Analysis of Active Portfolio Management

Table of Contents
1. READING 40
2. ANALYSIS OF ACTIVE PORTFOLIO MANAGEMENT
3. MODULE 40.1: VALUE ADDED BY ACTIVE MANAGEMENT
4. MODULE 40.2: THE INFORMATION RATIO VS. THE SHARPE RATIO
5. MODULE 40.3: THE FUNDAMENTAL LAW
6. MODULE 40.4: ACTIVE MANAGEMENT
7. KEY CONCEPTS
8. ANSWER KEY FOR MODULE QUIZZES
9. TOPIC QUIZ: PORTFOLIO MANAGEMENT
View more

READING 40

ANALYSIS OF ACTIVE PORTFOLIO MANAGEMENT

EXAM FOCUS

The information ratio is used to evaluate active managers and can be used to make portfolio allocation decisions for an investor. There are several formulas and linkages to understand in this reading. Know the differences between the Sharpe ratio and the information ratio. Be able to describe the fundamental law of active management and what influences each of its component terms. Understand the application of the fundamental law in the context of market timing and sector rotation strategies. Finally, be aware of the practical limitations of the fundamental law.

MODULE 40.1: VALUE ADDED BY ACTIVE MANAGEMENT

Video covering this content is available online.

LOS 40.a: Describe how value added by active management is measured.

Active management seeks to add value by outperforming a passively managed benchmark portfolio. The amount of value added is called active return (also called value-added) and is measured as the difference between the return of the active portfolio and the return of its benchmark.

Benchmark qualities

An appropriate benchmark should:

• be representative of the investment universe from which the active manager may choose;
• be replicable at low cost;
• have weights that are available beforehand (ex-ante), and benchmark returns that can be obtained promptly afterwards (ex-post).

Professor's note

In this topic review we assume that the systematic risk of the active portfolio is the same as the systematic risk of the benchmark portfolio (that is, the beta of the active portfolio relative to the benchmark is 1). If the beta of the actively managed portfolio differs from the beta of the benchmark, then active return is computed as the difference in risk-adjusted returns and is known as alpha.

Active return

Active return (R_A) is the value added by active management. Active return may be measured ex-ante (based on expectations) or ex-post (after the fact).

The ex-ante active return is the difference between the expected return of an actively managed portfolio and the expected return of its benchmark:

E(R_A) = E(R_P) - E(R_B)

Active weight is the difference between a security's weight in the actively managed portfolio and its weight in the benchmark. Securities that are overweighted (underweighted) relative to the benchmark have positive (negative) active weights. Active weights must sum to zero.

For an active portfolio of N securities, the portfolio active return can be written as the weighted sum of individual security returns, where each weight is the active weight:

Active return = Σ (active weight of security i × return of security i)

The ex-post active return is the realized return of the actively managed portfolio minus the realized return of its benchmark portfolio.

EXAMPLE: Active return

The following information is available for an actively managed portfolio and its benchmark.

Calculate the ex-ante active return.

Answer:

Active return = E(R_P) - E(R_B)

Given an investment strategy involving multiple asset classes, expected returns on the active and benchmark portfolios can be computed as the weighted average of the component asset returns. The ex-ante active return is the expected return on the active portfolio minus the expected return on the benchmark.

Active return can be decomposed into two parts:

• Asset allocation return - return due to deviations of asset class portfolio weights from benchmark weights;
• Security selection return - return due to active returns within asset classes.

For example, assume a benchmark composed of 25% stocks and 75% bonds. If the manager overweights stocks (and underweights bonds), the difference in returns from the benchmark due to that change is the asset allocation return. Within the stock allocation, choosing a specific stock over another contributes to the security selection return.

EXAMPLE: Active return - Optoma Fund

Optoma Fund invests in three asset classes: U.S. equities, U.S. bonds, and international equities. The asset allocation weights of Optoma and the expected performance of each asset class and the benchmark are shown in a supporting table.

