CFA Level III · Portfolio Management Pathway

Index-Based Equity Strategies: formula sheet

Full chapter guide

Key formulas

Active return: Active return = Portfolio return − Benchmark return; Positive means outperformance. Compare after fees and costs.
Tracking error (tracking risk): Tracking error = standard deviation of active returns; Passive aims for very low values. Active tolerates higher values.
Information ratio: IR = Average active return ÷ Tracking error; Measures active return earned per unit of active risk.
Net-of-cost view: Net active return = Gross active return − Management fees − Trading and other costs; Active must earn gross alpha above its extra costs to beat passive.
Market-cap weight: wᵢ = (Pᵢ × Sᵢ) ÷ Σ(Pⱼ × Sⱼ); Use free-float shares (Sᵢ) for a free-float-adjusted index.
Price weight: wᵢ = Pᵢ ÷ ΣPⱼ; Index value = ΣP ÷ divisor. Adjust the divisor after a split so the index value does not change.
Equal weight: wᵢ = 1 ÷ N; Weights drift as prices move, so rebalance to restore 1/N.
Fundamental weight: wᵢ = Fᵢ ÷ ΣFⱼ; F is the chosen fundamental (sales, earnings, cash flow, book value or dividends). A composite averages several measures.
Index return: R = Σ(wᵢ × Rᵢ); Use start-of-period weights.
New divisor after a split: New divisor = (sum of prices after split) ÷ (old index value); This keeps the index value unchanged on the split date.
Tracking error (ex post): Tracking error = standard deviation of (portfolio return − index return) over time; Also called tracking risk. Measures how closely the fund follows the index. Lower is better for an index fund.
Tracking error drivers: Portfolio return − index return = (cost drag) + (sampling/holding differences) + (cash drag and timing) + (securities lending and other income); A qualitative breakdown, not a numeric rule. Use it to explain why a fund's return differs from its index.
Method selection rule: Fewer, liquid constituents → full replication; many or illiquid constituents → sampling or optimization; A guide, not an absolute. Full replication gives the lowest tracking error but the highest trading and holding cost for broad indexes.
Active return (tracking difference) in one period: Active return = R_portfolio − R_benchmark; Calculate for each period. The average of these is the tracking difference.
Tracking error (ex post): TE = √[ Σ (AR_t − mean AR)² ÷ (n − 1) ]; Sample standard deviation of active returns. Use n − 1 unless the question says otherwise. Annualize by multiplying periodic TE by √(periods per year).
Annualizing tracking error: TE_annual = TE_monthly × √12 (quarterly: × √4; weekly: × √52); Scale by the square root of time, not by time itself.
Ex ante tracking risk from weights: Active weight_i = w_portfolio,i − w_benchmark,i; Larger active weights in volatile or highly correlated names raise expected tracking risk. Risk models turn these weights into an ex ante figure.
Approximate expected tracking difference: Expected gap ≈ −(fees + trading costs + cash drag) + securities lending income; Use this to explain why index funds usually trail the index by roughly their costs.
ETF premium or discount: Premium/discount = (Market price − NAV) ÷ NAV; Positive means premium: APs create shares. Negative means discount: APs redeem shares.
Futures fair value: F = S × (1 + r)^T − FV of dividends; FV of dividends means the dividends expected before expiry, each compounded forward to the futures expiry date (time T). Equivalent view: F ≈ S × (1 + (r − d) × T) for small T, where d is the dividend yield. The exam usually gives the inputs. Carry cost is financing less dividends.
Number of futures contracts: N = Exposure to add ÷ (Futures price × Multiplier); Round to a whole number of contracts. Use beta adjustment if the exposure is not the index itself.
Total return swap payoff to the receiver: Net payment to receiver = Notional × (Index return − (Floating rate + Spread) × period fraction); The receiver gets the index return, whether positive or negative, and pays floating plus spread. The receiver gains if the index return exceeds the financing leg. If the result is positive, the receiver receives that amount. If the result is negative, the receiver pays that amount. When the index return is negative, the result is negative, so the receiver pays the index decline plus the floating-plus-spread amount.
Tracking difference: Tracking difference = Fund return − Index return; Fees, trading costs and cash drag make it usually negative; securities lending can offset it.
Excess return (alpha vs index): Active return = Portfolio return − Benchmark return; Enhanced indexing targets a small positive active return after all costs.
Tracking error: TE = standard deviation of (Rp − Rb); Enhanced indexing keeps TE low, typically well below traditional active management.
Information ratio: IR = Active return ÷ Tracking error; Measures alpha earned per unit of active risk. Use it to judge if the extra risk is worth it.
Securities lending revenue (cash collateral): Gross revenue = Reinvestment return on collateral − Rebate paid to borrower; Applied to the collateral amount. Express in percent of collateral, then convert to the fund's assets.
Net lending revenue to fund: Net revenue = Gross revenue × (1 − Agent's share); Only if the agent's cut is a share of revenue. Check how the question defines the fee.
Revenue as a return on fund assets: Added return = Net revenue ÷ Fund assets; Scale by the share of assets on loan. Depending on assumptions, lending adds from a fraction of a basis point to roughly ten basis points, not percent.

