Portfolio Management

Question 1

Arbitrage Pricing Theory (APT) - value of R_i

Answer 1

• the intercept is the expected return of asset _i given that all the other factors take on a value of 0

Question 2

APT assumptions

Answer 2

• A factor model describes asset returns
• There are many assets, so investors can form well-diversified portfolios that eliminate asset-specific risk, not factor risk
• No arbitrage opportunities exist among well-diversified portfolios

Question 3

APT E(R_p) given that the 3 assumptions of the model hold

Answer 3

Question 4

Pure factor portfolio (or factor portfolio)

Answer 4

A portfolio with a sensitivity of 1 to factor j and a sensitivity of 0 to all other factors

Question 5

Carhart four-factor model notation (extension of the Fama and French model)

Answer 5

• RMRF = the return on a value-weighted equity index in excess of the one-month T-bill rate
• SMB = small minus big, a size (market capitalization) factor; SMB is the average return on three small-cap portfolios minus the average return on three large-cap portfolios
• HML = high minus low, the average return on two high book-to-market portfolios minus the average return on two low book-to-market portfolios
• WML = winners minus losers, a momentum factor; WML is the return on a portfolio of the past year’s winners minus the return on a portfolio of the past year’s losers

Question 6

Carhart four-factor model / R_p - R_f

Answer 6

Question 7

Carhart four-factor model / E(R_p)

Answer 7

Question 8

Types of multifactor models

Answer 8

• Macroeconomic
• Fundamental
• Statistical:
  
  • Factor analysis - best explain historical return covariances;
  • Components models - best explain historical variances

Question 9

b_ik for the fundamental factor model

Answer 9

Question 10

Active return formula

• Asset allocation or sensitivity factor
• Security selection factor

Answer 10

Question 11

Information ratio (IR)

Answer 11

• = Active return / Active risk

Question 12

Active risk (also called Tracking error (TE)) - formula

Answer 12

• s(R_p - R_B)
• s(R_A)

Question 13

Active risk squared (active risk is also called tracking error or tracking risk) - formula

Answer 13

s² (R_p - R_B)

Question 14

Active risk squared decomposition

Answer 14

• Active risk squared = Active factor risk + Active specific risk
• Active factor risk and active specific risk refer to variances rather than standard deviation

Question 15

• Active factor risk
• Active specific risk or security selection risk

Answer 15

• The contribution to active risk squared resulting from the portfolio’s different-from-benchmark exposures relative to factors specified in the risk model
• The contribution to active risk squared resulting from the portfolio’s active weights on individual assets as those weights interact with assets’ residual risk

Question 16

Portfolio alpha

Answer 16

Question 17

Active specific risk formula

Answer 17

Question 18

Active specific risk notation

Answer 18

• w^a_i is the ith asset’s active weight in the portfolio (that is, the difference between the asset’s weight in the portfolio and its weight in the benchmark)
• σ²_εi is the residual risk of the ith asset (the variance of the ith asset’s returns left unexplained by the factors)

Question 19

Value added formula

Answer 19

• R_Ai = R_i - R_B
• R_i = return to asset i
• R_B = return on the benchmark portfolio
• Δw_i = w_P,i – w_B,i

Question 20

Value added as the sum of the active asset allocation decisions and the weighted sum of the value added from security selection, R_A,j = R_P,j – R_B,j, within each asset class

Answer 20

Question 21

Answer 21

The first (parenthetical) term above is the value added from the asset allocation decision. The second term is the value added from security selection within the stock and bond portfolios. The active weights in the first term refer to differences from the policy portfolio

Question 22

Sharpe ratio

Answer 22

Question 23

Sharpe ratio for leveraged portfolio

• STD of the risk-free cash (1 - w_p) is 0
• R_C = w_PR_P + (1 – w_P)R_F

Answer 23

Question 24

(Sharpe ratio)²_p

Answer 24

Question 25

The level of active risk that leads to the optimal result for unconstrained portfolios

Answer 25

Question 26

The level of active risk that leads to the optimal result for constrained portfolios

Answer 26

Question 27

Maximum value of the constrained portfolio’s squared Sharpe ratio

Answer 27

Question 28

The total risk of the actively managed portfolio is the sum of the benchmark return variance and active return variance

Answer 28

Question 29

Mean–variance-optimal active security weights for uncorrelated active returns

Answer 29

Question 30

Grinold rule

Answer 30

* S_i represents a set of standardized forecasts of expected returns across securities (also called scores) * σ_i are multipliers (separate values for individual securities) * μ_i are the forecasted active returns

Question 31

Mean–variance optimal active weights using the Grinold rule

Answer 31

Question 32

Information coefficient (IC) - formula / IC is the ex ante risk-weighted correlation

