Analysis Of Active Portfolio Management

Question 1

What are 3 key qualities a benchmark should possess to be a quality benchmark for active portfolio management?

Answer 1

• benchmark is representative of assets the active manager will select
• benchmark positions can be replicated at low cost
• benchmark weights are know in advance, and returns can be measured ex post

Question 2

What is formula for return of benchmark and return of portfolio?

Answer 2

return on benchmark = (weight of individual security in benchmark * return of individual security in benchmark) + (weight of individual security in benchmark * return of individual security in benchmark)

return on portfolio = (weight of individual security in portfolio * return of individual security in portfolio) + (weight of individual security in portfolio * return of individual security in portfolio)

Question 3

What is active return or value added and formula?

Answer 3

• active return: excess of portfolio return over benchmark return

Ra = Rp - Rb

Ra = active return
Rp = return of portfolio
Rb = return of benchmark

Question 4

What is alpha and formula?

Answer 4

• alpha: risk adjusted (beta) measure or excess risk afjusted return above benchmark

ap = Rp - (Bp*Rb)

ap = alpha of portfolio
Rp = return of portfolio
Bp = beta of portfolio
Rb = return of benchmark

Question 5

What is the formula to decompose returns and determine how much of returns in active manager were impacted by stocks and a deviation from the target weight and how much of returns in active management was impacted by bonds and a deviation from target weight?

Answer 5

Ra = (difference in stock weight and target stock weight * (stock market return - benchmark stock market return)) + (difference in bond weight and target bond weight * (bond market return - benchmark bond market return))

(difference in stock weight and target stock weight * (stock market return - benchmark stock market return)) = how much of alpha was impacted by stock weights

(difference in bond weight and target bond weight * (bond market return - benchmark bond market return)) = how much of alpha was impacted by bond weights

Question 6

What are the two common sources performance attribution decomposes value added, describe them.

Answer 6

• asset allocation: asset allocation decisions add value if high returning assets are overweighted vs benchmark or if low returning assets are underweighted vs benchmark
• security selection: choosing individual assets that outperform others in the same asset class

Question 7

What is asset allocation value added formula and security selection value added formula?

Answer 7

asset allocation portion = ((portfolio weight - benchmark weight) * (benchmark return)) + ((portfolio weight - benchmark weight) * (benchmark return))

security selection = (portfolio weight * (portfolio return - benchmark return)) + (portfolio weight * (portfolio return - benchmark return))

Question 8

What is the formula for total active return with asset allocation portion & security selection portion?

Answer 8

active return = asset allocation portion + security selection portion

Question 9

What is the difference between an absolute and a relative term or return? Is the sharpe ratio an absolute measure or relative measure, and is the information ratio an absolute measure or a relative measure?

Answer 9

• absolute return: actual return
• relative return: actual return vs a benchmark

sharpe ratio: absolute measure

information ratio: relative measure

Question 10

What is the sharpe ratio and formula?

Answer 10

• sharpe ratio: risk adjusted returns

SRp = (Rp - Rf) / (STD (Rp))

SRp = sharpe ratio of portfolio
Rp = return of portfolio
Rf = risk free rate
STD (Rp) = standard deviation of portfolio return

Question 11

What is the information ratio and formula?

Answer 11

• information ratio: measure active return relative to active risk

IR = (Rp - Rb) / (STD (Rp - Rb)) which is also equivalent to (Ra /(STD (Ra)))

IR = information ratio
Rp = return of portfolio
Rb = return of benchmark
STD Rp - Rb = standard deviation of active risk or standard deviation of return of portfolio - return of benchmark
Ra = active return
STD (Ra) = active risk

Question 12

What is the formula for constructing an optimal portfolio by investing in both the actively managed portfolio and the benchmark portfolio?

Answer 12

SRp^2 = SRb^2 + IR^2

SRp^2 = squared sharpe ratio of portfolio
SRb^2 = squared sharpe ratio of benchmark
IR^2 = square information ratio

Question 13

What is the formula for constructing an optimal portfolio with the optimal amount of active risk by investing in both the actively managed portfolio and the benchmark portfolio?

Answer 13

STD (Ra) = (IR / SRb) * STD (Rb)

STD (Ra) = active risk
IR = information ratio
SRb = sharpe ratio of benchmark
STD (Rb) standard deviation of benchmark portfolio

Question 14

What is the overall standard deviation formula of a portfolio that consists of the benchmark portfolio and the active fund?

Answer 14

STD (Rp)^2 = STD (Rb)^2 + STD (Ra)^2

STD (Rp)^2 = squared standard deviation of portfolio
STD (Rb)^2 = squared standard deviation of benchmark
STD (Ra)^2 = squared standard deviation of active risk

Question 15

What is the fundamental law of active management?

Answer 15

• framework to think about potential value added through active portfolio management

Question 16

What 4 things can the fundamental law of active management be used for?

Answer 16

• size active weights
• estimate the expected value added from active management
• measure realized value added after the fact
• evaluate active management strategies

Question 17

What are the 2 types of fundamental law of active portfolio management?

Answer 17

• basic fundamental law
• full fundamental law

Question 18

What is formula for active return of security and expected or forecasted return of security?

Answer 18

Rai = Ri - Rb

Rai = active return for each security
Ri = return on security
Rb = return on benchmark

expected or forecasted return on security:
Ui = E (Rai)

Ui = expected or forecasted return on security
E (Rai) = active return for each security

Question 19

What are the 3 main components of the correlation triangle?

