--- title: "ETFs, Futures and Index Arbitrage" book: "Interviews II: Markets, Trading Games and the Final Round" subject: quant language: en chapter: 6 exercises: 0 source: https://one-course.com/books/quant/19/en/chapter/6-etfs-futures-and-index-arbitrage license: CC-BY-NC-SA-4.0 credit: "One Quant Book, One Course (one-course.com)" --- # Chapter 6 — ETFs, Futures and Index Arbitrage The delta-one desk’s arithmetic: what an ETF should trade at, when creating or redeeming pays, what a future’s price contains, what a leveraged fund’s daily reset does to it and to the market, and how much a rebalance or a roll is worth. Rates here are simple unless a stem says otherwise. **Question 6.1 ★ QT, Trader • market maker, hedge fund, bank • junior.** For every 100 of its shares, an ETF holds 50 shares of a stock at $20, 30 of a stock at $50, 10 of a stock at $100, and $500 in cash. What is its net asset value per share? It trades at $40.10: what is its premium in basis points? **Answer to Question 6.1.** Net asset value is $40.00 per share, and the premium is 25 basis points. The holdings are worth $50 \times 20 + 30 \times 50 + 10 \times 100 + 500 = 1{,}000 + 1{,}500 + 1{,}000 + 500 = 4{,}000$, for 100 ETF shares. And $0.10/40 = 0.25\%$. **Classic wrong answer.** Leaving out the cash. It belongs to the shareholders, and it is often a large part of the difference for a fund near a dividend date. **Question 6.2 ★ QT, Trader • market maker, hedge fund • junior.** An ETF’s net asset value is $40.00. Buying the basket and creating costs a total of 6 cents per ETF share, and redeeming and selling the basket costs a total of 4 cents. Between what ETF prices is there no arbitrage? The bid is $40.09: what do you do, and how much do you make per share? **Answer to Question 6.2.** There is no arbitrage between $39.96 and $40.06. At a bid of $40.09, you sell the ETF, buy the basket and create, and you make 3 cents a share. To create you pay $\$40.00 + \$0.06$, so a sale above $40.06 is a profit. To redeem you buy the ETF, then sell the basket for $40.00 less $0.04, so a purchase below $39.96 is a profit. So the ETF’s fair range is wider than the fund’s own value, and it is set by creation and redemption costs. **Classic wrong answer.** Using the same cost on both sides, or comparing the bid with the ask. You sell at the bid. And creating and redeeming have their own, possibly very different, costs. **Question 6.3 ★ QT, Trader, QR • bank, hedge fund, market maker • junior.** A stock trades at $100. It pays a $1 dividend in three months. Interest is 4% a year, and you borrow and lend at that rate. What is the fair price of a six-month forward? Show the trades that lock in a profit if the forward trades at $102. **Answer to Question 6.3.** The fair price is $100 \times 1.02 - 1.01 = \$100.99$. At $102, sell the forward and buy the stock with borrowed money. Borrow $100 for six months to buy the stock, which costs $102 at the end. The $1 dividend, in three months, earns $1\%$ for the remaining quarter, so it is worth $1.01 at the end. You deliver the stock at $102, repay $102, and keep the $1.01. So your locked-in profit is $1.01, the amount by which $102 exceeds $100.99. **Classic wrong answer.** Using $S(1 + rT)$ and ignoring the dividend. A holder of the forward gets no dividend, so the forward is cheaper by what the dividend is worth at delivery. **Question 6.4 ★ QT, Trader, Risk • hedge fund, bank • junior.** An index stands at 4,000, and its future has a multiplier of $50 per point. What is one contract’s notional value? How many contracts hedge a $10 million portfolio with a beta of 1, and how many with a beta of 1.2? **Answer to Question 6.4.** One contract’s notional is $200,000. You need 50 contracts for a beta of 1, and 60 for a beta of 1.2. $4{,}000 \times \$50 = \$200{,}000$. Then $\$10{,}000{,}000/\$200{,}000 = 50$. A beta of 1.2 means the portfolio moves like $12 million of the index, which is 60 contracts. **Classic wrong answer.** Hedging notional when the beta differs from 1. With 50 contracts, a beta of 1.2 leaves $2 million of market exposure. **Question 6.5 ★ QT, Trader • bank, hedge fund, market maker • junior.** An index stands at 4,000. Its future, expiring in a quarter of a year, trades at 4,020. The dividend yield is 1% a year. What financing rate does the future imply? If you can borrow at 2.5%, what would you do? **Answer to Question 6.5.** The implied financing rate is 3% a year. At a borrowing rate of 2.5%, you buy the basket with borrowed money and sell the future. The future’s premium is $20/4000 = 0.5\%$ for a quarter, or 2% a year, and $r - q = 2\%$, so $r = 3\%$. This is the cash and carry: you are lending your money at the implied 3% and borrowing at 2.5%, a pick-up of 0.5% a year before costs. **Classic wrong answer.** Comparing the premium’s 2% with 2.5%, and concluding the future is cheap. The holder of the basket also gets the dividend, so you have to add it back. **Question 6.6 ★ QT, Trader • bank, hedge fund, market maker • junior.** An index stands at 4,000, and its quarter-year future trades at 4,030. The interest rate is 5% a year. What dividends, in index points and as an annual yield, does the future imply? **Answer to Question 6.6.** The future implies 20 index points of dividends, a yield of 2% a year. Fair value without dividends is $4000 \times (1 + 5\% \times \tfrac14) = 4050$. The future at 4,030 is 20 points lower, so that is the implied dividend. And $20/4000 = 0.5\%$ for a quarter. **Classic wrong answer.** Treating every gap between spot and futures as dividends. The gap also holds any financing spread, and in stress it can hold plenty of that. **Question 6.7 ★ QT, Trader, QR, Analyst • any • junior.