Portfolio Management puzzles, solved step by step

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100
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All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
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  1. 010An ETF trades at 101.5 on the exchange while its indicative NAV is 100.0, and creating new units costs 0.3% of NAV. What does an authorised participant do, and what does it make per unit?Funds, ETFs and implementationCoreMutual fundsPortfolio implementation

    Try it first

    What does the authorised participant do?

    Show the worked solution

    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.

    The creation route: buy the basket, swap it for units, sell the units1. Buy the basketshares at NAV 100.02. Deliver to fundplus 0.3 of costs3. Receive unitsnew ETF units4. Sell on exchangeat 101.5Per unit: 101.5 received - 100.0 basket - 0.3 costs = 1.2, about 1.2% of NAV99.0100.0101.0102.0no-profit band 99.7 to 100.3ETF at 101.5selling new units pulls the price backPrice
    The authorised participant buys the basket at 100, pays 0.3 in creation costs, receives new units and sells them at 101.5, keeping 1.2 per unit; the trade stops paying once the ETF price is back inside the band from 99.7 to 100.3.

    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 relationship
    gain=PETF−NAV−c⋅NAV=101.5−100−0.3=1.2\text{gain}=P_{ETF}-NAV-c\cdot NAV=101.5-100-0.3=1.2
    P_{ETF}the exchange price of the ETF unit, 101.5
    NAVthe indicative net asset value per unit, 100
    cthe creation cost, 0.3% of NAV
    What it says in wordsThe authorised participant keeps the premium over NAV less the cost of creating the unit.

    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.

    Where candidates lose it

    Candidates reach for the wrong direction, buying units and redeeming them, because redeem sounds like cashing in a profit. Name the cheap side and the dear side first, and the direction follows.

    The quieter miss is forgetting the cost. A premium of 1.5 is not the profit; 1.2 is, and a premium of 0.2 is not an opportunity at all.

    What the interviewer asks next

    • The ETF trades at 99.0. Walk through the trade.
    • Why do bond ETFs sometimes trade at large discounts in a stressed market?
    • Who bears the cost when an ETF persistently trades at a premium?
  2. 038An equity fund keeps 8% of its assets in cash earning 6.5% a year, while its equities return 14%. How much return does the cash cost the fund, and when is that cost largest?Funds, ETFs and implementationCorePortfolio implementationMutual funds

    Try it first

    Roughly how much does the 8% cash holding cost the fund this year?

    Show the worked solution

    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.

    Cash drag is a slope, not a fixed cost: 8% of the equity-cash gap14.0%Fully invested13.4%8% in cash-0.6drag0.08 x (14.0 - 6.5) = 0.6 points+2+1-1-2-20%-10%0%10%20%30%Equity return in the yearDrag in points (positive = cost)14%: costs 0.6-10%: cushions 1.32zero at 6.5%
    Holding 8% in cash takes the fund from 14.0% to 13.4%, a drag of 0.6 points. Because the drag is 8% of the equity-cash gap, it grows in strong markets, is zero when equities earn exactly 6.5%, and becomes a cushion of 1.32 points when equities fall 10%.

    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.

    The relationship
    drag=wc (Re−Rc)=0.08×(14−6.5)=0.6\text{drag} = w_c\,(R_e - R_c) = 0.08 \times (14 - 6.5) = 0.6
    w_cthe cash weight, 8%
    R_ethe equity return, 14%
    R_cthe return on cash, 6.5%
    What it says in wordsThe fund gives up the cash weight times whatever equities earned above cash.

    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.

    Where candidates lose it

    The common slip is multiplying the cash weight by the whole equity return and saying 1.1 points, which assumes cash earns nothing. The second is treating cash drag as a fixed annual fee rather than a bet against the market that pays off only in down years.

    Give the formula, the 0.6, and then the shape: bigger in rallies, negative in falls. That second half is what separates an implementation answer from arithmetic.

    What the interviewer asks next

    • How would you equitise the cash, and what does that cost?
    • A manager says the cash is a deliberate market call. How would you judge it over five years?
    • If the fund's benchmark returns 14% and the fund's stocks return 14.5%, does the fund beat the benchmark after the cash drag?
  3. 068An index fund charges 0.20% a year and keeps 1% of its assets in cash, which earns 6.5% in a year when the index returns 12%. It earns an extra 0.05% from lending securities. What is its tracking difference for the year?Funds, ETFs and implementationCoreMutual fundsPortfolio implementation

    Try it first

    What does the 1% cash holding cost the fund?

    Show the worked solution

    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.

    From index return to fund return: the fee is most of the gap11.7%11.8%11.9%12.0%12.00%Index-0.20Fee-0.055Cash drag+0.05Lending11.795%FundThe vertical axis starts at 11.7% so hundredths of a point are visible.Trackingdifference-0.21%fee share ofthe costs: 78%
    Starting from the index's 12%, the fee takes 0.20 points, cash drag 0.055 points and securities lending adds 0.05, leaving the fund at 11.795%, a tracking difference of -0.205% of which the fee is the largest part.

    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.

    The relationship
    TD=−fee−wc(Ridx−Rc)+lending=−0.20−0.01×5.5+0.05=−0.205%TD = -\text{fee} - w_c(R_{idx} - R_c) + \text{lending} = -0.20 - 0.01 \times 5.5 + 0.05 = -0.205\%
    w_cthe share of the fund held in cash, 1%
    R_idxthe index return, 12%
    R_cthe return on cash, 6.5%
    What it says in wordsTracking difference is the fee plus the cost of the cash, less any income from lending.

    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.

    Where candidates lose it

    The trap is charging the full index return on the cash holding, 1% of 12%, and overstating the drag at 0.12%. The cash earned 6.5%; only the gap is lost.

    The second loss is mixing up tracking difference with tracking error. The first is a steady shortfall in return, the second is the wobble around it. Say which one you are computing.

    What the interviewer asks next

    • If the index falls 10% and cash still earns 6.5%, what is the tracking difference?
    • Why might an index fund have a near-zero tracking difference but a high tracking error?
    • When would securities lending income be a reason for concern rather than comfort?