Unit 1 · The Quant Landscape — Session 02 of 05

Where Returns Come From

Every profitable trade is paid for by someone. Before you write a line of backtest code you should be able to say who that someone is, and why they will keep paying. There are only five honest answers — and most strategies that look like skill turn out to be none of them, just a familiar risk wearing a costume. The core move: for every return, name who is on the other side — then check whether it is alpha, or beta in disguise.

DURATION 1.5 h + lab LEVEL Foundations TRACKS QR · QT · QD · IND HANDS-ON Unmask three fake alphas
📓 Companion lab Open in Colab Notebook on GitHub Source (.py)
On this page
  1. Every return has a payer
  2. The five sources, side by side
  3. The shape of a return
  4. Alpha and beta
  5. Why most apparent edges are none of them
  6. The arithmetic of active management
  7. Try it
  8. Glossary · Check yourself · Go deeper

1.2.1 Every return has a payer

Markets do not pay you for being clever, or for being patient, or for owning a fast computer. They pay you because somebody, somewhere, is handing you money — and you should be able to say who, and why.

The definition this course uses: an edge is a durable reason your trades earn more than they cost — a counterparty who pays you, a reason they pay, and a reason they will keep doing it. Returns above what cash pays — the excess return of Session 2.3 — come from two kinds of transaction. Either you provide a service somebody wants — you take a risk off their hands, or let them trade the instant they need to — or you profit from somebody else's mistake or delay. Fig 1.2.1 sorts the five honest sources of edge into those two families.

WHO PAYS YOU — five honest sources of edge, in two families PAID FOR A SERVICE — the payer gets something it wants, so the premium can last INVESTORS SHEDDING RISK hedgers and the risk-averse $ to shed risk, above all in bad times equity · carry · value · selling volatility RISK PREMIUM lasts: someone has to bear the risk TRADERS IN A HURRY forced sellers, big orders $ for immediacy — to trade right now market making · short-term reversal LIQUIDITY PREMIUM lasts: holding inventory is risky PAID FOR A MISTAKE OR A DELAY — the payer gets nothing, so competition erodes it BIASED TRADERS over- or under-reacting to news they trade at the wrong price momentum · overreaction · earnings drift BEHAVIOURAL ANOMALY lasts while arbitrage is costly LESS-INFORMED TRADERS trading on yesterday's picture they trade before the news is priced alternative data · better, faster analysis INFORMATIONAL EDGE lasts until others catch up SLOWER TRADERS stale quotes left in the book they get picked off in microseconds latency arbitrage · sniping stale quotes LATENCY EDGE lasts only while you are fastest Service premia are paid willingly and last; mistake premia get competed away. No row, no edge.
fig 1.2.1 — the five honest sources of edge: who pays you, what they pay for, and why the payment lasts

The difference between the families is the most useful idea on this page. When the payer gets something it wants — insurance against a crash, the ability to sell now — the payment can last indefinitely: it is a fair price for a real service. When the payer gets nothing — it sold in a panic, traded on stale information, was a microsecond slow — every competitor who notices takes a slice, and the edge shrinks towards zero. The first family is a business; the second is a race.

📌 Key idea — an edge is a customer, not a pattern

A pattern in historical prices is not an edge. An edge is a reason a particular group of market participants will keep transferring money to you. If you cannot name who pays you and why they keep paying, you do not have an edge — you have a backtest. Session 1.3 makes this the first gate of the research lifecycle.

🧭 Margin note — the engineering analogy

Treat a strategy like a business plan: revenue has to come from customers. A risk premium is an insurance business — steady premiums, occasional huge claims. Liquidity provision is a convenience store — a thin margin on every sale. Informational and latency edges are product races — great margins until a competitor ships something better. You met the firms that run each business in Session 1.1 — What Quants Actually Do.

1.2.2 The five sources, side by side

For each source, ask who pays you, why it persists, how much money it can absorb and what its returns look like. The answers decide how you test, size and watch it.

