Index Funds in a Portfolio: The Role and the Trade-Offs

An index fund gives a portfolio the result of a stated rule rather than the result of somebody's selection, so what separates the account from the rule is almost entirely the cost of delivering it. The trade-off fits in one line: the selection decision goes away, and so does any excess to measure, leaving the cost as the thing worth arguing about.

A question that sounds practical and is actually structural sits over this subject: does a rule beat a person? No arithmetic settles that one, and the arrangement is not really about it. The useful question is narrower and far more answerable: what changes inside a portfolio when the choosing of holdings stops being a judgement and becomes a rule, and what does not change at all.

The running example throughout is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for an invented charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its stated shape is equity 60.0 per cent at Rs 300 crore, fixed income 30.0 per cent at Rs 150 crore and cash 10.0 per cent at Rs 50 crore. The three figures are policy weights, and the actual weights drift away from them between one rebalancing and the next. Its benchmark is a composite of a broad equity index and a broad bond index, neither of which has to be named for the arithmetic to work. Every figure belongs to that mandate and to one stated twelve month period.

One line has to be drawn before anything else: what a rule-following holding does to a portfolio is a different matter from how the scheme behind it is put together, valued or run. How units are created and what a unit is worth on any day are covered separately. How an index itself is constructed is published by the exchanges.

What does a rule-following holding actually do to a portfolio?

A rule-following holding fixes the exposure to a stated ruleA written procedure that decides which securities are held and in what proportion, published in advance and applied the same way whoever is reading it., so the account holds what the rule holds, in the proportions the rule sets, for as long as the rule says so. None of that is a claim about returns. It is a claim about where one particular decision now lives.

Think about how a school lunch gets decided. One version has a cook who looks at what came in that morning and chooses. Another version has a printed menu on the wall: Monday is this, Tuesday is that, and whoever is in the kitchen follows it. The second version is not obviously better food. The printed menu is a different answer to who decides, and everything downstream follows from that one difference. Parents can read Monday's meal off the wall before Monday. Nobody has to trust the cook's morning. And if Monday's meal disappoints, the argument is about the menu, not about the cook.

An index fund is a decision about who decides, not a decision about what to hold, and every property described here falls out of that one distinction. The selection decisionThe judgement about which individual securities to hold and how much of each, taken by a person or a team against their own view of value. has moved out of the account and into a published procedure. The judgement has not been abolished. Somebody wrote the rule. But the account no longer takes it, cannot vary it, and does not answer for it.

The bottom right cell in that picture is empty. The available record carries a cost of delivery for the mandate as it was actually run and carries no cost at all for any rule-following alternative. The empty cell is not an oversight in the drawing. Nothing fills it, and that emptiness is what blocks any verdict about the alternative.

What role does a stated rule play that a selected sleeve cannot?

Three roles, and all three are structural. The first is that the exposure is knowable in advance. The rule can be read, the securities it names and their proportions seen, and what the account will hold known before a single rupee moves. A selected sleeve yields the same information only afterwards, by reading the account. For the Anantara mandate that reading produces 28 names across the Rs 300 crore equity sleeve and a largest holding of Rs 23 crore. The largest holding is 4.6 per cent of the Rs 500 crore portfolio and 7.7 per cent of the Rs 300 crore sleeve. All of that is true and none of it was knowable at the start of the period.

The second role is that nobody has to be right. A selected sleeve delivers what it delivers because a person formed a view and acted on it, and the account inherits the quality of that view. No judgement is applied period by period, so a rule-following holding delivers what the rule holds whether the judgement behind the rule was sharp or dull. The dependence is of a very different kind, and precision is worth the trouble here: the account has not escaped being wrong, it has changed what it is exposed to being wrong about.

The third role is that the result becomes checkable against something published. If the rule says hold these securities in these proportions and the account holds something else, that is a fact anybody with the rule and the account can establish. Checking a selected sleeve means forming a view about somebody's judgement, a slower and much less settled activity. All three roles are properties of the structure and none of them is a statement about performance. All three hold in a good period and a bad one alike.

Why does cost become the only thing left to explain the gap?

The mechanical heart of the matter is a short argument. When a holding follows a stated rule, the difference between the rule's result and the account's result is whatever delivering it consumed. The selection decision is the rule and the account followed the rule, so no selection decision is left over to explain any part of that difference.

Set that beside a selected sleeve, where a gap between the account and its benchmark has at least two plausible sources sitting inside it. Some of the gap came from the choices, and some of it came from what the choices cost to carry out. Untangling those two is real work, and the monitoring sequence takes it up separately. Under a rule, one of the two boxes is empty by construction, so the untangling is finished before it starts.

The empty box is exactly why cost arguments are sharper on a rule-following holding than anywhere else: with every other explanation removed by construction, the remaining one is arithmetic. Somebody arguing about the cost of a selected sleeve is arguing about one of several things. Somebody arguing about the cost of a rule-following holding is arguing about the only thing there is.

