Overview of Capital Asset Pricing Model (CAPM)

Capital Asset Pricing Model (CAPM)

Expected Return & Beta

• The Capital Asset Pricing Model (CAPM) relates expected return on an asset to its risk as measured by beta (B2).

Capital Allocation to Risky Assets

• Capital allocation involves choosing between risky assets and risk-free assets.

• The central question is: What fraction of a complete portfolio should an investor allocate to risky assets?

• Investors must evaluate the trade-off between risk and return.

Expected Return Formula

• Expected Return of the Complete Portfolio (E(r)):

  • $E(r) = E(r<em>r imes C</em>p) + (1 - C<em>p) imes R</em>f$

  • where:

  • $E(r_r)$ = Expected return of the risky portfolio

  • $C_p$ = Percentage of assets in the risky portfolio

  • $R_f$ = Return of the risk-free asset

• Standard Deviation of the Complete Portfolio (σ_p):

  • $ext{Standard Deviation: } ext{σ}<em>C = y imes ext{σ}</em>P$

  • where:

  • $ ext{σ}_P$ = Standard deviation of the risky portfolio

  • $y$ = a scalar that determines the risk allocation weight

Capital Allocation Line (CAL)

• Represents the risk-return combinations available by varying investments in risky and risk-free assets.

• Graphical representation includes:

  • Risk-free rate $r_f = 7\%$

  • Expected return of portfolio $E(r_p) = 15\%$

  • Slope of CAL indicates reward-to-variability ratio.

Sharpe Ratio

• Definition: The Sharpe ratio represents the slope of the Capital Allocation Line (CAL).

  • $ext{Sharpe Ratio} = \frac{E(r<em>p) - r</em>f}{σ_p}$

  • This ratio indicates the increase in expected return per unit of extra risk taken.

Risk Aversion

• Risk aversion is a preference to avoid risk; it influences capital allocation.

• Represented in the equation:

  • $y = ext{preferred capital allocation}, \ A = ext{coefficient of risk aversion}$

Passive Strategy / Indexing / CML

• The passive strategy is based on the premise that securities are fairly priced; it avoids detailed security analysis.

• Indexing: Involves holding a diversified portfolio that mirrors a market index (e.g., S&P 500).

• Capital Market Line (CML): This is the CAL using a market index portfolio as the risky asset.

Triumph of the Optimists

• Historical evidence suggests that stocks outperform bonds over time.

• However, the equity risk premium may not be as large as the data from 1926 would suggest.

• Analysis is impacted by survivorship bias, as data from countries like Russia and China are often excluded.

Passive Investing Benefits

• Benefits:

  • Simplicity and low costs. An example given: Fidelity index fund at 0 basis points (bps).

  • In contrast, the expense ratio for active mutual funds averages 40-50 bps and hedge funds typically have a 1.5% management fee and 15% profit-sharing.

  • Active management offers potential for higher returns but comes with increased costs.

CAPM Defined

• The model is mathematically expressed as:

  • E(Ri) = Rf + eta [Rm - Rf]

  • where:

  • $E(R_i)$ = expected return of the security

  • $R_f$ = risk-free rate

  • $eta$ = beta, the security's sensitivity to market risk

  • $R_m$ = expected return of the market portfolio

• Beta (B2) represents the responsiveness of a security's return to the market's return and is derived from the regression coefficient relating the security's return to the market return.

CAPM Assumptions

• Markets are perfectly competitive and provide equal opportunities.

• Investors are homogeneous apart from their initial wealth and risk aversion levels.

CAPM Implications

• All investors theoretically hold the market portfolio.

• Every stock in a portfolio is weighted according to its proportion of the market value.

• The market portfolio is located on the efficient frontier, showing optimized return for risk levels.

Efficient Frontier, CAL vs. CML

• Capital Allocation Line (CAL): Represents portfolios containing a risky and a risk-free asset.

• Capital Market Line (CML): Accounts for the risk-free asset and the market portfolio, depicting a higher risk-return ratio.

Equilibrium Risk Premium

• The equilibrium risk premium is based on:

  1. The variance of market returns.

  2. The risk aversion level of the average investor, represented mathematically by:

  • $A ext{ (risk aversion degree)}, \ ext{Variance} ext{ (of market returns)}$

Mean-Beta Relationship

• The relationship defines that the beta of a portfolio is the weighted average of the betas of its underlying assets.

• Under CAPM, only the security's beta is pertinent for determining expected returns.

Security Market Line (SML)

• A graphical representation that illustrates the expected return versus beta relationship as described by the CAPM.

Applications of CAPM

• Estimating expected returns for investing or capital budgeting decisions.

• Comparing alternate expected return estimates against CAPM for security selection.

• Appraising portfolios by comparing actual rates of return to expected returns.

Predicting Betas

• When predicting betas they tend to regress toward 1, indicating mean reversion.

• Adjusted Beta formula:

  • $ext{Adjusted Beta} = \frac{2}{3} \times ext{historical beta} + \frac{1}{3} \times 1$

  • Betas can be complex due to non-stationarity in the markets.

CAPM and The Real World

• The CAPM may not hold true based on its assumptions, yet it remains a useful predictor of expected returns.

• The theory of CAPM is considerately untestable as a principle, but investors can gain insights from its framework:

  • Advocates logically for diversification.

  • Highlights that systematic risk is the only relevant risk for well-diversified portfolios, suitable for various investors.