Index Construction, ETFs and the Mechanics of Passive and Active Equity

Asset Allocation & Performance (Finance-Plus ring), Branch AA · Node AA1.08 · ~16 focused hours · Mastery gate ≥ 85%

A unit of an index fund and a unit of an exchange-traded fund can track the same index, hold the same shares and charge almost the same fee. One of them can be bought, on a bad afternoon, at a price 7.00% above the last net asset value its manager published. The other cannot, ever, because the only price at which it transacts is a net asset value struck after the market closes.

That asymmetry is the whole subject. An exchange-traded fund is a mutual fund with a second market bolted on, and the second market is where every advantage and every hazard of the wrapper lives. You get intraday liquidity, a visible spread, an in-kind creation mechanism that pushes tax and trading costs onto the people who cause them, and a price that usually sits within a basis point or two of fair value. You also get a price set by whoever is willing to trade at three in the afternoon. And the mechanism that pulls that price back to fair value can be switched off, by a regulator, by a market halt, or by a limit on how much foreign currency the industry may send abroad this quarter.

Most of the passive world is taught as a fee comparison. Fees matter, and they are the largest single line in the arithmetic that follows, but they are not what separates a good tracker from a bad one. A fund charging 60 basis points can miss its index by 118. The worked example that shows exactly that also shows where the other 58 went: cash the fund could not invest, dividends a foreign tax authority kept, trading the rulebook forced at reconstitution, and a securities-lending programme that gave 8 basis points back. None of those appears in the expense ratio, and all of them are recoverable from published documents once you know what to subtract from what.

The rulebook is the first thing to take seriously, because a tracker does not choose its portfolio. The index provider does. Where a rulebook puts its buffer zone, how it defines free float, how many days of notice it gives before a change, and whether it reconstitutes on a fixed calendar or whenever a committee decides all land directly on the fund's turnover, its trading costs and its tracking error. A tracking fund is a firm whose entire investment policy is written by someone else and published as a PDF.

The second half of what follows crosses to the other side of the same street. An active equity manager faces the identical mechanics from the opposite direction: she also has a benchmark, she also pays impact costs, she also trades at reconstitution, and the risk system she reports to measures her deviations with the same tracking-error arithmetic a tracker uses to measure its failures. What changes is the sign. Where a tracker's deviation is a defect to be minimised, hers is the product, and the question becomes whether the deviations she is paid for survive the constraints her mandate imposes and the costs her trading incurs. A 30-name book with a 444 basis point tracking error and a name limit of 2.50 percentage points is a different product from the same book without the limit, and the difference is computable to the basis point.

Two habits carry through all of it. Compute the number before you argue about it, because almost every dispute in this territory ("the ETF is expensive", "sampling is risky", "the premium is a bargain") dissolves into arithmetic once someone writes the arithmetic down. And always say what your number is measured against: a premium to a stale net asset value and a premium to a live fair value are different quantities, and confusing them is how a retail investor buys 4.19% of pure loss and calls it an index fund.

Learning objectives

You can:

1. Read an index rulebook as a binding constraint on a fund, identify the four provisions that drive a tracker's turnover, and compute what a buffer zone costs in index purity and saves in trading.
2. Describe the exchange-traded fund wrapper in terms of its four parties and two markets, and state precisely which party bears which risk in each market.
3. Walk a creation and a redemption through in creation units, using a portfolio composition file, an in-kind basket and cash in lieu, and compute the authorised participant's economics on both legs.
4. Derive the no-arbitrage band around an exchange-traded fund's price from stated transaction costs, and compute the price at which an authorised participant restarts creating.
5. Compute a premium or discount from live quotes and decompose it into a stale-valuation component and a genuine dislocation, and name the conditions under which the second one persists.
6. Distinguish tracking difference from tracking error, compute both on the same fund, and decompose a tracking difference into fees, cash drag, withholding tax, securities-lending revenue and rebalancing cost so the parts reconcile exactly.
7. Compare full replication, stratified sampling and optimisation on one universe, computing the ex-ante tracking error, the reconstitution turnover and the build cost of each, and state which method wins for which mandate.
8. Build a total-cost-of-ownership comparison across four routes to the same exposure and solve for the holding period at which the ranking changes.
9. Compute the compounding gap on a daily-rebalanced leveraged or inverse fund over a stated path, reconcile it against the variance-drag approximation, and explain why the gap is a property of volatility rather than of fees.
10. Classify an active equity approach by process, breadth and style, and state what a factor or smart-beta mandate is actually selling.
11. Compute active share, ex-ante tracking error and the ex-ante information ratio of a 30-name book, apply position and sector limits, and price what those limits cost in expected active return and information ratio.
12. Decompose the implementation shortfall of one rebalance into delay, execution, opportunity and explicit costs, and read the decomposition as evidence about the trading process rather than about the market.

The gated skill throughout is the by-hand arithmetic: a premium decomposed, a tracking difference that reconciles, an active share computed from a weight vector. A spreadsheet does all of it faster once you can do it slowly, and the two augmentation blocks below say exactly where a tool helps and what it can never check for you.

Prerequisites & connections

Builds on. M0.02 supplies the order book, price-time priority and the impact-cost calculation, all used here without re-deriving them. EC1.02 owns index construction proper: the four weighting schemes, free-float adjustment and the Investible Weight Factor, the Nifty 50 and S&P 500 methodologies as their providers publish them, and the turnover a weighting scheme forces. Read that treatment first, because what follows starts where it ends, at the point where a real fund has to hold the thing the rulebook describes.

AA1.01 supplies the policy portfolio, constrained mean-variance optimisation and the idea that a constraint has a price in forgone expected return. AA1.03 supplies tracking error, the seven benchmark-validity properties and the price-return-versus-total-return defect. QM1.01 and QM1.02 supply variance, covariance and the factor-model algebra behind every tracking-error computation below.

Feeds forward. QD2.01 owns execution and microstructure in code: order types, the limit order book dataset, time-weighted and volume-weighted schedules, the square-root impact law, and the full implementation-shortfall decomposition. Section 13 applies that decomposition to one index-fund rebalance and hands the machinery straight back. QD1.04 builds the portfolio optimiser that Section 8 and Section 12 describe by hand. M0.06 owns the retail decision between an index fund and an exchange-traded fund, including the costs and taxes a household actually pays, and cross-links here for the mechanics behind that decision. AL1.04 takes the wrapper into alternatives. ES1.02 owns the regulatory perimeter around fund distribution and disclosure.

Deliberate non-overlaps matter here because six neighbours own material that borders this one, and none of it is rebuilt below. EC1.02 owns how an index is built, so the weighting schemes appear below only as inputs to a tracker's problem. M0.02 owns the order book, so impact cost is used rather than derived. AA1.03 owns benchmark validity and the price-return-versus-total-return trap, so a benchmark's legitimacy is assumed here and its mechanics are not re-argued. QD2.01 owns implementation shortfall, so the four-term decomposition is applied once and cited, not developed. M0.06 owns the household-level product choice and its tax treatment. AA1.02 owns risk budgeting and component risk contribution, which is the risk-side twin of the active-weight arithmetic in Section 12. When a question is about one of those six, that is where the answer lives.

Core teaching content

1. The tracker's problem: a portfolio somebody else designed

An active manager who dislikes a stock sells it. A tracking fund cannot. Its holdings are dictated by a document it did not write, revised on a calendar it does not control, according to eligibility tests it cannot appeal. The fund's investment skill, such as it is, consists entirely of holding that portfolio more cheaply and more accurately than the next fund does.

