Active Equity Investing: Portfolio Construction: formula sheet

Key formulas

Active share

Active share = ½ × Σ |w_p,i − w_b,i|

Sum over all securities in the portfolio and benchmark. Weights are portfolio weight and benchmark weight. Result runs from 0% to 100% when long-only and fully invested.

Active return

Active return = R_p − R_b

Return of portfolio minus return of benchmark in the same period.

Tracking risk

Tracking risk = standard deviation of (R_p − R_b)

Also called tracking error or active risk. Annualise by multiplying a periodic figure by √(periods per year) when returns are independent.

Active share (equivalent form)

Active share = Σ of positive differences (w_p − w_b) = Σ of absolute negative differences

For a fully invested long-only portfolio, overweights equal underweights, so either side gives the answer.

Basic fundamental law

IR = IC × √BR

Ex-ante IR when TC = 1, so there are no constraints.

Constrained information ratio

IR = TC × IC × √BR

Use this once constraints apply. TC below 1 lowers the IR.

Full fundamental law (expected active return)

E(RA) = TC × IC × √BR × σA

σA is active risk (tracking risk). Use the same time basis for BR and σA, normally annual.

Information ratio from active return

IR = E(RA) ÷ σA

Active return over active risk.

Transfer coefficient meaning

TC = correlation(unconstrained active weights, actual active weights)

Lies between 0 and 1 in normal use. Lower TC means more constraint drag.

PEG ratio

PEG = (P/E) ÷ expected earnings growth rate (in % points)

Used in GARP. A lower PEG suggests growth is cheaper. Use the same growth rate basis when comparing stocks.

Returns-based style analysis

Rp = b1·R(style 1) + b2·R(style 2) + … + bn·R(style n) + e

Exposures b are estimated by regression, typically constrained to be non-negative and to sum to 1. The unexplained part e reflects selection and other effects.

Style drift test

Compare current style exposures or holdings characteristics with the stated style over time

A persistent, unplanned shift in exposures or characteristics signals drift.

Equal weight

w(i) = 1 ÷ N

Each of N holdings gets the same weight at rebalancing. Weights drift between rebalances.

Market-cap weight

w(i) = Market cap(i) ÷ Σ Market cap(all stocks)

Use free-float market cap when the index is float-adjusted.

Net exposure

Net exposure = Long % − Short %

A 130/30 portfolio has net 100% and gross 160%.

Gross exposure

Gross exposure = Long % + Short %

Gross measures total capital at work and leverage.

Active weight

Active weight(i) = Portfolio weight(i) − Benchmark weight(i)

In a long-only portfolio the active weight cannot be lower than −benchmark weight.

Market-neutral beta condition

Σ (w(i) × β(i)) ≈ 0 across longs and shorts

Dollar neutral does not guarantee beta neutral.

Risk parity condition

w(i) × σ(i) × correlation with portfolio is equalised, so risk contributions are equal

Simplified case with equal correlations: w(i) is proportional to 1 ÷ σ(i).

Active return

Active return = Rp − Rb

Rp is portfolio return and Rb is benchmark return.

Active weight

Active weight of asset i = wp,i − wb,i

Active weights sum to zero for a fully invested long-only portfolio.

Ex-post tracking risk

TR = standard deviation of (Rp,t − Rb,t)

Uses realized history. Do not confuse it with ex-ante.

Ex-ante tracking risk (two-source, general)

TR = √(σ1² + σ2² + 2 × ρ12 × σ1 × σ2)

σ1 and σ2 are the active risk contributions of two sources, ρ12 is their correlation. For uncorrelated sources, drop the last term.

Ex-ante tracking risk (weights)

TR = √(a′ Σ a)

a is the vector of active weights and Σ is the covariance matrix of asset returns. Exam versions usually use two or three assets.

Information ratio

IR = E(active return) ÷ tracking risk

Higher IR means more active return per unit of risk and supports a larger budget.

Combined managers, uncorrelated active returns

TR = √(Σ wi² × TRi²)

wi is the fraction of the portfolio given to manager i and TRi is that manager's tracking risk.

Risk contribution share

Share of variance of source i = (variance contribution of i) ÷ (total active variance)

Budget by variance, not by standard deviation, because standard deviations do not add.

Active return

Active return = Portfolio return − Benchmark return

Use for each manager and for the total portfolio. Diversification is judged on this series.

Tracking error (active risk)

TE = standard deviation of active returns

Annualize consistently. It measures how far a portfolio's return strays from its benchmark.

Two-manager active risk

TE_p = √(w1² × TE1² + w2² × TE2² + 2 × w1 × w2 × ρ × TE1 × TE2)

ρ is the correlation of the two managers' active returns. Lower ρ means more diversification of active risk.

Portfolio active return

α_p = w1 × α1 + w2 × α2

Weights apply to expected active returns, usually net of fees.

Information ratio

IR = Active return ÷ Tracking error

Use net-of-fee active return to compare managers or combinations.

Net alpha after fees

Net alpha = Gross alpha − Management fee − Performance fee

Compute the performance fee on the stated base, such as the excess over a hurdle.

