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The beta coefficient for Stock \(C\) is \(b_{C}=0.4,\) whereas that for Stock \(D\) is \(b_{D}=-0.5\) (Stock D's beta is negative, indicating that its rate of return rises whenever returns on most other stocks fall. There are very few negative beta stocks, although collection agency stocks are sometimes cited as an example. a. If the risk-free rate is 9 percent and the expected rate of return on an average stock is 13 percent, what are the required rates of return on Stocks \(C\) and \(D\) ? b. For Stock \(C\), suppose the current price, \(P_{0}\), is \(\$ 25 ;\) the next expected dividend, \(D_{1}\), is \(\$ 1.50 ;\) and the stock's expected constant growth rate is 4 percent. Is the stock in equilibrium? Explain, and describe what will happen if the stock is not in equilibrium.

Step by step solution

01

Determine Required Return Using CAPM Formula

To find the required rate of return, we use the Capital Asset Pricing Model (CAPM) formula: \( r = r_f + b \cdot (r_m - r_f) \). Here, \( r_f = 9\% \) is the risk-free rate, and \( r_m = 13\% \) is the expected return on the market. For Stock \( C \), \( b_C = 0.4 \), so:\[ r_C = 9\% + 0.4 \cdot (13\% - 9\%) = 9\% + 0.4 \cdot 4\% = 9\% + 1.6\% = 10.6\% \]For Stock \( D \), \( b_D = -0.5 \), so:\[ r_D = 9\% + (-0.5) \cdot (13\% - 9\%) = 9\% - 0.5 \cdot 4\% = 9\% - 2\% = 7\% \]

02

Calculate Expected Return for Stock C

The expected return for Stock \( C \) can also be calculated using the Gordon Growth Model:\[ r = \frac{D_1}{P_0} + g \]where \( D_1 = \\(1.50 \), \( P_0 = \\)25 \), and \( g = 4\% \). Substituting these values gives:\[ r = \frac{1.50}{25} + 0.04 = 0.06 + 0.04 = 0.10 = 10\% \]

03

Determine if Stock C is in Equilibrium

Stock \( C \) is in equilibrium if the required return calculated from the CAPM is equal to the expected return given by the Gordon Growth Model. From Steps 1 and 2, the required return is 10.6%, and the expected return is 10%. Since these two values are not equal, Stock \( C \) is not in equilibrium.

04

Analyze Consequences of Disequilibrium

If Stock \( C \) is not in equilibrium, investors may assess that they are not being compensated adequately for the risk they are taking at the current price and expected return. Therefore, they might demand a higher return, causing the price of the stock to decrease until the expected return matches the required return, re-establishing equilibrium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Beta Coefficient

The beta coefficient is a measure that indicates how the price of a stock is expected to change relative to changes in the overall market. In the context of the Capital Asset Pricing Model (CAPM), it quantifies the stock's systemic risk:

• Stock C: With a beta of 0.4, it means Stock C is expected to be less volatile than the market. For every 1% move in the market, Stock C's return is expected to move by 0.4%.
• Stock D: Displays a negative beta of -0.5, suggesting an inverse relationship with the market. When most stocks decline, Stock D's value might rise, which is rare and common in certain sectors like collection agencies.

Beta is crucial for investors as it indicates the risk level attached to the stock compared to the broader market. A higher beta suggests more risk and the potential for higher returns, while a lower beta indicates less risk.

Examining the Risk-Free Rate

The risk-free rate is the theoretical return of an investment with zero risk, reflecting the interest an investor expects from an absolutely risk-free investment. In the CAPM formula,

• This rate is crucial as it represents the minimum return expected by investors as compensation for deferring their consumption.
• For the exercise, the risk-free rate is 9%, set as a benchmark against which riskier investments like stocks are assessed.

As it is often based on government bonds, which are considered safe, the risk-free rate helps set a foundational level in financial models. It effectively aids in determining the additional return required for investors to take on more risk.

Defining Equilibrium of Stock

Equilibrium in the stock market means that all forces are balanced and the market price reflects all available information. For a stock to be in equilibrium, the expected return should match the required return determined by CAPM:

• Stock C: In the given case, the required return is 10.6% from CAPM calculations, but the expected return from Gordon Growth Model is 10%. This mismatch indicates disequilibrium.
• When stocks are not in equilibrium, prices change as investors react to discrepancies, driving the price until the stock reaches an equilibrium state.

The lack of equilibrium points towards potential inefficiencies in pricing, suggesting investors might not be adequately compensated for the risk, pushing them to reassess their investments.

Exploring the Gordon Growth Model

The Gordon Growth Model, also known as the Dividend Discount Model (DDM), helps calculate a stock's expected return based on its future dividends, assuming constant growth:

• This model uses the formula: \[ r = \frac{D_1}{P_0} + g \]where \( D_1 \) is the expected dividend, \( P_0 \) is the current stock price, and \( g \) is the growth rate.
• For Stock C, with a dividend of \(1.50, price of \)25, and growth of 4%, the expected return was calculated as 10%.

This model is essential as it provides insights into the intrinsic value of a stock based on projected dividend growth. It is particularly useful for assessing investments in companies with stable growth and reliable dividend policies.