This changes what "good" means. A tracker is not trying to beat the index and is not trying to avoid its mistakes. It is trying to reproduce the index's return, minus the smallest possible amount, with the smallest possible variance around that shortfall. Both halves matter and they are different objectives. A fund that misses by exactly 22 basis points every year is doing a better job than one that misses by an average of 10 basis points with a standard deviation of 60, at least for a client who has to explain the number to a committee.

The index provider's incentives are worth understanding because they are not the tracker's. A provider wants an index that is investable, representative and stable enough to license, and it makes those trade-offs in public. Every provision that makes the index more representative tends to make it more expensive to track, and every provision that makes it cheaper to track tends to make it less representative. Buffer zones are the clearest case, and Worked example 1 prices the trade both ways on a single universe.

There is one structural gift in the standard design. A free-float market-cap-weighted index is self-adjusting under price moves: when one constituent rises and another falls, the index's own weights shift in exactly the proportion a holder's portfolio does, so a fund that owns the whole index needs to trade nothing at all. Every other weighting scheme forces trading merely to stand still. That is why the world's largest passive pools are cap-weighted, and why an equal-weighted or factor-weighted product carries structurally higher costs than its expense ratio suggests.

The trading a cap-weighted tracker cannot avoid comes from four events, and they are worth separating because they have different sizes and different warning times. Membership changes at reconstitution force real buying and selling. Free-float revisions, when a promoter sells a block or a lock-in expires, change a constituent's weight without any price moving. Corporate actions, meaning mergers, demergers, rights issues and share-count changes, force adjustments on their own timetable. And the fund's own cash flows, subscriptions and redemptions, force trading that has nothing to do with the index at all.

A fund's annual report will not label these separately. You can usually recover them anyway, by taking the disclosed portfolio turnover, subtracting the turnover the index itself published for the same period, and treating the difference as flow-driven. It is a rough split and it is much better than no split.

9. Total cost of ownership, and the crossover

An expense ratio is a rate. What an investor pays is a rate multiplied by a holding period plus a one-off charge, and the ranking of two products flips at whichever holding period equates them. Write each route as a straight line in time:

cost(T) = f_0 + f_y × T

where f_0 is the round-trip cost of getting in and out, f_y is the annual carrying cost, and both are in basis points of notional. Setting two routes equal gives the crossover directly: T* = (f_0^B - f_0^A) / (f_y^A - f_y^B).

Four routes to the same large-cap exposure, with the costs each one actually incurs:

An index mutual fund transacts at net asset value, so there is no spread and no impact to pay on the way in, and its f_0 is close to zero. Its expense ratio is typically higher than an exchange-traded fund's on the same index, so its f_y is higher.

An exchange-traded fund costs a half-spread, a commission and some impact on each side, so f_0 is real but small for a liquid fund. Its f_y is the expense ratio net of any securities-lending credit passed back.

Index futures are the cheapest thing to trade and among the most expensive things to hold. The holding cost is the roll: a futures price embeds an implied financing rate, and if that rate exceeds what the investor's own cash earns, the difference is a carrying charge paid four times a year, plus the transaction cost of each roll. The implied financing spread moves with dealer balance-sheet conditions and can swing by tens of basis points, so this is the input to check before quoting a comparison rather than after.

A direct basket of the constituent shares has the highest entry cost, because the tail is thin and the impact is real. It also has the lowest carrying cost: no expense ratio, all the securities-lending revenue kept, and only custody and the reconstitution trading the rulebook forces as recurring charges.

Worked example 5 puts numbers on all four and solves the three crossovers. The results are worth previewing because they are not what most investors assume. Futures beat the exchange-traded fund only for holds shorter than about twelve calendar days. The index mutual fund beats the exchange-traded fund for anything under about nine and a half months, because its zero entry cost outweighs a three basis point annual difference for that long. And the direct basket, despite its low carrying cost, does not overtake the exchange-traded fund for more than eight and a half years.

Three cautions about using this apparatus. The costs are all perishable: expense ratios fall, spreads widen in stress, and the futures financing spread is a market price rather than a constant, so date every input. The model is linear in time and reality is not, since a five-year holder will face flows, corporate actions and at least one stressed market. And it prices only cost, while the routes differ in things cost does not capture, including intraday liquidity, the ability to short, collateral treatment, tax and operational burden. A futures position that is 30 basis points a year more expensive can still be the right answer for a fund that needs to move $500 million of exposure this afternoon.

13. What implementation costs an active book

A rebalance is not free and the bill has four lines. The explicit costs are commissions, exchange and clearing charges and transaction taxes, and they are the only ones that appear on a contract note. The impact cost is the price the order moves against itself as it consumes the book, which is the walk down the order book the markets node builds and which scales roughly with the square root of the order's size as a fraction of daily volume. The delay cost is the market's drift between the decision and the moment the order reaches the market. The opportunity cost is what the unfilled portion of the order would have made or lost.

The accounting that makes these comparable is the implementation shortfall, which anchors everything to the price at the moment of the decision and refuses to move that anchor afterwards. The execution node owns the machinery and the code. Applied to one index-fund rebalance it looks like this. A fund decides at 100.00 to buy 400,000 shares, a decision notional of $40,000,000. The order reaches the market at 100.15, so the delay cost is $60,000, or 15.0000 basis points, spent before a single share traded. It fills 340,000 shares at an average of 100.42, so the execution cost is $91,800, or 22.9500 basis points. The remaining 60,000 shares never fill and the price closes at 100.60, so the opportunity cost is $36,000, or 9.0000 basis points. Fees are $3,400, or 0.8500 basis points. The total is $191,200, or 47.8000 basis points of the decision notional, on a fill rate of 85.0000%.

Read that decomposition as evidence about a process rather than about a market. Fifteen basis points of delay on an order that took hours to reach the desk is an operations finding, not a trading one. Nine basis points of opportunity cost on a 15% shortfall says the trader was too passive for the urgency of the order, or that the order was too large for the day. And a fund that reports only the 22.95 basis points of execution cost is reporting the one line its trader controls and omitting the two that its process controls.

Two implications matter for what came before. For a tracker, this whole bill is a tracking-difference component and belongs in the decomposition of Section 7. For an active manager it is the toll on every deviation she takes, so her alpha must be forecast net of it.

Otherwise the optimiser will happily build a portfolio whose expected active return is smaller than the cost of reaching it. Moving the 30-name book of Worked example 8 from its unconstrained optimum to its constrained solution turns over 20.331142% of net assets one way, and at 25 basis points of all-in cost per side that rebalance spends about 10 basis points of net asset value, against a 119.73 basis point expected active return.

14. Using the wrapper inside a real portfolio

Five uses, and each one has a failure mode worth stating alongside it.

Core exposure puts the strategic allocation into the cheapest broad tracker available and builds everything active around it. The failure mode is choosing on expense ratio alone when the tracking difference, which carries the four other components, is the number that compounds.

Tactical tilts express a view for a stated period through a sector, country or factor fund, without disturbing the core. The failure mode is that a tilt with no exit rule becomes a permanent allocation nobody chose, which is how a policy portfolio quietly drifts.

Cash equitisation buys index exposure the day an inflow lands, so the fund is not out of the market while it deploys. That is the cash-drag line of Section 7, attacked with the wrapper. The failure mode is forgetting to unwind as the cash goes to work and running double exposure.

Factor and thematic access buys a published rules-based tilt at a stated fee. The failure modes are paying an active fee for a mechanical exposure, and buying a theme whose index was constructed after the theme had already worked.

Transition management parks the assets in an index fund or a futures position while a mandate moves from one manager to another, so the portfolio keeps its market exposure instead of sitting in cash. The failure mode is measuring the transition against the wrong benchmark and calling the market's move during the gap a cost or a saving.

