Equity Valuation, CAPM, and Market Efficiency

Foundational Assumption: The valuation of shares and calculation of prices are based on the assumption of an efficient market.
Applicability: Market efficiency is generally reliable in large economies and is further supported in local contexts (e.g., the New Zealand economy) by regulators who aim to ensure market integrity.
Predictive Correlation: In an efficient market, different approaches to calculating the expected return on a share should demonstrate correlation or a level of consistency.

Historical Data Analysis: CAPM utilizes significant volumes of historical data for both a specific share and the market from which it originates.
Pairwise Comparison: Returns for the share and the market are compared on a pairwise basis across various time periods to identify correlated movements.
Beta (\beta) Definition: Beta is the numerical output of this analysis, representing the responsiveness of a share's returns to changes in market returns. * Market Index: The market itself is assumed to have a $\beta = 1$ . * Interpretation of Beta: The beta coefficient indicates whether a share's price (and consequently its return) changes at a greater or smaller rate than the market.
Mathematical Derivation: Beta is the slope coefficient of the Ordinary Least Squares (OLS) Regression mapping the share's returns against the market's returns. * If market data is plotted against itself, it results in a perfect $45^{\circ}$ line where the "rise and run" is $1$ over $1$ , resulting in a $\beta$ of exactly $1$ .
Predictive Use: Historical beta is used to predict future performance: if the market changes (observed via a stock exchange), beta helps estimate if the share will be more or less volatile (risky) than the market.

Required vs. Observable Return: While CAPM provides a required return, an observable return can be derived by comparing the current share price with expected dividends and growth. These two results are often slightly different.
Components of Expected Return: The return on any investment is the combination of two factors: 1. Price Change: The capital gain or loss over the holding period. 2. Dividend Cash Flow: The periodic income received from the share.
Formula Logic: * If an investor knows the dividend payment ( $D$ ), the expected rate of return ( $r$ or $k$ , potentially from CAPM), and the current price ( $P_0$ ), they can predict the future price ( $P_1$ ) by rearranging the valuation equation.

Aggressive Shares: A share with a \beta > 1. These are volatile and change in value/return at a faster rate than the market. Investors typically buy these when they expect the market to move upward.
Defensive Shares: A share with a \beta < 1. These are less volatile than the market.
Bull Market: An optimistic market environment where investors are "exuberant" and think things will go well. Policies in a bull market are typically aggressive.
Bear Market: A pessimistic market environment where the economy or market value is shrinking.

Risk-Free Rate ( $R_f$ ): Typically associated with government bonds. * Technically, the shortest-term bonds are used because they carry virtually no price risk ( $\text{risk-free}$ ). * Practitioners often match the tenor of the bond to the investment horizon, though this produces different results if there is a steep yield curve (where short-term and long-term rates differ significantly).
Market Return ( $R_m$ ): The return expected on a market portfolio or index (e.g., if an investor held all shares in the market).
Dividend Zero ( $D_0$ ): The dividend that has just been paid. It is NOT part of the future cash flow (it has already occurred).
Dividend One ( $D_1$ ): The next available dividend. This is the critical starting point for valuations. It is calculated as: $D_1 = D_0 \times (1 + g)$
Growth Rate ( $g$ ): Represents the assumed constant growth of the company. It applies simultaneously and at the same rate to the company's earnings and its dividend payments. In the Constant Growth Model, this growth is assumed to continue indefinitely.

Scenario Data:

Step 1: Calculate the Required Return ( $r_s$ ) using CAPM

Formula: $r_s = R_f + \beta \times (R_m - R_f)$
Calculation: $r_s = 7\% + 1.2 \times (12\% - 7\%)$
Calculation: $r_s = 7\% + 1.2 \times (5\%)$
Calculation: $r_s = 7\% + 6\% = 13\%$
Note: Using decimal equivalents ( $0.07$ , $0.12$ , etc.) is recommended to avoid conversion errors.

Step 2: Calculate Current Stock Market Value ( $P_0$ )

Step 3: Calculate Stock Market Value in One Year ( $P_1$ )

Method 1 (Gordon Growth for Year 1): Using $D_2$ in the formula.
Method 2 (Growth Application): Apply the growth rate to the current price ( $P_0$ ).
Calculation: $30.29 \times 1.06 = \$32.11$

The total return ( $13\%$ in the example) is comprised of two distinct yields: 1. Dividend Yield: The income component, calculated as the next dividend divided by current price. * Formula: $\text{Dividend Yield} = \frac{D_1}{P_0}$ * In the example: $\frac{2.12}{30.29} = 7\%$ 2. Capital Gains Yield: The growth in the value of the share. * Formula: $\text{Capital Gain} = \frac{P_1 - P_0}{P_0}$ * In the constant growth model, the capital gains yield is equal to the growth rate ( $g$ ). * In the example: $6\%$
Total Return Identity: $\text{Total Return} = \text{Dividend Yield} + \text{Capital Gains Yield}$
Example Check: $7\% \, (\text{Dividend Yield}) + 6\% \, (\text{Capital Gain}) = 13\% \, (\text{Total Return})$

Question: What is $R_f$?
Response: $R_f$ is the risk-free rate, normally associated with government bonds (ideally shortest-term) where there is no price risk.
Question: What is $D_0$?
Response: $D_0$ is the dividend just paid. It provides the basis to calculate $D_1$, but is not itself part of future cash flows.
Question: What determines if a share is aggressive or defensive?
Response: It is based on the Beta value. A Beta greater than one is aggressive/volatile; a Beta less than one is defensive.