---
title: "Exchange-Traded Funds"
book: "Markets I: The Ecosystem and Exchange-Traded Markets"
subject: quant
language: en
chapter: 14
exercises: 8
source: https://one-course.com/books/quant/1/en/chapter/14-exchange-traded-funds
---

# Chapter 14 — Exchange-Traded Funds

A fund’s shares trade at $50.10 on the exchange while the five hundred stocks inside it are worth $50.00 a share. Within a second or two a trading firm sells the fund, buys the five hundred stocks, and tonight hands those stocks to the fund in exchange for newly printed fund shares that close its short. It has earned a few cents a share; the fund has grown; the price is back in line. Nobody at the fund decided anything. At the end of August 2026 some twenty-four trillion dollars were invested in vehicles kept honest by this one mechanism. This chapter explains the mechanism, what happens to it when the stocks inside are closed or cannot be priced, and the funds that promise three times the index and deliver something else.

## 14.1 A fund with two markets

**Definition 14.1 (Exchange-traded fund).**

An *exchange-traded fund* (ETF) is an open-ended investment fund whose shares are listed and trade on exchanges throughout the day like any stock, and can in addition be created or redeemed with the fund itself, in large blocks, by designated intermediaries.

**Definition 14.2 (Authorised participant, creation unit, creation basket).**

An *authorised participant* (AP) is a [broker-dealer](https://one-course.com/books/quant/1/en/chapter/2-the-sell-side#def-m1-the-sell-side-sell-side) that has contracted with an ETF for the right to create and redeem its shares. It does so in *creation units*, blocks of a fixed number of ETF shares, by delivering to the fund (or receiving from it) the *creation basket*: the list of securities and cash, published daily by the fund, that one creation unit is exchanged for.

**Definition 14.3 (In-kind transfer).**

A creation or redemption is *in kind* when the AP and the fund exchange ETF shares for the securities themselves, not for cash. The fund then never trades in the market: the cost of turning cash into a portfolio, and back, is borne by the investors who come and go, not by those who stay.

![The two markets of an ETF. Investors meet only the exchange. The number of ETF shares in existence changes only on the right-hand side, in creation units, at the day’s net asset value, against the basket.](https://one-course.com/images/onecourse/chapters/quant-1/m1-exchange-traded-funds/fig-77d2d47f8c96.svg)

***Figure 14.1.** The two markets of an ETF. Investors meet only the exchange. The number of ETF shares in existence changes only on the right-hand side, in [creation units](#def-m1-exchange-traded-funds-ap), at the day’s [net asset value](#def-m1-exchange-traded-funds-nav), against the basket.*

**As of September 2026 — Size, and the rule that standardised the structure.**

Assets in [exchange-traded funds](#def-m1-exchange-traded-funds-etf) worldwide reached $24.03 trillion at the end of August 2026, a record, by an industry consultancy’s count. In the United States, Rule 6c-11 under the Investment Company Act, adopted on 25 September 2019, lets a transparent ETF operate without an individual exemptive order, provided it publishes its portfolio daily, and allows *custom baskets* that differ from a pro-rata slice of the portfolio. Leveraged and inverse ETFs cannot rely on the rule.

## 14.2 The arbitrage that holds the price

**Definition 14.4 (Net asset value, indicative value, premium).**

The *net asset value* (NAV) of a fund share is the value of the fund’s holdings less its liabilities, divided by the number of shares, computed once a day at official closing prices. The *indicative NAV* is an estimate of the same quantity recomputed during the day from current prices of the basket. The *premium* of an ETF is $P/\mathrm{NAV} - 1$, with $P$ its market price; a negative premium is a *discount*.

**Proposition 14.5 (The arbitrage band).**

Let $c_B$ and $c_E$ be the costs of trading the basket and the ETF (half the spread plus impact, as fractions of value), $\phi$ the creation or redemption fee per unit of value, and $f$ the cost of financing both legs until the creation settles. An [authorised participant](#def-m1-exchange-traded-funds-ap) profits from creating when the [premium](#def-m1-exchange-traded-funds-nav) exceeds $b = c_B + c_E + \phi + f$ and from redeeming when the discount exceeds $b$. In a market with competing APs the [premium](#def-m1-exchange-traded-funds-nav) therefore stays within $[-b, +b]$.