For a household the same menu compresses into one decision, and the retail products node owns it. An index fund and an exchange-traded fund on the same index differ in how you transact, in what a broker charges, in the tax treatment of the exit, and in whether a bad afternoon can sell you a 7% premium. The mechanics above make that comparison decidable rather than a matter of preference.

Common mistakes & how experts think differently

1. Comparing trackers on expense ratio alone. The expense ratio is one of five lines and often not the largest. A fund at 20 basis points with a poor lending programme, a large cash balance and clumsy reconstitution trading can cost more than a fund at 35 basis points that does all three well. An expert asks for the tracking difference over three and five years, then asks what the difference between it and the expense ratio consists of.

1. Reading a premium against a stale net asset value. A published net asset value can be a full session out of date, and for a fund holding overseas securities it almost always is. Quoting a 7.00% premium without splitting out the 2.52 points that are simply yesterday's prices is the single most common error in retail commentary on exchange-traded funds. An expert computes fair value from futures and the currency before saying the word premium.

1. Believing the arbitrage is automatic. The tether between price and value is a trade that somebody has to want to do. It costs money, it needs a functioning primary market, and it can be switched off entirely by a creation halt. An expert checks the fund's creation status before diagnosing a wide premium, because that single fact changes the reading from expensive to unbounded.

1. Treating tracking error as the measure of a passive fund's quality. Tracking error is the volatility of the shortfall, not its size. A fund that misses by exactly 40 basis points every year has near-zero tracking error and costs its holder 40 basis points a year. An expert asks for tracking difference for cost and tracking error for predictability, and never lets one stand in for the other.

1. Adding tracking-difference components as if they were independent. Each friction is charged on an asset base that compounds. The additive presentation understates the total in a rising year, by 16.61 basis points on the fund in Worked example 3, and overstates it in a falling one. An expert states the convention and shows the multiplicative chain when the numbers are large enough to matter.

1. Assuming a sampled tracker's risk is small because its holdings look like the index. The tracker in Worked example 4 holds the 30 largest of 50 names, which covers 88.38% of the index by weight and sounds safe. Its ex-ante tracking error is 139.29 basis points a year, and 55.63% of that variance comes from a single systematic exposure: the small-size factor it is structurally short. An expert decomposes the tracking error by source before deciding whether it is acceptable.

1. Expecting a leveraged fund to deliver its multiple over any period longer than a day. The prospectus says daily, and the daily promise is kept precisely. An expert computes the compounding gap on the actual path before forming any view, and knows that the gap is a function of realised volatility, so a leveraged position in a quiet market and one in a violent market are different products wearing the same name.

1. Optimising to the lowest tracking error without pricing the turnover. The optimiser in Worked example 6 delivers the lowest tracking error of the three methods and trades more than three times as much at reconstitution. An expert reports tracking error, turnover and cost together, because a tracking error bought with turnover is not free and the client pays for both.

1. Writing constraints without checking they are feasible. A sector band that is arithmetically unreachable given the position cap and the names held will produce either a solver failure nobody investigates or a silently infeasible answer. An expert computes the reachable range before writing a limit, which takes one line of arithmetic and prevents a mandate that cannot be complied with.

1. Reading implementation shortfall as a verdict on the trader. Delay cost is an operations finding, opportunity cost is usually a sizing or urgency finding, and only the execution term is mostly about trading. An expert reads all four lines and assigns each to the part of the process that produced it.

Worked examples

The figures are computed at full precision and rounded only for display. Where a check is available it is shown, because a decomposition that does not reconcile is a wrong decomposition rather than a rounding issue.

Worked example 1: A buffer zone, priced in both directions (India, ₹)

The Sundaram Broad 10 Index (synthetic) holds the ten largest names by free-float market capitalisation from a stated fourteen-name universe. At the current review the incumbents are A through J. Free-float capitalisations at review-date prices, in ₹ crore:

All the capitalisation changes since the last review are price-driven, so no free-float revision is in play. Compare a strict rank rule against a buffered rule under which a non-constituent enters only at rank 8 or better and an incumbent leaves only at rank 13 or worse.

Solution. Step 1, apply the strict rule. The top ten are A, C, B, E, D, G, F, H, K, I. K enters at rank 9 and J leaves from rank 12. One change.

Step 2, apply the buffer. K ranks 9, which is not 8 or better, so it does not enter. J ranks 12, which is not 13 or worse, so it does not leave. The index is unchanged. Zero changes.

Step 3, price the strict rule's turnover. The tracker currently holds the incumbents, whose combined free-float cap at review prices is ₹5,83,000 crore. The new index's combined cap is ₹5,88,800 crore. K's weight in the new index is 38,900 / 5,88,800 = 6.606658%, and J's weight in the old index is 33,100 / 5,83,000 = 5.677530%. The nine survivors each fall slightly in weight because the denominator rose, giving up 0.929128 percentage points between them.

Step 4, reconcile the trade. Buys are K's 6.606658% of net assets. Sells are J's 5.677530% plus the survivors' 0.929128%, which is 6.606658%. Buys equal sells to the twelfth decimal, as they must in a fully invested fund. One-way turnover is 6.606658% of net assets, two-way traded value is 13.213315%, and at an all-in trading cost of 25 basis points of traded value the reconstitution costs 0.13213315 × 0.0025 = 3.303329 basis points of net asset value. On a ₹4,200 crore tracker that is ₹138.74 lakh.

Step 5, price what the buffer gives up. The universe's total free-float cap is ₹7,16,100 crore. The strict index covers 5,88,800 / 7,16,100 = 82.223153% of it; the buffered index covers 5,83,000 / 7,16,100 = 81.413210%. The buffer costs 0.809943 percentage points of coverage and saves 6.606658 points of one-way turnover, a ratio of 8.156944 points of turnover saved per point of coverage given up.

Read it. The buffer sets an exchange rate between two goods that do not naturally convert, and the provider has priced it at 8.16 points of forced trading per point of representativeness. That judgement is defensible for a broad index and much less so for a narrow one, where a single name's absence changes what the index means. What the arithmetic does not capture is the second-order effect: because the buffer makes changes rarer and more predictable, the changes that do happen are more heavily front-run, so the cost per change rises even as the number of changes falls. A rulebook cannot make forced trading free, only rarer and more visible.

Worked example 2: A premium decomposed, and where the arbitrage restarts (India, ₹)

The Sanchit Global Tech ETF (synthetic) is India-domiciled, listed on an Indian exchange, and tracks a United States technology index. Its published net asset value per unit, struck on last night's American close and the central bank's reference rate, is ₹60.0000. During today's Indian session the American market is closed. Futures on the tracked index are up 2.5200% from that close and the rupee is unchanged against the dollar. The fund's last trade on the exchange is ₹64.2000. One creation unit is 50,000 units. An authorised participant faces basket acquisition costs of 35 basis points of basket value, an overnight hedge and funding carry of 3 basis points, exchange, clearing and statutory charges of 32 basis points on the sale of units, and a flat creation fee of ₹15,000 per creation unit.

Solution. Step 1, the quoted premium. (64.2000 - 60.0000) / 60.0000 = 7.000000%.

Step 2, live fair value. 60.0000 × 1.025200 = ₹61.512000 per unit.

Step 3, split the premium. The stale-valuation component is (61.512000 - 60.0000) / 60.0000 = 2.520000 percentage points. The genuine dislocation is (64.2000 - 61.512000) / 60.0000 = 4.480000 points. They sum to 7.000000 exactly, which is the identity to check before reading anything. Measured against live fair value rather than the published net asset value, the dislocation is 2.688000 / 61.512000 = 4.369879%.