SourceWho pays youWhy it persistsCapacityTypical shapeBuilt in
Risk premiumhedgers, the risk-averse, the leverage-constrainedthe risk is real and someone must hold itvery largesteady, with rare deep lossesSession 8.4, Session 8.5, Session 10.7
Behavioural anomalybiased or rule-bound traderslimits to arbitragemedium; shrinks as tradedvaries; some crashSession 7.1, Session 8.2, Session 8.6
Liquidity provisiontraders who must trade nowinventory risk; scarce capital in a panicsmall to mediummany small gains; bad days in stressSession 8.2, Session 14.5
Informational edgeless-informed tradersdata or skill is rare — until it isn'tsmall to medium; decayslumpy, around newsSession 8.6, Session 13.1
Latency edgeslower traders with stale quoteswinner-takes-most races, large fixed costssmall; a few firms take mosttiny, frequent, very smoothSession 14.6

Risk premia — paid for bearing risk others want to shed

A risk premium is the extra expected return for holding something that loses money at the worst possible time. The equity premium is the textbook case: in US data from 1889 to 1978, stocks beat safe short-term securities by about six percentage points a year (Mehra & Prescott, 1985). The usual explanation is that stocks fall in recessions, exactly when people most need their savings — though Mehra & Prescott's point was that standard models of risk aversion cannot justify a premium that large, the "equity premium puzzle", still debated. Carry, value and the volatility risk premium share the same logic: a risk premium persists because it has bad times. Without them it would be a free lunch — and free lunches get eaten.

Behavioural anomalies — paid by other people's biases

An anomaly is a return pattern that known risks do not explain. The classic is momentum: stocks that rose over the past three to twelve months have tended to keep rising (Jegadeesh & Titman, 1993), usually read as investors under-reacting to news. It survives because betting against a mispricing is itself risky and capital-hungry — the limits to arbitrage of Session 7.1 — but it can be arbitraged: published anomalies weaken once investors read about them, the post-publication decay of Session 1.3. Whether value and momentum are premia or anomalies is argued both ways (Session 7.4); the label matters because a risk premium should survive being known and a mispricing should not.

Liquidity provision — paid for immediacy

Someone who must sell now — a fund meeting redemptions, a trader whose stop was hit — accepts a slightly worse price to trade immediately. The liquidity premium is what you earn on the other side: a market maker earning the bid-ask spread, or a short-term reversal strategy buying last week's biggest losers. Nagel (2012) shows that reversal returns can be read as the returns to supplying liquidity, and that they spike in turmoil. You are paid for holding inventory while prices move and for trading against people who may know more (adverse selection). The same phrase also names the extra return on assets that are hard to sell at all.

Informational edge — paid for knowing sooner or better

An informational edge means your forecast is better than what the price already reflects: better data, or better processing of the same data. Katona, Painter, Patatoukas & Zeng (2025) show that investors with satellite counts of cars in retailers' car parks traded profitably ahead of earnings, with less-informed individual buyers on the other side. The limit is legal as well as competitive: trading on material non-public information is insider trading (insider dealing, in UK terms), and Session 13.1 draws the line for alternative data.

Latency edge — paid for being first

A latency edge is being faster, not smarter: trading against stale quotes before they are updated. Aquilina, Budish & O'Neill (2022) find that such speed races account for about a fifth of trading volume in FTSE 100 stocks, and that the top six firms account for over 80% of race wins and losses. For most readers the honest lesson is in Session 14.6: this is a winner-takes-most arms race, not a place to start.

1.2.3 The shape of a return

Two strategies with the same Sharpe ratio can be completely different animals. How the returns are distributed — steady with rare crashes, or slowly bleeding with rare jackpots — tells you which source you are really harvesting.