Quick revision

Passive aims to match the index at low cost; active aims to beat it and carries higher fees and manager risk.
Market-cap weighting follows the market, has low turnover, and can overweight overvalued stocks.
Equal weighting tilts to smaller companies and needs more rebalancing, so it has higher turnover.
Fundamental and other alternative weightings add rules and potential factor tilts that differ from the market.
Full replication holds every index security and gives the lowest tracking error, but costs more for large or illiquid indexes.
Sampling and optimization hold a subset to cut costs, at the price of more tracking error.
Tracking error is the standard deviation of the active return, the difference between portfolio and index returns.
Fund costs, cash drag, rebalancing and index changes cause tracking difference.
Futures and swaps give index exposure with little cash but bring roll, margin or counterparty considerations.
ETFs trade intraday and are often tax efficient; check spreads and premiums or discounts to NAV.
Enhanced indexing seeks a small excess return with tightly limited tracking risk.
Securities lending adds income but brings counterparty and collateral risk, and needs governance.

Common mistakes

Saying active managers cannot add value in an efficient market. Fix: Say alpha is harder to find and less likely to cover costs. Skilled managers may still exist.
Treating passive as zero tracking error. Fix: Passive funds still have small tracking error from fees, sampling, cash and rebalancing timing.
Using total shares instead of free-float shares when the question gives float. Fix: If free-float data is provided, use it. Weight = price × float shares ÷ total of that column.
Saying price-weighted indexes reflect company size. Fix: Price depends on the number of shares outstanding and on split history, so it says nothing about total company value. Two firms of equal value can have very different prices and weights.
Saying full replication always has the lowest total cost. Fix: Say it has the lowest tracking error from holdings, but high trading and rebalancing costs for broad or illiquid indexes. Costs can raise its net tracking error.
Confusing stratified sampling with optimization. Fix: Sampling builds cells and picks securities by hand to match cell weights. Optimization uses a risk model to minimize expected tracking error mathematically.
Treating tracking error and tracking difference as the same thing. Fix: Tracking difference is the average gap. Tracking error is the standard deviation of the gap. Check which one the question defines.
Dividing by n instead of n − 1, or forgetting to subtract the mean active return. Fix: Compute the mean active return first, then use deviations from it and divide by n − 1 unless told otherwise.
Saying ETF shares are created and redeemed by ordinary investors with the fund. Fix: Only authorized participants create and redeem ETF shares, usually in large blocks in kind. Other investors trade on the exchange.
Reversing the arbitrage direction when an ETF trades away from NAV. Fix: Premium: sell the expensive ETF, buy the basket, deliver it and create shares. Discount: buy the cheap ETF, redeem it for the basket.

Exam tips

Match the command word. If told to justify, give the reason from the vignette, not a generic definition.
Avoid absolute words like always and never when judging active or passive.
Use the client's stated constraints as your evidence. Quote the details.
Show the subtraction and division for active return and information ratio so a slip still earns method credit.
Show the weight calculation line by line. A correct number typed alone earns credit, but the working protects you if an input is wrong in your head.
For a comparison, name the tilt, one risk and one cost. Equal weight: small and value tilt, high turnover, low capacity.
Answer only the number of points asked for, in the order given.
Link the scheme to the client. A large, low-cost mandate fits cap weighting; a client seeking a value or small-cap tilt may prefer equal or fundamental weights.