Step 4, the participant's economics if creations are open. Per unit, the basket costs ₹61.512000, acquisition costs 0.0035 × 61.512000 = ₹0.215292, carry costs 0.0003 × 61.512000 = ₹0.018454, and the creation fee is 15,000 / 50,000 = ₹0.300000. Total before the sale leg is ₹62.045746. The sale nets 64.2000 × (1 - 0.0032) = ₹63.994560. Profit per unit is ₹1.948814, so per creation unit 1.948814 × 50,000 = ₹97,440.72 on a basket notional of 50,000 × 61.512000 = ₹30,75,600.00, a return of 3.168186% on capital committed for one settlement cycle.

Step 5, the restart level. Set profit to zero: P × (1 - 0.0032) = 62.045746, so P* = ₹62.244929. That is a premium of 1.191523% to live fair value and 3.741549% to the published net asset value. The price would have to fall ₹1.955071, or 3.045281%, before the arbitrage stopped being worth doing.

Read it. Three separate things are worth taking from those numbers. The band is not zero: even with creations wide open, a 1.19% premium to live fair value is inside the participant's costs and will simply sit there. The band measured against the published net asset value is much wider still, at 3.74%, purely because of the time zone, so a screener showing a 3% premium on this fund on a quiet day is showing a fund working correctly. And if creations are suspended, none of Step 4 happens at any price, the upper bound disappears, and the premium is whatever buyers will pay. That is not hypothetical: Indian fund houses paused fresh subscriptions into overseas schemes in early 2022 when the industry-wide overseas investment limit was reached, and India-listed overseas trackers traded at large premiums for months afterwards. Check the current limit and any successor arrangement with the market regulator and the industry association before quoting today's position. For the household on the other side of this trade the arithmetic is brutal and simple: a buyer at ₹64.2000 is down 4.186916% the moment the position is marked to live fair value, and 6.542056% against the published net asset value, before the index has moved at all.

Worked example 3: One year's tracking difference, decomposed to reconcile (India, ₹)

The same fund, over a full year. The tracked index returned 18.0000% in dollars on a total-return basis gross of withholding tax, and the rupee weakened 3.0000%, so the fund's stated benchmark returned 1.18 × 1.03 - 1 = 21.540000% in rupees. Average net assets were ₹1,050 crore.

The disclosed frictions are these. Average cash of 1.50% of net assets earning 6.00%. A gross dividend yield on the index of 0.90% with treaty withholding at 25%. A total expense ratio of 0.60% accrued daily. Gross securities-lending revenue of ₹1.40 crore of which the fund keeps 60%. And reconstitution trading of 20.00% of net assets in two-way traded value at an all-in 20 basis points.

Solution. Step 1, size each friction. Cash drag is 0.0150 × (21.5400% - 6.0000%) = 23.3100 basis points. Withholding is 0.9000% × 25% = 22.5000 basis points. The expense ratio is 60.0000 basis points as a rate. Lending credits 1.40 × 0.60 = ₹0.84 crore on ₹1,050 crore, which is 8.0000 basis points. Rebalancing is 0.2000 × 0.0020 = 4.0000 basis points.

Step 2, cross-check two of them independently. The annual report's total expense line of ₹6.30 crore on average net assets of ₹1,050 crore gives 0.600000%, matching the stated ratio. The lending note's ₹1.40 crore gross and 60% split gives ₹0.8400 crore, matching Step 1.

Step 3, run the chain. Each friction is levied on an asset base that has already compounded, so apply them multiplicatively in a stated order and record each step's effect on the return.

Step 4, reconcile. The fund returned 20.355840% against a benchmark of 21.540000%, so the tracking difference is −118.4160 basis points. The five step effects sum to −118.4160, with a residual of zero to ten decimal places. That is the identity, and a decomposition that misses it is wrong rather than approximate.

Step 5, contrast the linear presentation. Adding the five frictions as simple rates gives 23.3100 + 22.5000 + 60.0000 + 4.0000 - 8.0000 = 101.8100 basis points, which understates the chain by 16.6060 basis points. The ratio of the two is 1.163107, which sits between one and the index's growth factor of 1.215400, because each friction is charged part way through a year in which the base grew by that factor.

Step 6, tracking error, on the same fund. The twelve monthly relative returns, in basis points, are −2.15, −12.15, +2.85, −24.15, −6.15, −15.15, +5.85, −29.15, −3.15, −11.15, +0.85 and −4.15. Their product is −97.4248 basis points, which reproduces the geometric relative return (1 + R_fund) / (1 + R_index) - 1 implied by the annual figures, −97.4296, to within five thousandths of a basis point; the monthly figures are printed to two decimals and the gap is that rounding. Their standard deviation with the sample convention is 10.6600 basis points a month, so the annualised tracking error is 10.6600 × sqrt(12) = 36.9274 basis points. With the population convention it is 35.3553, a difference of 1.57 basis points that comes entirely from the divisor.

Read it. A fund charging 60 basis points missed its index by 118, and every one of the missing 58 is recoverable from published documents. The expense ratio itself cost 72.62 basis points of return rather than 60, because the fee is charged on a base that grew 21.54% while the return is measured against the base at the start. In a flat year the two would coincide; in a falling year the fee would cost less than its rate.

Withholding tax at 27.29 basis points is the second-largest line and is not a failure of the manager at all. The benchmark is gross of a tax the fund must pay, so the gap is permanent and belongs in a conversation about benchmark choice rather than manager selection. The tracking error of 37 basis points, meanwhile, says the shortfall is highly predictable, which is what a passive client should want. The same 118 point difference with a 200 basis point tracking error would be a much worse product carrying an identical headline number.

Worked example 4: A 30-name sampled tracker, risk and turnover (US, $)

A 50-name United States index, generated by three stated rules so the whole exhibit is reproducible. Name i for i = 1 to 50 has free-float capitalisation 1000 × 0.94^(i-1) in dollars billion, a market beta of 0.85 + 0.30 × ((3i) mod 7) / 6, and a small-size factor loading running linearly from −1.00 for the largest name to +1.00 for the smallest, namely (i − 25.5) / 24.5. Idiosyncratic volatility is 18.00% + 0.30% × (i − 1) a year. The market factor has a volatility of 16.00% a year, the size factor 8.00%, and the two are uncorrelated. Index weights are free-float capitalisation weights: name 1 carries 6.284899%, name 30 carries 1.044735%, name 50 carries 0.303084%, and the top 30 together carry 88.380806%.

A tracker holds the top 30 names at pro-rata capitalisation weights and nothing else. Compute its ex-ante tracking error and its turnover at one reconstitution.

Solution. Step 1, the active weight vector. Scaling the top 30 to sum to one multiplies each of their weights by 1 / 0.883808 = 1.131467, so name 1 goes from 6.284899% to 7.111158% and name 30 from 1.044735% to 1.182083%. Active weights are positive on all 30 held names and equal to minus the benchmark weight on all 20 excluded ones. The largest positive active weight is +0.826259 percentage points on name 1 and the largest negative is −0.982051 on name 31. The vector sums to zero to fifteen decimal places.

Step 2, active share. 0.5 × Σ|a_i|. Since the positives sum to the excluded tail's weight and the negatives sum to minus the same, active share equals that tail exactly: 11.619194%, which is the weight of names 31 to 50.

Step 3, the factor exposures. a'β = +0.001645 and a'z = −0.129863. The beta bet is negligible, which is what a cap-weighted sample should produce. The size bet is not: the tracker is structurally short the small end of its own index.

Step 4, the tracking error. With two uncorrelated factors and diagonal idiosyncratic risk,

TE² = (a'β)² σ_m² + (a'z)² σ_z² + Σ a_i² σ_i² = 0.0000000693 + 0.0001079314 + 0.0000860110 = 0.0001940117

so TE = 139.2881 basis points a year. The market factor contributes 0.0357% of the variance, the size factor 55.6314% and idiosyncratic risk 44.3329%. A two-standard-deviation year is 278.5762 basis points either side of the index.