FOUR SHAPES OF CUMULATIVE P&L — what each source typically looks like over time STEADY, THEN A CRASH negative skew selling volatility · carry most risk premia BLEED, THEN A JACKPOT positive skew trend following buying options MANY TINY WINS smooth, very high Sharpe market making · latency but small capacity LUMPY, AROUND NEWS jumps at events informational edges earnings · alternative data Illustrative, not to scale. Most risk premia look like the first panel: calm for years — which is why they pay.
fig 1.2.2 — four typical shapes of cumulative profit and loss, sketched: the source of an edge leaves a fingerprint on its return distribution

Risk premia usually look like the first panel: paid to hold what others fear, most months are good and a few are terrible — a long left tail, or negative skewness (Session 3.2). Currency carry trades crash when traders unwind them together (Brunnermeier, Nagel & Pedersen, 2008); momentum crashes, rarely and violently, when a falling market rebounds (Daniel & Moskowitz, 2016). Selling volatility is the purest case: the market's price of future S&P 500 variance has on average exceeded the variance that followed (Carr & Wu, 2009), so sellers collect a premium — until a day like 5 February 2018, when XIV, an exchange-traded note short VIX futures, lost more than 80% of its value and was wound up (Credit Suisse, 2018). Trend following, by contrast, loses small amounts often and wins big in sustained moves: the second panel.

The lab's Insurer makes the first panel concrete. Every day it collects a premium, and it pays 3 times the amount by which the day's fall exceeds 3% — a one-day put struck that far below the market, sold that many times over — priced at 2 times its expected payout. That markup is the risk premium. Over 1,000 simulated years it averages 5.0% a year with a Sharpe ratio of 0.53, makes money in 74% of months, and has a monthly skewness of −5.0.

The 20-year sample used in the rest of this session was unkind to it: 14 market crash days where 8 were expected cut its average to 2.1% a year (Sharpe ratio 0.19), with a worst month of −23% — though it still made money in 75% of months. Same strategy, same premium: whether its record looks brilliant or dismal depends on whether the bad times turned up (fig 1.2.3).

EVERY MONTHLY RETURN, SORTED FROM WORST (LEFT) TO BEST (RIGHT) Market Picker Insurer −30% −20% −10% 0% 10% 20% 50 100 150 200 Market Picker Insurer
fig 1.2.3 — every month of the sample, sorted from worst to best: the market and the Picker trace roughly symmetric S-curves; the Insurer's line is flat and just above zero, then falls off a cliff in its worst handful
⚠️ Pitfall — "a smooth equity curve means low risk"

A strategy that sells insurance looks safest just before it isn't. Volatility and the Sharpe ratio are measured on the months you happen to have seen; if your sample holds few crashes, both flatter the strategy — and if it holds many, as ours did, they damn it. Look at the worst month, the maximum drawdown and the skewness, and ask what a crash twice as bad as the worst you have seen would do. A high Sharpe ratio with strongly negative skew is a warning, not a trophy; Session 9.8 builds the tools.

1.2.4 Alpha and beta

Most of any strategy's return is exposure to things you could buy cheaply. Alpha is what is left over — and measuring it is a regression.

Beta measures how much a strategy moves when a factor moves: a market beta of 2 means that on a day the market rises 1%, the strategy tends to rise 2%. Beta is cheap: an index fund, which simply holds an index's stocks at index weights, sells you market beta for a small fraction of a percent a year. Alpha is the average return left after you subtract what the strategy's betas would have earned anyway. Only alpha is worth paying skill prices for.

In words: each day's excess return is a constant, plus beta times the market's excess return, plus noise that averages to zero. With RM,tR_{M,t} the market's return and rf,tr_{f,t} the risk-free rate over the same day:

Rt−rf,t=α+β(RM,t−rf,t)+εtR_t - r_{f,t} \;=\; \alpha \;+\; \beta\,(R_{M,t} - r_{f,t}) \;+\; \varepsilon_t

With several factors, each gets its own beta, and ordinary least squares splits the average excess return exactly into alpha plus each beta times its factor's average — a factor model, built properly in Session 7.2 — CAPM, APT & Factor Regressions, with honest standard errors in Session 5.2. Here it is a lie detector.