Calculate the expected active return.

Answer:

Alternatively, by decomposing expected active return between asset allocation and security selection components, it can be seen that all of the expected active return is attributable to security selection. The active weights do not contribute to any asset allocation return in this case.

MODULE QUIZ 40.1

1. When measuring value added by active management, it is most accurate to state that the active weights in an actively managed portfolio:

A. must add to 100%.

B. are the differences between an individual asset's weight in the actively managed portfolio versus the corresponding weight in an equally-weighted portfolio.

C. must be positively correlated with realized asset returns for value added to be positive.

MODULE 40.2: THE INFORMATION RATIO VS. THE SHARPE RATIO

Video covering this content is available online.

LOS 40.b: Calculate and interpret the information ratio (ex post and ex ante) and contrast it to the Sharpe ratio.

The Sharpe ratio (SR) and the information ratio (IR) are two different measures of risk-adjusted return.

The Sharpe ratio is defined as excess return per unit of total risk (standard deviation):

SR = (R_P - R_F) / σ_P

An important attribute of the Sharpe ratio is that it is unaffected by the addition of cash (the risk-free asset) or leverage to the portfolio. For example, a 50% allocation to the risk-free asset reduces both the excess return and the standard deviation by half, leaving the Sharpe ratio unchanged.

The information ratio is the ratio of active return to the standard deviation of active returns (the volatility of the return relative to the benchmark). Active risk is also called benchmark tracking risk:

IR = E(R_A) / σ_A

Some important points

In this topic review, the information ratio discussed is typically the ex-ante information ratio (based on expectations). The ex-ante information ratio is generally positive (otherwise active management is not worth pursuing), while ex-post information ratios will often turn out to be negative.

A closet index fund is a fund that is purported to be actively managed but in reality closely tracks the underlying benchmark index. Such funds will have a Sharpe ratio similar to that of the benchmark index, a very low information ratio, and little active risk. After fees, the information ratio of a closet index fund is often negative.

A fund with zero systematic risk (for example, a market-neutral long-short equity fund) that uses the risk-free rate as its benchmark would have an information ratio equal to its Sharpe ratio. This is because active return would be equal to the portfolio return minus the risk-free rate, and active risk would be equal to total risk.

Unlike the Sharpe ratio, the information ratio will change with the addition of cash or the use of leverage. The numerator (active return) of the information ratio is measured relative to a non-cash benchmark. Adding cash to a portfolio is likely to lower active return, while active risk (the volatility of active return) should not change much; therefore adding cash will most likely decrease the information ratio.

The information ratio of an unconstrained portfolio is unaffected by the aggressiveness (scale) of the active weights. If the active weights of a portfolio are tripled, the active return and the active risk both triple, leaving the information ratio unchanged.

If an actively managed portfolio is combined with a portion invested in the benchmark (a blend), the blended portfolio will have the same information ratio as the original active portfolio. As the weight of the benchmark increases, active return and active risk decrease proportionately, leaving the information ratio unchanged.

Investors can select an appropriate amount of active risk by investing a portion of their assets in the active portfolio and the remaining portion in the benchmark. For example, if the active risk of a fund is 10% and an investor wants to limit active risk to 6%, the investor can allocate 60% to the active fund and 40% to the benchmark.

For an unconstrained active portfolio the optimal amount of active risk is the level of active risk that maximises the portfolio's Sharpe ratio. The resulting Sharpe ratio of the portfolio that combines the benchmark and the optimally scaled active portfolio has the following relation:

SR_P = √(SR_B² + IR²)

where SR_B is the benchmark Sharpe ratio and IR is the information ratio of the active strategy. This identity shows the incremental contribution of active management (IR) to total portfolio Sharpe.

