Unit 2 · Markets & Instruments — Session 02 of 07

Equities, Indices & ETFs

A share is a claim on whatever a company has left over; an index is a portfolio with a published rule; an ETF is a basket held in line by an arbitrage trade; a short sale is a loan you repay in shares. Each has plumbing that quietly moves your returns — dividends missing from price charts, splits that fake crashes, discounts that open in a panic, borrow fees that turn a right call into a loss. The core move: measure what you actually own and what it costs to hold — total returns on adjusted prices, net of every fee — never the quoted price alone.

DURATION 1.5 h + lab LEVEL Foundations TRACKS QR · QT · QD · IND HANDS-ON Build three indices; stress an ETF; short a stock on 20,000 paths
📓 Companion lab Open in Colab Notebook on GitHub Source (.py)
On this page
  1. What a share is: the residual claim
  2. Dividends, splits & corporate actions
  3. Price return vs total return
  4. An index is a weighting rule
  5. ETFs: a basket with an arbitrage loop
  6. Short selling: selling what you borrowed
  7. Short P&L, squeezes & GameStop
  8. Where people go wrong
  9. Try it
  10. Glossary · Check yourself · Go deeper

2.2.1 What a share is: the residual claim

Before prices or indices, ask what a share entitles you to. The answer — whatever is left over — explains why equity is the riskiest claim on a company, and why it should pay the most.

A company funds its assets with debt and equity. Lenders are promised fixed payments and are paid first. A share of common stock is a slice of ownership: a vote, dividends when the board pays them, and — the defining feature — a claim on whatever is left once every fixed claim has been paid. Liability is limited: you can lose what you paid, and no more. Picture the cash flowing down a queue of claims in order of seniority (fig 2.2.1): in a good year the fixed claims take their fixed amounts and equity keeps the rest; in a bad year equity is the first to run dry.

WHO GETS PAID FIRST — a company's value flows down its claims in order of seniority THE FIRM its assets and all its future cash flows 1 · SECURED DEBT paid first · backed by specific assets 2 · BONDS & LOANS fixed interest, then the principal 3 · PREFERRED STOCK a fixed dividend, ahead of common stock 4 · COMMON STOCK the residual claim: whatever is left votes · dividends · limited liability GOOD YEAR claims 1–3 get their fixed amounts; common stock keeps all of the upside BAD YEAR common stock absorbs the losses first — it can fall to zero, but never below Equity is the residual claim: all of the upside, and first in line for the losses.
fig 2.2.1 — the capital structure as a payment queue: fixed claims are paid first, and common stock owns whatever is left over

That is why stocks should, and on average did, pay more than bonds — the equity premium of Session 1.2 — and why equity looks like an option on the firm's assets with the debt as the strike, the Merton model of Session 11.4. A company's market capitalisation is its price times its shares outstanding. The count changes as companies issue shares, buy them back and split them, and not every share trades: the free float is the fraction that does, excluding blocks held by founders, families or governments — the part an index fund can actually buy (§2.2.4).

📌 Key idea — a price is the equity divided by a share count

Anything that changes the share count without changing the firm — a split, a stock dividend — changes the price but not your wealth, and anything that moves cash from the firm to you — a dividend — lowers the price by about the cash you received. Your data has to know the difference.

2.2.2 Dividends, splits & corporate actions

Companies pay cash, split shares, merge and spin off pieces. Each moves the quoted price, few change your wealth, and every one is a data bug waiting for a careless pipeline.

A dividend is cash paid per share, set by the board. Four dates matter (fig 2.2.2): the declaration; the ex-dividend date, the first day the shares trade without the dividend; the record date, when the company reads its register; and the payment. The ex-date follows the settlement cycle of Session 2.1: in US markets, since T+1 settlement began on 28 May 2024, it is normally the record date itself (FINRA Regulatory Notice 24-04); under T+2 — still the EU's cycle until its planned move to T+1 on 11 October 2027 (ESMA) — it falls one business day before.

THE FOUR DATES OF A DIVIDEND — and what corporate actions do to a price series 1 · DECLARATION board sets the amount and the dates 2 · EX-DIVIDEND DATE buyers from today don't get it: price drops ≈ D 3 · RECORD DATE company reads its shareholder register 4 · PAYMENT DATE cash reaches holders, often weeks later In the US, since T+1 settlement (28 May 2024), the ex-date is normally the record date itself. WHAT THE EXCHANGE QUOTES raw closing prices time → ex-date: drop ≈ dividend a split: a fake crash ADJUSTED FOR BOTH past prices rescaled, so every return is true time → earlier prices ÷ N (the split) and × (1 − D/P) (the dividend); the latest prices are unchanged: no cliff, no step The raw series records what traded; only the adjusted series records what an owner earned.
fig 2.2.2 — a dividend's four dates (top) and what a dividend and a split do to a raw price series versus an adjusted one (bottom; a conceptual sketch, not data)

On the ex-date the price should open lower by about the dividend: cash has moved from the company to you. In Elton and Gruber's (1970) data the drop was noticeably smaller than the dividend, a gap they tied to taxes; others blame trading frictions. You cannot see it in one stock on one day anyway: in the lab the standard deviation of a day's move is 2.6 times the average dividend, and across 2,000 ex-dates a regression measures the drop at 0.99 ± 0.06 dividends (the truth is 1).