Implementation shortfall (total)

Implementation shortfall = Paper portfolio return − Actual portfolio return

Includes explicit costs, market impact, delay and opportunity cost. Express in basis points or % of trade value.

Implementation shortfall components

Shortfall = Explicit costs + Realized (execution) cost + Delay cost + Missed-trade opportunity cost

Sum the components in the same unit. Delay cost is the price move between the decision and the arrival of the order at the market. If the arrival price equals the decision price, delay cost is zero, as in the worked example. Opportunity cost applies only to the unfilled part.

Execution cost per share (buy)

Execution cost = (Average execution price − Decision price) × Shares executed

This uses the decision price as the start point, which is correct when the arrival price equals the decision price. If the two differ, measure execution cost from the arrival price and count the move from decision to arrival as delay cost. For a sell, reverse the sign: (Decision price − Average execution price) × Shares sold.

Opportunity cost of unfilled shares (buy)

Opportunity cost = (Closing price − Decision price) × Shares not filled

Positive when the price rose after you failed to buy. Use the benchmark price stated in the question.

Annual cost drag

Cost drag ≈ Annual turnover × Round-trip cost

This pairing fits turnover defined as the lesser of purchases or sales ÷ average assets, which counts a round trip once. If turnover is (purchases + sales) ÷ average assets, use the one-way cost (half the round-trip cost). Check the definition given.

Net active return

Net active return = Gross active return − Trading and implementation costs − Management fees

Compare managers on a net basis.

Rebalancing rule of thumb

Rebalance when expected benefit of reducing drift > expected transaction cost

A guide to the trade-off, not a fixed rule.

Quick revision

• Active share measures the portion of the portfolio that differs from the benchmark; tracking risk measures the volatility of active returns.
• Higher active share with high tracking risk points to a more concentrated, stock-picking approach; low values point to closet indexing.
• Information ratio = expected active return ÷ active risk.
• Fundamental law: expected active return depends on the information coefficient, breadth and active risk, scaled by the transfer coefficient when constraints reduce implementation.
• Constraints such as long-only rules lower the transfer coefficient and so reduce realised value added.
• Match style to client: value, growth, GARP, quality and momentum differ in holdings, risk and market conditions.
• Equal weighting tilts toward smaller stocks and value; market-cap weighting is the neutral, benchmark-like approach, but it overweights stocks that have risen in price, so it can hold overvalued stocks.
• Risk budgeting allocates total active risk across managers, strategies or bets, and should reflect where you have skill.
• Core-satellite keeps a low-cost core and adds higher-conviction active satellites.
• Implementation shortfall captures explicit costs, market impact and delay or opportunity cost.
• Rebalancing trades off tracking error drift against transaction costs; wider ranges mean fewer trades but more drift.
• Always tie the answer to the client's objectives and constraints.

Common mistakes

• Forgetting the ½ in the active share formula. Fix: Remember that every overweight is matched by an underweight, so the raw sum counts activity twice. Halve it.
• Omitting benchmark securities the portfolio does not hold. Fix: Include every security in either list. A benchmark stock with portfolio weight 0 contributes its full benchmark weight as a difference.
• Forgetting the square root on breadth. Fix: Write √BR on the first line every time. Doubling bets does not double IR.
• Counting correlated positions as separate breadth. Fix: Count only independent decisions per year. If bets share a common driver, reduce breadth.
• Treating value as always lower risk or always better in downturns. Fix: Remember value stocks can be cheap for cause and can be value traps. Judge risk from the evidence given.
• Confusing style with top-down or bottom-up. Fix: Style is what kind of stocks are held. Approach is where the process starts. They are independent.
• Treating market-neutral as risk-free or as equal dollars long and short. Fix: Neutral applies to market beta only. Check beta-weighted exposure. Residual risks include stock selection, factor, short squeeze and leverage risks.
• Saying a long-only manager can underweight any stock by any amount. Fix: Maximum underweight equals the benchmark weight. Small stocks give little room, so negative views are harder to express.
• Adding tracking risks directly instead of variances. Fix: Square first, combine with weights and correlation, and take the square root last. Only perfectly correlated risks add directly.
• Forgetting to weight a manager's tracking risk by its share of the portfolio. Fix: Multiply each manager's tracking risk by its portfolio weight before squaring.

Exam tips

• Write the formula with the ½ first, then show the sum of absolute differences. A correct number typed alone earns credit, but showing the working protects you.
• For a classification question, state both measures versus the thresholds, then name the category. Keep it to one or two sentences.
• If the question asks you to justify a recommendation, link the classification to fees and the client's objective, such as paying active fees only for high active share.
• Watch command words: 'calculate' needs a number, 'identify' needs a label, 'explain' needs a reason. Answer only the number of responses asked for.
• Read the command word. Calculate means show a number. Explain or justify means state the driver and its effect in a sentence.
• Write the formula, substitute, then give the answer. A bare correct number can earn credit, but shown work protects you.
• If a constraint is mentioned, such as long-only or a turnover cap, expect TC to fall below 1 and explain that link.
• For recommendations, tie the change to the client's risk budget and constraints, not to the formula alone.