EXAMPLE: Optimal active risk (Omega fund)

Omega fund has an information ratio of 0.2 and active risk of 9%. The benchmark portfolio has a Sharpe ratio of 0.4 and a total risk (σ_B) of 12%. If a portfolio (portfolio P) with an optimal level of active risk has been constructed by combining Omega fund and the benchmark portfolio, calculate:

1. Portfolio P's Sharpe ratio.

2. Portfolio P's excess return (return above the risk-free rate).

3. The proportion of benchmark and Omega fund in portfolio P.

Answer:

1. The Sharpe ratio for portfolio P with an optimal level of active risk is:

SR_P = √(0.4² + 0.2²) = √(0.16 + 0.04) = √0.20 = 0.4472

2. The expected active return given an active risk of 6% is:

E(R_A) = IR × σ_A = 0.2 × 0.06 = 0.012 = 1.2% = (R_P - R_B)

Given the benchmark Sharpe ratio of 0.4 and benchmark total risk of 12%:

(R_B - R_F) = 0.4 × 12% = 4.8%

Therefore, portfolio P's excess return is:

(R_P - R_F) = (R_P - R_B) + (R_B - R_F) = 1.2% + 4.8% = 6.0%

The total variance of portfolio P is the sum of the variances of benchmark and active components (assuming orthogonality between benchmark return and active return):

σ_P² = 0.12² + 0.06² = 0.0144 + 0.0036 = 0.018

σ_P = √0.018 = 0.134164 ≈ 13.4%

The Sharpe ratio is then:

SR = 6.0% / 13.4% ≈ 0.4472

3. The optimal level of active risk is 6% and Omega fund has active risk of 9%, so the proportion of portfolio P allocated to Omega is:

Allocation to Omega = 6% / 9% = 66.666...% ≈ 67%

Allocation to benchmark = 100% - 67% = 33%

Professor's note

Unconstrained active portfolios have optimal weights for each security determined by ex-ante expectations of active return and active risk. Constraints (for example, long-only positions) may force actual weights away from optimal weights and reduce achievable performance. The Sharpe ratio of a portfolio with the optimal level of active risk is given by the identity shown above and the total risk of the portfolio equals the root of the sum of squares of benchmark risk and active risk when the active return is uncorrelated with benchmark return.

MODULE QUIZ 40.2

1. Which of the following statements regarding the ex-post and ex-ante information ratio and Sharpe ratio is most accurate?

A. The Sharpe ratio measures reward per unit of risk in benchmark relative returns.

B. The information ratio measures reward per unit of absolute risk.

C. The information ratio can be applied either ex ante to expected returns or ex post to realised returns.

MODULE 40.3: THE FUNDAMENTAL LAW

Video covering this content is available online.

LOS 40.c: Describe and interpret the fundamental law of active portfolio management, including its component terms-transfer coefficient, information coefficient, breadth, and active risk (aggressiveness).

There are three principal factors that determine the information ratio:

• Information coefficient (IC) - a measure of a manager's skill. The IC is the ex-ante, risk-weighted correlation between active returns and forecasted active returns. The ex-post information coefficient (IC_R) measures the realised correlation between active returns and forecasted active returns.
• Breadth (BR) - the number of independent active bets taken per year. For example, if a manager takes 10 independent active positions each month, then BR = 10 × 12 = 120.
• Transfer coefficient (TC) - the correlation between actual active weights and the optimal active weights. The optimal active weight for a security is positively related to its expected active return and negatively related to its expected active risk. For an unconstrained active portfolio, the active weights equal the optimal weights and TC = 1. For constrained portfolios (for example, constraints on short positions or limits on active risk), TC may be less than 1.

The fundamental law of active management relates these components to the information ratio. A commonly used representation (the Grinold rule) is:

IR = TC × IC × √BR

The Grinold rule allows computation of the expected information ratio based on the information coefficient, breadth, and transfer coefficient. The expected value added by active management can then be obtained using the relation E(R_A) = IR × σ_A.

For an unconstrained portfolio, TC = 1 and optimal values are denoted by an asterisk (*). For constrained portfolios, the actual active weights Δw_i will differ from the optimal active weights Δw_i* and the transfer coefficient will be less than 1.