In a stock split each share becomes several — 4-for-1 means four shares at a quarter of the price — and nothing economic happens; a reverse split does the opposite. One mechanical consequence (§2.2.4): a split changes a stock's weight in a price-weighted index. When Apple split 4-for-1 at the end of August 2020, S&P Dow Jones Indices replaced three members of the Dow Jones Industrial Average the same day, in part to offset the fall in its technology weight (S&P DJI, 24 August 2020).

A corporate action is any company-initiated event that changes its shares or pays its holders. Session 4.3 builds the pipeline, with an adjustment factor for each event; this is the minimum to recognise:

ActionQuoted priceWhat your data must do
Cash or special dividenddrops by about the payout on the ex-dateadd it back: a total return (§2.2.3)
Split N-for-1 (reverse: 1-for-N)divides (multiplies) by Nrescale earlier prices by the same factor
Spin-offdrops by about the value of the new company's sharesadjust by that value; start a history for the new company
Merger or acquisitionmoves toward the offer; the ticker may endrecord the payout; map the old identifier to the new one
Rights issuedrops on the ex-rights dateadjust by the value of the right
Buyback or share issuanceno mechanical changeupdate shares outstanding: index weights move
Ticker or name changenonekey every series on a permanent identifier, never the ticker
💡 Looking ahead

A missed split is a fake crash, a missed spin-off a fake loss, and a merger can drop a company from your history altogether. Session 4.2 handles the companies that vanish (delisting returns, survivorship bias); Session 4.3 — Corporate Actions, Security Master & Continuous Futures builds adjusted prices and a security master.

2.2.3 Price return vs total return

A price chart leaves out the dividends. Over a year that is a rounding error; over 20 years at a 2% yield it is 33% of your wealth.

With DtD_t a dividend that goes ex during period tt, the price return ignores it and the total return counts it, as if reinvested on the ex-date:

Rtprice=PtPt−1−1,Rt=Pt+DtPt−1−1=(1+Rtprice)(1+DtPt)−1R^{\text{price}}_t = \frac{P_t}{P_{t-1}} - 1, \qquad R_t = \frac{P_t + D_t}{P_{t-1}} - 1 = \bigl(1 + R^{\text{price}}_t\bigr)\Bigl(1 + \frac{D_t}{P_t}\Bigr) - 1

The total return is the simple return of Session 2.3, which builds compounding and leverage on it. Compounded, the two returns differ by the product of the ex-date yields; when dividends are a fixed fraction of the price, as in the lab, that product is known in advance:

∏t(1+Rt)∏t(1+Rtprice)=∏t(1+DtPt)\frac{\prod_t (1 + R_t)}{\prod_t \bigl(1 + R^{\text{price}}_t\bigr)} = \prod_t \Bigl(1 + \frac{D_t}{P_t}\Bigr)

The lab simulates 20 years of an index-like asset: expected total return 8%, volatility 18%, dividend yield qq of 2% paid quarterly. On this (below-median) path the price compounded at 2.98% a year and the total return at 5.07%: 2.1 percentage points a year, or $2.69 against $1.80 per dollar. Over YY years of quarterly dividends the ratio is (1−q/4)−4Y=(1 - q/4)^{-4Y} = 1.49: a price-only view misses 33% of the owner's wealth, whatever the market does (fig 2.2.3).

GROWTH OF $1 OVER 20 SIMULATED YEARS: TOTAL VS PRICE RETURN (LOG SCALE) total return (dividends reinvested) price return $1.00 $2.00 0 5 10 15 20 total return (dividends reinvested) total return (dividends reinvested): $2.69 price return price return: $1.80 total: $2.69 price: $1.80
fig 2.2.3 — 20 simulated years with a 2% dividend yield: the two lines share every wiggle, but the gap between them compounds steadily — the level is luck, the gap is not

The split, mishandled and handled

Now the classic bug. In the lab's four-stock market (§2.2.4) the stock Pricey splits 4-for-1 after 5 years. Returns computed from quoted prices show a −75.0% day (the true return was −0.2%), and every statistic downstream inherits it:

raw_r = raw[1:, i_pricey] / raw[:-1, i_pricey] - 1         # the bug: returns from quoted prices
adj_r = R[:, i_pricey]                                     # handled: split-adjusted returns
naive_split_day = raw_r[SPLIT_DAY - 1]
true_split_day = adj_r[SPLIT_DAY - 1]
assert np.isclose(1 + naive_split_day, (1 + true_split_day) / SPLIT_RATIO)
Pricey, 10 simulated yearsFrom quoted prices (the bug)Split-adjusted (correct)
Compound annual growth (CAGR)−4.74%9.42%
Maximum drawdown−88.3%−53.1%
Annualised volatility36.5%27.8%
📌 Key idea — returns are total returns on adjusted prices

Every return in this course is a total return on prices adjusted for corporate actions, unless it says otherwise. A price series records what traded; only an adjusted, dividend-inclusive series records what an owner earned. Headline index levels are usually price indices; providers commonly publish a total-return version alongside, and that is the one to compare a strategy with — it is also what a collateralised index future earns (Session 2.4).

2.2.4 An index is a weighting rule

"The market rose 1% today" means some index rose 1% — and an index is a portfolio with a published rule. Change the rule and you change the portfolio, its risk and its return.