**Proof.** At [premium](#def-m1-exchange-traded-funds-nav) $\pi > 0$ the AP sells ETF shares at $P(1 - c_E)$ and buys the basket at $\mathrm{NAV}(1 + c_B)$, pays the fee and the financing, and delivers the basket for ETF shares at NAV. Its profit per unit of value is $\pi - c_E - c_B - \phi - f$ to first order, positive iff $\pi > b$. The redemption trade is symmetric. ∎

**Example 14.6 (Seven basis points).**

For a large-capitalisation equity ETF take $c_B = 4$, $c_E = 1.5$, $\phi = 1$ and $f = 0.5$ basis points: $b = 7$ basis points, three and a half cents on a $50 share. For a fund of small or foreign stocks, $c_B$ alone can be thirty basis points; for a fund of corporate bonds, more. The band is not a defect of the ETF: it is the cost of trading the underlying, which the ETF investor avoids so long as the price stays inside it ([Figure 14.2](#fig-m1-exchange-traded-funds-premium)).

![A simulated premium. Order flow pushes it around; whenever it reaches the band of (dashed) an authorised participant creates or redeems and brings it halfway back. Inside the band nothing forces it to zero. Data: the tutorial’s simulation.](https://one-course.com/images/onecourse/chapters/quant-1/m1-exchange-traded-funds/fig-9e7fa3ccfbbb.svg)

***Figure 14.2.** A simulated [premium](#def-m1-exchange-traded-funds-nav). Order flow pushes it around; whenever it reaches the band of [Example 14.6](#ex-m1-exchange-traded-funds-band) (dashed) an [authorised participant](#def-m1-exchange-traded-funds-ap) creates or redeems and brings it halfway back. Inside the band nothing forces it to zero. Data: the tutorial’s simulation.*

Most ETF trading never reaches the primary market. A [market maker](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-market-maker) who sells ETF shares hedges with futures, with the basket or with a correlated ETF, and lets buyers and sellers net against each other; only the residual, often at the end of the day, becomes a creation. Shares outstanding change little on a day when turnover is many times the fund’s size. Market making in ETFs, the business of several of the firms in [Chapter 1](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#ch-m1-what-a-trading-firm-does), is treated in One Quant Book 11.

## 14.3 When the inside cannot be priced

The arbitrage argument needs a basket price. Two kinds of ETF do not have one. For a fund of Japanese stocks trading in New York, the basket’s market is closed all day. For a fund of corporate bonds, most of the holdings have not traded today at all: their “prices” are a vendor’s estimates.

**Proposition 14.7 (A stale NAV produces a spurious premium).**

Let the true value $V_t$ of the holdings move, and let the published NAV adjust only partly each day, $N_t = N_{t-1} + (1-\theta)(V_t - N_{t-1})$ with $0 < \theta < 1$. If the ETF trades at $V_t$, then after a sustained fall of $V$ the ETF shows a discount to NAV, which closes without any trade in the ETF being a bargain.

**Proof.** $V_t - N_t = \theta(V_t - N_{t-1})$. After a fall $V_t < N_{t-1}$, so $V_t <
N_t$: the ETF, at $V_t$, is below NAV. When $V$ stabilises the difference decays geometrically at rate $\theta$. ∎

**Example 14.8 (March 2020).**

During the sell-off of March 2020, investment-grade corporate bond ETFs closed about 365 basis points below their NAVs, on an asset-weighted basis, on 12 March and again on 19 March; the largest of them went from a discount of 2.8% on 20 March to a [premium](#def-m1-exchange-traded-funds-nav) of 2.9% on 23 March, the day a central-bank purchase programme was announced. A central-bank study of the episode read the discounts as the ETFs’ prices leading stale bond valuations, in a market where dealers had stopped making prices: the ETF was the price, and the NAV the estimate.

![A simulated ten-day sell-off in an illiquid asset class. The NAV lags; the ETF appears to trade at a discount that reaches 4% and then “closes” as the valuations catch up. Data: the tutorial’s simulation, with = 0.75.](https://one-course.com/images/onecourse/chapters/quant-1/m1-exchange-traded-funds/fig-2240b2fe26c0.svg)

***Figure 14.3.** A simulated ten-day sell-off in an illiquid asset class. The NAV lags; the ETF appears to trade at a discount that reaches 4% and then “closes” as the valuations catch up. Data: the tutorial’s simulation, with $\theta = 0.75$.*

The practical rule: a [premium](#def-m1-exchange-traded-funds-nav) is evidence of mispricing only to the extent that the basket can be traded at the prices used in the NAV. For domestic equities it can, and seven basis points is the whole story. For bonds and for closed markets the ETF is often the best available estimate of value, and the right comparison is with futures, with other ETFs and with the few bonds that did trade.

