Porfolio management part 2 Flashcards
Active portfolio management
What consists active management
Active management seeks to add value by outperforming a passively managed benchmark porfolio
What are the 3 main qualities an appropriate benchmark should represent.
- Be representative of investment universe from which the active manager may choose.
- Be replicable at low cost
- Weights available ex-ante and ex-post.
Alpla
the difference in risk-adjusted returns. (difference in beta of actively managed portfolio and the benchmark)
How is active return calculated and what is it?
The value added by active management.
Can be measure based on expectation (ex-ante) or after the fact (ex-post).
Ex-ante active return: difference between the expected return of an actively managed portfolio and expected return of its benchmark.
What are active weights?
- It determines the amount of value added - the difference between a security weight in an actively managed portfolio and benchmark portfolio.
How can we decompose active return?
What are the Sharpe and the information (ex post and ex ante) ratio used for?
The information ratio and the Sharpe ratio are two different methods of measuring a portfolio’s risk-adjusted rate of return.
How is the calculated the Sharpe ratio (SR)
- Calculated as the excess return per unit of risk (standard deviation)
Why is the Sharpe ratio unaffected by adding cash or leverage to a portfolio?
Because both excess return and standard deviation scale proportionally when cash or leverage is added.
For example:
Allocating 50% to a risk-free asset (cash) reduces both:
Portfolio excess return
Portfolio standard deviation
… by half.
If both the numerator and denominator are halved, the ratio stays the same.
➡️ Sharpe Ratio is unchanged.
✨ Key Insight:
Sharpe ratio measures risk-adjusted performance, and is invariant to portfolio scaling.
How is calculated the information ratio (IR).
- ratio of the active return to the standard deviation of active returns, known as active risk or benchmark tracking risk
The optimal active risk is calculated?
You scale active risk based on how good your alpha is (IR) relative to the Sharpe performance of the benchmark.
Sharpe Ratio of the Optimal Portfolio
How to calculate the total portfolio risk with active management?
What does the Sharpe Ratio (SR) measure?
Excess return per unit of total risk: SR = (Rp - Rf) / σp
What does the Information Ratio (IR) measure?
Active return per unit of active risk: IR = (Rp - Rb) / σ(Rp - Rb)
What is active return?
RA = Rp - Rb (portfolio return minus benchmark return)
What is active risk?
Standard deviation of active return: σA = std.dev(Rp - Rb)
Is the Sharpe Ratio affected by adding cash or leverage?
No, both return and risk scale proportionally.
Is the Information Ratio affected by adding cash or leverage?
Yes, cash lowers active return but not active risk, reducing IR.
When does IR equal SR?
When benchmark = risk-free rate (e.g., market-neutral portfolio).
What kind of fund has low IR and low active risk?
A closet index fund — mimics benchmark with little true active management.
What is the ex-ante Information Ratio?
IR based on expected active return and risk — usually positive.
What is the ex-post Information Ratio?
IR based on realized past return and risk — can be negative.
What happens to IR if active weights are increased?
Nothing. Active return and active risk scale equally, IR stays the same.
What happens to IR if we blend active portfolio with benchmark?
IR remains constant as both RA and σA shrink proportionally.
How can investors target specific active risk levels?
By adjusting weight in active vs. benchmark portfolio.
What is the formula for optimal active risk?
σA* = (IR / SRB) × σB
What is the formula for Sharpe ratio with optimal active risk?
SRP = sqrt(SRB² + IR²)
How is total portfolio risk calculated with active and benchmark risk?
σP² = σB² + σA²
If IR = 0.2, SRB = 0.4, σB = 12%, what is optimal active risk?
σA* = (0.2 / 0.4) × 12% = 6%
If σA* = 6% and fund’s active risk = 9%, what % goes to active fund?
6% / 9% = 67% to active fund, 33% to benchmark.
How is portfolio excess return calculated using IR and σA?
E(RA) = IR × σA, then add (RB - RF) to get total excess return.
What’s the Sharpe Ratio of portfolio P with 6% excess return and 13.4% risk?
SR = 6% / 13.4% ≈ 0.4472
What are the 3 factors determining the information ratio
- Information coefficient (IC)
- Transfer coefficient (TC)
- Breadth (BR)
Explain what does the information coefficient (IC) represent?