An index tracks a basket chosen and weighted by a published rule. Every common rule fits one formula: quantities sis_i times prices, over a divisor δt\delta_t that the provider adjusts so that non-economic events leave the level unchanged:

It=1δt∑isiPi,t,wi,t=siPi,t∑jsjPj,t,RtI=∑iwi,t−1Ri,tI_t = \frac{1}{\delta_t} \sum_i s_i P_{i,t}, \qquad w_{i,t} = \frac{s_i P_{i,t}}{\sum_j s_j P_{j,t}}, \qquad R^{I}_t = \sum_i w_{i,t-1} R_{i,t}

The index return is the start-of-period-weighted average of its members' returns (Session 2.3). Only sis_i differs:

  • Market-cap weighting: sis_i is the share count, so each weight is the company's share of market value. The S&P 500 counts only the free float.
  • Equal weighting: the same value in every stock, reset at each rebalance — the S&P 500 Equal Weight Index gives each company 0.2% at every quarterly rebalance (S&P DJI).
  • Price weighting: si=1s_i = 1, so the weight is the share price. In the Dow Jones Industrial Average (30 stocks) a $1 move in any member moves the index the same number of points, whatever the company's size. Fig 2.2.4 puts the rules side by side.
SAME FOUR STOCKS, FOUR RULES — who gets the money (illustrative proportions) MARKET-CAP ∝ price × shares A B C D FLOAT-ADJUSTED CAP ∝ price × tradable shares A B C D EQUAL the same share each, reset A B C D PRICE ∝ share price (the Dow) A B C D A · a huge company with a low share price B · a large company with a high share price C · mid-sized, mostly family-owned (low float) D · a small company with a high share price Same companies, four different portfolios: the weighting rule is itself an investment decision.
fig 2.2.4 — one universe, four weighting rules (bar widths are illustrative, not the lab's): float adjustment shrinks the family-owned stock, and price weighting hands the most money to whichever share happens to be dearest

The lab's toy market has four companies with the same expected return, 8% a year, so any difference between the indices is either the rule or luck:

StockPriceShares (m)Free floatMarket cap ($bn)Float-cap weightEqual weightPrice weight
Mega$502,00095%10066.2%25%8.3%
Pricey$40010090%4025.1%25%66.7%
Family$3080040%246.7%25%5.0%
Small$12025100%32.1%25%20.0%

Price weighting hands 66.7% of the index to Pricey, which is 24.0% of the market's value, and 20.0% to Small, worth 1.8%. Size is not in a share price, so a pure relabelling moves the index: at Pricey's split its price weight falls from 73.1% to 40.4% overnight, while its cap weight barely moves (24.2% to 24.3%). Forget to reset the divisor and the index "falls" −55.0% that day, turning a gain over the period into a CAGR of −3.5%.

SAME FOUR STOCKS, THREE INDEX RULES: GROWTH OF $1 OVER 10 SIMULATED YEARS cap-weighted (float) equal-weighted price-weighted $0.50 $0.75 $1.00 $1.25 $1.50 $1.75 $2.00 0 2 4 6 8 10 cap-weighted (float) cap-weighted (float): $1.53 equal-weighted equal-weighted: $1.56 price-weighted price-weighted: $1.56 equal: $1.56 price: $1.56 cap: $1.53
fig 2.2.5 — one simulated decade of the same four stocks under three rules: here they finish close together, and which one wins is mostly luck

On this path (fig 2.2.5) the float-cap index compounded at 4.38% a year, equal weighting at 4.53% and price weighting at 4.53%. Across 2,000 simulated decades the float-cap and equal-weight indices have the same expected wealth, eμY=e^{\mu Y} = 2.23 per dollar for Y=Y = 10 years (simulated: 2.25 ± 0.04 and 2.22 ± 0.04, ± one standard error), but not the same growth. The volatility drag of Session 2.3 applies to portfolios, which compound at about μ−12w⊤Σw\mu - \tfrac12 w^\top \Sigma w, with Σ\Sigma the covariance matrix. Equal weighting puts a quarter of the money in the wildest stock (portfolio volatility 23.2% against 20.9%), so its typical decade compounds at 5.40% (theory 5.46%) against 6.00% — yet it still wins 46% of decades. Both beat their average stock's 2.6%: diversification cuts the drag.

Rebalancing and reconstitution

A cap-weighted index never needs to trade: yesterday's weights drift into today's exactly as a buy-and-hold portfolio's do, and the lab asserts it:

drifted = w_prev * (1 + R)
drifted /= drifted.sum(axis=1, keepdims=True)
assert np.allclose(drifted, float_mv[1:] / float_mv[1:].sum(axis=1, keepdims=True), atol=1e-12)

So an index fund trades only when the index changes. Equal weighting sells winners and buys losers every quarter — 17% one-way turnover a year here — which costs money (Session 6.5) and tilts toward small companies (Session 7.3). Providers also add and delete members (reconstitution) and update share counts (rebalancing); tracking funds then trade, typically at the close of the effective day, and the price pressure is the index-rebalance trade of Session 8.6. And today's members are survivors: backtest them over the past and you inherit survivorship bias (Session 4.2).

📌 Key idea — an index is a portfolio with a rule

Cap weighting is the one rule that holds itself — no trading, no opinion, the average investor's portfolio — which is why it is the natural benchmark and why index funds can run it so cheaply (Session 1.2). Every other rule is an investment decision with turnover and tilts attached.

2.2.5 ETFs: a basket with an arbitrage loop

An ETF is a fund that trades like a share. A trade, not a rule, keeps its price near the value of its holdings — and trades stop when they get too expensive.