Step 5, turnover at one reconstitution. Suppose the year's returns are +40.00% for odd-numbered names and +5.00% for even-numbered ones. Recomputing capitalisations and re-ranking, name 31 rises into the top 30 and name 30 falls out. The tracker's drifted weights need one purchase and one sale plus the pro-rata adjustment, giving one-way turnover of 1.261086% of net assets, two-way traded value of 2.522173%, and a cost of 0.151330 basis points at 6 basis points of traded value. Post-reconstitution the ex-ante tracking error is 136.9852 basis points.

Read it. What matters is where the 139 basis points comes from rather than the number itself. Over half the variance is one systematic exposure, the small-size bet the sampling created without anyone deciding to take it, and it is exactly the exposure a smarter sample would hedge. Note also what full replication costs at the same reconstitution: nothing at all. Since no name entered or left the fifty-name index and all the capitalisation changes were price-driven, a fund holding all fifty needed to trade zero. The sampled fund's 1.26% of turnover exists purely because the boundary of its sample moved, which is a cost sampling creates rather than one it inherits. That is the trade the next example prices properly.

Worked example 5: Four routes to the same exposure, and three crossovers (US, $)

An institution wants $25,000,000 of exposure to the same large-cap index. Four routes, with costs stated as a one-off round-trip charge and an annual carrying charge, both in basis points of notional. Every figure is a market price rather than a constant and must be dated before use.

Solution. Step 1, evaluate cost(T) = f_0 + f_y × T at three horizons.

Step 2, name the winner at each horizon. At one month the index mutual fund wins at $1,250, because it charges nothing to enter. At one year the exchange-traded fund wins at $13,250, ahead of the mutual fund's $15,000. At five years the exchange-traded fund wins by a wide margin, at $41,250 against the basket's $57,500 and the mutual fund's $75,000.

Step 3, solve the crossovers. T* = (f_0^B - f_0^A) / (f_y^A - f_y^B).

Futures against the exchange-traded fund: (2.50 - 1.50) / (33.00 - 2.80) = 0.033113 years, which is 12.0861 calendar days or 8.3444 trading days. Index mutual fund against the exchange-traded fund: (2.50 - 0.00) / (6.00 - 2.80) = 0.781250 years, which is 9.375000 months. Direct basket against the exchange-traded fund: (18.00 - 2.50) / (2.80 - 1.00) = 8.611111 years.

Read it. Each crossover contradicts a piece of received wisdom. Futures are widely described as the cheap institutional route to index exposure, and on these inputs they are cheaper only for holds under a fortnight, because a 30 basis point implied financing spread is an enormous carrying charge against a 3 basis point fund. The index mutual fund, usually presented as the retail compromise, wins for anything under nine and a half months on the strength of transacting at net asset value with no spread. And the direct basket, which is the only route with no manager and no fee, takes more than eight and a half years to pay for its entry cost. What the arithmetic cannot decide is everything the routes differ on besides cost: futures can be put on in minutes and unwound the same way, the basket can be tax-managed and voted, and the funds can be bought by anyone. Cost picks the route only when the other properties are equal, which for a five-year strategic holding they usually are.

Worked example 6: Three replication methods on one universe (US, $)

The same 50-name universe as Worked example 4, now with a ten-sector map. Sector membership, benchmark weight and the names in each sector:

Three methods, each with a 30-name budget where a budget applies. Full replication holds all fifty at index weights. Stratified sampling holds the three largest names in each sector, weighted pro-rata within the sector so that every sector's aggregate weight matches the index exactly. Optimisation minimises forecast tracking error over all fifty names subject to weights summing to one and non-negativity, dropping the smallest optimal position and re-solving until thirty names remain. Build cost is 4 basis points of traded value on the top 30 names and 14 on the thin tail; reconstitution turnover is measured over the same stated year as Worked example 4.

Solution.

Step 1, why stratification does worse than the naive sample. Matching each sector's weight forces the three largest names in a small sector to carry that whole sector's weight, which produces active positions of up to 2.811644 percentage points against 0.982051 for the naive sample. Idiosyncratic risk accounts for 61.3146% of the stratified sample's active variance. Controlling the sector bet is worth having when sectors drive the covariance; here the size factor drives it, and stratifying on sector leaves the size exposure worse, at a'z = −0.147576 against −0.129863 for the naive sample.

Step 2, what the optimiser does instead. Backward elimination lands on names 1 to 30, the same set the naive sample holds, and then re-weights them. Its size exposure is −0.079681, roughly 40% smaller than the naive sample's, and its market beta bet is +0.000796. Tracking error falls to 120.3740 basis points, and idiosyncratic risk rises to 71.9459% of a smaller total variance, which is the signature of a portfolio whose systematic bets have been hedged out.

Step 3, what the optimisation costs. Reconstitution turnover is 4.192170% of net assets, more than three times the naive sample's 1.261086%, because the optimiser's weights move whenever the risk model moves, not merely when membership changes. At 6 basis points of traded value that is 0.5031 basis points a year against 0.1513.

Read it. Ranking the methods on tracking error alone gives full replication, optimisation, the naive sample, stratification. Ranking on total cost gives a different order at every fund size, because the 19 basis points of tracking error the optimiser saves against the naive sample is a risk reduction while the 0.35 basis points of extra turnover cost is a certain payment. For a fund whose client cares about the level of the shortfall, the naive sample is arguably the better product; for one whose client cares about the variance around it, the optimiser is. The genuinely important caveat sits underneath all four rows: every tracking error here is a forecast from a stated covariance model, and if that model understates the size factor, the optimiser is the method that suffers most, because it is the only one that deliberately loaded up on the model's judgement.

Worked example 7: The compounding gap on a leveraged fund (US, $)

A fund promises two times the daily return of an index, rebalanced at each close. Over ten trading days the index returns, in order: +3.20%, −2.10%, +4.50%, −5.30%, +1.80%, −3.60%, +5.10%, −4.40%, +2.70%, −3.00%.

Solution. Step 1, the index. Π(1 + r_t) - 1 = −1.789877% over the ten days. Twice that is −3.579753%, which is what a buyer who read the fund's name expects.

Step 2, the fund. The fund earns 2r_t each day and compounds: Π(1 + 2r_t) - 1 = −4.912423%.

Step 3, the gap. −4.912423 - (−3.579753) = −1.332669 percentage points, a loss of one and a third points that neither the index nor any fee produced.

Step 4, check it against theory. For a fund at daily leverage k, the drag over T days with daily volatility σ is approximately −0.5 × k × (k - 1) × σ² × T. The path's realised daily standard deviation is 3.751386%, so the approximation gives −0.5 × 2 × 1 × 0.03751386² × 10 = −1.407290 percentage points, within 0.074621 points of the exact −1.332669. The formula is an approximation because it drops higher-order terms, and it is close enough to use as a sanity check.

Step 5, the inverse fund. A fund at k = −1 returns Π(1 - r_t) - 1 = +0.397029% while minus the index return is +1.789877%, a gap of −1.392848 points. The formula gives −0.5 × (−1) × (−2) × σ² × T = −1.407290, the same drag as the two-times fund, which is exactly what k(k-1) predicts since 2 × 1 = (−1) × (−2).

Step 6, show it is volatility and not fees. Scale every daily move to one tenth its size. The index now returns −0.116975%, the two-times fund −0.247873%, and the gap is −0.013922 points. In magnitude that is 1.332669 / 0.013922 = 95.7229 times smaller for a tenth of the volatility, close to the factor of 100 the squared-volatility term predicts. Now take a path that only trends, +2.00% every day for ten days. The index returns 21.899442%, the fund returns 48.024428%, and the gap is positive 4.225544 points.