** An index rises 2% and then falls 2%. What is its total return over the two days? A fund resets each day to three times the index’s daily return. What is its return? How does that compare with three times the index’s return? **Answer to Question 6.7.** The index returns $-0.04\%$, the fund returns $-0.36\%$, and three times the index’s return is $-0.12\%$. The index is $1.02 \times 0.98 = 0.9996$. The fund is $1.06 \times 0.94 = 0.9964$. Each day’s multiple is exact, but over two days, the return compounds differently, and it is lower by the extra volatility. **Classic wrong answer.** Thinking a three-times fund gives three times the return over any horizon. It promises the daily multiple only. **Question 6.8 ★ QT, Trader, Analyst • any • junior.** An index goes from 100 to 110 and back to 100. A fund resets each day to minus one times the index’s daily return. What is the fund’s total return? Someone holds it to hedge a long index position for those two days. What is the combined result? **Answer to Question 6.8.** The fund loses about 1.82%. Combined with the long index for those two days, you lose about 0.91% on the $2 you put in. Up 10% means the fund loses 10%. Then the index falls $10/110 \approx 9.09\%$, and the fund gains 9.09%. So $0.9 \times 1.0909 \approx 0.9818$. The index holding is flat, so with $1 in each, you end with $1.9818. The one-day hedge is exact, but it stops being exact as soon as the two positions drift, since the fund’s exposure is reset daily and you have not done the same to your position. **Classic wrong answer.** Treating the inverse fund as a long-horizon hedge. To keep it a hedge, you have to rebalance between the two, as often as the fund resets. **Question 6.9 ★ QT, QR, Analyst • hedge fund, bank • junior.** A fund holds two stocks equally, $100 million in each. One rises 10%, and the other falls 10%. What trades bring it back to equal weights, and what is the one-way turnover as a share of the fund? What trades would a cap-weighted fund need? Why? **Answer to Question 6.9.** Sell $10 million of the winner and buy $10 million of the loser, a one-way turnover of 5%. A cap-weighted fund needs no trades, because its weights move with the prices by themselves. The holdings become $110 million and $90 million. Equal weight needs $100 million of each. The total is $200 million, so $10 million is 5%. So cap-weighting has low turnover, and equal-weighting systematically sells winners and buys losers, and pays the trading cost to do so. **Classic wrong answer.** Assuming that a cap-weighted fund must trade after a price move. It trades on changes in the index’s membership or share counts. **Question 6.10 ★★ QT, Trader, Risk • hedge fund, bank • junior.** You are long 10 index futures at 4,000 with a $50 multiplier, and you post 5% of notional as initial margin. The index falls to 3,900 by the close. What is your loss, and how much cash moves today? What is your margin if the initial margin is still required at the new price? What is the difference between this position and the same exposure in a forward? **Answer to Question 6.10.** Your loss is $50,000, and $50,000 moves in cash today. The initial margin is $100,000 at the old price and $97,500 at the new one, so you must have $47,500 more than the $50,000 left in the account. Your position’s notional is $10 \times 50 \times 4000 = \$2{,}000{,}000$, and 5% of that is $100,000. A 100-point fall on 10 contracts at $50 is $50,000. So after the settlement, the account has $50,000, and the new requirement of $5\% \times \$1{,}950{,}000 = \$97{,}500$ needs $47,500 more. A forward with the same exposure would have no cash moving until settlement. Its gain or loss is carried as a credit risk. Futures turn market risk into a liquidity risk, and you need the cash today. **Classic wrong answer.** Thinking a hedged position cannot cause trouble. A futures hedge against an illiquid long position can be margin-called in cash while the long cannot be sold. **Question 6.11 ★★ QT, Trader, Analyst • any • junior.** A fund stays long by holding the nearest monthly oil future and rolling it each month. The front future trades at 70 and the next at 71.40. Spot and the whole curve stay exactly the same for a year. What does the fund earn from the roll in a month? What over a year? Why does an unchanged oil price not give it an unchanged result? **Answer to Question 6.11.** The fund loses about 2% a month from the roll, for a compounded loss of about 21% a year on the unchanged curve, before interest on its cash. An unchanged spot does not give an unchanged result because the fund keeps buying a higher-priced future that then falls toward spot. Each month, it sells at 70 and buys at 71.40, so it holds $70/71.40 = 0.9804$ as many contracts. In a year, that is $0.9804^{12} \approx 0.788$, a loss of 21.2%. This curve is in contango: the future is above spot. It pays for storage and financing. The holder of a contract gets none of the benefit of having oil, and so the cost is charged to that holder. **Classic wrong answer.** Saying the fund loses on the trade at each roll because it “pays 1.40”. The roll itself is at market. The loss comes as the higher-priced contract falls to the front price over the month. **Question 6.12 ★★ QT, Trader, QR • market maker, hedge fund • junior, experienced.** A fund with $1 billion of assets resets each day to three times the index. The index rises 1% today. What are its assets and exposure before the reset, and how much must it buy or sell at the close? Write a formula for a leverage $L$, a fund size $A$ and a return $r$. What does a minus-three fund do on the same day? **Answer to Question 6.12.