An ETF (exchange-traded fund) holds a basket, usually tracking an index, and its shares trade on an exchange all day. A mutual fund deals with investors once a day at its net asset value (NAV), the value of its holdings per share; an ETF publishes a NAV too, but you pay the market price. Creation and redemption ties the two together. Authorised participants (APs), large dealers under contract with the fund, can deliver the basket for new ETF shares in big blocks called creation units, or hand shares back for the basket — usually in kind (fig 2.2.6). Everyone else simply trades existing shares (Lettau & Madhavan 2018).

CREATION AND REDEMPTION — the two trades that pull an ETF's price back to its NAV PREMIUM ETF price above NAV + AP cost 1 · BUY BASKET the index's stocks, in the market 2 · DELIVER the basket to the issuer, in kind 3 · CREATE receive a creation unit of new shares 4 · SELL the new ETF shares on the exchange more ETF shares for sale → the price falls back toward NAV INSIDE THE BAND (|premium| ≤ AP cost): no trade — investors simply swap existing shares on the exchange DISCOUNT ETF price below NAV − AP cost 1 · BUY ETF shares cheaply on the exchange 2 · REDEEM hand a creation unit back to the issuer 3 · RECEIVE the basket, in kind 4 · SELL BASKET the stocks, in the market fewer ETF shares for sale → the price rises back toward NAV Each trade closes the gap that pays for it — as long as the APs can afford to trade.
fig 2.2.6 — the creation/redemption loop: a premium beyond the AP's cost triggers a creation, a discount a redemption, and each trade pushes the ETF's price back toward its NAV until the gap no longer pays

Write the premium as πt=PtETF/NAVt−1\pi_t = P^{\text{ETF}}_t / \text{NAV}_t - 1 (negative: a discount) and an AP's all-in round-trip cost as cAPc_{\text{AP}}. Any |πt|>cAP|\pi_t| \gt c_{\text{AP}} pays an AP to create or redeem, and the trade shrinks the gap, so in normal markets |πt|≤cAP|\pi_t| \le c_{\text{AP}}. Because

1+R0→TETF=(1+R0→TNAV)1+πT1+π0,1 + R^{\text{ETF}}_{0 \to T} = \bigl(1 + R^{\text{NAV}}_{0 \to T}\bigr)\,\frac{1 + \pi_T}{1 + \pi_0},

an ETF earns its NAV's return to within about 2cAP2c_{\text{AP}} over any holding period. In the lab's toy (illustrative parameters) order flow moves the premium by 10 bp shocks that fade slowly: each day's premium is 0.98 times yesterday's plus a fresh shock. With no APs its standard deviation is 50.3 bp in theory and 50.5 bp over 2,000 simulated years, and it sits more than 50 bp from NAV, in either direction, on 31.9% of days (theory 32.0%) — roughly a closed-end fund, whose share count is fixed. APs with a 15 bp cost cut the average absolute premium from 40 to 10 bp and keep every simulated year's ETF return within 30 bp of its NAV's.

When the loop breaks: March 2020

The band holds only while cAPc_{\text{AP}} is small. In the lab's 20-day stress episode sellers hit the ETF every day while the cost of arbitrage jumps to 400 bp — a redeeming AP must sell the basket into a market with no buyers. The cost of arbitrage now caps the discount only at that level; inside the cap, the selling sets it, and it reaches 3.4% without the band ever binding (fig 2.2.7).

ETF PRICE VS NAV OVER ONE SIMULATED YEAR, WITH A 20-DAY STRESS EPISODE premium (+) / discount (−) AP band, lower edge −4.0% −3.0% −2.0% −1.0% 0.0% 1.0% 0 0.2 0.4 0.6 0.8 1 premium (+) / discount (−) AP band, lower edge
fig 2.2.7 — one simulated year of an ETF's premium: APs pin it inside a narrow band until the band itself blows out in a 20-day stress episode (grey: the lower edge of the band), then snap it back

That is a cartoon of March 2020. Comparing Vanguard bond ETFs with mutual-fund shares of the same portfolios, Haddad, Moreira and Muir found discounts of up to 10%, with an average peak near 5%, in mid-March — in investment-grade corporate, municipal and Treasury funds, while high-yield funds showed little and an S&P 500 ETF no substantial gap (their Figure 7; NBER w27168, version of May 2020). APs redeemed about $30 billion, yet the gaps faded only after the Federal Reserve announced on 23 March 2020 that it would buy investment-grade corporate bonds and the ETFs that hold them. What the discounts meant is contested: stale NAVs built on bonds that were barely trading, with the ETF price the more honest number (ETFs as a venue for price discovery, Lettau & Madhavan 2018) — or real forced selling, since the least liquid bonds, high yield, showed no such gap (Haddad, Moreira & Muir).

Beyond the expense ratio you met in Session 1.1, judge an ETF by its tracking difference — the fund's total return, measured on NAV, minus its index's — which collects the fee, cash drag, sampling (bond ETFs hold part of their index) and trading costs, partly offset by income from lending its securities to short sellers. The premium or discount you trade at comes on top. In the US, in-kind redemptions also keep an ETF's taxable distributions low (Lettau & Madhavan) — a feature of US tax law, not of ETFs everywhere. Leveraged ETFs add the daily-rebalancing drag of Session 2.3.

⚠️ Pitfall — "a 3% discount is a free 3%"

Only for someone who can redeem, and you cannot: redemption is for APs, in creation units, and in a crisis their cost of selling the basket can exceed the discount — which can widen before it closes, and may partly be a stale NAV. Treat premiums and discounts as a liquidity signal, not an arbitrage: the limits to arbitrage of Session 7.1.