## 14.4 Leveraged and inverse funds

**Definition 14.9 (Leveraged ETF).**

A *leveraged ETF* with factor $\beta$ promises $\beta$ times the *daily* return of an index, before fees; an *inverse* ETF has $\beta < 0$. It obtains the exposure with swaps and futures and resets it at each close.

**Proposition 14.10 (The daily rebalance).**

A fund with net assets $N$ and exposure $\beta N$ experiences an index return $r$ during the day. To be $\beta$ times leveraged at the close it must increase its exposure by

$$
\Delta = \beta(\beta - 1)\,N\,r .
$$

Since $\beta(\beta-1) > 0$ for every $\beta$ outside $[0,1]$, leveraged *and* inverse funds all buy after a rise and sell after a fall.

**Proof.** After the move the assets are $N(1+\beta r)$ and the exposure $\beta N(1+r)$. The target exposure is $\beta N(1+\beta r)$. The difference is $\beta N(\beta
r - r)$. ∎

**Example 14.11 (Six percent of assets per percent).**

For $\beta = 3$, $\Delta = 6Nr$; for $\beta = -1$, $\Delta = 2Nr$; for $\beta
= -3$, $\Delta = 12Nr$. A $10 billion three-times fund must buy $3 billion of exposure at the close of a day on which its index rose 5%, whatever anyone at the fund thinks. The flow is public arithmetic, is concentrated in the final minutes, and has the sign of the day’s move: it amplifies closing moves, and anticipating it is one of the event strategies of One Quant Book 8 ([Figure 14.4](#fig-m1-exchange-traded-funds-rebalance)).

![Exposure a fund must buy at the close after a +1\% day, as a percentage of its net assets: ( -1). Inverse funds trade more than leveraged funds of the same magnitude, and in the same direction. Data: .](https://one-course.com/images/onecourse/chapters/quant-1/m1-exchange-traded-funds/fig-d541f6501de3.svg)

***Figure 14.4.** Exposure a fund must buy at the close after a $+1\%$ day, as a percentage of its net assets: $\beta(\beta-1)$. Inverse funds trade more than leveraged funds of the same magnitude, and in the same direction. Data: [Proposition 14.10](#prop-m1-exchange-traded-funds-rebalance).*

**Definition 14.12 (Volatility decay).**

*Volatility decay* is the shortfall of a leveraged fund’s multi-day return relative to $\beta$ times (or the $\beta$-th power of) the index’s return over the same period, caused by daily resetting in a volatile market.

**Proposition 14.13 (How much decay).**

If the index follows $\dd S/S = \mu\dd t + \sigma\dd W$ and the fund is continuously rebalanced to leverage $\beta$ with no financing cost, then

$$
\frac{L_T}{L_0} \;=\; \left(\frac{S_T}{S_0}\right)^{\beta}
\exp\!\Bigl(-\tfrac12\,\beta(\beta-1)\,\sigma^2 T\Bigr).
$$

**Proof.** $\dd L/L = \beta\,\dd S/S$, so by Itô’s formula $\dd\ln L = \beta\mu\dd t +
\beta\sigma\dd W - \tfrac12\beta^2\sigma^2\dd t$, while $\beta\,\dd\ln S =
\beta\mu\dd t + \beta\sigma\dd W - \tfrac12\beta\sigma^2\dd t$. Subtract and integrate. ∎

With $\sigma = 30\%$ and $T = 1$ year the factor is $\exp(-0.27) = 0.76$ for $\beta = 3$ and $\exp(-0.09) = 0.91$ for $\beta = -1$. The fund is not cheating: it delivered $\beta$ times each day’s return. It is the investor who holds it for a year, expecting $\beta$ times the year’s return, who has misread the contract.