DECOMPOSING A RETURN — peel off every priced exposure; alpha is what is left 1 · EXCESS RETURN subtract what cash would have paid 2 · MARKET BETA regress on the market: β × market return 3 · OTHER FACTORS style, carry, crash risk: Σ β × factor return 4 · ALPHA + NOISE what is left over: α + ε, day by day 5 · IS α REAL? t-stat, costs, shape and story ONE STRATEGY'S AVERAGE RETURN, SPLIT TWO WAYS — illustrative, not to scale MARKET-ONLY MODEL what most people run market β "alpha" — a significant t-stat, and it looks like skill FULL FACTOR MODEL every priced factor market β style or crash-insurance β × that factor's return α ≈ 0, t-stat ≈ 0 Same returns, same total. The model you regress on decides how much of it you call alpha.
fig 1.2.4 — the decomposition workflow: against the market alone a hidden exposure shows up as alpha; against every priced factor it is beta

Four strategies in a world where we know the truth

On real data you never know the true alpha, so the lab builds a synthetic market with quantnb.synth.factor_model_returns: 300 stocks over 20 years, driven by the market (a 6% premium, with rare one-day crashes), a value-like STYLE factor (a 4% premium), an industry factor that pays nothing, and the Insurer's crash-insurance factor. Cash pays 0%. Only one of four strategies has skill:

  • Rocket buys the 30 most market-sensitive stocks and adds leverage: 1.2× the capital. Market beta in a costume.
  • Picker is factor-neutral (market, style and industry hedged) with a genuine informational edge: its monthly forecasts of stock-specific returns are correlated just 0.01 with the truth.
  • Insurer sells crash insurance on the index (§1.2.3). A risk premium.
  • Bargain is long the 30 most STYLE-exposed stocks and short the 30 least, market hedged. A style factor in disguise.
GROWTH OF $1 OVER 20 SIMULATED YEARS (LOG SCALE; X-AXIS IN YEARS) Market Rocket Insurer Picker $1.00 $2.00 $5.00 $10.00 $20.00 0 5 10 15 20 Market Market: $5.21 Rocket Rocket: $27.25 Insurer Insurer: $1.33 Picker Picker: $1.74 Rocket: $27.25 Market: $5.21 Picker: $1.74 Insurer: $1.33
fig 1.2.5 — growth of $1 over the sample: the Rocket looks like the winner, the crash-hit Insurer like the loser, and the Picker — the only one with skill — like the dullest
StrategyAverage (arithmetic) return a yearCAGRVolatilitySharpe ratioMonthly skewMax drawdown
Market9.5%8.6%16.0%0.60−0.2−34%
Rocket22.2%18.0%33.4%0.66−0.2−65%
Picker2.9%2.8%5.0%0.570.1−15%
Insurer2.1%1.4%11.5%0.19−4.0−34%
Bargain4.9%4.6%8.6%0.57−0.1−29%

Judged like a fund factsheet, the Rocket is the star: it averaged 22.2% a year against the market's 9.5%. The crash-hit Insurer looks like the dud, and the Picker, the only one with skill, looks dull. Averages are shown because the alpha-plus-beta split is exact only for them; the money compounded at the lower CAGR, the gap being the volatility drag of Session 2.3.

Now the regressions. Each alpha comes with its t-statistic — the estimate divided by its standard error; beyond about 2 it is conventionally called significant. The long-run column is what the market-only model would report with unlimited data (the lab's 1,000-year simulation).

StrategyMarket betaAlpha vs market only (t-stat)Alpha vs full model (t-stat)Market-only alpha, long runThe truth
Rocket2.062.5% (2.0)2.1% (1.7)0.3%market beta 2.06, no alpha
Picker0 (hedged)2.9% (2.5)2.9% (2.6)3.0%alpha 3.0% from information
Insurer0.25−0.2% (−0.1)0.6% (0.6)4.0%3× short insurance, no alpha
Bargain0 (hedged)4.8% (2.5)0.1% (0.1)4.0%style beta 1.00, no alpha

The Rocket's costume is the thinnest: the market explains 97% of its daily variance, and beta times the market's average explains 19.7% a year out of its 22.2%. The leftover has a t-statistic of 2.0 — borderline "significant" — yet its long-run value is 0.3%: a sample this short can manufacture a t of 2 (Session 1.3). The Bargain is the clean case: a market-only alpha with a t of 2.5, the kind of number that gets a strategy funded, falls to 0.1% once the style factor is added. The Insurer shows one sample misleading the other way: its crash-heavy sample left nothing for either model to call alpha, but in the long run a market-only regression credits it with 4.0% a year — the premium for crash risk that a straight line through the market cannot see. Only the Picker keeps its alpha under every model, close to its true 3.0%.