2.2.6 Short selling: selling what you borrowed

To profit from a fall you sell shares you do not own — which means borrowing them, with a fee, obligations, and a lender who can want them back at the worst moment.

Short selling is selling borrowed shares, then buying them back (covering) and returning them; the profit is the fall in price minus the costs of the loan. People short to bet on a fall, to hedge (the short side of a long-short portfolio) and, as market makers, to sell to a buyer before they own the stock (SEC staff 2021). Fig 2.2.8 is the lifecycle.

THE LIFE OF A SHORT SALE — and the two ways it can be ended for you 1 · LOCATE broker confirms the shares can be borrowed 2 · BORROW lender delivers them; you leave collateral 3 · SELL at today's price; the cash stays as collateral 4 · CARRY daily: borrow fee, dividends, margin 5 · COVER buy the shares back and return them WHAT IT COSTS TO STAY SHORT fee and dividends, every day it is open gain capped at 100% · loss has no ceiling LENDER RECALLS return the shares within days, or re-borrow elsewhere MARGIN CALL the price rises: add cash or be bought in by the broker either one can force you to cover at the worst possible price A short pays rent every day it is open — and the lender or the broker can end it for you.
fig 2.2.8 — the short-sale lifecycle: locate, borrow, sell, carry, cover — with the two exits someone else can force on you, a recall and a margin call

In the US a broker may not accept a short sale until it has borrowed the shares, arranged to, or has reasonable grounds to believe they can be borrowed in time for delivery — the locate of Regulation SHO Rule 203(b)(1). The lender, often an index fund or another long-term holder, takes cash collateral of about 102% of the shares' value, marked daily (D'Avolio 2002); you pay it every dividend as a "manufactured" dividend; and it can recall the shares at any time, leaving you only days to return them or borrow elsewhere — three in D'Avolio's T+3-era data, and less under today's US T+1 settlement cycle (as of 2026).

The borrow fee is the annual rate, charged daily on the loan's value, that the lender earns. With cash collateral it is netted from the rebate the lender pays on your cash: a 5% cash rate and a 1.5% fee give a 3.5% rebate (D'Avolio's example). In D'Avolio's data (April 2000 to September 2001), 91% of the stocks on loan cost under 1% a year — general collateral, with a value-weighted mean fee of 17 basis points — while the 9% of specials averaged 4.3%, a few reaching 50%. Recalls hit 2% of loaned stocks in an average month, but fees and recalls both rise with disagreement about a stock, exactly where shorts cluster. Engelberg, Reed and Ringgenberg (2018) show that this short-selling risk is priced: stocks with more of it have lower returns and less short selling.

Margin works as in Session 2.4, with one difference: securities margin is a percentage of the position's value, not 2.4's dollars per contract.

Cost or ruleSize or requirement (source; rules as of 2026)
Borrow feeunder 1% a year for 91% of stocks; specials 4.3% on average; extremes near 50% (D'Avolio 2002); GameStop about 25% in January 2021 (SEC staff 2021)
Initial margin (US)the sale proceeds plus 50% of the position — 150% in all (Regulation T)
Maintenance margin (US)equity of at least 30% of the shares' current value or $5 a share, whichever is greater, for stocks at $5 or more; brokers may ask for more (FINRA Rule 4210)
Circuit breaker (US)after a 10% fall from the prior close, short sales only above the best bid, that day and the next (Regulation SHO Rule 201)
EUno uncovered short sales of shares (Art. 12); net short positions reported to the regulator from 0.1% of share capital (Art. 5, as amended by Delegated Regulation 2022/27, from 31 January 2022) and published from 0.5% (Art. 6) — Regulation (EU) No 236/2012
🧭 Margin note — short-selling bans, and what they did

Regulators reach for bans in crises. Studying the bans of 2008–09 across 30 countries, Beber and Pagano (2013) found they hurt liquidity (most for small, volatile stocks without listed options), slowed price discovery, and failed to support prices except possibly for US financial stocks. Session 6.5 turns fees and constraints into the borrow cost line of a strategy's P&L.

2.2.7 Short P&L, squeezes & GameStop

A short's payoff is a long's turned upside down, and the flip is not symmetric: at most you make 100%, and the loss has no ceiling.

Short one share at P0P_0 for nn trading days, paying a borrow fee κ\kappa and dividends at a yield qq, both accruing daily on the current price:

P&L=P0−Pn−q+κ252∑t=0n−1Pt\text{P\&L} = P_0 - P_n - \frac{q + \kappa}{252} \sum_{t=0}^{n-1} P_t

The profit is capped at P0P_0; PnP_n is not. If the stock's expected total return is μ\mu, its price is expected to grow at μ−q\mu - q, and with a=e(μ−q)/252a = e^{(\mu - q)/252}:

𝔼[P&L]=P0(1−an)−q+κ252P0∑t=0n−1at\E[\text{P\&L}] = P_0\bigl(1 - a^{n}\bigr) - \frac{q + \kappa}{252}\, P_0 \sum_{t=0}^{n-1} a^{t}

A short pays the stock's expected total return plus the fee, in excess of any rebate your collateral earns (zero here): it needs a genuine negative forecast to pay its way. The lab's crowded short: price $50, expected total return 8%, dividend yield 2%, volatility 50%, fee 5%, 20,000 one-year paths, with a risk-free rate of zero so the collateral earns nothing. World A is a geometric Brownian motion. World B adds short squeezes — sudden rallies that force shorts to buy back, which pushes the price higher still: 0.3 a year on average, each lifting the price 80% in a day, with the drift lowered so the expected price path is identical.

