Porfolio management part 2

Question 1

What consists active management

Answer 1

Active management seeks to add value by outperforming a passively managed benchmark porfolio

Question 2

What are the 3 main qualities an appropriate benchmark should represent.

Answer 2

1. Be representative of investment universe from which the active manager may choose.
2. Be replicable at low cost
3. Weights available ex-ante and ex-post.

Question 3

Alpla

Answer 3

the difference in risk-adjusted returns. (difference in beta of actively managed portfolio and the benchmark)

Question 4

How is active return calculated and what is it?

Answer 4

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.

Question 5

What are active weights?

Answer 5

• It determines the amount of value added - the difference between a security weight in an actively managed portfolio and benchmark portfolio.

Question 6

How can we decompose active return?

Answer 6

Question 7

What are the Sharpe and the information (ex post and ex ante) ratio used for?

Answer 7

The information ratio and the Sharpe ratio are two different methods of measuring a portfolio’s risk-adjusted rate of return.

Question 8

How is the calculated the Sharpe ratio (SR)

Answer 8

• Calculated as the excess return per unit of risk (standard deviation)

Question 9

Why is the Sharpe ratio unaffected by adding cash or leverage to a portfolio?

Answer 9

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.

Question 10

How is calculated the information ratio (IR).

Answer 10

• ratio of the active return to the standard deviation of active returns, known as active risk or benchmark tracking risk

Question 11

The optimal active risk is calculated?

Answer 11

You scale active risk based on how good your alpha is (IR) relative to the Sharpe performance of the benchmark.

Question 12

Sharpe Ratio of the Optimal Portfolio

Answer 12

Question 13

How to calculate the total portfolio risk with active management?

Answer 13

Question 14

What does the Sharpe Ratio (SR) measure?

Answer 14

Excess return per unit of total risk: SR = (Rp - Rf) / σp

Question 15

What does the Information Ratio (IR) measure?

Answer 15

Active return per unit of active risk: IR = (Rp - Rb) / σ(Rp - Rb)

Question 16

What is active return?

Answer 16

RA = Rp - Rb (portfolio return minus benchmark return)

Question 17

What is active risk?

Answer 17

Standard deviation of active return: σA = std.dev(Rp - Rb)

Question 18

Is the Sharpe Ratio affected by adding cash or leverage?

Answer 18

No, both return and risk scale proportionally.

Question 19

Is the Information Ratio affected by adding cash or leverage?

Answer 19

Yes, cash lowers active return but not active risk, reducing IR.

Question 20

When does IR equal SR?

Answer 20

When benchmark = risk-free rate (e.g., market-neutral portfolio).

Question 21

What kind of fund has low IR and low active risk?

Answer 21

A closet index fund — mimics benchmark with little true active management.

Question 22

What is the ex-ante Information Ratio?

Answer 22

IR based on expected active return and risk — usually positive.

Question 23

What is the ex-post Information Ratio?

Answer 23

IR based on realized past return and risk — can be negative.

Question 24

What happens to IR if active weights are increased?

Answer 24

Nothing. Active return and active risk scale equally, IR stays the same.

Question 25

What happens to IR if we blend active portfolio with benchmark?

Answer 25

IR remains constant as both RA and σA shrink proportionally.

Question 26

How can investors target specific active risk levels?

Answer 26

By adjusting weight in active vs. benchmark portfolio.

Question 27

What is the formula for optimal active risk?

Answer 27

σA* = (IR / SRB) × σB

Question 28

What is the formula for Sharpe ratio with optimal active risk?

Answer 28

SRP = sqrt(SRB² + IR²)

Question 29

How is total portfolio risk calculated with active and benchmark risk?

Answer 29

σP² = σB² + σA²

Question 30

If IR = 0.2, SRB = 0.4, σB = 12%, what is optimal active risk?

Answer 30

σA* = (0.2 / 0.4) × 12% = 6%

Question 31

If σA* = 6% and fund’s active risk = 9%, what % goes to active fund?

Answer 31

6% / 9% = 67% to active fund, 33% to benchmark.

Question 32

How is portfolio excess return calculated using IR and σA?

Answer 32

E(RA) = IR × σA, then add (RB - RF) to get total excess return.

Question 33

What’s the Sharpe Ratio of portfolio P with 6% excess return and 13.4% risk?

Answer 33

SR = 6% / 13.4% ≈ 0.4472

Question 34

What are the 3 factors determining the information ratio

Answer 34

1. Information coefficient (IC) 2. Transfer coefficient (TC) 3. Breadth (BR)

Question 35

Explain what does the information coefficient (IC) represent?

Answer 35

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

Question 36

Explain the transfer coefficient (TC)

Answer 36

correlation between actual active weights and optimal active weight

Question 39

What is the Information Coefficient (IC)?