LONG-RUN AVERAGE RETURN A YEAR, SPLIT TWO WAYS (POPULATION VALUES) α, market-only model β × factor premia, full model α, full model (the truth) 0% 5% 10% 15% Rocket · α, market-only model: 0% 0% Rocket · β × factor premia, full model: 13% 13% Rocket · α, full model (the truth): 0% 0% Rocket Picker · α, market-only model: 3% 3% Picker · β × factor premia, full model: 0% 0% Picker · α, full model (the truth): 3% 3% Picker Insurer · α, market-only model: 4% 4% Insurer · β × factor premia, full model: 5% 5% Insurer · α, full model (the truth): 0% 0% Insurer Bargain · α, market-only model: 4% 4% Bargain · β × factor premia, full model: 4% 4% Bargain · α, full model (the truth): 0% 0% Bargain
fig 1.2.6 — long-run values: the market-only model's "alpha" (orange) against the full model's split into factor exposure (grey) and alpha (blue); three of four strategies are almost pure beta

The regression, as the lab runs it:

from quantnb.stats import factor_regression

fit = factor_regression(strategies["Bargain"], factors[["MKT"]], names=["MKT"])
fit["alpha_annual"], fit["alpha_t"]        # what most people check: the market-only "alpha"

cols = ["MKT", "STYLE", "IND", "PUT"]         # every priced factor in this world
fit = factor_regression(strategies["Bargain"], factors[cols], names=cols)
fit["alpha_annual"], fit["betas"]["STYLE"]   # alpha ≈ 0; the STYLE beta ≈ 1
📌 Key idea — alpha is always relative to a model

There is no such thing as the alpha of a strategy, only its alpha against a stated set of factors. Against the market alone, selling insurance and tilting to cheap stocks look like skill; against the factors they load on, they are beta. When someone quotes you an alpha, the first question is: relative to what?

1.2.5 Why most apparent edges are none of them

Most strategies that look like alpha fail for one of five boring reasons. Rule out all five before you believe in skill.

DisguiseWhat it looks likeHow to unmask itTaught in
Hidden market betabeats the index in a rising marketregress on the market; compare Sharpe ratios, not raw returnsSession 7.2
Hidden factor beta"market-neutral alpha" with a steady tiltregress on the known factors: value, momentum, carry, qualitySession 7.3, Session 7.4
A risk premium with a crash in its futurehigh Sharpe, smooth curve, negative skewlook at skew, worst months and drawdown; add a volatility or crash factorSession 3.2, Session 10.7
Luck and overfittinga great backtest after many triescount the trials, deflate the Sharpe ratio, use a holdout onceSession 1.3, Session 6.7
Costsan edge smaller than the cost of trading itbacktest with realistic transaction costs and capacitySession 6.5
"IS THIS ALPHA?" — SAME FOUR STRATEGIES, TWO WAYS TO JUDGE THEM THE NAIVE WAY compare the raw average return with the index trust a high Sharpe ratio and a smooth equity curve never ask who is on the other side of the trade Buys the Rocket, the Insurer and the Bargain. Pays skill prices for cheap, replicable risks. THE DISCIPLINED WAY 1 · regress on the market and every priced factor 2 · alpha's t-stat above a hurdle, after costs 3 · check the shape: skew, worst month, drawdown 4 · name who pays you, and why it will persist Keeps only the Picker — and prices the rest as beta. Pays for alpha; buys beta at beta prices.
fig 1.2.7 — two ways to judge a track record: the naive reviewer funds three costumes; the disciplined one finds the single genuine edge
⚠️ Pitfall — "it's market-neutral, so its return must be alpha"

Neutral to the market is not neutral to everything. The Bargain's market beta was zero by construction, yet its return came from a style factor; the Insurer's long-run market beta is only 0.16, and its return comes from selling crash insurance. Paying alpha-level fees for value, momentum, carry or short-volatility exposure that factor products sell cheaply is a common and expensive mistake.