** Before the reset, assets are $1.03 billion and exposure is $3.03 billion. So it must buy $60 million at the close. The flow is $L(L - 1)rA$. A minus-three fund of the same size buys $120 million. Its exposure was $3 billion, and with 1% up it is $3.03 billion. Its assets rose by $30 million. The target is $3 \times 1.03 = \$3.09$ billion, so it buys $60 million. In general, exposure after the move is $LA(1 + r)$, assets are $A(1 + Lr)$, and the target is $LA(1 + Lr)$. The difference is $LAr(L - 1)$. At $L = -3$, $L(L - 1) = 12$, so it trades $12 \times 1\% \times \$1$ billion: it must also buy when the market rises. As a check at $L = 1$: nothing to do, and that is right. **Classic wrong answer.** Thinking a fund that is short the market sells when the market rises. The index’s rise has increased its short exposure and reduced its assets, so it must buy. **Question 6.13 ★★ QR, QT • hedge fund, market maker • junior, experienced.** A fund resets daily to $L$ times an index with annual volatility $\sigma$, with no rates or fees. Over a year, how does its return relate to the index’s? If the index has 30% volatility and finishes the year flat, what do the three-times fund and the minus-one fund return? **Answer to Question 6.13.** Over a time $T$, the fund’s value is $(S_T/S_0)^L\exp\left(-\tfrac12L(L - 1)\sigma^2T\right)$. With 30% volatility and a flat index, the three-times fund returns about $-23.7\%$, and the minus-one fund about $-8.6\%$. The log return of a daily-reset fund is about $L$ times the index’s log return, with the volatility drag $-\tfrac12L^2\sigma^2T$ of the fund against $-\tfrac12\sigma^2T$ on the index. So the difference is $-\tfrac12(L^2 - L)\sigma^2T$. For $L = 3$: $e^{-3 \times 0.09} = e^{-0.27} \approx 0.763$. For $L = -1$: $L(L - 1) = 2$, so $e^{-0.09} \approx 0.914$. **Classic wrong answer.** Calling this a fee or a mistake in the fund. It is the arithmetic of rebalancing, and the fund does exactly what it says. **Follow-up.** *And if the index has trended?* A trend can make the fund beat $L$ times the return: if the index rises 50% with little volatility, the three-times fund is up nearly $1.5^3 - 1 = 237\%$. It depends on path. **Question 6.14 ★★ QT, QR, Dev • bank, hedge fund • junior.** A price-weighted index has three stocks at $100, $50 and $30, and a divisor of 3, so it stands at 60. The first stock splits 2 for 1. What must the new divisor be, and why? If all three stocks then rise 10%, what is the index, and what would it have been without the split? **Answer to Question 6.14.** The new divisor is $130/60 \approx 2.167$. After a 10% rise, the index is 66, and it would have been 66 without the split. So the split has no effect on the level, but it changes the weights. The split takes the first stock to $50, so the sum of prices is $50 + 50 + 30 = 130$. The index must stay at 60, so the divisor is $130/60$. If each stock rises 10%, the sum is 143, and $143/2.167 = 66$. Without a split, the sum is 198, and $198/3 = 66$. If only the first stock rose 10% after the split, the index gains $5/2.167 \approx 2.31$, while before the split it would have gained $10/3 \approx 3.33$. So the split cut the stock’s weight. **Classic wrong answer.** Thinking the split leaves the index’s sensitivity the same. A price-weighted index’s weights are the share prices, so a split reduces that stock’s importance. **Question 6.15 ★★ QT, Trader • bank, hedge fund, market maker • junior.** A price index stands at 4,000. A stock with a 2% weight goes ex-dividend today, paying 1% of its price. How many points does the index fall at the open, with all else unchanged? What happens to a total-return version of the index, and to the fair value of the future expiring after the date? **Answer to Question 6.15.** The index falls 0.8 points. A total-return version does not fall. The fair value of the future does not move on the day, because the dividend was already in the forecast. $4000 \times 2\% \times 1\% = 0.8$. A price index does not reinvest, so it drops by the dividend. A total-return index adds the dividend back, so it is unchanged. The future’s fair value is spot, less dividends to come. Spot falls by 0.8, and the dividends yet to come fall by 0.8, so the two cancel, up to interest. **Classic wrong answer.** Calling a fall on the ex-date a loss. A holder gets the cash, so the total position is unchanged. **Question 6.16 ★★ QT, Trader • bank, hedge fund • junior, experienced.** An index stands at 4,000. You buy the basket and sell a one-year future at a price assuming dividends of 2% of spot. Dividends turn out to be 2.5%. What is your gain in points? Who is on the other side of this risk? What would you trade if you only wanted this dividend exposure? **Answer to Question 6.16.** You gain 20 points. The other side is whoever holds the future long; often that is a structured-product dealer laying off the long dividend exposure its book carries. For the exposure alone, you would trade a dividend future or a dividend swap. You receive $2.5\%$ of 4,000 instead of 2%, which is 100 points rather than 80. The price of the future did not change, so the 20 points is your profit. Dealers who sell structured products to investors often get long dividends as a by-product, and they sell them on. So implied dividends can sit below the expected level for long periods. The interest on the dividend has been ignored here. **Classic wrong answer.** Treating a basis trade as risk-free. Your forecast of dividends is one of its main risks, and it matters more at long maturity. **Question 6.17 ★★ QT, Trader, Analyst • hedge fund, bank • junior, experienced.