Read it. The gap is what daily rebalancing does rather than a defect or a hidden charge, and it runs in both directions: choppy markets punish the holder and trending markets reward them. What makes these products dangerous is not the mathematics but the mismatch between a daily promise and the holding period of the person who buys it, and the mismatch grows with volatility, which is precisely the environment in which somebody reaches for leverage. The regulatory positions differ: Indian mutual funds are not permitted to offer leveraged or inverse schemes, so an Indian investor reaches them only through overseas products, while they are freely available in the United States. Verify the current Indian position with the market regulator before relying on it.

Worked example 8: Active share, sector limits and what a constraint costs (US, $)

The same 50-name universe. A manager forecasts an alpha for each name by the stated rule α_i = 0.50% × ((7i) mod 11) − 2.50%, which produces forecasts between −2.50% and +2.50%, and holds the three highest-alpha names in each of the ten sectors, breaking ties toward the larger name, giving 30 holdings. She maximises a'α − (λ/2) a'Σa with λ = 8.0, long-only and fully invested, using the two-factor covariance matrix of Worked example 4. Compare that book with the same book under a cap of +2.50 percentage points on any name's active weight and a band of ±4.50 points on any sector's.

Solution. Step 1, the two solutions.

Step 2, price the constraints. Expected active return falls by 44.4703 basis points and tracking error by 70.0457, so the information ratio falls by about 0.0496, from 0.369872 to 0.320226. Risk fell by more in basis points and return by more in proportion, 27.08% against 15.78%, which is why a constraint's price belongs in a report as a ratio rather than as a return.

Step 3, find which constraint binds. The position cap binds on fourteen names: 1, 3, 6, 9, 14, 15, 17, 20, 23, 25, 28, 31, 36 and 47. Only one of the ten sector bands binds, Industrials at exactly +4.500000 points.

Step 4, the constrained book by sector.

The active column sums to zero. Two sectors carry only two funded positions, because the optimiser drove two of the thirty permitted holdings to zero weight; a mandate with a minimum-holdings rule would have to add them back and pay for it.

Step 5, test the sector band for feasibility. The manager holds three Financials names, numbers 6, 28 and 47, with benchmark weights of 4.6125%, 1.1824% and 0.3649%, summing to 6.159779%. At the +2.50 point position cap their maximum combined weight is 6.159779 + 3 × 2.50 = 13.659779%, against a sector benchmark weight of 17.9117%. The tightest achievable Financials underweight is therefore 4.251943 points. A sector band of ±4.00 points is infeasible rather than tight, and no solver can satisfy it without breaking the position cap or changing the names held.

Read it. Three things in that output would change a committee conversation. The information ratio give-up of 0.0496 is the honest price of the mandate's limits, and it is small enough that a committee could reasonably decide the governance is worth it, which is a decision they can only make once someone computes the number. The binding constraint is the position cap, not the sector bands everybody argues about, so tightening sector limits further would cost nothing until they start to bind. And the largest active position in the constrained book belongs to no holding at all. It is −5.9078 percentage points on the index's second-largest constituent, a name the manager simply does not own. A limit framed as a cap on overweights leaves that position entirely unregulated, which is how a long-only book ends up with its biggest single bet in a name nobody discussed.

Verify block

Every headline figure above, restated at the precision it was computed to, so a reader who rebuilds the arithmetic can find the exact line where a discrepancy starts. Where a figure came from a numerical optimiser rather than closed-form algebra it is marked, because those carry solver noise beyond the fourth decimal and should not be quoted more tightly.

Two reconciliations should hold exactly wherever you rebuild them. In the premium decomposition, the stale component plus the dislocation equals the quoted premium to fourteen decimal places. In the tracking-difference chain, the five step effects sum to the tracking difference with a residual of zero to ten decimal places. A rebuild that misses either identity has a wrong input rather than a rounding problem.

Practice set

Work each problem fully before reading its solution. A calculator is assumed. Numeric answers within 2% of the stated figure score as correct, except where a reconciliation or a sum check is demanded, in which case it must be exact. Currencies alternate deliberately; the arithmetic does not care and the conventions do.

P1 (guided). Premium and discount from quotes. An Indian equity exchange-traded fund last traded at ₹482.10 while its indicative net asset value was ₹478.55. Compute the premium in percent and in basis points. Then, for a second fund that last traded at ₹231.40 against an indicative net asset value of ₹233.05, compute the discount.

Solution. (482.10 - 478.55) / 478.55 = 0.741824%, which is 74.1824 basis points. For the second fund, (231.40 - 233.05) / 233.05 = −0.708003%, a discount of 70.80 basis points. Both are quoted against an indicative net asset value rather than a published one, which is the right denominator for a domestic fund whose constituents are trading live. For a fund whose underlying market is closed, this arithmetic would be measuring the time zone rather than the fund.

P2 (guided). Creation-unit economics. A creation unit of an Indian equity fund is 100,000 units. The fund's net asset value per unit is ₹248.60 and the exchange price is ₹250.35. The composition file specifies ₹1,18,000 of cash in lieu for one suspended constituent. An authorised participant faces acquisition and cash-in-lieu charges of 25 basis points on the creation unit's full net asset value and a flat creation fee of ₹12,000. Compute the basket's market value, the participant's gross spread, its costs, its net profit, and the exchange price at which the arbitrage stops being worth doing.

Solution. The full creation unit is worth 100,000 × 248.60 = ₹2,48,60,000, of which ₹1,18,000 is cash, so the securities basket is ₹2,47,42,000. The gross spread is 100,000 × (250.35 - 248.60) = ₹1,75,000. Costs are 0.0025 × 2,48,60,000 + 12,000 = ₹74,150. Net profit is ₹1,00,850, or 40.5672 basis points of the creation unit's notional. Break-even is 248.60 + 74,150 / 100,000 = ₹249.341500, a premium of 0.298270% to net asset value. Below that price the trade is a loss, which is why a persistent premium of a quarter of a percent on a liquid domestic fund is normal rather than evidence of anything.

P3 (guided). Tracking difference against tracking error. A fund's twelve monthly returns, in percent, are 1.82, −0.94, 2.31, 0.77, −1.55, 3.02, 0.41, −2.18, 1.66, 0.95, −0.37, 2.04. Its benchmark's are 1.90, −1.06, 2.56, 0.80, −1.64, 3.18, 0.45, −2.36, 1.88, 1.00, −0.44, 2.15. Compute the twelve-month tracking difference and the annualised tracking error, and say what the two numbers together tell a client.

Solution. Chain-linking, the fund returned 8.078587% and the benchmark 8.570030%, so the tracking difference is −49.1443 basis points. The twelve monthly active returns in basis points are −8, +12, −25, −3, +9, −16, −4, +18, −22, −5, +7 and −11, with a mean of −4.0000 and a sample standard deviation of 13.5042. Annualising, 13.5042 × sqrt(12) = 46.7799 basis points of tracking error. Using the population divisor instead gives 44.7884, a 2 basis point difference that comes entirely from the convention; state which you used. Together the two say the fund costs about half a percentage point a year against its index and that the shortfall in any single year could plausibly land anywhere in a band roughly a percentage point wide.

P4. Cash drag, both directions. A fund holds an average 2.50% of net assets in cash earning 4.50% while its index returns 11.80%. Compute the cash drag in basis points. Then recompute it for a year in which the index falls 6.00%.

