5. Single Index Model

Question 1

What is the formula for the regression equation for excess return in the single-index model?

Answer 1

Ri(t) = αi + βi * RM(t) + εi(t)

Question 2

What does the expected return-beta relationship express?

Answer 2

E(Ri) = αi + βi * E(RM)

Question 3

How is total risk decomposed in the single-index model?

Answer 3

σ²i = βi² * σ²M + σ²(εi)

Question 4

What is the formula for the covariance between two securities using the index model?

Answer 4

Cov(Ri, Rj) = βi * βj * σ²M

Question 5

How is the Sharpe ratio of a portfolio calculated in the single-index model?

Answer 5

SP = E(RP) / σP

Question 6

How does the single-index model simplify the estimation of the covariance matrix?

Answer 6

By assuming that security returns are linearly related to a single market index, it reduces the number of covariance terms from n(n-1)/2 to n, requiring only βi and σ²(εi) for each security.

Question 7

Why is the residual variance important in the single-index model?

Answer 7

It measures the firm-specific risk, which is the variance of the portion of the return unexplained by the market index.

Question 8

How does diversification affect residual variance in an equally-weighted portfolio?

Answer 8

Residual variance decreases as the number of securities (n) increases, following the formula σ²(εp) = σ²(εi) / n.

Question 9

What is the role of the security characteristic line (SCL) in the index model?

Answer 9

The SCL describes the relationship between a security’s excess return and the market’s excess return, with the slope representing βi and the intercept representing αi.

Question 10

How does the single-index model determine systematic risk and firm-specific risk?

Answer 10

Systematic risk is represented by βi² * σ²M, while firm-specific risk is represented by σ²(εi).

Question 11

How can the single-index model be used to explain the correlation between two securities?

Answer 11

By showing that the correlation is driven by their systematic risks, using the formula Corr(Ri, Rj) = (βi * βj * σ²M) / (σi * σj), which ties their returns to the market index.

Question 12

What are the implications of a high βi for a security in a portfolio?

Answer 12

A high βi indicates that the security is highly sensitive to market movements, contributing more to systematic risk and amplifying the portfolio’s response to changes in the market index.

Question 13

How is the optimal weight for an active portfolio calculated using the single-index model?

Answer 13

The initial weight is given by w0i = αi / σ²(εi), adjusted by incorporating market risk using the formula wA* = w0A / (1 + (1 - βA) * w0A).

Question 14

Why does the single-index model favor using the information ratio for portfolio optimization?

Answer 14

The information ratio measures the reward-to-risk tradeoff of active portfolio management by comparing the alpha (excess return over the market) to the residual risk (σ(εA)), enabling the selection of optimal active portfolio weights.

Question 15

How does the single-index model assist in constructing a tangency portfolio?

Answer 15

It identifies the weights of the market and active portfolios that maximize the Sharpe ratio, balancing market risk and alpha contribution using wM* = 1 - wA* and other related formulas.

Question 16

Analyze the trade-offs between the single-index model and the full-covariance model in portfolio construction.

Answer 16

The single-index model simplifies calculations by assuming a linear relationship with the market index, reducing computational complexity, but it sacrifices accuracy by ignoring direct interdependencies between securities, which are captured in the full-covariance model.

Question 17

How would you interpret a negative alpha (αi) for a security in the context of portfolio management?

Answer 17

A negative alpha suggests the security is expected to underperform relative to its risk-adjusted benchmark, indicating it may not be a good candidate for inclusion in an active portfolio unless its systematic risk or diversification benefit compensates.

Question 18

In what ways does the single-index model influence the diversification strategy for a portfolio?

Answer 18

By breaking down total risk into systematic and firm-specific components, the model emphasizes diversifying firm-specific risk while accepting systematic risk as an inherent characteristic tied to the market.

Question 19

Evaluate the impact of βi variability among securities on the design of an equally-weighted portfolio.

Answer 19

Variability in βi leads to unequal contributions to systematic risk, making the portfolio more sensitive to securities with higher βi, which may skew the expected market responsiveness and risk-return profile.