Answer 32

Question 33

E(R_A)\* - optimal expected active return

Answer 33

Question 34

IR\* - information ratio of the unconstrained optimal portfolio

Answer 34

Question 35

E(R_A) of the full fundamental law

Answer 35

Question 36

Transfer coefficient (TC) - definition

Answer 36

The TC is the cross-sectional correlation between the forecasted active security returns and the actual active weights, adjusted for risk

Question 37

Transfer coefficient (TC) - formula

Answer 37

Question 38

Information ratio (IR) of the full fundamental law

Answer 38

Question 39

The expected value added conditional on the realized information coefficient, IC_R

Answer 39

Question 40

Realized active return R_A considering noise - formula

Answer 40

Question 41

The IR for a constrained portfolio generally decreases

Answer 41

With the aggressiveness of the strategy

Question 42

Active return does not equal alpha

Answer 42

R_A = R_P – R_B, while α_P = R_P – β_P × R_B

Question 43

Unconstrained Portfolio

Answer 43

An investment style that does not require a fund or portfolio manager to adhere to a specific benchmark. Unconstrained investing allows managers to pursue returns across many asset classes and sectors

Question 44

μi - the security's expected active return

Answer 44

μ_i = E(R_Ai)

Question 45

Active weights

Answer 45

Question 46

R_Ai

Answer 46

Question 47

R_Ai as the residual return in a single-factor statistical model

Answer 47

Question 48

Aggressiveness of the active weights impact on the information ratio

Answer 48

* The information ratio is unaffected by the aggressiveness of the active weights (deviations from benchmark weights) because both the active return and the active risk increase proportionally * The aggressiveness will only decrease the IR of a constrained portfolio

Question 49

Portfolio variance

Answer 49

Question 50

Portfolio variance for a portfolio with equal weights 1/N

Answer 50

* σ² *overlined* = average variance * Cov *overlined =* average covariance

Question 51

Capital allocation line (CAL)

Answer 51

Question 52

Capital market line (CML)

Answer 52

Question 53

Capital market line (CML) versus capital allocation line (CAL)

Answer 53

The capital market line (CML) is the capital allocation line formed when the risky asset is a market return rather than a single-asset return

Question 54

Portfolio variance for a portfolio of two assets

Answer 54

Question 55

CAL and the Sharpe ratio

Answer 55

The Sharpe ratio is the slope of the CAL

Question 56

Markowitz efficient frontier

Answer 56

* The graph of the set of portfolios offering the maximum expected return for their level of risk (standard deviation of return) *

Question 57

Market model (single-index model) - return, variance and covariance

Answer 57

Question 58

* Factor portfolio * Tracking portfolio

Answer 58

* A portfolio with a factor beta equal to one for one factor and zero for all other factors. Factor portfolios have higher active factor risk (which refers to the deviation of a portfolio's factor betas from those of the benchmark) * A portfolio with factor betas equal to those of the benchmark

Question 59

Macroeconomic models

Answer 59

* The intercept a_i is the expected return to stock _i * All other factors have an expected value of 0 and represent a *surprise*

Question 60

Ex post IR

Answer 60

* = t_b0 / (number of observations)^1/2 * = t-value for the intercept / square root of the number of observations

Question 61

Value added from α

Answer 61

* α = IR \* ω * VA = α - (λ - ω²) * ω = residual risk * λ = risk aversion * α = expected value of the active return \**tested in 2015*

Question 62

Information ratio from α

Answer 62

* IR = α / ω * ω = residual risk * α = expected value of the active return

Question 63

Information ratio additivity

Answer 63

* IR_new² = IR_a² + IR_b² * The IR of the new stocks can simply be added to the IR of those already managed

Question 64

Portfolio constraint

Answer 64

* Time horizon * Liquidity * Tax concerns * Legal and regulatory factors * Unique circumstances

Question 65

The alpha and information ratio of a portfolio consisting of only cash and the benchmark

Answer 65

Both the alpha and the information ratio are 0

Question 66

IC using the number of successful decision

Answer 66

* IC = 2(N_c / N) - 1 * N_{c =} number of correct decisions * N = total number of decisions

Question 67

Sharpe ratio and cash

Answer 67

The Sharpe ratio is unaffected by the addition of cash or leverage in a portfolio and would thus not be appropriate to evaluate a portfolio in which an allocation to cash was a key part of the investment decision process

Question 68

The Fundamental Law of Active Management separates the expected value added, or portfolio return relative to the benchmark return, into the basic elements of the strategy

Answer 68

* Skill (information coefficient) * Portfolio construction (transfer coefficient) * Breadth (number of independent decisions per year) * Aggressiveness (benchmark tracking risk)