if th

inefficient (e.g. noise traders are present) then the price might rise not only in

period but also in subsequent periods. Tests of the stock price response to anno

are known as event studies.

At the moment (4.23) is â€˜non-operationalâ€™ since it involves unobservable ex

terms. We cannot calculate fundamental value (i.e. the RHS of (4.23)) and he

see if it corresponds to the observable current price P,.We need some ancillar

hypotheses about investorsâ€™ forecasts of dividends and the discount rate. It i

straightforward to develop forecasting equations for dividends (and hence prov

ical values for E,D,+j ( j = 1,2, . . .). For example on annual data an AR(1)

model for dividends fits the data quite well. The difficulty arises with the equil

+

of return k,(S, = 1/(1 k , ) ) which is required for investors willingly to hold

There are numerous competing models of equilibrium returns and next it will

how the CAPM can be combined with the rational valuation formula so tha

price is determined by a time varying discount rate.

folio). Merton (1973) developed this idea in an intertemporal framework and s

the (nominal) excess return over the risk-free rate, on the market portfolio, is p

to the expected variance of returns on the market portfolio

The expected return can be defined as comprising a risk-free return p

premium r p t :

+

ER): = r, r p ,

where r p , = h E r a ˜ , + ,Comparing (4.21) and (4.25) we see that according to

.

the required rate of return on the market portfolio is given by:

The equilibrium required return depends positively on the risk-free interest rat

the (non-diversifiable) risk of the market portfolio, as measured by its condition

If either:

agents do not perceive the market as risky (i.e. Eta:,+, = 0), or

0

agents are risk neutral (i.e. h = 0)

0

then the appropriate discount factor used by investors is the risk-free rate, r,. N

determine price using the RVF, investors in general must determine k, and hen

future values of the risk-free rate and the risk premium.

Individual Securities

Consider now the price of an individual security or a portfolio of assets which

of the market portfolio (e.g. shares of either all industrial companies or all bank

The CAPM implies that to be willingly held as part of a diversified portfolio th

(nominal) return on portfolio i is given by:

+ Bit (ErR:+l

EtRit+l = rt - rt )

Substituting from (4.24) for Erair+, have('):

we

+ AEr(oim)t+l

ErRir+l = rt

where a i m is the covariance between returns on asset i and the market portf

comparing (4.21) and (4.29) the equilibrium required rate of return on asset i

The covariance term may be time varying and hence so might the future disco

6 , + j in the RVF for an individual security (or portfolio of securities).

an individual stock as part of a wider portfolio is equal to the risk-free rate plu

for risk or riskpremium r p ,

Summary: The RVF

From the definition of the expected return on a stock it is possible via

equation and rational expectations to derive the rational valuation formula

stock prices.

Stock prices in an efficient and well-informed market are determined by t

expected future dividends and expected future discount rates.

If equilibrium expected returns are constant then the discount rate in t

constant.

If equilibrium (nominal) expected returns are given by the CAPM then th

factor in the RVF may be time varying and depends on the (nominal) ris

and a variance/covariance term.

4.1.2 The Consumption CAPM

In the one-period CAPM the individual investorâ€™s objective function is assu

fully determined by the standard deviation and return on the portfolio. The in

to earn the highest expected return for any given level of portfolio risk.

An alternative view of the determination of equilibrium returns in a well

portfolio is provided by the intertemporal consumption CAPM (denoted

Here, the investor maximises expected utility which depends only on current

consumption (see Lucas (1978) and Mankiw and Shapiro (1986)). Financial as

role in this model in that they help to smooth consumption over time. Securiti

to transfer purchasing power from one period to another. If an agent had no

was not allowed to accumulate assets then his consumption would be determi

current income. If he holds assets, he can sell some of these to finance consum

his current income is low. A n individual asset is therefore more â€˜desirableâ€™ if i

expected to be high when consumption is expected to be low. Thus the system

the asset is determined by the covariance of the assets return with respect to co

(rather than its covariance with respect to the return on the market portfolio

â€˜basicâ€™ CAPM). The C-CAPM is an intertemporal model unlike the basic

CAPM. In the C-CAPM, dividends and prices are all denominated in consum

and hence all analysis is in terms of real variables. In the C-CAPM the individu

maximises

where D,= dividends received (e.g. see Sheffrin, 1983), X , = holdings of s

+

time t which will yield dividend payments at t 1. The first term on the RHS

dividend income and the second term represents receipts from the sale of the

(shares). The individual must choose the amount of consumption today and the

taneously choose the quantity of risky assets to be carried forward into the ne

First-Order Condition

ñòð. 10 |