Delivering a rule consumes charges taken from the account and the frictions of buying and selling to stay in line with the rule, and how those charges reach a value struck per unit is covered under fund valuation. Whatever the total comes to is the whole of the gap, and the available record carries no number for it.

What does the account give up when it follows the rule?

Four things, and the honest way to state them is as things given up rather than as risks. Calling them risks would make them sound like accidents that might not happen. None of the four is an accident. Each one is the other side of the same choice, and each one happens every time.

The first is that the account cannot exclude a holding. If the rule holds a security the holder would rather not hold, for any reason at all, leaving it out means no longer following the rule. Drop the rule and the property the whole arrangement was chosen for goes with it. The second is closely related and is the one that catches mandates out: the account cannot write a limit at holding level without leaving the rule. The Anantara mandate carries a cap of 5 per cent of the portfolio on any single holding. The cap is the mandate's own constraint rather than anything a regulator sets. Under a selected sleeve that cap is a live instruction to a person. Under a rule it is a test run on what the rule already holds, and if the answer comes back over the line the account has to choose between the constraint and the rule.

Testing that cap is where a base error walks straight back in. The mandate's largest holding is Rs 23 crore. Against the Rs 500 crore portfolio that is 4.6 per cent and it sits inside the cap. Against the Rs 300 crore equity sleeve the same Rs 23 crore is 7.7 per cent. Neither number is wrong and they answer different questions, and the cap in the Anantara document is written against the portfolio, so the portfolio base is the one the test uses. Moving between the two bases without saying so can report a breach where there is none, or miss one that is real.

The third thing given up is the one people notice last and it is the cleanest. There is no gross excessThe amount by which a portfolio's return sits above its benchmark's return before the cost of delivery has been taken out of it. to measure. The thing being compared and the thing being compared against have become the same object. A rule-following holding is not trying to sit above the rule and cannot be praised or blamed for failing to. The fourth is that the account takes whatever the rule concentrates. If the rule carries a very large weight in one place, so does the account, and the mandate's own concentration reporting has to be run against what the rule holds rather than against a manager's list. Every one of the four is a consequence of the choice rather than a defect in it, and dressing them up as risks would be dishonest in the opposite direction from pretending they are not there.

What did the mandate's own decisions cost against what they produced?

The comparison that follows runs entirely on figures already established. Start with what the delivery cost. The Anantara mandate carries a management fee of 1.25 per cent of assets, or Rs 6.25 crore on Rs 500 crore, and a performance fee of 15 per cent of the return above a 10 per cent hurdle. In the stated year the return was 14.2 per cent, so 4.2 points sat above the hurdle. On Rs 500 crore that is Rs 21 crore, and 15 per cent of Rs 21 crore is Rs 3.15 crore. Total Rs 9.40 crore, or 1.88 per cent of Rs 500 crore. The terms are the mandate's own commercial arrangement, not a market rate, an industry level or anything a regulator sets.

Now take what the decisions produced. Over the same stated twelve month period the portfolio returned 14.2 per cent against the benchmark's 12.6 per cent, a gross excess of plus 1.6 points. On Rs 500 crore that is Rs 8.00 crore. Splitting that under the beta decomposition, at a beta of 1.08 against a benchmark sitting 6.1 points above the risk free rate, about 0.49 points of the gross excess is simply carrying more market exposure and about 1.11 points is the residualThe part of a return that is left after subtracting what the exposure carried would have been expected to produce on its own. The leftover carries Michael C. Jensen's name., the part that is anything else. The residual is Michael C. Jensen's alpha, and on Rs 500 crore it is Rs 5.55 crore. Two conventions travel with that figure. The subtraction written out in full gives 1.112 points and the stated record locks the leftover at 1.11 points, both describing the same gross quantity before any cost is taken out. At this size the distance between the two is Rs 1,00,000/-, and which convention governs is settled under the rounding conventions.

One warning that costs nothing to heed. A second decomposition of the same 1.6 points of gross excess exists, an allocation effect and a selection effect, and it answers a different question on a different base. The two are never mixed, and no term from one is ever set beside a term from the other. The decomposition running here is the beta decomposition.

Now put the two side by side. The cost of deliveryEverything the holder pays to have a portfolio run, expressed as a rate on the assets and as an amount for a stated period. was Rs 9.40 crore. The gross excess was Rs 8.00 crore. The residual was Rs 5.55 crore. Divide 1.88 by 1.6 and the cost is 1.175 times the gross excess. Divide 1.88 by 1.11 and it is about 1.69 times the residual. The cost was larger than both of the things it was charged against.

And then the consequence, stated with both labels, because a gross figure and a net figure are different quantities and neither is safe to read without the other. The gross return was 14.2 per cent and the cost was 1.88 per cent, so the net excessThe amount by which a portfolio's return sits above or below its benchmark after the cost of delivery has been subtracted. The net figure can carry the opposite sign from the gross one. is computed on 12.32 per cent against a benchmark of 12.6 per cent. Plus 1.6 points of gross excess became minus 0.28 points net. In rupees, Rs 8.00 crore of gross lead against Rs 9.40 crore of cost is a net shortfall of Rs 1.40 crore. The portfolio beat its benchmark and the holder did not.