![One simulated year with a daily volatility of 2% and daily returns that sum to zero. The index ends 5.8% lower; three times that would be -17.4\%; the three-times fund ends 41.8% lower. Data: the tutorial’s simulation.](https://one-course.com/images/onecourse/chapters/quant-1/m1-exchange-traded-funds/fig-7b07722b5c2f.svg)

***Figure 14.5.** One simulated year with a daily volatility of 2% and daily returns that sum to zero. The index ends 5.8% lower; three times that would be $-17.4\%$; the three-times fund ends 41.8% lower. Data: the tutorial’s simulation.*

## 14.5 Tutorial: bands, stale values and decay

**Goal.** Compute an arbitrage band, reproduce a spurious discount from a stale NAV, and measure [volatility decay](#def-m1-exchange-traded-funds-decay) against the formula. **End state:** the four data figures of this chapter.

1. **The band** is a sum of four costs. `class ArbCosts : basket_half_spread_bp: float # cost of trading the basket, one way etf_half_spread_bp: float # cost of trading the ETF, one way creation_fee_bp: float # fixed creation/redemption fee spread over one unit financing_bp: float # carrying both legs until settlement @property def band_bp (self ) -> float : """Premium (or discount) beyond which creating (redeeming) is profitable.""" return (self .basket_half_spread_bp + self .etf_half_spread_bp + self .creation_fee_bp + self .financing_bp)` **Listing 14.1.** The costs an authorised participant must cover before an arbitrage pays. code/markets-1/14-exchange-traded-funds/python/etf_demo.py
2. **Stale NAV.** One line of smoothing is enough to manufacture a 4% discount in a ten-day sell-off. `def stale_nav (true_value: np.ndarray, staleness: float ) -> np.ndarray: """NAV computed from quotes that adjust only partly each day: an exponential lag.""" nav = np.empty_like(true_value) nav[0 ] = true_value[0 ] for t in range (1 , len (true_value)): nav[t] = nav[t - 1 ] + (1.0 - staleness) * (true_value[t] - nav[t - 1 ]) return nav` **Listing 14.2.** A net asset value computed from quotes that adjust only partly each day. code/markets-1/14-exchange-traded-funds/python/etf_demo.py
3. **The rebalance and the decay.** `def rebalance_trade (beta: float , nav: float , index_return: float ) -> float : """Exposure the fund must add at the close to be `beta` times leveraged again: beta (beta - 1) N r.""" return beta * (beta - 1.0 ) * nav * index_return def leveraged_path (index_returns: np.ndarray, beta: float ) -> np.ndarray: return np.cumprod(1.0 + beta * index_returns) def decay_factor (beta: float , sigma_annual: float , years: float ) -> float : """exp(-beta (beta - 1) sigma^2 T / 2): leveraged fund versus (index ratio)^beta.""" return math.exp(-0.5 * beta * (beta - 1.0 ) * sigma_annual**2 * years)` **Listing 14.3.** The closing trade of a leveraged fund, its path, and the continuous-time decay factor. code/markets-1/14-exchange-traded-funds/python/etf_demo.py
4. **Check the formula.** The tests simulate 4 000 years at 30% volatility and compare the mean log ratio of the three-times fund to the cube of the index with $-\tfrac12\cdot6\cdot0.09 = -0.27$ .

**What to change next.** Add a financing spread and a 0.95% annual fee to the leveraged fund. Then make the index trend (positive autocorrelation of daily returns) and find the regime in which the three-times fund *beats* three times the index.

## 14.6 Build: the premium monitor

**Purpose.** The miniature firm makes markets in ETFs on its simulated exchange (One Quant Book 11). It needs a live fair value for each fund and an alarm when the market price leaves the arbitrage band.

**Interface.** `Basket(etf, shares_per_unit, components, cash)` with components as (symbol, quantity per [creation unit](#def-m1-exchange-traded-funds-ap)); `indicative_nav(basket, mids)`; `premium_bp(price, inav)`; `band_bp(costs)`; `signal(price, inav, costs)` returning `create`, `redeem` or `none`; and for each component a `stale` flag when its last quote is older than a threshold.

**Rules.** Integer prices; the indicative value uses mids from the consolidated quote of [Section 9.8](https://one-course.com/books/quant/1/en/chapter/9-us-equity-market-structure#bld-m1-us-equity-market-structure-nbbo). If components worth more than a configurable fraction of the basket are stale the monitor reports `unreliable` instead of a [premium](#def-m1-exchange-traded-funds-nav). Corporate actions change the basket overnight ([Section 8.7](https://one-course.com/books/quant/1/en/chapter/8-shares-corporate-actions-and-indices#bld-m1-shares-corporate-actions-indices-adjuster)).

