Analysis of Active Portfolio Management Flashcards
Active weights
Difference between the weight of the security between the portfolio and the benchmark.
Δ portfolio - benchmark
What is the components / formula for asset allocation
(ΔWeight of stocks X Return benchmark stocks ) + (ΔWeight bonds X Returns benchmark bonds)
∑(Active weights x Benchmark Return)
What is the components / formula for Security selection
(Portfolio weight stocks X Active return stocks) + (Portfolio bond weights X Active return bonds)
Where does active return come from?
Overweighing securities that will do better than benchmark, and underweighting stocks that will perform poorly than the benchmark
How to calculate cash in the Sharpe ratio?
- New desired standard deviation / previous standard deviation
σ₁ / σ₀ = Weight in portfolio = Wp
(1- Wp) = Cash
How to calculate weight in the Sharpe ratio?
New desired standard deviation / previous standard deviation
σ₁ / σ₀ = Weight in portfolio = Wp
The formula for combined return in Sharpe ratio
(Rp * Wp) + (1-Wp)*RfR
Does cash or leverage change the Sharpe ratio?
No, it does not.
Does cash or leverage change the Information Ratio?
Yes, it does!!
How can we change the risk in the Sharpe ratio?
With cash and leverage
But this does not change the Sharpe Ratio
How can we change the risk in the Information ratio?
Through the aggressiveness of active weights in the portfolio
We can change the risk by investing in the active portfolio and the benchmark portfolio
Property of active management theory
Implies that the active portfolio with the highest IR will also have the highest Sharpe Ratio
Optimal amount of active risk (Active risk) Formula
(Information Ratio X Standard deviation of benchmark) / Sharpe Ratio of benchmark
Optimal expected active return is a function of …
Forecasting ability
Breath
Active risk
The basic fundamental law formula
IC* √BR * σA
The full fundamental Law formula
TC * IC* √BR * σA
In regards to fundamental law, what if we assume all securities have the same standard deviation
Then, the correlation does not have to be risk-adjusted
What does it mean if we have a Transfer Coefficient = 1?
Unconstrained portfolio.
Our information ratio is invariant to changes in active risk
What is the relationship with IR and a constrained portfolio?
If we have constraint in our portfolio our information ratio drops
Formula to calculate combined Sharpe Ratio
SRP^2=(SRB^2+IR^2)
SRP=(SRB^2+IR^2)^0.5
What is the formula to determine a portfolio managers ability to achieve active return?
Information ratio.
Using Full fundamental law
IR=(TC)(IC)√BR
Basic Fundamental law
IR=(IC)*√BR
Does adding cash to the portfolio change the portfolio’s information ratio?
Yes it does!
The information ratio for a portfolio of risky assets will generally shrink if cash is added to the portfolio.
If we add cash, our Information ratio will decrease.
Does increasing the aggressiveness of active weights change the portfolio’s information ratio?
No it doesn’t !!
Because the diversified asset portfolio is an unconstrained portfolio, its information ratio would be unaffected by an increase in the aggressiveness of active weights.
Characteristics of A closet index
- Low active risk
- Sharpe Ratio close to benchmark
- Information ratio can be Inconclusive because of low active risk
- IR can be negative due to management fee
A closet index will have a very low active risk and will also have a Sharpe ratio very close to the benchmark.
A closet index’s information ratio can be indeterminate (because the active risk is so low) and is often negative due to management fees.
What does the Information coefficient measure?
Signal Quality
The IC measures an investment manager’s ability to forecast returns
How can we measure which factor most influences our active returns?
Return from factor tilts = Sum of the absolute contribution to active return
= ∑[(Portfolio sensitivity) − (Benchmark sensitivity)] × (Factor return)
Formula for active Value
∑ Weights of security ( Return of security - Return of Benchmark)
Formula for Information Ratio
(Portfolio Return - Benchmark Return) / Standard deviation (Portfolio Return - Benchmark Return)
Is the Information Ratio affected by the aggressiveness by active weights?
No, it is not!!!
What does the transfer coefficient measure?
Portfolio Construction
The correlation between forecast and actual weights of the portfolio
What Is Breath? (BR)
The number of independent decision made in a year in constructing a portfolio
Combined portfolio Sharpe Ratio (Including TC)
(Combined portfolio^2) = (IR^2)x (TC^2) + (Benchmark Sharpe Ratio^2)
Combined portfolio Sharpe Ratio = ROT(Combined portfolio^2)
Formula for proportion of benchmark in combined portfolio
1 - (Optimal level of risk / portfolio active risk)
Formula for optimal IR
Expected Return / Optimal volatility
Formula for optimal active weights
Δwi∗ = μi / σi^2
μi = Forecasted active return
σi^2 = Forecast volatility of active return
How can the The transfer coefficient be expressed as?
The transfer coefficient can be expressed as the risk-weighted correlation between the optimal active weights and the actual active weights, which is
𝐶𝑂𝑅(Δ𝑤𝑖∗𝜎𝑖,Δ𝑤𝑖𝜎𝑖)
For a constrained portfolio, how will aggressiveness of active weights afect the IR?
Aggressiveness of active weights will decrease IR
For a unconstrained portfolio, how will aggressiveness of active weights afect the IR?
Aggressiveness of active weights does not affect IR
