Portfolio Management puzzles, solved step by step

It creates new units and sells them, keeping about 1.2 per unit, or 1.2% of NAV. The authorised participant buys the underlying basket at the NAV of 100, delivers it to the fund, receives new ETF units and sells them on the exchange at 101.5. Revenue of 101.5 less 100 for the basket and 0.3 of creation costs leaves 1.2. Its selling pushes the ETF price back toward NAV until the gap no longer covers the cost.

Why can anyone profit from the gap at all?

Picture a shop selling gift hampers for Rs 101.5 when the items inside cost Rs 100 at the market next door, and assembling a hamper costs Rs 0.3. Someone will buy the items, pack hampers and sell them until the hamper price falls. An ETF unit is a claim on a basket, and the creation and redemption window lets a large dealer convert one into the other, so the unit's price cannot drift far from the basket's value. The authorised participantA large dealer appointed by the fund to create and redeem ETF units in bulk directly with the fund. is that hamper maker.

Where does the arbitrage stop?

Every new unit sold adds supply on the exchange and every basket bought adds demand for the underlying shares, so the gap closes from both sides. The trade keeps paying until the premium falls to the creation cost of 0.3, so in a calm market the ETF price sits inside a band of roughly 99.7 to 100.3. Below 99.7 the trade reverses: buy cheap units, redeem them for the basket and sell the shares. The band is only as tight as the cost: an ETF holding illiquid bonds or foreign shares that trade in another time zone can show a wider gap for days, and that gap is a cost to whoever trades against it.

The portfolio management point: this machinery is why an ETF can trade close to its holdings without the fund itself selling anything. Mutual fund investors transact at NAV once a day; ETF investors transact at a market price that stays near NAV only because this arbitrage is open.

Rs 23,200, or 2.32% of the Rs 10 lakh intended trade. Against a paper portfolio that bought all 10,000 shares at the decision price of Rs 100, the 6,000 filled at Rs 101.2 cost Rs 7,200 in execution. The 4,000 unfilled shares missed a rise to Rs 104, an opportunity cost of Rs 16,000. The order you did not fill cost more than the one you did.

What is the shortfall measured against?

Think of deciding to buy a train ticket at the counter price, then finding the queue slow: you pay a bit more for some of the group's tickets, and the rest miss the train and take a dearer taxi. The cost of the trip is measured against the plan, not against what you ended up doing. Implementation shortfall compares the real portfolio with a paper portfolio that traded everything instantly at the decision price, so both the extra paid and the move missed count as cost. The idea is due to Andre Perold, and the benchmark here is Rs 100, the price when the decision was made.

What should the manager take from the split?

That trading patiently is not free. A desk that works an order slowly to keep execution cost down can lose more to the price running away, so the right trading speed depends on how fast the manager's idea is likely to be priced in. Here the price moved 4% in a day, a sign the idea was urgent, or that the buying itself pushed the price. Say the limitations: explicit costs such as commission and taxes should be added on top, and the unfilled shares are charged at the closing price by convention; had the order been cancelled deliberately, a different end point might be fairer.

About 0.6 points a year here. Cash drag is the cash weight times the gap between what equities and cash earn: 0.08 x (14 minus 6.5) is 0.6. The fund returns 13.4% instead of 14%. The drag is largest in strong markets and turns into a cushion when equities fall: in a year equities lose 10%, the same cash saves about 1.3 points.

Why is the cost the gap, not the whole equity return?

Suppose you keep a little of your salary in a savings account instead of a higher-paying deposit. You have not lost the whole deposit rate on that money, only the difference between the two rates. Cash in an equity fund still earns something, so the return given up is the cash weight times the gap between the equity return and the cash return. Here that is 8% of 7.5 points, which is 0.6 of a point for the whole fund.

Why does it matter most in strong markets?

The drag scales with the equity return. In a 30% year the same 8% of cash costs about 1.9 points, in a 14% year 0.6, and in a year equities lose 10% it adds about 1.32 points. That is why cash drag shows up most in the years a fund is judged hardest: a fund that trails its benchmark in a strong rally often simply held more cash than the index, which holds none.

Funds hold cash for real reasons: redemptions, new money not yet invested, and dry powder for opportunities. The implementation answer is not zero cash but equitising it, for example by holding index futures against the cash balance so that the fund keeps market exposure while holding liquidity. Say that as the practical close, and note that futures carry their own costs and margin needs.