Because transfer coefficients are always less than or equal to one (TC ≤ 1), the information ratio of a constrained portfolio must be less than or equal to the optimal information ratio for an unconstrained portfolio.

Recall that the optimal level of active risk (in an unconstrained portfolio) is that which maximises the portfolio Sharpe ratio. For a constrained portfolio the optimal level of active risk σ_CA* will be less than the optimal active risk for an unconstrained portfolio. Similarly, the Sharpe ratio of a constrained portfolio is lower than the Sharpe ratio of an unconstrained portfolio.

Ex-post performance measurement

Realised value added from active management is the ex-post active return the manager achieves. Using the ex-post information coefficient IC_R, the fundamental law can be written in ex-post terms substituting IC_R for IC in the IR expression.

The actual return on the active portfolio can be expressed as its conditional expected return and a noise term:

R_A = E(R_A | IC_R) + noise

where E(R_A | IC_R) represents the expected value added given the realised skill of the investor in that period, and the noise term captures constraint-induced noise and other unpredictable components.

The proportion of realised active return variance attributed to variation in the realised information coefficient is TC². The remaining proportion, (1 - TC²), is attributed to constraint-induced noise.

LOS 40.d: Explain how the information ratio may be useful in investment manager selection and choosing the level of active portfolio risk.

Portfolio theory states that investors choose a combination of the risk-free asset and an optimal risky portfolio, where the optimal risky portfolio is the one with the highest Sharpe ratio. Because the Sharpe ratio of a portfolio that combines benchmark and active strategies increases with the information ratio of the active strategy according to SR_P = √(SR_B² + IR²), the active manager with the highest information ratio produces the highest possible Sharpe ratio when combined with the benchmark.

Therefore, investors seeking the best active contribution will select the active manager with the highest information ratio. The investor can then combine that manager's active portfolio with the benchmark to achieve the desired overall risk level.

The information ratio can also be used to determine the expected active return for a given target level of active risk:

E(R_A) = IR × σ_A

MODULE QUIZ 40.3

1. Investors that are constrained by regulation or investment policy may find that some of the important variables identified by the fundamental law of active portfolio management are out of their control. The element that is most likely to still be within the investor's control is the:

A. information coefficient.

B. transfer coefficient.

C. benchmark tracking risk.

2. The information ratio is least appropriate as a criterion for:

A. quantifying an actively managed portfolio's return in excess of the risk-free rate.

B. constructing an actively managed portfolio.

C. evaluating the past performance of actively managed portfolios.

MODULE 40.4: ACTIVE MANAGEMENT

Video covering this content is available online.

LOS 40.e: Compare active management strategies, including market timing and security selection, and evaluate strategy changes in terms of the fundamental law of active management.

Market timing is a bet on the direction of the market or a segment of the market. For example, a market timer may move money out of equities and into cash in anticipation of falling stock prices. For a market timer, the information coefficient is related to the proportion of correct calls as follows:

IC = 2 × (% correct) - 1

If the manager is correct 50% of the time, IC = 0.

EXAMPLE: Market timer vs. security selector

Darsh Bhansali is a manager with Optimus Capital. Bhansali, a market timer, makes quarterly asset allocation decisions based on his forecast of the direction of the market. Bhansali's forecasts are right 55% of the time.

Mike Neal is an equity analyst focusing on technology stocks. Neal, a security selector, typically makes 50 active stock selections annually. Neal has an information coefficient of 0.04.

Compute the information ratios of Bhansali and Neal assuming both managers construct unconstrained portfolios.

Answer:

Because both portfolios are unconstrained, TC = 1.