Short, one year (return on the initial short value)World A: no squeezesWorld B: squeezes
Average return (theory −13.4% in both)−13.8%−13.0%
P(any profit)49.3%55.2%
P(lose more than the whole initial value)8.0%9.9%
P(the price doubles by year-end) — theory6.7% — 6.5%8.6% — 8.7%
P(margin call within the year), price test only71.8% (theory 71.5%)67.6%
P(margin call), counting accrued fees and dividends73.8%69.6%
P(called, yet the price ended below entry)29.4%30.2%
1st percentile of returns−213%−304%

The averages match the formula in both worlds, and world A's price-test call rate matches the barrier formula of Session 2.4 (the lab asserts both at four standard errors). Squeezes change the shape, not the mean. World B wins more often and loses bigger (fig 2.2.9) — the negative skewness that makes shorting feel easy until it isn't. Then margin: equity is the cash (150% of the entry value) minus the shares owed, and a call comes when it falls below 30% of their value, 1.5P0−P=0.3P1.5P_0 - P = 0.3P, so P⋆=1.51.3P0P^\star = \tfrac{1.5}{1.3}P_0 — a rise of only 15.4%, to $57.69, a move this volatile stock makes routinely. That ignores the accrued fees and dividends, which lower the call price as the year goes on: counting them, 73.8% of world-A paths get a call.

ONE−YEAR RETURN OF A SHORT, WORLD B (20,000 PATHS; LOSSES BELOW −300% SHOWN AT −300%) −300% −200% −100% 0% −300% to −294% −294% to −288% −288% to −282% −282% to −276% −276% to −270% −270% to −263% −263% to −257% −257% to −251% −251% to −245% −245% to −239% −239% to −233% −233% to −227% −227% to −221% −221% to −215% −215% to −209% −209% to −202% −202% to −196% −196% to −190% −190% to −184% −184% to −178% −178% to −172% −172% to −166% −166% to −160% −160% to −154% −154% to −148% −148% to −142% −142% to −135% −135% to −129% −129% to −123% −123% to −117% −117% to −111% −111% to −105% −105% to −99% −99% to −93% −93% to −87% −87% to −81% −81% to −74% −74% to −68% −68% to −62% −62% to −56% −56% to −50% −50% to −44% −44% to −38% −38% to −32% −32% to −26% −26% to −20% −20% to −14% −14% to −7% −7% to −1% −1% to 5% 5% to 11% 11% to 17% 17% to 23% 23% to 29% 29% to 35% 35% to 41% 41% to 47% 47% to 54% 54% to 60% 60% to 66% 66% to 72% 72% to 78% 78% to 84% 84% to 90% lost more than the initial value ←
fig 2.2.9 — a crowded short's one-year return in world B: gains stop short of +100%, losses do not; coral bars lost more than the entire initial short value (the 1.1% of paths below −300% are stacked at −300%)

One path, chosen to illustrate rather than because it is typical: the short was winning, the price down 28.0%, when on trading day 77 a squeeze lifted it 73.0% in one session and breached maintenance. Bought in at the close, the trade returned −26.4%; held, it would have returned +37.8%, as the price ended the year 44.1% below entry (fig 2.2.10). The broker does not care that you would have been right.

SHORT 100 SHARES AT $50 WITH 150% COLLATERAL: ONE SQUEEZE PATH (X = TRADING DAYS) equity, if you could hold on maintenance requirement $0 $1.0k $2.0k $3.0k $4.0k $5.0k 0 50 100 150 200 250 equity, if you could hold on equity, if you could hold on: $4.4k maintenance requirement maintenance requirement: $839 equity: $4.4k maintenance: $839
fig 2.2.10 — 100 shares shorted at $50 with 150% collateral: one squeeze day drives the account's equity below the maintenance requirement, months before the price collapses

GameStop, January 2021

The textbook squeeze is GameStop (GME), documented in the SEC staff report of 14 October 2021. Short interest — shares sold short as a share of the float — reached 122.97% in January 2021, possible because a borrowed share, once sold, can be lent and shorted again, and borrowing cost about 25% a year. The price surged from 13 January, closed at a high of $347.51 on 27 January — more than 1,600% above its 11 January close — and touched $483.00 intraday on the 28th, when several retail brokers restricted trading after the clearing house's margin calls (Session 2.1 — Market Anatomy & the Life of an Order). Some funds closed their shorts at large losses; by 19 February the price had fallen as low as $40.59. The staff found that short covering added to the rise at times, but concluded that "it was the positive sentiment, not the buying-to-cover, that sustained the weeks-long price appreciation."

📌 Key idea — a short is a callable loan with unlimited liability

You pay a variable fee and every dividend, the lender can end the loan at will, and your broker will end it after a modest rise. Size a short by the squeeze you can fund, not by how sure you are. The same frictions explain why overpriced stocks can stay overpriced — the limits to arbitrage of Session 7.1.

2.2.8 Where people go wrong

None of these mistakes raises an exception. Each one changes a backtest's verdict.

