5. Single Index Model Flashcards
What is the formula for the regression equation for excess return in the single-index model?
Ri(t) = αi + βi * RM(t) + εi(t)
What does the expected return-beta relationship express?
E(Ri) = αi + βi * E(RM)
How is total risk decomposed in the single-index model?
σ²i = βi² * σ²M + σ²(εi)
What is the formula for the covariance between two securities using the index model?
Cov(Ri, Rj) = βi * βj * σ²M
How is the Sharpe ratio of a portfolio calculated in the single-index model?
SP = E(RP) / σP
How does the single-index model simplify the estimation of the covariance matrix?
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.
Why is the residual variance important in the single-index model?
It measures the firm-specific risk, which is the variance of the portion of the return unexplained by the market index.
How does diversification affect residual variance in an equally-weighted portfolio?
Residual variance decreases as the number of securities (n) increases, following the formula σ²(εp) = σ²(εi) / n.
What is the role of the security characteristic line (SCL) in the index model?
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.
How does the single-index model determine systematic risk and firm-specific risk?
Systematic risk is represented by βi² * σ²M, while firm-specific risk is represented by σ²(εi).
How can the single-index model be used to explain the correlation between two securities?
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.
What are the implications of a high βi for a security in a portfolio?
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.
How is the optimal weight for an active portfolio calculated using the single-index model?
The initial weight is given by w0i = αi / σ²(εi), adjusted by incorporating market risk using the formula wA* = w0A / (1 + (1 - βA) * w0A).
Why does the single-index model favor using the information ratio for portfolio optimization?
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.
How does the single-index model assist in constructing a tangency portfolio?
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.
Analyze the trade-offs between the single-index model and the full-covariance model in portfolio construction.
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.
How would you interpret a negative alpha (αi) for a security in the context of portfolio management?
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.
In what ways does the single-index model influence the diversification strategy for a portfolio?
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.
Evaluate the impact of βi variability among securities on the design of an equally-weighted portfolio.
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.
How can the concept of the security characteristic line (SCL) guide decisions about over- or underweighting securities in a portfolio?
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.
What is the main theoretical assumption of the single-index model?
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.
How does the single-index model simplify portfolio analysis?
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.
What does the regression equation in the single-index model represent?
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).
Why is alpha (αi) significant in evaluating securities?
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.