** You want index exposure. A rolling future costs 15 basis points a year of implied financing above your funding rate, plus 4 basis points a year in roll costs. An ETF has a 9-basis-point annual fee, earns 2 basis points a year from lending, and costs 1 basis point in spread to buy and sell. Over one year, which is cheaper? At what holding period are they equal? **Answer to Question 6.17.** Over one year, the future costs 19 basis points and the ETF 8. They are equal at one month, so below that the future is cheaper. The future costs $19H$ for a holding period of $H$ years. The ETF costs $7H + 1$. Setting them equal gives $12H = 1$. So the fixed spread of the ETF is what matters on a short horizon, and the running cost on a long one. **Classic wrong answer.** Comparing only the fees. Implied financing is hidden in the futures price, and it moves. In stress it can rise far above 15 basis points. **Question 6.18 ★★ QT, Trader • bank, hedge fund, market maker • junior, experienced.** An index future expiring in three months trades at 4,020, and the next one, three months later, trades at 4,050. Rates are 5% a year, and dividends over that later quarter run at 2% a year. Is the calendar spread fair, using 4,020 as the base? What if the later future trades at 4,060? What do you do if you are long the near one and must stay long? **Answer to Question 6.18.** Fair value for the spread is about 30 points, so 4,050 is fair. At 4,060, the later future is about 10 points rich. Then, if you must stay long, you could buy spot through the basket at expiry, or take the near one to expiry and enter another way, since rolling costs an extra 10 points, about 0.25%. The spread is $4020 \times (5\% - 2\%) \times \tfrac14 \approx 30.15$. A calendar spread is the market price of financing for that quarter. If you hold the near future, you can sell the spread (sell near, buy far) at 40 points, paying 10 over fair. Instead you could look for other exposure: hold to expiry, then buy the basket or an ETF, with its own costs. If those are under 10 points, you skip the roll. **Classic wrong answer.** Treating the roll as a free event at the month’s end. Everyone rolls in the same few days, so the calendar spread can carry a premium. **Question 6.19 ★★ Analyst, QT • hedge fund, bank • junior.** An index fund has a fee of 0.20% a year and earns 0.08% from lending. It keeps 1% of its assets in cash, and over the year the index returns 8% while cash returns 4%. It holds a sample of the index whose return misses it by 5 basis points a year, with mean zero. What is its expected return gap with the index? Distinguish tracking difference from tracking error. **Answer to Question 6.19.** The expected gap is $-0.16\%$ a year. Tracking error is the 0.05% variation. The fee costs 0.20%, and lending earns 0.08%. A cash holding of 1% in a year when the index beats cash by 4% costs 0.04%. So $-0.20 + 0.08 - 0.04 = -0.16$. Tracking difference is the average gap, which is what you actually lose. Tracking error is its volatility. A fund can have low tracking error and a poor tracking difference. **Classic wrong answer.** Using the fee as the whole cost of holding a fund. Lending can more than pay the fee, and the drag from cash reverses in a falling market. **Question 6.20 ★★ QT, Trader • bank, hedge fund • junior, experienced.** An index stands at 4,000, with a half-year future and dividends of 2% a year. You borrow at 5% and lend at 4.5%. Selling the basket short costs a further fee of 0.3% a year. Entering and leaving the trade costs 0.1% of spot in total. Between what future prices is there no arbitrage for you? Why is the band not symmetric? **Answer to Question 6.20.** The no-arbitrage band is 4,040 to 4,064. Above the band, you sell the future and buy the basket with borrowed money: $4000 \times (1 + (5\% - 2\%) \times \tfrac12) = 4060$, and with costs of 4 points, that gives 4,064. Below the band, you buy the future and short the basket, and you lend the proceeds at 4.5%, with a borrow fee of 0.3% to pay. So $4000 \times (1 + (4.5\% - 0.3\% - 2\%) \times \tfrac12) = 4044$, and with the costs, 4,040. It is not symmetric because the two sides face different rates and only the short side pays a fee. Its width, 24 points, has two parts: 8 from the costs, and 16 from the gap in financing rates. So a firm with better financing has a tighter band and can arbitrage where you cannot. **Classic wrong answer.** Thinking there is one fair value for a future. There is one for each cost of funding, so a future can sit between many firms’ bands. **Question 6.21 ★★ QT, Trader, Risk • market maker, hedge fund, bank • junior, experienced.** In a credit sell-off, a bond ETF trades at 95 while its published net asset value is 100. You would receive bonds that, sold right now, would really bring 95.50, and it costs 0.20 to redeem. What do you make per ETF share if everything happens at once? Selling the bonds takes three days, with a daily volatility of 0.5% in their price. Is it arbitrage? Which number is the better estimate of value, the 95 or the 100? **Answer to Question 6.21.** You would make 0.30 on the ETF share if it all happened at once. But selling takes three days, with a standard deviation of about 0.87, so it is not arbitrage. The 95 is the better estimate. The published 100 is stale. Buying at 95 and receiving bonds worth 95.50, less 0.20, leaves 0.30. The risk is $0.5\% \times \sqrt 3 \approx 0.87\%$. So one standard deviation is almost three times the profit. Bonds trade rarely. The net asset value is made from stale prices or models, while the ETF trades now. In stress, the ETF is often the price discovery venue, and the “discount” is largely the true loss not yet in the marks. An ETF’s discount is at most a modest arbitrage. **Classic wrong answer.