CFA Level III · Level III Core

Overview of Equity Portfolio Management: formula sheet

Full chapter guide

Key formulas

Market capitalization: Market cap = share price × number of shares outstanding; Used to sort companies into large, mid and small cap. Cut-offs vary by index provider, so use the thresholds given in the question.
Free-float market cap: Free-float market cap = share price × shares available to public investors; Many indexes weight by free float, not full market cap. Closely held shares are excluded.
Equity total return: Total return ≈ dividend yield + price change; Shows that equity return comes from both income and capital growth.
Market-cap weight: wᵢ = (Pᵢ × Sharesᵢ) ÷ Σ(Pⱼ × Sharesⱼ); Uses all shares outstanding. Weights sum to 100%.
Float-adjusted weight: wᵢ = (Pᵢ × Float sharesᵢ) ÷ Σ(Pⱼ × Float sharesⱼ); Float shares exclude strategic, insider and government holdings.
Equal weight: wᵢ = 1 ÷ N; N is the number of constituents. Weights drift and must be reset at each rebalance.
Price-weighted weight: wᵢ = Pᵢ ÷ Σ Pⱼ; Based on price only, not company size.
Fundamental weight: wᵢ = Fundamentalᵢ ÷ Σ Fundamentalⱼ; Fundamental is sales, earnings, book value, dividends or a blend.
Qualities of a valid benchmark: Unambiguous, investable, measurable, appropriate, reflective of current investment opinions, specified in advance, accountable, owned by the investment manager; Use this eight-item list as a checklist when judging a benchmark.
Active return: Active return = Portfolio return − Benchmark return; Calculate it for each period. Use the same period and the same currency for both.
Tracking error (tracking risk): TE = √[ Σ (Active return_t − Mean active return)² ÷ (n − 1) ]; Sample standard deviation of active returns. Use n − 1 for a sample unless the question says otherwise. It is not the mean active return.
Tracking error from risk and correlation: TE = √(σp² + σb² − 2 × ρ × σp × σb); Use when you are given portfolio and benchmark standard deviations and their correlation.
Annualizing tracking error: Annual TE = Periodic TE × √(periods per year); For example, monthly TE × √12. Do this only for returns that are independent over time.
Futures contracts to equitize cash: Number of contracts = (Cash amount ÷ (Futures price × Multiplier)) × (Beta target ÷ Beta of futures); For an index future with beta of 1 and a target beta of 1, the beta ratio is 1. Round to the nearest whole contract.
Active share: Active share = ½ × Σ |w(portfolio,i) − w(benchmark,i)|; Sum over all securities. A value of 0% is a pure index copy; 100% means no overlap with the benchmark.
Active return: Active return = Portfolio return − Benchmark return; Also called excess return versus the benchmark.
Active risk (tracking error): Active risk = standard deviation of (Rp − Rb); Measured over time using the active returns.
Information ratio: IR = Average active return ÷ Active risk; Return per unit of active risk. Higher is better.
PEG ratio: PEG = P/E ÷ expected earnings growth rate (in % points); Used by GARP managers. A lower PEG suggests growth is cheaper.
Active share: Active share = ½ × Σ |wp,i − wb,i|; Sum over all securities held by the portfolio or the benchmark. Weights are portfolio and benchmark weights. Result ranges from 0% to 100% for long-only portfolios.
Active share (overweights only): Active share = Σ of positive (wp,i − wb,i) = Σ of negative differences in absolute value; This holds when the portfolio weights and the benchmark weights each sum to 100% (long-only, no cash, no leverage). Then overweights equal underweights, so you can sum one side only.
Active return: Active return = Rp − Rb; Return of the portfolio minus return of the benchmark for the same period.
Tracking risk (tracking error): Tracking risk = standard deviation of (Rp − Rb); Use the sample standard deviation of periodic active returns, then annualise if asked (monthly × √12).
Information ratio: IR = average active return ÷ tracking risk; Active return per unit of tracking risk. Use the same time basis for both.
Tracking risk of a core-satellite portfolio: TE_total = √(w_core² × TE_core² + w_sat² × TE_sat² + 2 × w_core × w_sat × ρ × TE_core × TE_sat); Two-component case, where each TE is measured against the same benchmark and ρ is the correlation of active returns. With a passive core, TE_core ≈ 0, so TE_total ≈ w_sat × TE_sat.
Portfolio active return: Active return = Σ (w_i × active return_i); Weights are each manager's share of the total portfolio. Subtract fees to get net value added.
Information ratio: IR = active return ÷ tracking risk; Use net-of-fee active return when comparing managers on what the client keeps.
130/30 exposure: Long 130% − Short 30% = Net 100%; Net exposure is 100% by construction, and gross exposure is 130% + 30% = 160%. Market beta is near 1 only if the long and short books have similar betas.
Performance fee with hurdle: Fee = base fee × assets + incentive rate × max(0, return − hurdle) × assets; Check whether the fee is charged on excess over the hurdle only, or on all returns once the hurdle is cleared. A high-water mark means that losses must be recovered first.
Active return: Active return = Rp − Rb; Portfolio return minus benchmark return for the same period.
Allocation effect (sector i): (wp,i − wb,i) × (Rb,i − Rb); Compares sector weight difference with the sector benchmark return relative to the total benchmark return.
Selection effect (sector i): wb,i × (Rp,i − Rb,i); Uses benchmark weight. Interaction is then (wp,i − wb,i) × (Rp,i − Rb,i), shown separately.
Interaction effect (sector i): (wp,i − wb,i) × (Rp,i − Rb,i); Allocation + selection + interaction = total active return across sectors.
Information ratio: IR = (Rp − Rb) ÷ Active risk; Active risk is the standard deviation of active returns (tracking error).
Sharpe ratio: (Rp − Rf) ÷ σp; Excess return over the risk-free rate per unit of total risk.
Implementation shortfall: Paper portfolio return − Actual portfolio return; Sign convention: shortfall is positive when the actual return is below the paper return. It captures explicit costs, market impact, delay costs and opportunity cost of unfilled orders.