MistakeSymptom (in the lab)Fix
Returns from quoted prices across a splita −75.0% day; CAGR −4.74%, not 9.42%adjust for every corporate action (Session 4.3)
A price index used as a total return2.1 percentage points a year missing at a 2% yielduse total-return series
A price-weighted index without a divisor reseta −55.0% "crash" on the split dayreset the divisor at every corporate action
Backtesting today's index membersonly survivors; past losers silently droppedpoint-in-time membership (Session 4.2)
Treating ETF price and NAV as one numberphantom arbitrage; in stress you trade at the priceknow who can arbitrage, at what cost
Short P&L = −(long P&L)fees, dividends, recalls and margin calls ignoredmodel the borrow cost and the margin path
⚠️ Pitfall — "a short is a long with a minus sign"

Short 1,000 shares at $50; a year later they are at $45. You were right: $5,000 on the price. But you owe $1,000 of dividends plus a fee on the $50,000 borrowed (on the starting value, for simplicity). As general collateral at 0.25% the fee is $125 and you keep $3,875; as a special at 5%, $2,500, leaving $1,500; hard to borrow at 15%, $7,500 — and you lose $3,500 on a correct call.

2.2.9 Try it

Open the companion lab, predict before you run, and break things on purpose.

🔧 Try it (15 min, in the lab)

1. In section 2, set SPLIT_RATIO = 10. First predict the naive split-day "return" and Pricey's new price weight: at price xx, with the other three summing to yy, its weight goes from x/(x+y)x/(x+y) to x/10y+x/10\tfrac{x/10}{y + x/10}.
2. In section 7, raise FEE_S from 0.05 to 0.25, GameStop's January 2021 borrow cost. Predict the new average return with the 𝔼[P&L]\E[\text{P\&L}] formula, then compare with short_theory_ret and the simulation.
3. In section 5, set STRESS_C_BP = 15.0 so the APs keep working through the stress. What happens to the maximum discount?
Success: step 1 lands near −90%; step 2's simulation lands within four standard errors of your prediction; and step 3 shows that the cost of arbitrage caps the discount, while inside the cap the selling sets it.

2.2.10 Glossary · Check yourself · Go deeper

Common stock — a share of ownership: a vote, dividends when paid, and the residual claim on whatever is left after all fixed claims, with limited liability.
Free float — the fraction of a company's shares that can actually trade, excluding blocks held by founders, families, governments or other companies.
Dividend — cash paid per share to holders; the price drops by about the dividend on the ex-dividend date.
Ex-dividend date — the first trading day on which a share trades without the right to its next dividend; under US T+1 settlement, normally the record date itself.
Stock split — each share becomes N (or N become one, a reverse split); the price divides (multiplies) by N and the value is unchanged.
Corporate action — any company-initiated event that changes its shares or pays its holders: dividends, splits, spin-offs, mergers, rights issues, buybacks.
Total return — price change plus dividends reinvested, (Pt+Dt)/Pt−1−1(P_t + D_t)/P_{t-1} - 1; the return this course always means.
Index — a number tracking a rule-defined basket: quantity × price summed, over a divisor adjusted so corporate actions leave the level unchanged.
Market-cap weighting — weighting each stock by its market value (usually float-adjusted); the one rule that needs no trading between index changes.
ETF — an exchange-traded fund: a basket whose shares trade all day, held near its NAV by creation and redemption.
Net asset value — a fund's holdings minus its liabilities, per share (NAV); what a mutual fund deals at and what an ETF's price is compared with.
Authorised participant — a large dealer under contract with an ETF, allowed to create and redeem its shares in creation units.
Creation and redemption — authorised participants swapping the basket for new ETF shares, or ETF shares for the basket, usually in kind; the arbitrage that bounds the premium.
Short selling — borrowing shares, selling them, and later buying them back to return; the gain is capped at 100%, the loss is not.
Borrow fee — the annual rate, charged daily on the loan's value, that a short seller pays the lender; under 1% for most stocks, far more for hard-to-borrow ones.
Short squeeze — a rally that forces short sellers to buy back, whose buying pushes the price higher and forces yet more buying.

Check yourself

1. A company has $10bn of assets, $6bn of debt and 400 million shares. What is a share worth, and what happens to it if the assets lose 20%?
Equity is the residual, $4bn: $10 a share. Assets of $8bn leave $2bn of equity — $5 a share, a 50% fall on a 20% fall in the firm. The residual claim magnifies the firm's moves: the leverage of Session 2.3, inside every stock.
2. A friend plans to buy the day before the ex-date, collect a $1 dividend and sell next morning. Why is that not free money?
The price opens lower by about the dividend, since the share no longer carries it, so the cash is roughly offset by a capital loss (often taxed differently). And one day's random move is several times a quarterly dividend — its standard deviation is 2.6 times in the lab — so a single trade is a coin flip plus costs.
3. In a price-weighted index, one member's weight halves overnight while prices are flat. What happened, and what should your code do?
A split: in the lab Pricey's weight fell from 73.1% to 40.4% at a 4-for-1 split, because the index weights by quoted price. Reset the divisor so the level does not jump, and accept the new weight — that is the rule, not a bug.
4. A bond ETF trades at a 3% discount to NAV in a panic. Can you buy it, redeem it and pocket 3%?
No: only APs redeem, in creation units, and they will not if selling the basket costs more than 3% — as in March 2020, when discounts persisted until the Fed's 23 March announcement. Part of the gap may also be a stale NAV. Buying is a bet on the gap closing, not an arbitrage.
5. You short a $50 stock with 150% collateral and 30% maintenance. When does the first margin call come, and why does it matter for a volatile stock?
On price alone, at P⋆=1.5P0/1.3=P^\star = 1.5P_0/1.3 = $57.69, a 15.4% rise; accrued fees and dividends bring it a little lower over time. At 50% volatility the lab's short is called within a year on 74% of paths, often while it would have ended the year right. Keep spare cash, or size smaller.
🔧 Homework (~1 h before Session 2.3)