**Acceptance tests.** `code/firm/etfmonitor/tests/`: a three-stock basket priced by hand; signals at and beyond the band; the stale-component rule.

**Stretch.** Replace components whose market is closed by a regression on instruments that are open (futures, currency, a related ETF): the fair value of an international fund.

Sources and further reading

- ETFGI, press releases on global ETF industry assets, July–September 2026 (as reported by Markets Media, “Global ETF Assets Top $24 Trillion for First Time”).
- US Securities and Exchange Commission, *SEC Adopts New Rule to Modernize Regulation of Exchange-Traded Funds* , press release 2019-190, September 2019.
- S. Aramonte and F. Avalos, “The recent distress in corporate bond markets: cues from ETFs”, *BIS Bulletin* no. 6, 14 April 2020; K. Todorov, “The anatomy of bond ETF arbitrage”, *BIS Quarterly Review* , March 2021.
- M. Cheng and A. Madhavan, “The dynamics of leveraged and inverse exchange-traded funds”, *Journal of Investment Management* 7 (2009).
- A. Madhavan, *Exchange-Traded Funds and the New Dynamics of Investing* , Oxford University Press, 2016.

## 14.7 Exercises

**Exercise 14.1 ★.**

An ETF’s NAV is $82.40 and it trades at $82.31. Give the [premium](#def-m1-exchange-traded-funds-nav) in basis points. With a band of 9 basis points, does an [authorised participant](#def-m1-exchange-traded-funds-ap) act, and how?

**Solution of Exercise 14.1.**

$82.31/82.40 - 1 = -10.9$ basis points: a discount, outside the band of 9. The [authorised participant](#def-m1-exchange-traded-funds-ap) buys ETF shares on the exchange, sells the basket (short if need be), and redeems the ETF shares with the fund for the basket, which closes its short. It earns about 1.9 basis points after costs.

**Exercise 14.2 ★.**

A [creation unit](#def-m1-exchange-traded-funds-ap) is 50 000 ETF shares at a NAV of $40, and the fund charges $500 per creation. Express the fee in basis points of the unit’s value. What is it if the AP creates ten units in one order for the same fee?

**Solution of Exercise 14.2.**

The unit is worth $50\,000 \times 40 = \$2$ million; $500/2\,000\,000 = 2.5$ basis points. On ten units it is 0.25 basis point: the fee is fixed per order, so the band is narrower for those who create in size, and small funds with small creations have wider bands.

**Exercise 14.3 ★.**

A $-2\times$ fund has net assets of $800 million and its index falls 4% today. Give the fund’s return, its new net assets, and the trade it must make at the close, with its direction.

**Solution of Exercise 14.3.**

The fund returns $-2 \times -4\% = +8\%$; net assets become $864 million. $\Delta = \beta(\beta-1)Nr = 6 \times 800 \times (-0.04) = -\$192$ million: the exposure must fall by 192 million. For an inverse fund “less exposure” means a larger short: it sells $192 million of the index, after a fall. Check: the short was 1 600, it shrank with the index to 1 536, and the target is $2
\times 864 = 1\,728$.

**Exercise 14.4 ★★.**

An index rises 10% on day one and falls 9.09% on day two, ending where it started. Compute the two-day return of a $2\times$ fund, a $3\times$ fund and a $-1\times$ fund. Which loses most, and why?

**Solution of Exercise 14.4.**

$2\times$: $1.20 \times 0.8182 - 1 = -1.82\%$. $3\times$: $1.30 \times 0.7273
- 1 = -5.45\%$. $-1\times$: $0.90 \times 1.0909 - 1 = -1.82\%$. The three-times fund loses most: the loss grows like $\beta(\beta-1)$, which is 6 for $\beta = 3$ and 2 for both $\beta = 2$ and $\beta = -1$.

**Exercise 14.5 ★★.**

For $\sigma = 40\%$ a year compute the decay factor of [Proposition 14.13](#prop-m1-exchange-traded-funds-decay) over one year for $\beta = 2$, $3$ and $-2$. Over what horizon does a $3\times$ fund lose 10% to decay at that volatility?

**Solution of Exercise 14.5.**

$\tfrac12\beta(\beta-1)\sigma^2 = 0.16$, $0.48$ and $0.48$: factors $\mathrm{e}^{-0.16} = 0.852$ for $\beta = 2$ and $\mathrm{e}^{-0.48} = 0.619$ for both $\beta = 3$ and $\beta = -2$. A 10% loss: $0.48\,T = -\ln 0.9 =
0.105$, so $T = 0.22$ year, about 55 trading days.