About -0.21%: the fund returns about 11.79% against the index's 12%. The fee takes 0.20 points. The 1% in cash earns 5.5 points less than the index, which costs 0.01 times 5.5, or 0.055 points. Securities lending adds back 0.05. Together: minus 0.20, minus 0.055, plus 0.05, a tracking difference of -0.205%. The expense ratio is most of the story.

What is tracking difference, and how does it differ from tracking error?

Think of a tailor copying a suit. The tracking differenceThe fund return minus the index return over a period. A steady shortfall, mostly the costs of running the fund. is how much shorter the copy comes out on average; the tracking errorThe standard deviation of the gap between fund and index returns. How unevenly the fund follows the index. is how uneven the stitching is. Tracking difference is the fund's return less the index's, and for a well-run index fund it is mostly the costs of running the fund. Investors feel it directly: it is the return they lose for holding the fund instead of the index itself.

Why is the cash drag so small?

Because the cash is not idle; it earns 6.5%. Cash drag is the cash weight times the gap between the index return and the cash return, not the cash weight times the index return. Here the gap is 5.5 points and the weight 1%, so the drag is 0.055 points. In a falling market the sign flips: if the index lost 10%, the 1% in cash would add about 0.165 points and the fund would lag by less than its fee.

Other small items sit in the same bucket and are worth naming: trading costs when the index changes its members, the timing of dividends, and taxes the index ignores but the fund pays. None usually rivals the fee. When two index funds on the same index are compared, a lower expense ratio is the first thing to check, then the reported tracking difference over several years, which the fund documents disclose; confirm the current disclosure rules for the market in question.

About 0.48% a year. Turnover of 120% means the fund sells and replaces the equivalent of its whole portfolio 1.2 times a year. At 40 basis points per round trip, the drag is 1.2 x 40, or 48 basis points. On a Rs 1,000 crore fund that is Rs 4.8 crore a year, taken from returns rather than charged as a fee.

What exactly is a round trip, and why count it that way?

Trading in a car costs you twice: the dealer pays less than it is worth when you sell, and charges more than it is worth when you buy the next one. A fund switching one stock for another pays the same two-sided cost, so the natural unit is the round trip: one sale and one purchase together. Reported turnover is usually the lesser of purchases and sales over average assets, which counts each switch once, so 120% maps to 1.2 round trips.

Why does this matter if the expense ratio looks fine?

The expense ratio covers the manager's fee and running costs. Trading costs are paid inside the portfolio, through worse prices and brokerage, so they reduce the return without ever appearing in the expense ratio. A fund with a modest fee and high turnover can cost more in total than a pricier fund that trades little. For a Rs 1,000 crore fund, 48 basis points is Rs 4.8 crore a year.

Say the limitation too. The 40 basis points is an average; impact rises with trade size, so a fund that grows while keeping the same turnover usually pays more per round trip, not less.

About 40 basis points per rupee, and 8 times the total cost. Under a square-root law, cost per rupee rises with the square root of size: four times the size is twice the cost per rupee, 40 basis points. Total cost is size times cost per rupee, so it rises four times two, or eight times. On Rs 50 crore of daily volume, Rs 1 lakh becomes Rs 8 lakh.

Why does a bigger order cost more per rupee?

Selling one flat in a building is easy at the going price; selling ten at once means working down to buyers who would only pay less. A large order uses up the willing counterparties near the current price and has to reach further, so each extra rupee traded moves the price more than the one before. Empirical studies of trading costs across markets find that the cost per rupee grows roughly with the square root of the order's share of volume, a rule practitioners use widely as a first estimate.

What does the portfolio manager do with this?

Two things. Because total cost grows faster than size, spreading a large order over several days is usually cheaper: four days at 10% each costs about Rs 4 lakh against Rs 8 lakh in one go, provided the price impact of one day's trading has largely faded by the next and the price does not drift away meanwhile. And a strategy's capacity is limited: a fund that doubles in size pays about 2.8 times the rupee impact on each rebalance, which eats into returns quickly.

Say the limits. The square-root law is an empirical average, with a coefficient that differs by stock and market conditions, and splitting an order adds timing risk: if the information behind the trade leaks or the price moves, waiting costs more than impact would have.