Question 20

How can the concept of the security characteristic line (SCL) guide decisions about over- or underweighting securities in a portfolio?

Answer 20

The SCL helps identify securities with positive alpha (intercept above zero), which should be overweighted to exploit expected excess returns, while underweighting those with negative alpha to minimize their drag on portfolio performance.

Question 21

What is the main theoretical assumption of the single-index model?

Answer 21

The single-index model assumes that a security’s return is linearly related to the return of a market index, with deviations explained by firm-specific risk.

Question 22

How does the single-index model simplify portfolio analysis?

Answer 22

It reduces the number of inputs required by assuming returns are driven by a single factor, the market index, simplifying the estimation of covariances and portfolio risk.

Question 23

What does the regression equation in the single-index model represent?

Answer 23

The regression equation models a security’s excess return as the sum of its alpha (intercept), market sensitivity (beta multiplied by market return), and firm-specific deviation (residual term).

Question 24

Why is alpha (αi) significant in evaluating securities?

Answer 24

Alpha represents the portion of a security’s return not explained by the market index, highlighting its potential for outperformance or underperformance relative to market movements.

Question 25

What role does beta (βi) play in portfolio construction?

Answer 25

Beta measures a security’s sensitivity to market movements, guiding how much systematic risk it contributes to a portfolio and informing asset allocation decisions.

Question 26

How does the single-index model address covariance between securities?

Answer 26

Covariance is derived using the relationship Cov(Ri, Rj) = βi * βj * σ²M, relying on shared sensitivity to the market index rather than direct inter-security relationships.

Question 27

What is the significance of residual variance (σ²(εi)) in the single-index model?

Answer 27

Residual variance quantifies firm-specific risk, which is independent of market movements and diversifiable in larger portfolios.

Question 28

How is systematic risk defined in the context of the single-index model?

Answer 28

Systematic risk is the portion of total risk attributed to market factors, calculated as βi² * σ²M for each security.

Question 29

What theoretical advantage does the single-index model provide for portfolio optimization?

Answer 29

It focuses on market-related drivers of risk and return, allowing for easier optimization of portfolios by reducing computational complexity compared to the full-covariance model.

Question 30

How does the Sharpe ratio relate to the single-index model’s approach to risk and return?

Answer 30

The Sharpe ratio evaluates the efficiency of a portfolio by comparing excess return to total risk, incorporating systematic risk and residual variance derived from the model.

Question 31

What is the information ratio, and how is it used in the single-index model?

Answer 31

The information ratio measures the reward-to-risk tradeoff of active management, calculated as alpha divided by the residual standard deviation, aiding in the assessment of active portfolio contribution.

Question 32

Why is the assumption of a single market index both a strength and a limitation of the model?

Answer 32

It simplifies analysis by focusing on one explanatory factor but may oversimplify reality by ignoring additional factors or interdependencies influencing security returns.

Question 33

How is the regression equation for excess return represented in the single-index model?

Answer 33

Ri(t) = αi + βi * RM(t) + εi(t), where Ri(t) is the excess return of security i, RM(t) is the excess market return, and εi(t) is the residual.

Question 34

What is the formula for total risk in the single-index model?

Answer 34

σ²i = βi² * σ²M + σ²(εi), where total risk is the sum of systematic risk (βi² * σ²M) and firm-specific risk (σ²(εi)).

Question 35

How is covariance between two securities calculated using the single-index model?

Answer 35

Cov(Ri, Rj) = βi * βj * σ²M, where the covariance depends on their betas and market variance.

Question 36

What is the formula for the Sharpe ratio of a portfolio in the single-index model?

Answer 36

SP = E(RP) / σP, where SP is the Sharpe ratio, E(RP) is the portfolio’s expected excess return, and σP is its standard deviation.

Question 37

How is the variance of an equally-weighted portfolio determined in the single-index model?

Answer 37

σ²p = βp² * σ²M + σ²(εp), where systematic variance is βp² * σ²M and residual variance is σ²(εp) = σ²(εi) / n.