Does that comparison settle whether the rule would have been better?

No, and this is a position rather than a hedge. Everything in the previous section is arithmetic and every line of it holds. The step that does not hold is the next one: concluding that the holder would have been better off with a rule-following holding instead. Three separate reasons block that conclusion, and every one of the three is load bearing rather than decorative.

The first is that the stated record contains no alternative arrangement at all. There is no rule-following holding in it, no cost rate for one and no return for one. Not a low figure, not a high figure: no figure. The second is that a rule-following alternative would have had its own cost of delivery, and no figure for that is here either, so the only cost in the comparison belongs to one side of it. The third is that one stated twelve month period is one drawA single observed period, one outcome out of the many that could have happened. A draw shows what did happen and cannot show what usually happens., and even a comparison with all four cells filled would describe that period rather than the next one.

The missing thing has a name, and naming it helps. A verdict about whether an arrangement was worth having is a claim about a counterfactualWhat would have happened under a course of action that was not taken. A counterfactual is never observed, so it has to be estimated, and an estimate needs inputs., a course that was not taken. The observed course has numbers. The unobserved course has none, and cannot acquire them by being compared with the observed one.

A conclusion reached from this record would be built out of missing information, and building one is a worse fault than declining to conclude. A built conclusion looks like an answer. Stating the comparison in full and then declining the verdict is the whole of the honest position, and it is taken deliberately rather than out of caution.

Two readings from that control are worth writing out as sentences that can be carried away without the control. At an alternative rate of 1.00 per cent, Rs 500 crore carries a cost of Rs 5.00 crore, so the difference against Rs 9.40 crore is Rs 4.40 crore. At 0.50 per cent the cost is Rs 2.50 crore and the difference is Rs 6.90 crore. The relationship is a straight line, and its slope is the only thing about it worth remembering: on Rs 500 crore, every one point of rate is Rs 5 crore.

What does a rule-following holding leave completely unchanged?

The section most easily skipped is the one that stops the whole idea being oversold. An index fund removes the selection decision and removes nothing else. Everything else the portfolio had to decide, it still has to decide, and everything the portfolio had to test, it still has to test.

The allocation decision remains, exactly as it was. Somebody still has to say equity 60.0 per cent, fixed income 30.0 per cent and cash 10.0 per cent of Rs 500 crore, and choosing a rule for the equity part settles nothing whatever about how large the equity part should be. The constraints remain. The equity range of 50 to 70 per cent, the 5 per cent cap on a single holding, the ban on unlisted holdings and the minimum credit standing on the fixed income sleeve are all still in the mandate, and all still have to be tested, now against what the rule holds. The concentration remains and is whatever the rule's own concentration happens to be, a figure the holder has to look up rather than assume. And the cost of delivery remains, the one item that ends up carrying the whole difference.

Here is the household version. The shape is identical at any size. A household that stops picking individual savings products and moves everything into one arrangement that follows a written rule has not decided how much to keep in savings, how much risk the household can carry, or what it is saving toward. The household has decided only who picks the contents, and that decision answers none of the other three. Choosing a rule instead of a selector changes who decides, and changes nothing at all about whether the portfolio was designed well.

How does a holder actually put this arithmetic to work?

Not by reaching a verdict, the one thing the arithmetic cannot do, but by turning a vague argument into four answerable questions. An investment committee like Rukmini Deshpande's would put them in a fixed order, and each one has a place the answer comes from rather than a person whose opinion is sought.

A lender or an analyst reading somebody else's portfolio runs the same four in reverse and gets more out of the fourth than the other three. Given a reported cost and a reported result, what does turning that into a judgement about the arrangement require? A cost for the alternative, a result for the alternative, and more than one period. A report that supplies the first three answers and stays silent on the fourth has done the easy part.

The practical value of this arithmetic is that it converts an argument about opinions into a list of items, some of which can be looked up and some of which cannot, and the second list is the useful one. The Anantara committee can compute Rs 9.40 crore, Rs 8.00 crore and Rs 5.55 crore in an afternoon. The committee cannot compute anything about a course it did not take, and knowing which of the two lists a question belongs to is most of the skill.

What is an index fund not?

An index fund is not a claim that rules beat people. Whether rules beat people is a question about evidence, and no arithmetic of this kind is evidence. It is not a way of removing risk. The account holds securities and those securities move. It is not a way of avoiding the work of designing a portfolio. The allocation, the constraints and the concentration testing all survive intact. And it is not free. Something is always consumed in delivering it, even when the available record carries no figure for what.

An index fund is, in the end, a change in where one decision lives, with a set of consequences that follow from that change and nothing else. The most useful sentence to leave with is the one that sounds least dramatic: an index fund settles who chooses the holdings, and leaves every other question about the portfolio exactly where it found it.