Answer 19

top = forecasted active return (Ui)
bottom right = realized active returns (Rai)
bottom left = active weights (Wi)

Question 20

In terms of the correlation triangle how are the forecasted active returns (Ui) and the realized active returns (Rai) correlated, how are the realized active returns (Rai) and the active weights (Wi) correlated, how are the active weights (Wi) and the forecasted active returns (Ui) related?

Answer 20

forecasted active returns - realized active returns are correlated by signal quality (aka information coefficient). active management will only work if manager can to some degree predict realized active returns

realized active returns - active weights are correlated by value add (value is added when overweighting well performing and underweighting poor performing securities)

active weights - forecasted active returns are correlated by portfolio construction (aka transfer coefficient) measure efficiency with what manager translates their insights into their portfolio bets

Question 21

What are the 2 steps to get the mean variance optimal active security weights and formulas?

Answer 21

step 1: use grinold rule

Ui = IC * Oi * Si

Ui = active return forecast
IC = expected information coefficient
Oi = forecasted volatility of active returns
Si = set of standardized forecasts of expected returns across securities (aka scores)

step 2: formula to get mean variance optimal active security weights

mean variance optimal active security weights = (Ui /Oi ^2) * (Oa/(IC * sqrt BR))

Ui = forecasted active return
Oi = forecasted volatility of active return
Oa = active portfolio risk or standard deviation of the active return
IC = information coefficient (aka managers ability to predict actual returns)
BR = breadth (number of independent investment decisions per year made by investor constructing portfolio)

Question 22

What does an information coefficient of 1, 0, and -1 mean?

Answer 22

1: active manager is effective or perfect at predicting or forecasting active returns
0: no relationship
-1: active manager is terrible at predicting or forecasting active returns (aka always wrong at prediction of active returns)

Question 23

What is the difference in formula between the expected active portfolio return and expected active portfolio return if portfolio is constructed using the optimal active security weights?

Answer 23

E (Ra) = Wi * Ui

E (Ra) = expected active portfolio return
Wi = active security weights
Ui = expected security returns

formula if using optimal active security weights:
E (Ra) = IC * sqrt BR * Oa

E (Ra) = expected active portfolio return
IC = assumed information coefficient
BR = breadth but BR must = number of securities
Oa = portfolio active risk

Question 24

What is formula for expected active portfolio return if portfolio is unconstrained (basic fundamental law) and is constructed using the optimal active security weights and how can you use that information to attain the Information ratio (IR)?

Answer 24

formula if using optimal active security weights:
E (Ra) = IC * sqrt BR * Oa

E (Ra) = expected active portfolio return
IC = assumed information coefficient
BR = breadth but BR must = number of securities
Oa = portfolio active risk

IR can be calculate as:

IR = E (Ra) / Oa = IC * sqrt BR

IR = information ratio
E (Ra) = expected active portfolio return
IC = assumed information coefficient
BR = breadth but BR must = number of securities
Oa = portfolio active risk

Question 25

What is formula for expected active portfolio return if portfolio is constrained (full fundamental law) and how can you use that information to attain the Information ratio (IR)?

Answer 25

E (Ra) = TC * IC * sqrt BR * Oa

E (Ra) = expected active portfolio return
TC = transfer coefficient
IC = assumed information coefficient
BR = breadth but BR must = number of securities
Oa = portfolio active risk

IR can be calculate for constrained portfolio as:
IR = E (Ra) / Oa = TC * IC * sqrt BR

IR = information ratio
E (Ra) = expected active portfolio return
IC = assumed information coefficient
BR = breadth but BR must = number of securities
TC = transfer coefficient
Oa = portfolio active risk

Question 26

What is the formula for maximum possible value of the constrained portfolio’s squared Sharpe ratio? How do you remove a square root from the sharpe ratio?

Answer 26

SRp^2 = SRb^2 + ((TC)^2 * (IR)^2)

SRp^2 = squared sharpe ratio of portfolio
SRb^2 = squared sharpe ratio of benchmark
TC = transfer coefficient
IR = information ratio

remove square root from SRp^2 by square rooting SRb^2 + ((TC)^2 * (IR)^2) the other side of the formula

Question 27

What is the ex post performance measurement and what is the expected value added conditional on realized information coefficient?

Answer 27

• ex post performance measurement: once returns or IC have been realized instead of assumption

E [Ra | ICr] = TC * ICr * sqrt BR * Oa

E [Ra | ICr] = expected value added conditional on realized information coefficient
TC = transfer coefficient
ICr = realized information coefficient
sqrt BR = breadth
Oa = portfolio active risk

Question 28

What is the actual active return formula on an ex post performance measurement and how can the actual active return formula be broken up into portfolio active return variance?

Answer 28

Ra = E [Ra | ICr] + Noise

E [Ra | ICr] = expected value added conditional on realized information coefficient (skill of investor)
Noise = constraints on optimal portfolio

Ra = E [Ra | ICr] + Noise
is equal to
Ra = (TC^2 + 1-TC^2)

TC ^2 = variance due to realized information coefficient
1 - TC^2 = variance due to constraint induced noise

Question 29

What is information coefficient?

Answer 29

The information coefficient is the correlation between the portfolio investor’s forecasts and actual outcomes. This is a measurement of the investor’s skill.