** Saying the ETF “broke” because of a discount. The discount reflects bonds that could not be sold at their marks. **Question 6.22 ★★ QT, Trader • market maker • junior, experienced.** You quote an ETF at $40.00 and $40.02, fair value being $40.01. When filled, you hedge at once with a future, paying half of its spread, 0.5 basis points. If your hedge and your ETF move apart while you hold them, what is the risk? What do you net per share on one fill? For 100,000 shares what is that in dollars, and what does a basis move of 1 basis point against you do? **Answer to Question 6.22.** You net about 0.8 cents a share, or $800 on 100,000 shares. A basis move of 1 basis point against you is 0.4 cents a share, or $400. You earn 1 cent against fair value on the fill. The hedge costs half the future’s spread: $0.5 \text{ bp} \times \$40 = 0.2$ cents. So $1 - 0.2 = 0.8$. The risk is the gap between the ETF and the future: the future may not match the ETF’s basket, and the future’s value includes financing and dividend assumptions. So you are holding an extra risk until you offset it. Half your profit can go on a basis move of 1 basis point. **Classic wrong answer.** Treating the hedge as perfect. A market maker’s profit lies largely in how small this risk stays relative to the half-spread. **Question 6.23 ★★ QT, Trader, Analyst • market maker, hedge fund • junior, experienced.** Buying an ETF’s basket costs half a spread of 2 basis points. Over a day, investors buy $100 million of the ETF and sell $90 million, and a market maker takes the other side of all of it. What must the market maker hedge in the basket? What does that cost, per dollar of ETF traded? Why can an ETF’s spread be narrower than the spread of what it holds? **Answer to Question 6.23.** The market maker has only the net $10 million to hedge in the basket, which costs about $2,000 in total. Per dollar of ETF traded, on $190 million, that is about 0.1 of a basis point. So an ETF’s spread can be narrower than the basket’s, because most trades are offset by other investors. $\$10$ million at 2 basis points is $2,000. Over $190 million, that is about 0.105 basis points. The offsetting trades never go to the basket, so only a little is needed to cover that part. The rest of the spread pays for risk. **Classic wrong answer.** Assuming that an ETF cannot be cheaper than what it holds. Its liquidity is more than the sum of the parts as long as trading is two-way. In a one-way market, the spread goes back toward the basket’s. **Question 6.24 ★★ QT, Trader, Analyst • bank, hedge fund • junior, experienced.** You want exposure to a stock yielding 2% a year. Holding it in cash costs you funding plus 30 basis points, and you lose 15% of the dividend to withholding tax. Holding it through a total return swap costs funding plus 40 basis points, and the dealer passes on 95% of the dividend. Which is cheaper, and by how much? What risk do you take on with the swap, and what does the dealer do to hedge? **Answer to Question 6.24.** The swap is cheaper by 10 basis points a year: cash costs funding plus 60, and the swap costs funding plus 50. In cash you pay 30 and lose $15\% \times 2\% = 30$ on dividends. With the swap, you pay 40 and lose $5\% \times 2\% = 10$. With a swap, you take on the dealer’s credit risk, usually reduced by daily margin, and the dealer may end the trade. You have no vote, and the terms can change. The dealer hedges by holding the stock, and it may get better tax treatment or financing than you, and that is the source of the gap. Check the legal position, since tax rules on such swaps vary by country. **Classic wrong answer.** Comparing only the two financing spreads. The dividend leak is the same size as the financing here, and it decides the answer. **Question 6.25 ★★★ QT, Trader, Analyst • hedge fund, market maker • junior, experienced.** A stock with a market value of $10 billion, 80% of it free float, joins a cap-weighted index. Funds that track the index together own 20% of the free float of every member. What must they buy? Daily volume is $100 million: how many days is that at 100% of volume, and at 20%? With the rule that the cost is $\sigma\sqrt{Q/V}$ and 2% daily volatility, what does the price do, and why should you doubt that number? **Answer to Question 6.25.** They must buy $1.6 billion. That is 16 days at 100% of volume, and 80 days at 20%. The rule gives a price pressure of 8%. You should doubt it, because 16 days of volume is far outside where the rule was measured. Free float is $\$10$ billion $\times 80\% = \$8$ billion, and 20% of that is $1.6 billion. Then $1.6/0.1 = 16$, and $16/0.2 = 80$. The rule says $2\% \times \sqrt{16} = 8\%$. Reasons to doubt it: this demand is announced, and that draws in sellers who would not trade otherwise, as well as arbitrageurs who hold stock in advance. Volume on the effective day is many times normal. So you may see only part of this price pressure, and part of that will reverse. **Classic wrong answer.** Saying the funds must buy over 80 days. They want the closing price on the effective date, so they will buy nearly all of it at one moment. **Question 6.26 ★★★ QR, QT • hedge fund, market maker • junior, experienced.** A colleague notices that a three-times and a minus-three-times fund both lose value in volatile markets, and proposes selling $1 of each short and holding for a year, ignoring borrow costs. Daily returns have mean zero and an annual volatility of 30%. What is your expected profit? What is your median profit, and how often do you lose money? What does this say about the “decay”? **Answer to Question 6.26.