Question

Show that the characteristic lines of all securities intersect at a common point in the CAPM. What are the coordinates of this point?

Short answer

The characteristic lines of all securities intersect at the point \((R_f, R_f)\).

Step-by-step solution

- Understand the Characteristic Line

The characteristic line of a security in the Capital Asset Pricing Model (CAPM) describes the relationship between the expected return of the security and the expected return of the market. It is typically represented as: \[ E(R_i) = \beta_i E(R_M) + (1 - \beta_i) R_f \] where \( E(R_i) \) is the expected return of the security, \( \beta_i \) is the beta of the security, \( E(R_M) \) is the expected return of the market, and \( R_f \) is the risk-free rate.

- Setting Up the Characteristic Line for Two Securities

Consider the characteristic lines for two securities: \[ E(R_i) = \beta_i E(R_M) + (1 - \beta_i) R_f \] and \[ E(R_j) = \beta_j E(R_M) + (1 - \beta_j) R_f \]

- Equating the Characteristic Lines

To find the intersection point, set the expected returns equal to each other: \[ E(R_i) = E(R_j) \] \( \beta_i E(R_M) + (1 - \beta_i) R_f = \beta_j E(R_M) + (1 - \beta_j) R_f \)

- Solving for the Intersection

Rearrange and solve the equation from Step 3: \[ (\beta_i - \beta_j) E(R_M) = (1 - \beta_i - (1 - \beta_j)) R_f \] \[ (\beta_i - \beta_j) (E(R_M) - R_f) = 0 \] This simplifies to either \( \beta_i = \beta_j \) or \( E(R_M) = R_f \). However, \( E(R_M) = R_f \) would imply no risk premium, which is not typical in CAPM. Therefore, consider the non-trivial solution at a specific point:

- Identify the Common Intersection Point

The non-trivial intersection point occurs when the market return \( E(R_M) \) equals the risk-free rate \( R_f \). Thus, substituting \( R_f \) for \( E(R_M) \) in the characteristic line: \[ E(R_i) = \beta_i R_f + (1 - \beta_i) R_f = R_f \] Therefore, all characteristic lines intersect at the point \((R_f, R_f)\).

Key concepts

Characteristic Line

The characteristic line in the Capital Asset Pricing Model (CAPM) is a straight line that depicts the relationship between the expected return of a security and the expected return of the market. This relationship can be described with the formula: \[ E(R_i) = \beta_i E(R_M) + (1 - \beta_i) R_f \]Here:

• E(R_i): Expected return of the security.
• E(R_M): Expected return of the market.
• β_i: Beta of the security, which measures its sensitivity to market returns.
• R_f: Risk-free rate, the return of an investment with zero risk.

This line helps investors understand how the return of a security changes with the market. The slope of the line is the beta, indicating how much the security's return moves with the market.

Expected Return

The expected return is a key concept in finance, representing the anticipated return on an investment. In CAPM, the expected return of a security is given by the formula: \[ E(R_i) = \beta_i E(R_M) + (1 - \beta_i) R_f \]Using this, investors can assess how much return they can expect, considering both market performance and the risk-free rate. The expected return is crucial for decision-making as it provides a benchmark for comparing different investments. By adjusting their portfolios according to expected returns, investors aim to maximize their returns while managing risk.

Beta

Beta (β) measures a security's sensitivity to market movements. It tells us how much the return of a security is expected to change given a change in the market return. It's found using: \[ \beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)} \]Where:

• Cov(R_i, R_M): Covariance between the return of the security and the market.
• Var(R_M): Variance of the market return.

A beta greater than 1 means the security is more volatile than the market, while a beta less than 1 means it's less volatile. Beta helps investors understand risk: higher beta means higher risk and potentially higher returns. It's a fundamental part of CAPM, shaping the characteristic line and affecting the expected return.

Market Return

The market return, denoted as E(R_M), represents the average return of the market portfolio, which comprises all available securities. It's an essential component in CAPM, used to calculate expected returns of individual securities. Market return reflects the overall performance of the market and is typically measured by benchmarks like the S&P 500. By understanding market return, investors can compare individual security performance against the broader market. It serves as a gauge for what's achievable given the market conditions, allowing investors to adjust their strategies to align with market performance. In CAPM, market return is combined with beta and the risk-free rate to determine the security's expected return.

Risk-Free Rate

The risk-free rate (R_f) is the return on an investment with zero risk, typically represented by government bonds from stable countries. It's a crucial element in CAPM, which uses it as a benchmark for the minimum expected return from an investment. In the formula: \[ E(R_i) = \beta_i E(R_M) + (1 - \beta_i) R_f \]It signifies the part of the return that isn’t influenced by the market’s movements. The risk-free rate anchors the characteristic line at the x-axis intercept, with all lines intersecting at \((R_f, R_f)\). It helps investors understand the baseline return for any investment, beyond which they encounter varying levels of market risk. Understanding this rate enables more informed investment decisions and balancing of portfolios.