Solution. Drag is w_cash × (r_index - r_cash) = 0.0250 × (11.80% - 4.50%) = 18.2500 basis points. In the falling year, 0.0250 × (−6.00% - 4.50%) = −26.2500 basis points, meaning the cash added 26.25 basis points to the fund's relative return. A tracker's tracking difference therefore improves in bad years for a reason that has nothing to do with how well it is run, which is why a single year's figure is close to uninformative.

P5. A buffer zone on a five-name index. An eight-name universe has review-date free-float capitalisations, in ₹ crore, of P 540, Q 470, R 410, S 360, T 262, U 305, V 244, W 231. The index holds the top five and its current constituents are P, Q, R, S and T. Compute the one-way turnover under a strict rank rule and under a buffer in which a non-constituent enters only at rank 4 or better and a constituent leaves only at rank 7 or worse.

Solution. Ranking gives P, Q, R, S, U, T, V, W. Under the strict rule U enters at rank 5 and T leaves from rank 6. The new index's capitalisation is 540 + 470 + 410 + 360 + 305 = ₹2,085 crore against the incumbents' ₹2,042 crore. U's weight in the new index is 305 / 2,085 = 14.628297%, and since the denominator rose every survivor's weight fell, so buys equal that figure and one-way turnover is 14.628297% of net assets. Under the buffer U ranks 5, which is not 4 or better, and T ranks 6, which is not 7 or worse, so nothing changes and turnover is 0.000000%. Turnover per change is large here only because a five-name index gives each constituent a fifth of the fund; on a fifty-name index the same single swap moves a fiftieth as much.

P6. Ex-ante tracking error on a three-name bet. Using the 50-name universe of the worked examples, a manager is +3.00 percentage points on name 1, +2.00 on name 10 and −5.00 on name 45, with everything else at benchmark. Compute the active beta, the active size loading and the ex-ante tracking error.

Solution. Betas are 1.00, 0.95 and 0.95, so a'β = 0.03(1.00) + 0.02(0.95) − 0.05(0.95) = +0.001500. Size loadings are −1.000000, −0.632653 and +0.795918, so a'z = −0.082449. Idiosyncratic volatilities are 18.00%, 20.70% and 31.20%. The three variance terms are 0.0015² × 0.16² = 0.0000000576, 0.082449² × 0.08² = 0.0000435061 and 0.0009(0.0324) + 0.0004(0.042849) + 0.0025(0.097344) = 0.0002896596, summing to 0.0003332233. The square root is 182.5441 basis points. Idiosyncratic risk is 87% of the variance, which is what a three-name bet always looks like.

P7. A leveraged fund over two days. An index returns +10.00% then −8.00%. Compute the cumulative index return, the return of a two-times daily fund, the return of a minus-one-times daily fund, and the compounding gap on each.

Solution. The index returns 1.10 × 0.92 - 1 = 1.200000%. The two-times fund earns +20.00% then −16.00%, so 1.20 × 0.84 - 1 = 0.800000%, against an expected 2.400000%, a gap of −1.600000 percentage points. The inverse fund earns −10.00% then +8.00%, so 0.90 × 1.08 - 1 = −2.800000%, against an expected −1.200000%, a gap of −1.600000 points. The two gaps are identical because the drag scales with k(k − 1), and 2 × 1 equals (−1) × (−2).

P8. A crossover holding period. A futures position costs 1.20 basis points round trip and 41.00 basis points a year. An exchange-traded fund costs 3.20 basis points round trip and 4.50 basis points a year. Find the holding period at which they cost the same, and say which is cheaper at three months.

Solution. T* = (3.20 - 1.20) / (41.00 - 4.50) = 0.054795 years, which is 20.0000 calendar days. At three months the futures cost 1.20 + 41.00 × 0.25 = 11.4500 basis points and the fund 3.20 + 4.50 × 0.25 = 4.3250, so the fund is cheaper by 7.12 basis points. Anything the desk describes as a tactical position for a quarter should not be expressed in futures at these prices.

P9. Active share from a sector table. A portfolio's sector weights are 34, 15, 14, 6, 13, 7, 0, 6, 5 and 0 percent against benchmark weights of 28, 19, 14, 11, 9, 7, 5, 4, 2 and 1. Compute the sector-level active share and say what it does and does not tell you.

Solution. Both columns sum to 100%. The absolute deviations are 6, 4, 0, 5, 4, 0, 5, 2, 3 and 1, summing to 30, so active share is 0.5 × 30 = 15.000000%. That figure is a floor on the name-level active share, never a substitute for it, because a portfolio can match every sector weight exactly and still hold entirely different companies. It also says nothing about risk: a 15% sector-level deviation concentrated in one volatile sector and the same deviation spread across nine sectors are very different portfolios.

P10. Securities lending and the tracking difference. A fund with ₹900 crore of average net assets earns ₹3.60 crore of gross securities-lending revenue and keeps 70% after the agent's share. Its tracking difference before the lending credit is −38.00 basis points. Compute the credit and the tracking difference after it, then name two risks the credit is paying you to bear.

Solution. The fund keeps 3.60 × 0.70 = ₹2.52 crore, which on ₹900 crore is 28.0000 basis points. The tracking difference improves from −38.00 to −10.00 basis points. The risks are borrower default against insufficient collateral, and loss on the reinvestment of cash collateral, which is the one that has actually broken funds. A third, worth naming, is that shares on loan cannot be voted, so a lending programme quietly sells the fund's governance rights for basis points.

P11. Counterparty exposure in a synthetic fund. A swap-based fund has a net asset value of 100.0 and posted collateral of 105.5, both per 100 of net asset value. Compute the uncollateralised counterparty exposure. Then recompute it after the collateral basket falls 13.5% in value, and say what that scenario implies about collateral selection.

Solution. Exposure is max(0, 100.0 - 105.5) = 0.0000, fully covered. After the fall, collateral is 105.5 × 0.865 = 91.2575, so exposure is max(0, 100.0 - 91.2575) = 8.7425 per 100 of net asset value. European fund rules cap single-counterparty exposure at a fraction of net asset value and the current threshold should be checked in the rules themselves rather than in a summary. The scenario's point is that collateral and counterparty tend to fail together. A basket of the counterparty's own regional bank paper gives you a second helping of the same risk under the name of protection.

P12. Implementation shortfall on one rebalance. A fund decides to buy 250,000 shares at a decision price of ₹1,240.00. The order reaches the market at ₹1,243.10 and 200,000 shares fill at an average of ₹1,247.85. The remaining shares never fill and the closing price is ₹1,251.00. Explicit costs are ₹46,500. Decompose the shortfall and read the result.

Solution. Decision notional is 1,240.00 × 250,000 = ₹31,00,00,000. Delay is (1,243.10 - 1,240.00) × 250,000 = ₹7,75,000, or 25.0000 basis points. Execution is (1,247.85 - 1,243.10) × 200,000 = ₹9,50,000, or 30.6452 basis points. Opportunity is (1,251.00 - 1,240.00) × 50,000 = ₹5,50,000, or 17.7419 basis points. Fees are ₹46,500, or 1.5000 basis points. The total is ₹23,21,500, or 74.8871 basis points, on a fill rate of 80.0000%. The execution term at 30.6452 basis points is the trader's; the 25.0000 basis points of delay accrued before a single share traded and belongs to whoever sat on the instruction. Reporting only the execution line would hide that delay term, a third of the bill on its own.

Applied mini-project

Cost one real exposure across four routes, over about three hours. You will pick a single equity index you could actually buy, price four ways of owning it over three holding periods, and write a one-paragraph recommendation naming the holding period at which your answer changes. Every cost must be traceable to a named document with a pull date, and every number must be recomputable from what you write down.

