Portfolio Management puzzles, solved step by step

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  1. 023A portfolio manager decides to buy 10,000 shares when the price is Rs 100. The trading desk fills 60% of the order at an average of Rs 101.2, and the rest goes unfilled as the price closes the day at Rs 104. What is the implementation shortfall?Funds, ETFs and implementationHardPortfolio implementationInstitutional asset management

    Try it first

    Which part of the shortfall is larger?

    Show the worked solution

    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.

    Implementation shortfall: the order you did not fill cost more than the one you didExecution cost6,000 filled x (101.2 - 100)Rs 7,200 (0.72%)Opportunity cost4,000 unfilled x (104 - 100)Rs 16,000 (1.60%)Total shortfallof the Rs 10 lakh intendedRs 23,200 (2.32%)Benchmark: the decision price of Rs 100. Both costs are measured against it,so an unfilled order is charged for the move it missed.
    Against a decision price of Rs 100, the 6,000 shares filled at Rs 101.2 cost Rs 7,200 and the 4,000 unfilled shares cost Rs 16,000 as the price rose to Rs 104, a total shortfall of Rs 23,200, or 2.32% of the intended trade.
    ComponentSharesPer share, RsCost, RsShare of Rs 10 lakh
    Execution cost6,0001.27,2000.72%
    Opportunity cost4,0004.016,0001.60%
    Total10,00023,2002.32%
    The execution and opportunity costs add to a shortfall of Rs 23,200, 2.32% of the intended Rs 10 lakh purchase.

    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.

    Where candidates lose it

    Most candidates measure only the execution cost, 1.2% on the filled shares, and forget that shares never bought still cost the fund. The unfilled 40% is the bigger number here.

    The second slip is the base. Express the shortfall against the whole intended trade, Rs 10 lakh, not against the Rs 6 lakh filled, or the number cannot be compared across orders.

    What the interviewer asks next

    • What if the price had closed at Rs 99 instead?
    • How would you decide how fast to trade the order?
    • Why might a VWAP benchmark make the same desk look good?
  2. 093Trading 10% of a stock's daily volume costs 20 basis points in market impact. If impact follows a square-root law, what does trading 40% of daily volume cost per rupee traded, and how does the total cost compare?Funds, ETFs and implementationHardPortfolio implementationQuantitative asset management

    Try it first

    How much larger is the total impact cost of the 40% order?

    Show the worked solution

    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.

    Four times the size, twice the cost per rupee, eight times the total10%20%30%40%50%20 bp40 bp60 bp80 bpif cost rose in line with size20 bp40 bpOrder size, share of daily volumeDaily volume Rs 50 crore10%: Rs 5 crore x 20 bpRs 1 lakh40%: Rs 20 crore x 40 bpRs 8 lakhSplit over 4 days at 10%about Rs 4 lakhif impact fades overnight
    Cost per rupee rises along a square-root curve from 20 basis points at 10% of daily volume to 40 at 40%, so the total cost on a Rs 50 crore volume stock rises from Rs 1 lakh to Rs 8 lakh, eight times, while spreading the order over four days would cost about Rs 4 lakh.
    The relationship
    c(q)=20q10% bp,total∝q⋅c(q)∝q3/2c(q) = 20\sqrt{\frac{q}{10\%}}\ \text{bp}, \qquad \text{total} \propto q \cdot c(q) \propto q^{3/2}
    qorder size as a share of daily volume
    c(q)impact cost per rupee traded
    What it says in wordsCost per rupee grows with the square root of size, so total cost grows with size to the power one and a half.

    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.

    Where candidates lose it

    The common slip is answering two times, applying the square root to the total cost. The law describes cost per rupee; the total picks up the size again.

    The second loss is stopping at the number. Say what follows from it: large orders get split, and strategy capacity falls as assets grow.

    What the interviewer asks next

    • At what share of daily volume does cost per rupee reach 60 basis points?
    • Why might splitting the order over four days cost more, not less?
    • How would you estimate the capacity of a small-cap strategy from this rule?