1. A total-return function. Write total_return(prices, dividends) from raw closes and ex-date dividends; compounded over section 4's series it must reproduce tr[-1] / tr[0] to ten decimal places.
2. A fifth stock. Add "Giant" to UNIVERSE (price $2,000, 50 million shares, float 1.0, beta 1.0, volatility 0.25). Predict its price and float-cap weights, then check how vol_ew, vol_cw0 and the growth gap change.
3. A break-even short. With MU_S = -0.10, use the 𝔼[P&L]\E[\text{P\&L}] formula to find the borrow fee at which the short only breaks even; confirm it in the lab.
4. Read the creation/redemption section of Lettau & Madhavan (2018) and section 3 of the SEC staff report on GameStop.

📚 Go deeper

  1. Bodie, Z., Kane, A. & Marcus, A. J. (2024). Investments, 13th ed., §2.3 "Equity Securities", §2.4 "Stock and Bond Market Indexes", §3.9 "Short Sales", §4.6 "Exchange-Traded Funds". McGraw Hill. ISBN 978-1-264-41266-2
  2. Lettau, M. & Madhavan, A. (2018). "Exchange-Traded Funds 101 for Economists." Journal of Economic Perspectives 32(1), 135–154. Points here are drawn from the working-paper version, NBER w24250. doi:10.1257/jep.32.1.135
  3. D'Avolio, G. (2002). "The Market for Borrowing Stock." Journal of Financial Economics 66(2–3), 271–306. doi:10.1016/S0304-405X(02)00206-4
  4. Engelberg, J. E., Reed, A. V. & Ringgenberg, M. C. (2018). "Short-Selling Risk." Journal of Finance 73(2), 755–786. doi:10.1111/jofi.12601
  5. Duffie, D., Gârleanu, N. & Pedersen, L. H. (2002). "Securities Lending, Shorting, and Pricing." Journal of Financial Economics 66(2–3), 307–339. doi:10.1016/S0304-405X(02)00226-X

Sources for the numbers

  1. SEC staff (2021). "Staff Report on Equity and Options Market Structure Conditions in Early 2021." U.S. Securities and Exchange Commission, 14 October 2021. https://www.sec.gov/files/staff-report-equity-options-market-struction-conditions-early-2021.pdf
  2. Haddad, V., Moreira, A. & Muir, T. (2021). "When Selling Becomes Viral: Disruptions in Debt Markets in the COVID-19 Crisis and the Fed's Response." Review of Financial Studies 34(11), 5309–5351 doi:10.1093/rfs/hhaa145. Figures and Figure 7 quoted from NBER Working Paper 27168, version of May 2020 doi:10.3386/w27168
  3. Board of Governors of the Federal Reserve System (2020). "Federal Reserve announces extensive new measures to support the economy." Press release, 23 March 2020. https://www.federalreserve.gov/newsevents/pressreleases/monetary20200323b.htm
  4. Elton, E. J. & Gruber, M. J. (1970). "Marginal Stockholder Tax Rates and the Clientele Effect." Review of Economics and Statistics 52(1), 68–74. doi:10.2307/1927599
  5. FINRA (2024). Regulatory Notice 24-04, "FINRA Adopts Amendments to Conform its Rules to the T+1 Settlement Cycle," 26 February 2024. https://www.finra.org/rules-guidance/notices/24-04
  6. ESMA (2024). "ESMA proposes to move to T+1 by October 2027," 18 November 2024. https://www.esma.europa.eu/node/214007
  7. S&P Dow Jones Indices (2020). "Salesforce.com, Amgen and Honeywell International Set to Join Dow Jones Industrial Average." Press release, 24 August 2020. https://press.spglobal.com/2020-08-24-Salesforce-com-Amgen-and-Honeywell-International-Set-to-Join-Dow-Jones-Industrial-Average
  8. S&P Dow Jones Indices. "S&P U.S. Indices Methodology" (the S&P 500's float-adjusted market-cap weighting). https://www.spglobal.com/spdji/en/methodology/article/sp-us-indices-methodology/
  9. S&P Dow Jones Indices. "FAQ: S&P 500 Equal Weight Index" (0.2% per company at each quarterly rebalance). https://www.spglobal.com/spdji/en/education/article/sp-500-equal-weight-index-faq/
  10. US rules (as of 2026): Regulation SHO, 17 CFR 242.203 (locate) and 242.201 (circuit breaker); FINRA Rule 4210 (margin requirements). https://www.law.cornell.edu/cfr/text/17/242.203 · https://www.law.cornell.edu/cfr/text/17/242.201 · https://www.finra.org/rules-guidance/rulebooks/finra-rules/4210
  11. EU rules (as of 2026): Regulation (EU) No 236/2012 on short selling, Arts. 5, 6 and 12; Commission Delegated Regulation (EU) 2022/27 (the 0.1% notification threshold). https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex%3A32012R0236 · https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32022R0027
  12. Beber, A. & Pagano, M. (2013). "Short-Selling Bans Around the World: Evidence from the 2007–09 Crisis." Journal of Finance 68(1), 343–381. doi:10.1111/j.1540-6261.2012.01802.x
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Educational material only — not investment, legal or tax advice. Backtests and simulations are hypothetical. Disclaimer · Text CC BY 4.0 · Code MIT.