1. Choose the exposure and source the products (about 30 minutes). Pick one index with a liquid tracker in both jurisdictions if you can, or one Indian and one United States index if you cannot. For each, identify a real index mutual fund, a real exchange-traded fund, the listed index futures contract, and the constituent list you would need for a direct basket. Record the fund names, the scheme or prospectus documents, the exchange's contract specification, and the date you pulled each.
2. Build the cost inputs (about 45 minutes). For each route, record a one-off cost and an annual cost in basis points. The fund routes need the expense ratio from the scheme document or prospectus, plus any securities-lending credit disclosed in the annual report, plus the quoted spread from the exchange for the traded route. The futures route needs the current implied financing spread, which you back out from the futures price, the spot index, the dividend yield and the days to expiry, plus the round-trip commission and the number of rolls a year. The basket route needs a defensible impact estimate, for which the exchange's own published impact-cost measure or a square-root model on the tail names is enough.
3. Compute and tabulate (about 30 minutes). Produce the three-horizon table at one month, one year and five years, in basis points and in your chosen currency on a stated notional. Show the arithmetic for at least one cell by hand.
4. Solve every crossover (about 20 minutes). For each pair of routes, solve T* = (f_0^B - f_0^A) / (f_y^A - f_y^B), report it in days and in months, and mark any pair whose lines do not cross in a sensible range. State which route wins in each interval between crossovers.
5. Stress the two most uncertain inputs (about 25 minutes). Take the futures implied financing spread and the basket impact estimate, move each by a stated plausible amount in both directions, and recompute the crossovers. Report the range each crossover moves over, not just the central case.
6. Add what cost does not capture (about 20 minutes). In one short table, note for each route: tax treatment on entry, on income and on exit; intraday liquidity; collateral and margin treatment; voting rights; and the operational burden. Mark which of these would change your answer regardless of cost.
7. Write the recommendation (about 20 minutes). At most 350 words. Name the route you would use, the holding period assumption behind that choice, the holding period at which the answer changes, and the single input whose revision would change it fastest.

Deliverables and their tests. A sourcing table, which passes if every cost carries a document name and a pull date. A cost table, which passes if one cell is reproduced by hand and matches. A crossover table, which passes if each crossover is reproduced from the two lines it came from. A sensitivity table, which passes if both stressed inputs move the crossovers in the direction the algebra requires. A qualitative table. A recommendation.

Rubric, 20 points, passing at 16 with no zero on any row.

Any deliverable whose test fails scores zero on its row, however well the surrounding paragraphs are written. A cost comparison whose inputs cannot be traced is an opinion with decimal places.

Reading & resources

Core, do these first.

• The methodology document for one index you actually track, from the provider itself: NSE Indices for the Nifty family at niftyindices.com, S&P Dow Jones Indices for the S&P family at spglobal.com/spdji. Read the eligibility screens, the buffer provisions, the review calendar and the notice period, and check the revision date on the document before quoting any threshold. [Free] [Beginner]
• One exchange-traded fund's scheme information document or prospectus, plus its most recent annual report, from the asset manager's own site. The annual report is where the securities-lending revenue, the collateral policy and the portfolio turnover live, and none of the three appears on a factsheet. [Free] [Beginner]
• The Securities and Exchange Board of India's master circular for mutual funds, at sebi.gov.in, for the Indian rules on index funds and exchange-traded funds, including the tracking-difference and tracking-error disclosure requirements and the position on leveraged schemes. Check the current version; this text is amended often. [Free] [Intermediate]
• The United States Securities and Exchange Commission's exchange-traded fund rule, Rule 6c-11, at sec.gov, which sets the conditions under which most American funds operate, including the daily portfolio disclosure and the premium and discount reporting on the fund's own website. [Free] [Intermediate]
• Any large fund provider's published premium and discount history for a fund you own or follow. Rule 6c-11 requires American funds to publish it, and reading a year of one fund's daily record teaches more about arbitrage bands than any description of them. [Free] [Beginner]

Going deeper.

• Grinold and Kahn, Active Portfolio Management: the reference for the objective function, the fundamental law and the cost of constraints. Chapters on portfolio construction and implementation are the ones that bear directly on the second half here. [Paid] [Advanced]
• Cremers and Petajisto (2009), "How Active Is Your Fund Manager?", Review of Financial Studies: the paper that introduced active share and the source of the distinction between active share and tracking error. Read it for the measure's construction rather than for its performance claims, which have been argued over since. [Free/Paid] [Intermediate]
• Perold (1988), "The Implementation Shortfall: Paper versus Reality", Journal of Portfolio Management: the original statement of the accounting used in Section 13. Short, and clearer than its summaries. [Paid] [Intermediate]
• Cheng and Madhavan (2009), "The Dynamics of Leveraged and Inverse Exchange-Traded Funds": the standard treatment of the compounding gap, including the variance-drag approximation checked in Worked example 7. [Free/Paid] [Intermediate]
• A large provider's guide to the creation and redemption process, published by BlackRock, Vanguard, State Street or an Indian asset manager. These are marketing documents and are still the clearest available description of the primary market's plumbing; read two from competing firms and note where they disagree. [Free] [Beginner]

Data and tools.

• The exchange's own quote and impact-cost pages at nseindia.com and bseindia.com for India, and the listing venue's for the United States. NSE publishes a per-stock impact cost computed for a stated order size, which is a ready-made input for the basket route in the mini-project. [Free]
• The Association of Mutual Funds in India, at amfiindia.com, for scheme-level net asset value history and industry-level disclosures. [Free]
• The Federal Reserve Economic Data service at fred.stlouisfed.org for the short-rate series a futures financing spread is measured against. [Free]
• Python with pandas and NumPy for the weight, turnover and tracking-error arithmetic, and SciPy or cvxpy for the constrained optimisation of Section 12. The branch's quantitative-development nodes build the production versions. [Free]

Do this rather than only reading it, because five sessions of data teach what no description can. Pick one exchange-traded fund whose underlying market is closed while it trades, and for five consecutive sessions record the published net asset value, the closing price, the index futures move and the currency move. Decompose the premium each day. By the fifth day you will be able to tell a stale valuation from a dislocation at a glance, which is a skill almost nobody selling these products has.

Teach it back & journal

Feynman prompt. Explain to a friend who has just seen an exchange-traded fund quoted at a 7% premium why that number, on its own, tells them almost nothing, and what two numbers they would need to know instead. Use the actual mechanics. A net asset value struck on a market that has been closed for sixteen hours, a fair value estimated from index futures and the exchange rate, and an authorised participant who cannot profitably create until the price falls to a level still 3.74% above the published net asset value. Then, without changing register, explain why the friend should still not buy above that level, and what would have to be true about the fund's creation status for the premium to close at all. You have taught it if your listener asks, unprompted, whether the fund is still issuing units.

Journal prompt. Take one fund you own, or one you have recommended, or one whose factsheet you can download in the next ten minutes. Find its stated benchmark and its expense ratio, then find its three-year and five-year tracking difference. Subtract the expense ratio from the tracking difference and write down the number that is left over. Now go and find where that number went: the cash balance in the portfolio statement, the withholding tax note, the securities-lending disclosure in the annual report, and the portfolio turnover against the index's own published turnover. Write down which of the four you could source and which you could not, because the ones you could not source are the diligence that nobody in the distribution chain has done either. Then write a paragraph on the more uncomfortable question underneath all of this: the whole passive proposition rests on a fund reproducing a rulebook faithfully and cheaply, and almost everything that determines whether it does is disclosed once a year in a document written for auditors. What is the one figure you would want published monthly instead, and what would you stop believing about your own portfolio if it turned out to be twice what you assumed?

This module's flashcards and mastery quiz are wired into the app: see the node's Quiz and Reviews.