**Exercise 14.6 ★★.**

A bond ETF’s NAV follows [Proposition 14.7](#prop-m1-exchange-traded-funds-stale) with $\theta = 0.8$. The true value falls from 100 to 94 in one day and stays there. Give the NAV and the apparent discount on that day and on the next three days. After how many days is the discount below 0.5%?

**Solution of Exercise 14.6.**

The gap between value and NAV is $6 \times 0.8^{k+1}$ on the $k$-th day after the fall. NAV: 98.80, 97.84, 97.07, 96.46. Discount: 4.86%, 3.92%, 3.16%, 2.55%. It passes below 0.5% on the eleventh day after the day of the fall. A patient observer sees a discount of several percent “closing” over two weeks while the ETF’s price does not move at all.

**Exercise 14.7 ★★★.**

*Coding.* With `simulate_premium` and the costs of [Example 14.6](#ex-m1-exchange-traded-funds-band), run 5 000 steps with seed 1 and report the number of steps at which an [authorised participant](#def-m1-exchange-traded-funds-ap) acted and the mean absolute [premium](#def-m1-exchange-traded-funds-nav). Halve the basket cost $c_B$ and report the same. What does an investor in the ETF gain?

**Solution of Exercise 14.7.**

With a band of 7 basis points an [authorised participant](#def-m1-exchange-traded-funds-ap) acts at 1 638 of the 5 000 steps and the mean absolute [premium](#def-m1-exchange-traded-funds-nav) is 3.28 basis points. With $c_B =
2$ the band is 5: 2 219 actions and a mean absolute [premium](#def-m1-exchange-traded-funds-nav) of 2.42. The investor who buys or sells at a random moment pays, on average, a smaller deviation from fair value; cheaper underlying markets make tighter ETFs, and more primary-market activity is the sign of a tighter product, not of a stressed one.

**Exercise 14.8 ★★★.**

*Find the flaw.* A newsletter writes: “This high-yield bond ETF closed at a 3% discount to NAV in a falling market. Buy it: the discount always closes within two weeks, a riskless 3%.” Explain why the discount closing does not imply a profit, and describe the position that would be needed to capture it if it were real.

**Solution of Exercise 14.8.**

The “discount” compares a live price with a NAV built from stale bond valuations ([Proposition 14.7](#prop-m1-exchange-traded-funds-stale)). It can close entirely by the NAV falling to the ETF price, in which case the buyer of the ETF earns nothing, or loses if the market continues down. To capture a real discount one must buy the ETF *and sell the basket* at the prices used in the NAV, then redeem; in high-yield bonds in a falling market nobody will buy the basket at those prices, which is exactly why the discount is there. “Always closes” is true and irrelevant; “riskless” is false.

## 14.8 Problem: The Daily Rebalance of a Three-Times Fund

**Problem 14.1.**

Weekend problem — the last ten minutes of a leveraged fund

A $3\times$ fund and a $-3\times$ fund track the same equity index. At last night’s close the first had net assets of $8 billion and the second $3 billion. The index futures trade $300 billion a day, of which a tenth in the last ten minutes.

**Part I — A quiet day.**

1. The index rises 1%. Give each fund’s return and new net assets.
2. Give each fund’s closing trade with its direction, and the total.
3. Express the total as a fraction of the last ten minutes’ volume.
4. With the square-root rule of [Method 2.6](https://one-course.com/books/quant/1/en/chapter/2-the-sell-side#met-m1-the-sell-side-sqrt) , a daily volatility of 1% and $\kappa = 0.7$ , estimate the impact of that trade, taking $V$ to be the ten-minute volume and scaling $\sigma$ to ten minutes of a 390-minute day.

**Part II — A violent day.**

5. The index falls 5%. Give each fund’s return and new net assets.
6. Give the closing trades and the total.
7. As a fraction of the last ten minutes’ volume?
8. Estimate the impact as in question 4 with a daily volatility of 3%.
9. The impact feeds back: if the funds’ selling moves the index a further $x$ , they must sell a little more. Using your answer to question 8 as $x$ , by how much does the required trade grow?