Bhansali's IC = 2 × 0.55 - 1 = 0.10

Bhansali's IR = TC × IC × √BR. If Bhansali makes quarterly bets, BR = 4, therefore:

Bhansali's IR = 1 × 0.10 × √4 = 0.10 × 2 = 0.20

Neal's IR = TC × IC × √BR. If Neal makes 50 independent stock selections per year, BR = 50, therefore:

Neal's IR = 1 × 0.04 × √50 ≈ 0.04 × 7.0711 ≈ 0.2828

Sector rotation

Market timing can also be applied as sector rotation. For example, a manager may allocate assets into sectors expected to outperform. Consider a market composed of two sectors X and Y with expected returns E(R_X), E(R_Y) and volatilities σ_X, σ_Y, and correlation r_XY.

The active risk of rotating between two sectors (the standard deviation of the differential return R_X - R_Y) is:

σ_C = √(σ_X² + σ_Y² - 2 r_XY σ_X σ_Y)

The annualised active risk depends on the number of independent bets per year (breadth). For example, if bets are made quarterly then BR = 4 and the annualised active risk equals the per-bet active risk times √BR.

EXAMPLE: Sector rotation - Hwang Soi

Hwang Soi makes monthly allocation decisions between consumer discretionary and consumer staples based on a proprietary model. The historical correlation between returns of the two sectors is 0.30 and Soi's bets have been correct 60% of the time. Further information is given in a supporting table.

1. What is the annualised active risk of Soi's sector rotation strategy?

2. What is the expected annualised active return of Soi's sector rotation strategy?

3. What will be the allocation to the consumer discretionary sector if Soi feels that consumer staples will outperform consumer discretionary over the next month and if active risk is limited to 5.20%?

Answer:

1. Monthly active risk = σ_C (given as 5.0% in the supporting data).

Annualised active risk = 0.05 × √12 = 0.05 × 3.4641 = 0.1732 = 17.32%

2. IC = 2 × 0.60 - 1 = 0.20

Using the probability-weighted average method for monthly outcomes: expected monthly active return = (0.60)(0.05) + (0.40)(-0.05) = 0.01 = 1% per month.

Annual active return = 1% × 12 = 12%.

3. If active risk is limited to 5.20% for the portfolio and the full active position would have implied an active risk of 17.32% annually, then the permitted deviation (scale) relative to the full strategy is:

Scale factor = 5.20% / 17.32% = 0.30 = 30%

Assuming the benchmark allocation is 65% consumer staples and 35% consumer discretionary, when Soi expects staples to outperform she increases staples weight by 30% of the potential deviation: staples allocation = 65% + 30% = 95%; discretionary allocation = 5%.

LOS 40.f: Describe the practical strengths and limitations of the fundamental law of active management.

The fundamental law can be used to evaluate a range of active strategies, including security selection, market timing, and sector rotation. Practical limitations are primarily due to poor input estimates - "garbage in, garbage out". For unconstrained optimisation, the two central inputs that determine the information ratio are the information coefficient (IC) and the breadth (BR) of the manager's strategy.

Key limitations arise from:

• Ex-ante measurement of skill - the IC is an estimate of forecast accuracy. Managers tend to overestimate their skill, leading to upwardly biased IC estimates and overstated expected IR. The accuracy of the IC determines the accuracy of the ex-ante information ratio.
• Independence (breadth) - breadth is intended to measure the number of truly independent decisions the manager makes. If decisions rely on the same or similar information, they are not independent and the effective breadth is lower than the raw count of decisions. If individual decisions are correlated, the effective breadth can be estimated by adjusting for correlation.

Decision independence may be compromised by:

• Cross-sectional dependency - for example, applying the same value screen across many stocks in the same industry may produce correlated outcomes because the stocks share similar fundamentals;
• Time-series dependency - for example, monthly rebalancing decisions may be correlated with decisions in nearby months.

MODULE QUIZ 40.4

1. Breadth is most likely to be equal to the number of securities multiplied by the number of decision periods per year if active returns are correlated:

A. cross-sectionally.

B. over time.

C. with active weights.

2. Which of the following factors least accurately identifies one of the major limitations of the fundamental law of active management?