Measure of a manager skill
IC, is the ex-ante (forecast/expected), risk-weighted correlation between active returns and forecasted active returns.
The ex-post information coefficient, ICR measures actual correlation between active returns and expected active returns
Explain the transfer coefficient (TC)
correlation between actual active weights and optimal active weight
What is the Information Coefficient (IC)?
The IC is a measure of a manager’s skill, defined as the ex-ante risk-weighted correlation between active returns and forecasted active returns.
What is the ex-post Information Coefficient (ICR)?
ICR measures the actual correlation between active returns and expected active returns.
What is the Transfer Coefficient (TC)?
TC is the correlation between actual active weights and optimal active weights.
What is the formula for the Transfer Coefficient (TC)?
TC = CORR(μi / σi, ∆wiσi) = CORR(∆wi* σi, ∆wiσi)
Define Breadth (BR) in the context of active portfolio management.
BR is the number of independent active bets taken per year.
What is the Grinold rule?
The Grinold rule computes expected active return: μi = ICσiSi, where Si is the standardized score of security i.
How is the expected value added by active management calculated?
E(RA) = ∑ ∆wi μi
What is the optimal Information Ratio (IR*) for an unconstrained portfolio?
IR* = IC × √BR
How is the expected active return for an unconstrained portfolio calculated?
E(RA)* = IC × √BR × σA
What is the Information Ratio (IR) for a constrained portfolio?
IR = TC × IC × √BR
How is expected active return for a constrained portfolio calculated?
E(RA) = TC × IC × √BR × σA
What is the formula for optimal active risk in an unconstrained portfolio?
σA* = (IR / SRB) × σB
What is the formula for optimal active risk in a constrained portfolio?
σCA* = (TC × IR*) / SRB × σB
What is the Sharpe ratio of a constrained portfolio?
SRPC = √(SRB² + TC² × IR*²)
What is the formula for realized value added from active management?
E(RA | ICR) = TC × ICR × √BR × σA
How is actual active return expressed with noise?
RA = E(RA | ICR) + noise
What does TC² represent in terms of active return variance?
It is the proportion of realized active return variance attributed to variation in realized IC.
Why is the information ratio important for manager selection?
Because the active portfolio with the highest IR will also have the highest Sharpe ratio, making it optimal for all investors.
How is expected active return calculated for a given level of active risk?
E(RA) = IR × σA
What is market timing in active management?
Market timing is a bet on the direction of the market or its segments, such as shifting from equities to cash based on expected market declines.
How is the Information Coefficient (IC) for market timing calculated?
IC = 2(% correct) − 1
Given IC = 0.10 and BR = 4, what is the Information Ratio (IR)?
IR = IC × √BR = 0.10 × √4 = 0.20
Given IC = 0.04 and BR = 50, what is the IR?
IR = 0.04 × √50 ≈ 0.28
How is active risk (σC) in a two-sector market calculated?
σC = √[σX² − 2σXσY rXY + σY²]
How do you annualize active risk from a monthly strategy with σC = 0.05?
Annualized active risk = 0.05 × √12 ≈ 0.1732 or 17.32%
If IC = 0.20, σA = 0.1732, and BR = 12, what is the annualized active return?
E(RA) = IC × √BR × σA = 0.20 × √12 × 0.1732 ≈ 12%
What is the monthly active return if correct 60% of the time?
Monthly return = (0.60)(0.05) + (0.40)(-0.05) = 0.01 or 1%
How is the allocation changed if active risk is capped at 5.20% and base risk is 17.32%?
Deviation = 5.20% / 17.32% ≈ 30%, so sector overweight = benchmark + 30%
What does the phrase ‘garbage in, garbage out’ imply in the fundamental law?
It implies that inaccurate input estimates for IC and BR will lead to incorrect evaluations.
What are the two main inputs of the fundamental law of active management?
Information Coefficient (IC) and Breadth (BR)
Why is ex-ante measurement of IC a limitation?
Because managers may overestimate their forecasting skill, leading to inaccurate IR estimates.
How is effective breadth adjusted for correlated decisions?
BR = N / [1 + (N − 1)r], where r is the average correlation between decisions.
What compromises decision independence in breadth calculation?
Cross-sectional dependency and time-series dependency in decision-making.