**Part III — A bad week.** The index returns $-5\%$, $+4\%$, $-6\%$, $+5\%$, $+2.5\%$ over five days.

10. Give the index’s cumulative return.
11. Give the $3\times$ fund’s cumulative return, and three times the index’s.
12. Give the $-3\times$ fund’s cumulative return, and minus three times the index’s.
13. An investor held both funds in equal dollar amounts “to be hedged”. Give her return.
14. Explain her result with [Proposition 14.13](#prop-m1-exchange-traded-funds-decay) .

**Part IV — Judgement.**

15. Who is on the other side of the closing trades of Part II, and what do they need to know in advance?
16. Why does the $-3\times$ fund trade twice as much per dollar of assets as the $3\times$ fund?
17. What daily index move would wipe out the $3\times$ fund? How do such funds’ prospectuses deal with it?
18. A regulator asks whether these funds destabilise the close. Give the quantity that answers the question and its value on the violent day.
19. State the *named result* : the notional the two funds together must trade at the close after the 5% fall, in billions of dollars.
20. In one sentence: for whom is a [leveraged ETF](#def-m1-exchange-traded-funds-leveraged) a sensible instrument?

**Solution of Problem 14.1.**

**1.** $+3\%$ and $-3\%$: $8.24 billion and $2.91 billion. **2.** $6 \times 8 \times 0.01 = \$0.48$ billion and $12 \times 3 \times
0.01 = \$0.36$ billion, both to buy: $0.84 billion. **3.** The last ten minutes trade $30 billion: 2.8%. **4.** $\sigma_{10} = 1\% \times \sqrt{10/390} = 0.16\%$; impact $0.7
\times 0.16\% \times \sqrt{0.028} = 1.9$ basis points. Invisible. **5.** $-15\%$ and $+15\%$: $6.8 billion and $3.45 billion. **6.** $-\$2.4$ billion and $-\$1.8$ billion: $4.2 billion to sell. **7.** 14%. **8.** $\sigma_{10} = 0.48\%$; $0.7 \times 0.48\% \times \sqrt{0.14} =
12.6$ basis points. **9.** The trade is $(6 \times 8 + 12 \times 3) = 84$ billion per unit of return; a further $0.126\%$ adds $0.11 billion, 2.5% more. The feedback exists and converges at once at this size; it would not if the funds were ten times larger relative to the close. **10.** $0.95 \times 1.04 \times 0.94 \times 1.05 \times 1.025 - 1 =
-0.05\%$: flat. **11.** $0.85 \times 1.12 \times 0.82 \times 1.15 \times 1.075 - 1 =
-3.49\%$, against $-0.14\%$. **12.** $1.15 \times 0.88 \times 1.18 \times 0.85 \times 0.925 - 1 =
-6.11\%$, against $+0.14\%$. **13.** $-4.80\%$ in a week in which the index did nothing. **14.** Both funds decay at the rate $\tfrac12\beta(\beta-1)\sigma^2$: 3 and 6 times the index’s variance. The index exposures cancel; the decays add. The pair is a short position in realised variance, paid for in five days of 4–6% moves. (The mirror trade, short both funds, is long nothing and short decay: it earns the variance but is exposed to a trend, in which one leg compounds without limit, and to the cost of borrowing the shares.) **15.** [Market makers](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-market-maker) and proprietary firms, who know the funds’ assets (published daily) and the day’s return (public), hence the size and sign of the trade, well before the close. They buy or sell ahead of it and provide the other side in the closing auction ([Chapter 13](https://one-course.com/books/quant/1/en/chapter/13-auctions#ch-m1-auctions)); their competition is what keeps the impact near the estimate of question 8. **16.** $\beta(\beta-1)$ is 12 against 6. After a rise the inverse fund’s assets fall while its short grows in value: both effects push its leverage up, where for the long fund they partly offset. **17.** $-33.3\%$ in one day. US market-wide circuit breakers halt trading for the day when the S&P 500 has fallen 20%, which caps the one-day loss of a fund on a broad US index below that level; funds on single stocks, sectors or volatility have no such protection, and their prospectuses warn that an investor can lose the entire investment in a single day. **18.** The rebalancing trade as a fraction of closing volume: 14% here, 2.8% on the quiet day. It scales with assets times $\beta(\beta-1)$ times the move, so it is largest exactly on the days when the close is already strained. **19.** **$4.2 billion to sell.** **20.** For someone who wants leverage for a day or a few days without a margin account, and knows that beyond that horizon the product is a position in the path, not only in the destination.

