Valuation Models and Asset Returns We look at models that seek to determine how investors decide what is the fundamental or fair value V t for a particular stock, where the fundamentals are dividends and future required returns. A key idea is that stock returns and stock prices are inextricably linked. If P t V t, then unexploited profit opportunities exist in the market. If P t < V t, then investors would buy the share since they anticipate that they will make a capital gain as P t rises towards its correct value in the future. 1
As investors purchase the share with P t < V t, this would tend to lead to a rise in the current price as demand increases, so that it quickly moves towards its fundamental value. Now assume the investors subjective view of the probability distribution of fundamental value reflects the true underlying distribution; risky arbitrage is instantaneous, so investors set the actual market price P t equal to fundamental value V t. 2
Time-Varying Expected Returns Suppose investors require a different expected return in each future period in order that they will willingly hold a particular stock. Our model is therefore E t R t +1 = k t +1 (21) Involving forward substitution, gives P t = E t [δ t +1D t +1 + δ t +1δ t +2D t +2 + + δ t +N 1δ t +N(D t +N + P t +N)] (22) In fact, from the definition of the expected return: E R E P P E D P k 3
one obtains: P t = δ t+1 E t (P t +1 + D t +1) where δ t+1 =1/(1+k t+1 ) Leading this equation one period and taking expectations: E t P t+1 = δ t+2 E t (P t +2 + D t +2) E t P t+2 = δ t+3 E t (P t +3 + D t +3). By successive substitutions one obtains eq.(22). Now let N and, hence, δ t +N 0. If the expected growth in D is not explosive so that E t V t+n is also finite, then, 4
Lim n E t [ δ t +N 1δ t +N(D t +N + P t +N)] 0 This is known as a terminal condition or transversality condition. Eq. (22) can be written in a more compact form (assuming the transversality condition holds):, (23) where δ t +i = 1/(1 + k t +i) and δ t,t +j = δ t +1... δ t +j. The current stock price, therefore, depends on expectations of future discount rates and dividends. Note that 0 < δ t +j < 1 for all periods. 5
Note that in a well-informed ( efficient ) market, one expects the stock price to respond immediately and completely to the news. Tests of the stock price response to announcements are known as event studies. We cannot calculate fundamental value (i.e. the right-hand side of (23)) and hence cannot see if it corresponds to the observable current price P t. Forecast equations for dividends, for example, on annual data, an AR(1) or AR(2) model for dividends fits the data quite well. We need to model the equilibrium rate of return k t. The CAPM suggests the following relationships. 6
Individual Asset Returns Consider the price of an individual security or a portfolio of assets, which is a subset of the market portfolio. The CAPM implies that to be willingly held as part of a diversified portfolio, the expected return on portfolio-i is given by E t R it +1 = r t + β it (E t R m,t +1 r t ) (27) where β it = Et(σ im /σ 2m ) t +1 Comparing (21) and (27), the equilibrium required rate of return on asset-i is k t +1 = r t + λe t (σ im ) t +1 7
The covariance term may be time-varying and hence so might the future discount factors δ t +j in the RVF for an individual security (or portfolio of securities Market Portfolio The one-period CAPM predicts that in equilibrium, all investors will hold the market portfolio (i.e. all risky assets will be held in the same proportions in each individual s portfolio). Merton (1973) developed this idea in an intertemporal framework and showed that the excess return over the risk-free rate, on the market portfolio, is proportional to the expected variance of returns on the market portfolio. E t R m, t +1 r t = λ(e t σ 2m,t +1) (24) 8
Comparing (21) and (24), we see that according to the CAPM, the required rate of return on the market portfolio is given by k t = r t + λ(e t σ 2m,t +1) (26) Some special cases if the required rate of return is constant, then eq. (23) becomes (11) The Gordon model assumes that: E t D t+j = (1+g) j D t (18) 9
Substituting the forecast of future dividends from (18) in the rational valuation formula gives 1 (19) and hence (k-g)>0 (20) 10
Stock Price Volatility In this chapter, we discuss whether fundamentals such as changing forecasts about dividends or future returns can explain the observed volatility in stock prices. If a rational fundamentals model cannot explain such observed volatility, then we have the volatility puzzle (Campbell 2000). Volatility tests directly examine the rational valuation formula (RVF) for stock prices under a specific assumption about equilibrium expected returns. 11
The simplest assumption is that one-period returns are constant. If we had a reliable measure of expected dividends, we could calculate the above formula. A test of the RVF would then be to see if equals var(p t ). Shiller (1981) obviated the need for data on expected dividends. 12
He noted that under rational expectations (RE), actual and expected dividends only differ by a random (forecast) error and therefore so do the actual price P t and the perfect foresight price, defined as Note that uses actual dividends. Shiller demonstrated that the RVF implies var(p t ) var( ). A number of commentators often express the view that stock markets are excessively volatile prices alter from day to day or 13
week to week by large amounts that do not appear to reflect changes in news about fundamentals. If true, this constitutes a rejection of the efficient markets hypothesis (EMH). Shiller Volatility Tests Under constant (real) returns (2) Here, P t +n is the expected terminal price at time t + n, and all investors take the same view of the future. 14
At time t, we do not know what investors forecasts of expected future dividends would have been. However, Shiller (1981) proposed a simple yet very ingenious way of getting round this problem. Suppose we have data on actual dividends in the past, say from 1900 onwards, and we have the actual price P t +n today, say in 2004. We assume δ is a known value, say, 0.95, for annual data, implying a required return of k = 5.2% p.a. [δ = 1/(1 + k)]. 15
Then, using (2), we can calculate what the stock price in 1900 would have been, if investors had forecast dividends exactly, in all years from 1900 onwards. This is the perfect foresight stock price,, in 1900. By moving one year forward and repeating the above, we can obtain a data series for for all years from 1900 onwards using the following formula. (3) When calculating for 1900, the influence of the terminal price P t +n is fairly minimal since n is large and δ n is relatively small. 16
We assume that the actual price at the terminal date is close to its expected value E t P t +n and the latter is usually done in empirical work. Comparing the time series of P t and, we see that they differ by the sum of the forecast errors of dividends w t +i = D t +i E t D t +i, weighted by the discount factor δ i. From (2) and (3) and the definition of w t +i : 17
Assuming the final term is close to zero as n and noting that under RE, cov(p t,w t +i) = 0, then it follows that the RVF implies var( ) var(p t ). In fact, we know that investors made forecast errors η t = - P t. Hence = P t +η t (4) where 1 Under RE, η t is independent of the stock price at time t. From (4), we obtain 18
var(p ) = var(p t ) + var(η t ) + 2 cov(η t, P t ) (5) Informational efficiency (orthogonality) implies cov(η t, P t ) is zero, hence, var(p ) = var(p t ) + var(η t ) var(p ) / var(p t ) 1 (7a) Hence, under EMH/RE, and a constant discount factor, we would expect the variance of the perfect foresight price P to exceed that of the actual price P t. 19
Consider the results in Figure 1, which use Shiller s US data 1871 1995. 20
The solid line is the (real) stock price P t, which is clearly much more volatile than the perfect foresight stock price P calculated under the assumption that the (real) discount rate is constant. Eyeballing Figure 1 suggests rejecting the EMH-RVF under the constant (real) returns assumption. 21