Quick revision

Link every equity choice to the client's objectives and constraints.
A benchmark should be appropriate, investable and specified in advance.
Index construction choices, such as weighting method, change the exposures you get.
Full replication tracks closely but can be costly for broad or illiquid indexes.
Sampling and optimization reduce cost and trading but add tracking error.
Active managers differ by style, such as value, growth or other approaches, and each carries distinct risks.
Active share shows how far holdings differ from the benchmark; tracking risk shows how much returns differ.
High fees need enough active risk and skill to justify them.
Manager selection should combine quantitative evidence with judgment about process and fit.
Attribution splits return differences into their sources so you can judge the decision.
State recommendations briefly with a reason drawn from the vignette.
Check the command word before you write an essay answer.

Common mistakes

Treating the style label as fixed across index providers. Fix: Use the criteria in the question. Providers use different rules, so the same stock can be labelled differently.
Using full market cap when free-float cap is required. Fix: Check whether the question mentions shares held by insiders or the government. If so, exclude them for free-float cap.
Weighting by market cap when the question says float-adjusted. Fix: Underline the word 'float' and use float shares to compute value and weights.
Saying equal weighting needs less trading than market-cap weighting. Fix: Equal weights drift as prices move, so rebalancing means more turnover and cost.
Saying optimization always has the lowest tracking error. Fix: Full replication has the lowest expected tracking error before costs. Optimization relies on a model and past data, so it can drift.
Using the average active return as tracking error. Fix: Tracking error is the standard deviation of active returns. A portfolio can have a mean active return of zero and still have high tracking error.
Confusing active share with tracking error. Fix: Active share is about holdings weights. Tracking error is about volatility of return differences.
Calling any manager who holds growth stocks a growth manager. Fix: Judge the style from the whole portfolio and the selection discipline, not one stock.
Forgetting the ½ in the active share formula. Fix: Check the range. Active share cannot exceed 100% for long-only portfolios. If your sum is 80% from overweights and underweights combined, halve it to 40%.
Treating active share and tracking risk as the same thing. Fix: Active share uses holdings weights at a point in time. Tracking risk uses the standard deviation of return differences over time.

Exam tips

In essay sets, tie every statement to a client objective or constraint. A generic fact alone usually earns little.
Use the market cap thresholds and style definitions given in the vignette, not ones you memorised.
Calculations are scored on the correct number, but show steps in case you make a small slip elsewhere.
Answer only the number of points asked for, in the order given. Extra points are not evaluated.
Watch the command word: identify needs a name, justify needs a reason, recommend needs a choice plus a reason.
Practise weight calculations with float shares. Show each stock's value, the total, then the weight, so a slip still earns method credit.
Learn one tilt per weighting method. Exam questions often ask what a method overweights.
For benchmark questions, tie the answer to the mandate in the vignette. Generic checklist answers earn fewer points.