A. Ex ante measurement of skill using the information coefficient.

B. Assumption of independence in forecasts across assets and over time.

C. Attribution of value added to a small number of inputs.

KEY CONCEPTS

LOS 40.a

• Value-added = active return = active portfolio return - benchmark return.
• Portfolio active return = Σ (active weight of security i × return of security i).
• Active return is composed of two parts: asset allocation return plus security selection return.

LOS 40.b

• Sharpe ratio: SR = (R_P - R_F) / σ_P.
• Information ratio: IR = E(R_A) / σ_A.
• Unconstrained portfolio optimal active risk - the portfolio Sharpe ratio with optimal active risk satisfies SR_P = √(SR_B² + IR²).

LOS 40.c

• The three components of the information ratio are the information coefficient (IC) - a measure of manager skill; the breadth (BR) - the number of independent bets per year; and the transfer coefficient (TC) - the degree to which actual active weights match optimal active weights.
• For an unconstrained portfolio, TC = 1.

LOS 40.d

• An investor will prefer the active manager with the highest information ratio because that active portfolio combined with the benchmark yields the highest achievable Sharpe ratio, irrespective of the investor's risk aversion.
• The investor can scale the active portfolio relative to the benchmark to reach a desired level of active risk.

LOS 40.e

• The information coefficient of a market timer is IC = 2 × (% correct) - 1.
• The fundamental law can be used to evaluate market timing, sector rotation and security selection strategies by expressing IR as TC × IC × √BR.

LOS 40.f

• While the fundamental law is useful for evaluating active strategies, its practical limitations include bias in estimating the ex-ante information coefficient and the difficulty of establishing true independence when measuring breadth.

ANSWER KEY FOR MODULE QUIZZES

Module Quiz 40.1

1. C - Value added will be positive only when end-of-period realised asset returns are positively correlated with the asset weights that the manager selected at the beginning of the period. Active weights are defined as the differences between an asset's weight in a managed portfolio versus its weight in the benchmark portfolio. Active weights in a portfolio must add up to zero, not 100%. (LOS 40.a)

Module Quiz 40.2

1. C - The information ratio can be applied either ex ante to expected returns or ex post to realised returns. The Sharpe ratio measures reward per unit of absolute (total) risk. The information ratio measures reward per unit of risk relative to a benchmark. (LOS 40.b)

Module Quiz 40.3

1. A - The information coefficient represents an active manager's own skill and ability to forecast returns accurately. The other three elements of the fundamental law (transfer coefficient, breadth of the strategy, and benchmark tracking risk) may be beyond an investor's control if constrained by investment policy or regulation. (LOS 40.c)

2. A - The information ratio evaluates risk-adjusted return in relation to a benchmark investment baseline rather than relative to a risk-free investment. Expected information ratio is the single best criterion for building an actively managed portfolio. The ex-post information ratio is typically used to evaluate past performance of actively managed funds. (LOS 40.d)

Module Quiz 40.4

1. C - Breadth (BR) is intended to measure the number of independent decisions that an investor makes each year. Breadth equals the number of securities multiplied by the number of decision periods per year only if (1) active returns are cross-sectionally uncorrelated and (2) active returns are uncorrelated over time. (LOS 40.e)

2. C - The fundamental law's usefulness stems from its ability to separate expected value added into contributions from the few basic elements of the strategy. Limitations concern uncertainty about the ex-ante information coefficient and the definition of breadth as the number of independent decisions. (LOS 40.f)

TOPIC QUIZ: PORTFOLIO MANAGEMENT

You have now finished the Portfolio Management topic section. Please log into your online dashboard and take the Topic Quiz on this section. The Topic Quiz provides immediate feedback on the effectiveness of your study for this material. Questions are more exam-like than the Module Quiz questions; a score of less than 70% indicates that further review is likely needed. These tests are best taken under timed conditions; allow three minutes per question.