** Your expected profit is zero. Your median profit is about $0.41, and you lose money about 28% of the time. A loss of about $2 or more comes about once in 20 years. The decay is real for a typical path, and it is paid for with a rare, very large loss. Each fund’s value is $\prod_i(1 \pm 3r_i)$, and with independent, mean-zero daily returns, its expected value is exactly 1. So short positions in both have zero expected profit, despite each fund’s strong decay. By the formula, with drag $\exp(-\tfrac12L(L - 1)\sigma^2T)$ and $x$ the index’s multiple, the total value of the two funds is about $e^{-0.27}x^3 + e^{-0.54}x^{-3}$. At $x = 1$, that is about 1.35, so the profit is about 0.65 if the index ends flat. The most you can make, if realised volatility is exactly 30%, is about 0.67. You lose when the index’s log return is above about $+28\%$ or below about $-37\%$. The value then grows with the cube of the index’s gain, and that loss has no upper limit. A simulation of daily returns gives the median, the 28% and the tail. **Classic wrong answer.** Treating the decay as a yield. It is a typical-path result, and it has the same logic as selling a strangle: mean zero, and a positive median. Even that holds before the borrow fee, which is usually high on these funds. **Question 6.27 ★★★ QT, QR, Trader • market maker, hedge fund • junior, experienced.** Three-times funds hold $10 billion on an index, and minus-three-times funds hold $5 billion. The index falls 4% today. How much does each group have to trade at the close, and in which direction? Check each by computing exposures. If the market’s liquidity in the last 15 minutes is $20 billion, what share is this? Why does this look like the hedge of a short gamma position? **Answer to Question 6.27.** Each group sells $2.4 billion, $4.8 billion in all, which is 24% of the last 15 minutes’ liquidity. It looks like a short gamma hedge because the amount is proportional to the move and it trades with the move. By the formula $L(L - 1)rA$. For $L = 3$: $6 \times (-4\%) \times \$10$ billion $= -\$2.4$ billion. For $L = -3$: $12 \times (-4\%) \times \$5$ billion $= -\$2.4$ billion. Check the first by exposures: it was $30 billion and is now $28.8 billion, while the assets are $\$10 \times 0.88 = \$8.8$ billion, so the target is $26.4 billion. That is a sale of $2.4 billion. Check the second: it was $-\$15$ billion and is now $-\$14.4$ billion, the assets are $\$5 \times 1.12 = \$5.6$ billion, and the target is $-\$16.8$ billion. That is a sale of $2.4 billion. Someone short gamma has a delta that grows against them, and has to trade with the market. Here, the funds are short gamma in effect: their position per move is $L(L - 1)A$. So a late-day trend can be helped along by the funds’ own flow. **Classic wrong answer.** Thinking long and inverse funds cancel. They both trade with the move, so their flows add. **Question 6.28 ★★★ QR, QT • bank, hedge fund • experienced.** A stock pays nothing and is at 100. The one-period rate is 4%. Under the risk-neutral measure, next period it is either 120 with a one-period rate of 6%, or 84 with a one-period rate of 2%, and in each state it then grows at that rate for one more period. Find the risk-neutral probabilities. What are the two-period forward price and the two-period futures price? Why are they different, and what makes the difference zero? **Answer to Question 6.28.** The up-state probability is $5/9$. The forward price is about 108.35, and the futures price is about 108.75. They differ because the stock is positively correlated with interest rates. If the rate is not random, or not correlated with the stock, the difference is zero. The stock must grow at the rate on average: $100 \times 1.04 = 120p + 84(1 - p)$, which gives $p = 20/36 = 5/9$. The two-period zero-coupon bond is worth $\tfrac{1}{1.04}\left(\tfrac59/1.06 + \tfrac49/1.02\right) \approx 0.92292$, so the forward is $100/0.92292 \approx 108.35$. The futures price is the expected final price under this measure: $\tfrac59 \times 120 \times 1.06 + \tfrac49 \times 84 \times 1.02 \approx 108.75$. The long future gets daily gains in the up state, when the rate is high and they can be invested at it, and losses in the down state, when they can be financed cheaply. So the holder has an advantage over a forward, and the futures price must be higher to offset it. This is the convexity adjustment. **Classic wrong answer.** Treating forwards and futures as the same at all times. They are close for short equity contracts, but they differ materially for long-dated contracts and for rates themselves. **Question 6.29 ★★★ QR, QT, Risk • hedge fund, bank • junior, experienced.** A sector ETF has 18% annual volatility. You hedge it with two stocks, $A$ of 20% and $B$ of 30% volatility, with a correlation of 0.5 with each other. The ETF has correlations of 0.8 with $A$ and 0.7 with $B$. What minimum-variance hedge weights do you use per dollar of ETF, and what is the remaining volatility? Why not hedge with 100 names? **Answer to Question 6.29.** The weights are 0.54 on $A$ and 0.24 on $B$, per dollar of ETF. The remaining volatility is about 8.8%, a variance reduction of 76%. The covariances among the hedge stocks are $0.04$, $0.03$ and $0.09$. The ETF’s covariances with them are $0.8 \times 0.18 \times 0.2 = 0.0288$ and $0.7 \times 0.18 \times 0.3 = 0.0378$. Solving $\Sigma w = c$ gives $w = (0.54, 0.24)$. The explained variance is $w^\top c = 0.024624$ out of $0.0324$. What is left is $18\% \times \sqrt{0.24} \approx 8.8\%$. More names give a lower variance in the sample, but your costs and the error in the estimates both grow. A large covariance matrix fits noise, so out of sample, weights can be unstable, and the real variance is higher than the fit says. So choose a small number of stable hedges. **Classic wrong answer.