Download Modeling the Price of a Share of Common Stock - Notes | ECON 423 and more Study notes Financial Market in PDF only on Docsity!
Economics 423 Michael Salemi
Modeling the Price of A Share of Common Stock
1. Market Fundamentals Model The mainstream model of share prices is the “fundamentals” model. The model says that shares are valued because and only because they are a claim to expected future dividends.
Assume: I. Agents are risk neutral so that they care only about expected returns. II. Agents have common beliefs about what determines the value of a share. III. Agents forecasts are rational. The agents know as much as the economist. IV. Agents are price takers.
Let P (^) z , t be the price of a share of company Z at time t and let D^ z , t be the dividend paid by Z at time t. Then the fundamentals model says that the price of a share is the present value of expected future dividends.
One version of the fundamentals model adds hypotheses about how firms earn profits and pay dividends. Given those additional hypotheses, the fundamentals model predicts how the price of a share will related to the parameters of the dividend model. For simplicity of notation we drop the Z-subscript so that Pt = P (^) z, t. Suppose the firm pays dividends equal to a fixed proportion of profits and profits are equal to a “standard” return to capital plus a random component. The firm uses retained profits to increase its capital stock.
Let Q (^) t be profit (per share) at time t, K (^) t = capital (per share) at time t, and let U (^) t be an unforecastable random shock to profits.
D t = * C Qt
Q t = D Kt + Ut
K t+1 = Kt + (1! *) Qt
Plugging the equations into the present value expression for stock prices permits derivation of the following:
If R = .10, * = .25, and D =.
2. The "Bubbles" Model
Equation (1) implies:
Suppose we modify assumptions II and III to permit diversity of opinion. The fundamentals model says that the expected value of next period’s stock price E (^) t (P (^) t + 1 ) must be "justified" by forecasts of dividends. The bubbles model weakens that requirement and says that if anything happens that makes people think the stock will be more valuable in the future, then the current price will rise as a result. The expectations of the traders are self-fulfilling.
3. The “Random Walk” Model Reinstate II and III. Suppose we measure time in days rather than months or years. Then R, the per period rate of interest, is a small number. Because it is unlikely that a dividend will be paid in the next period E (^) t (D (^) t + 1) will be zero for most stocks and time periods. Then, the above equation reduces to:
This property is called the Martingale property. One example of a martingale is a random walk such as the following equation for stock prices where Vt is a "white noise" random variable.
4. The Capital Asset Pricing Model (CAPM) Drop assumption I and assume that people require compensation for bearing risk. Rearrange the bubbles- model equation to derive R (^) z , t , the expected one-period return from holding a share of Z.
How large should R (^) z , t be? In a risk neutral world, the required return on stocks would be approximately the required return on long-maturity treasury bonds. CAPM suggests that accounting for risk aversion provides and an alternative answer.
where R (^) t is the "risk free" rate (Treasury Bond yield), R (^) M , t is the average return on an optimally diversified
portfolio and $ Z C ( R M , t! R t ) is the risk premium built by the market into R z, t. The parameter $ Z
measures the correlation between returns on the Z share and the overall market return.