Answer 39

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.

Question 40

What is the ex-post Information Coefficient (ICR)?

Answer 40

ICR measures the actual correlation between active returns and expected active returns.

Question 41

What is the Transfer Coefficient (TC)?

Answer 41

TC is the correlation between actual active weights and optimal active weights.

Question 42

What is the formula for the Transfer Coefficient (TC)?

Answer 42

TC = CORR(μi / σi, ∆wiσi) = CORR(∆wi* σi, ∆wiσi)

Question 43

Define Breadth (BR) in the context of active portfolio management.

Answer 43

BR is the number of independent active bets taken per year.

Question 44

What is the Grinold rule?

Answer 44

The Grinold rule computes expected active return: μi = ICσiSi, where Si is the standardized score of security i.

Question 45

How is the expected value added by active management calculated?

Answer 45

E(RA) = ∑ ∆wi μi

Question 46

What is the optimal Information Ratio (IR*) for an unconstrained portfolio?

Answer 46

IR* = IC × √BR

Question 47

How is the expected active return for an unconstrained portfolio calculated?

Answer 47

E(RA)* = IC × √BR × σA

Question 48

What is the Information Ratio (IR) for a constrained portfolio?

Answer 48

IR = TC × IC × √BR

Question 49

How is expected active return for a constrained portfolio calculated?

Answer 49

E(RA) = TC × IC × √BR × σA

Question 50

What is the formula for optimal active risk in an unconstrained portfolio?

Answer 50

σA* = (IR / SRB) × σB

Question 51

What is the formula for optimal active risk in a constrained portfolio?

Answer 51

σCA* = (TC × IR*) / SRB × σB

Question 52

What is the Sharpe ratio of a constrained portfolio?

Answer 52

SRPC = √(SRB² + TC² × IR*²)

Question 53

What is the formula for realized value added from active management?

Answer 53

E(RA | ICR) = TC × ICR × √BR × σA

Question 54

How is actual active return expressed with noise?

Answer 54

RA = E(RA | ICR) + noise

Question 55

What does TC² represent in terms of active return variance?

Answer 55

It is the proportion of realized active return variance attributed to variation in realized IC.

Question 56

Why is the information ratio important for manager selection?

Answer 56

Because the active portfolio with the highest IR will also have the highest Sharpe ratio, making it optimal for all investors.

Question 57

How is expected active return calculated for a given level of active risk?

Answer 57

E(RA) = IR × σA

Question 59

What is market timing in active management?

Answer 59

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.

Question 60

How is the Information Coefficient (IC) for market timing calculated?

Answer 60

IC = 2(% correct) − 1

Question 61

Given IC = 0.10 and BR = 4, what is the Information Ratio (IR)?

Answer 61

IR = IC × √BR = 0.10 × √4 = 0.20

Question 62

Given IC = 0.04 and BR = 50, what is the IR?

Answer 62

IR = 0.04 × √50 ≈ 0.28

Question 63

How is active risk (σC) in a two-sector market calculated?

Answer 63

σC = √[σX² − 2σXσY rXY + σY²]

Question 64

How do you annualize active risk from a monthly strategy with σC = 0.05?

Answer 64

Annualized active risk = 0.05 × √12 ≈ 0.1732 or 17.32%

Question 65

If IC = 0.20, σA = 0.1732, and BR = 12, what is the annualized active return?

Answer 65

E(RA) = IC × √BR × σA = 0.20 × √12 × 0.1732 ≈ 12%

Question 66

What is the monthly active return if correct 60% of the time?

Answer 66

Monthly return = (0.60)(0.05) + (0.40)(-0.05) = 0.01 or 1%

Question 67

How is the allocation changed if active risk is capped at 5.20% and base risk is 17.32%?

Answer 67

Deviation = 5.20% / 17.32% ≈ 30%, so sector overweight = benchmark + 30%

Question 68

What does the phrase 'garbage in, garbage out' imply in the fundamental law?

Answer 68

It implies that inaccurate input estimates for IC and BR will lead to incorrect evaluations.

Question 69

What are the two main inputs of the fundamental law of active management?

Answer 69

Information Coefficient (IC) and Breadth (BR)

Question 70

Why is ex-ante measurement of IC a limitation?

Answer 70

Because managers may overestimate their forecasting skill, leading to inaccurate IR estimates.

Question 71

How is effective breadth adjusted for correlated decisions?

Answer 71

BR = N / [1 + (N − 1)r], where r is the average correlation between decisions.

Question 72

What compromises decision independence in breadth calculation?

Answer 72

Cross-sectional dependency and time-series dependency in decision-making.