** Using the two stocks’ market-cap weights in the ETF. A minimum-variance hedge is not the same as a slice of the holdings. **Question 6.30 ★★★ QR, QT, Trader • market maker, hedge fund • junior, experienced.** A future is 6 points rich to fair value, and it must converge to fair value at expiry, in three days. The gap moves up or down by 4 points each day with equal probability on days 1 and 2, then goes to 0 on day 3. You may hold a position of $+1$, $0$ or $-1$ in the gap. If you stay short from today until expiry, what do you make? What is the most you can expect with free flipping? What is that option worth if each flip costs 0.5 points in total? When do you use it? **Answer to Question 6.30.** If you stay short, you make 6. With free flipping, you can expect 7, so the option is worth 1 point. With 0.5 points per flip, it is worth 0.875 points. You use it only on day 2, if the gap has reached $-2$. Work back. On day 3 the gap goes to 0. So holding on day 2 is worth $|b|$: short if the gap is positive, long if it is negative. On day 1, the next move has zero mean, so the position does not matter in expectation, and the value is $E|b_2|$. From 10: $(14 + 6)/2 = 10$. From 2: $(6 + 2)/2 = 4$. From today: $(10 + 4)/2 = 7$. The gain comes only from the path 6, 2, $-2$, of probability $\tfrac14$. Staying short loses 2 there, and flipping long makes 2. That is 4 more, and $\tfrac14 \times 4 = 1$. With the 0.5 cost, it is $\tfrac14 \times 3.5 = 0.875$. This is a real part of the value of a basis trade: you hold an option on early exit and reversal, and that is one reason why people will hold a trade at a tight basis. **Classic wrong answer.** Saying a basis that has to converge is worth just its gap. If you hold the right to reverse, the convergence pays more than the opening gap. **Question 6.31 ★★★ QT, Trader, Risk • hedge fund, market maker • junior, experienced.** A commodity fund stops issuing new shares because it has reached a limit on its holdings, while the old shares keep trading. Demand is strong, and the shares reach a 20% premium to net asset value. You buy at that level. A month later the underlying is up 5%, and creation resumes, so the premium disappears. What do you return? Why does the normal arbitrage not close the premium? What would you do with a short? **Answer to Question 6.31.** You return about $-12.5\%$. The arbitrage does not close the premium because no one can create new shares. To short, you have to find a borrow, so you pay for one and you risk being squeezed. You would short small, sized for a further widening, and only with a catalyst that lets creation resume. Your value is $1.05/1.20 = 0.875$. The underlying rose 5%, and you still lost 12.5%. Normally a premium draws in creators, who sell the ETF against the basket. If creation is halted, the fund acts like a closed-end one, so its price can float away from value. The premium can widen before it closes, and the borrow may get expensive, so you can be right and still lose your position. **Classic wrong answer.** Buying the fund for the underlying’s move and not looking at the premium. You are buying the underlying and the premium, and the premium will not stay. **Question 6.32 ★★★ QT, Trader, Risk, Analyst • hedge fund, bank • junior, experienced.** An ETF has 100 million shares outstanding and a short interest of 120 million shares. How can more than 100% be sold short? How many shares do holders of record and those long through loans believe they own in total? What do market makers do when they deliver? What is the actual risk to investors if the ETF closes? **Answer to Question 6.32.** Short interest above 100% is possible because the same share can be lent, sold, then lent and sold again. The long holdings total 220 million: the 100 million outstanding plus 120 million from shorts. Market makers can create more shares to deliver or to meet demand to borrow. If the ETF closes, the main risk is that the fund’s assets are realised, and shorts must buy back. Holders are paid through their brokers, with the shorts having to cover what they owe. A lender holds a claim on a broker, and the buyer on the other side also counts as holding the share. So longs equal the shares outstanding plus the short interest. Each short is matched by an extra long. The creation process gives ETFs a way of relieving shortage: an authorised participant can create shares, and deliver them to whoever needs them. So a high short interest in an ETF is often a hedging tool. The risk is usually not on the assets, which are held for the fund. It is that an agent or a chain of lenders fails to deliver, or that in stress it takes time to unwind the chain. Investors can check how much is on loan. **Classic wrong answer.** Saying that a short interest over 100% means the fund has “phantom shares” and is about to collapse. The asset value is backed, and the extra longs are matched by shorts. **Question 6.33 ★★★ Analyst, QT • hedge fund, bank • junior, experienced.** A fund tracks a cap-weighted index. A stock joins it with a weight of 0.1%. Buying at the effective close is expected to cost 3% in impact on that stock. Buying over the five days before costs 1%, but the fund then holds an extra position, which you should take as the whole 0.1% for all five days. The stock has a daily volatility of 2%, hedged only by the rest of the index. What is the expected saving for the whole fund, in basis points? What is the tracking risk it takes on? Which would you do? **Answer to Question 6.33.