## 14.9 Interview questions

**Interview question 14.1 ★ trader, researcher, developer.**

How does an ETF’s price stay close to the value of what it holds?

**Solution of Interview question 14.1.**

[Authorised participants](#def-m1-exchange-traded-funds-ap) can exchange the ETF’s shares for the underlying basket with the fund, in either direction, at NAV. If the ETF is dear they sell it, buy the basket and create; if cheap, the reverse. The price stays inside a band set by the cost of trading the basket, the ETF, the fee and financing. Most of the time [market makers](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-market-maker) hedging with futures or the basket do the work and creations are only the net residual.

*What the interviewer is looking for: both directions, the band rather than “equal to NAV”, and the fact that the fund itself does not trade.*

**Interview question 14.2 ★ trader, researcher.**

A $2\times$ [leveraged ETF](#def-m1-exchange-traded-funds-leveraged)’s index is up 10% over the year. Is the ETF up 20%?

**Solution of Interview question 14.2.**

Almost never exactly. The fund delivers twice each *daily* return; over a year the result is the square of the index ratio times $\exp(-\sigma^2 T)$. With 20% volatility: $1.21 \times 0.961 = 1.163$, up about 16% before fees and financing; in a smooth trending year it can exceed 20%.

*What the interviewer is looking for: the word “daily”, a number, and the remark that trends help.*

**Interview question 14.3 ★★ trader, researcher.**

An ETF holding Japanese stocks trades in New York at a 1.5% [premium](#def-m1-exchange-traded-funds-nav) to its NAV at 11:00 New York time. Is that an arbitrage?

**Solution of Interview question 14.3.**

No. The NAV uses Tokyo’s close, ten hours old. Since then futures on the Japanese index, the yen and US markets have moved; the ETF price is the market’s estimate of where Tokyo will open. Compare the ETF with a fair value built from instruments that are open now. An arbitrage would require trading the basket at the stale prices, which is impossible.

*What the interviewer is looking for: refusal of the word “arbitrage”, and a concrete fair-value construction.*

**Interview question 14.4 ★★ trader, researcher.**

You make markets in an ETF of 500 stocks. A client buys $50 million from you. Walk me through your hedge and your end of day.

**Solution of Interview question 14.4.**

I am short $50 million of the ETF. Immediately buy index futures for the same delta: one trade, cheap, leaves basis risk between future and basket. During the day other clients sell to me and reduce the position. Near the close, for what remains, I either buy the basket (a program trade, often in the closing auction so that my cost matches the NAV) and create ETF shares to cover the short, selling the futures at the same time, or carry the short ETF against futures overnight if borrowing the ETF is cheap and creation is not worth the fee. My edge is the spread I charged minus hedge costs, fee and financing.

*What the interviewer is looking for: hedge first, create last; awareness of the NAV strike at the close and of inventory netting.*

**Interview question 14.5 ★★ researcher, mle.**

Derive the amount a leveraged fund must trade at the close as a function of its leverage and the day’s return. What is surprising about inverse funds?

**Solution of Interview question 14.5.**

Assets $N$, exposure $\beta N$. After return $r$: assets $N(1+\beta r)$, exposure $\beta N(1+r)$, target $\beta N(1+\beta r)$; trade $\beta(\beta-1)Nr$. The coefficient is positive for negative $\beta$ too: inverse funds buy after rises and sell after falls, like leveraged ones, and a $-1\times$ fund trades as much as a $2\times$ fund.

*What the interviewer is looking for: a clean three-line derivation and the sign result for inverse funds.*

**Interview question 14.6 ★★★ researcher, trader.**

In a crisis a bond ETF trades 4% below NAV. Build the case that the ETF is mispriced, the case that the NAV is, and say what data would decide.

**Solution of Interview question 14.6.**

ETF mispriced: [authorised participants](#def-m1-exchange-traded-funds-ap)’ balance sheets are constrained, so the band has widened; forced sellers use the ETF because it is the only liquid thing. NAV mispriced: bonds have not traded, valuations lag, and the ETF leads. Evidence: prices of the bonds that did trade against their valuations; the behaviour of NAV over the following days (does it fall to the ETF?); credit-default-swap indices and futures as independent fair values; redemption activity (if APs redeem heavily the discount is real to them); and whether redemption baskets are being negotiated at prices below the valuation.

*What the interviewer is looking for: both cases argued, and data that discriminates, especially the subsequent convergence of NAV to the ETF.*