** The expected saving is 0.2 basis points of the fund, and the tracking risk is about 0.45 basis points of the fund. You could reasonably do either, and a fund that is judged on tracking error in the month might not. The stock is 0.1% of the fund, so the saved 2% of impact is 0.2 basis points. The volatility over five days is $2\% \times \sqrt 5 \approx 4.47\%$, and on a 0.1% weight, that is about 0.45 basis points. A position built evenly over the five days would risk about 0.30. That is a fair Sharpe ratio in a single name. The problem is scale: the benefit is real only as long as few others do the same, and an index fund’s mandate often makes it give up that gain. Many such changes over a year diversify the risk, and that argues for doing it. **Classic wrong answer.** Comparing 3% and 1% without scaling to the fund. Both numbers on the fund’s return are tiny, and what counts is how often this happens across all changes. **Question 6.34 ★★★★ QR • hedge fund, market maker, bank • experienced.** A fund targets $L$ times the return on an index with constant volatility $\sigma$ and rate $r$, continuously rebalanced, with fee $f$. Using Ito’s lemma, derive its value at time $T$ in terms of the index. For $L = 3$, $\sigma = 20\%$, $T = 1$, no fee and a fund starting at 1, what is its value if the index is unchanged and rates are zero, and what if the index rises 10% and rates are 4%? With zero rates, what is an at-the-money one-year call on the fund worth, per 100 of fund value? Compare it with three one-year at-the-money calls on an index at 100. **Answer to Question 6.34.** The fund’s value satisfies $$ \frac{L_T}{L_0} = \left(\frac{S_T}{S_0}\right)^{\!L}\exp\left(-(L - 1)rT - \tfrac12L(L - 1)\sigma^2T - fT\right). $$ For $L = 3$, $\sigma = 20\%$ and a flat index with zero rates, it is about 0.887. With a 10% rise and rates of 4%, it is about 1.090. An at-the-money one-year call on the fund, with zero rates, is worth about 23.6 per 100. Three such calls on the index are worth about 23.9. The fund holds exposure $L$ and borrows $L - 1$ at $r$: $dL_t/L_t = L\,dS_t/S_t - (L - 1)r\,dt - f\,dt$. By Ito, $d\ln S = dS/S - \tfrac12\sigma^2dt$ and $d\ln L_t = L\,dS/S - (L - 1)r\,dt - f\,dt - \tfrac12L^2\sigma^2dt$. Multiply the first by $L$ and subtract, and the $dS/S$ term disappears. So the fund is a deterministic function of today’s index, and it has no other path dependence given a constant $\sigma$. With numbers: $e^{-3 \times 0.04} = e^{-0.12} \approx 0.887$ at $r = 0$. With the rise and the rate, $1.331 \times e^{-0.08 - 0.12} \approx 1.090$. The call: the fund has volatility $L\sigma = 60\%$, so its call is $100(2\Phi(0.3) - 1) \approx 23.6$. An index call is 7.97, and three of them make 23.9. The two are close, but the payoffs are different. Three index calls pay linearly in the index, and the fund’s call pays in its cube, with a scale factor. So the fund’s call pays less if the index rises a little, and it pays more for a large rise. **Classic wrong answer.** Assuming the formula holds if volatility changes. Then the exponent uses realised variance, and the fund depends on the path of volatility. So the fund’s own value, and with it the payoff of an option on it, depends on realised variance, which the payoff of an index option does not. **Question 6.35 ★★★★ QT, QR, Trader • hedge fund • experienced.** Funds must buy $1.6 billion of a stock at the close in five trading days, on an announced addition to an index. Daily volume is $100 million and daily volatility is 2%. Use a peak price pressure of $\sigma\sqrt{Q/V}$. Half of it was priced on the announcement. You buy $100 million over the five days at 20% of volume, at an average cost of $\sigma\sqrt{Q/V}$ on your own size, and you sell to the funds at the close. What is your expected profit, and what risk do you run? If others own $400 million by the close and do the same, what happens to the pressure? When would you pass? **Answer to Question 6.35.** You expect about $2 million on your $100 million, or 2%, with a standard deviation of about 4.5%, which is about $4.5 million. If others own $400 million, the funds’ net demand on the market is $1.2 billion, the pressure is about 6.9%, and half of that is left, so the profit falls to about 1.5% before the extra competition for your entry. You would pass when many have the same position and the stock has already gone beyond the rule’s price. The peak is $2\% \times \sqrt{16} = 8\%$. Half was priced on the announcement, so 4% is yet to come. Your entry costs $2\% \times \sqrt{1} = 2\%$: $100 million is one day’s volume. So the profit is $4\% - 2\% = 2\%$, or $2 million. The risk is the stock’s move over the five days, $2\% \times \sqrt 5 \approx 4.47\%$, or $4.47 million if you are unhedged. Then a hedge with its sector removes much of the market part. This is about 0.45 standard deviations per event, so it is a modest trade, even if it works. With others holding $400 million, net demand is $1.2 billion, and $2\% \times \sqrt{12} \approx 6.93\%$. If half is priced, about 3.46% remains, against your 2% cost, which now rises because they are buying with you. That is about 1.46%. I would pass when the stock’s move since the announcement is well above what this says, because arbitrageurs have already crowded it, or the funds’ demand can be met by a large holder who is likely to sell at the close. And the risk of a reversal is greater if others all exit at the same time. **Classic wrong answer.** Buying at the announcement, since “the funds have to buy”. They do, and everyone knows, so the trade can be priced before you. And after the close, the price often falls back.