🧭 Margin note — hidden beta is not a crime

Harvesting risk premia on purpose is a perfectly good business. The errors are paying for them as skill, sizing them as if they could not crash, and believing they diversify you when they are the bet you already own. Name the exposure, price it, size it.

1.2.6 The arithmetic of active management

Beating the market is a contest among active investors, and the scoreboard has an unforgiving property: before costs it sums to zero; after costs, to less than zero.

William Sharpe (1991) needed three pages. Divide every investor into passive — in practice, index funds holding every security at its market weight — and active, holding anything else. Passive investors earn the market return by construction, and the market return is the value-weighted average of everyone's return. So the average actively managed dollar earns the market return too, before costs — and less after, because active management costs more. With ss the passive share of the market:

RM=sRpassive+(1−s)Ractive,Rpassive=RM⟹Ractive=RMR_M = s\,R_{\text{passive}} + (1-s)\,R_{\text{active}}, \qquad R_{\text{passive}} = R_M \;\;\Longrightarrow\;\; R_{\text{active}} = R_M

That makes beating the market, measured against the market, a zero-sum game among active investors before costs, and a negative-sum one after. It does not say nobody can win; it says every winner's excess gain is somebody else's excess loss.

SHARPE'S ARITHMETIC — every share is owned by someone (widths illustrative) PASSIVE · share s holds every stock at market weight ACTIVE · share 1 − s holds anything else: every overweight is someone else's underweight 1 · PASSIVE EARNS THE MARKET it holds the market, so it earns the market, by definition 2 · SO ACTIVE EARNS IT TOO M = s·M + (1 − s)·A forces A = M, before costs 3 · THEN SUBTRACT COSTS fees, spreads, market impact: the average active dollar trails Before costs, beating the market is zero-sum among active investors; after costs, negative-sum.
fig 1.2.8 — Sharpe's arithmetic in three steps: M is the market return, A the return on the average active dollar, s the passive share

The lab checks it with 1,000 active managers who hold random tilted portfolios, re-drawn every year, which together with a 40% passive share own exactly the whole market. In every year of the sample, the average active dollar earned the market return to within 4e-16 — floating-point rounding. Individually, though, each manager is a coin toss: in a given year 48% beat the index fund before costs. Charge them 1.0% a year against the index fund's 0.05% — illustrative figures for fees plus trading costs (Session 1.1 explains how funds charge) — and only 35% beat it in a given year, and 8% over all 20 years.

HorizonManagers beating the index fund, before costsAfter costs
Any single year48%35%
All 20 years, compounded47%8%
1,000 ACTIVE MANAGERS: ANNUAL RETURN MINUS THE INDEX FUND, AFTER COSTS, 20 YEARS −3.0% −2.0% −1.0% 0.0% 1.0% −3.7% to −3.6% −3.4% to −3.2% −3.0% to −2.8% −2.8% to −2.7% −2.7% to −2.6% −2.6% to −2.4% −2.4% to −2.3% −2.3% to −2.2% −2.2% to −2.1% −2.1% to −1.9% −1.9% to −1.8% −1.8% to −1.7% −1.7% to −1.5% −1.5% to −1.4% −1.4% to −1.3% −1.3% to −1.1% −1.1% to −1.0% −1.0% to −0.9% −0.9% to −0.8% −0.8% to −0.6% −0.6% to −0.5% −0.5% to −0.4% −0.4% to −0.2% −0.2% to −0.1% −0.1% to 0.0% 0.0% to 0.2% 0.2% to 0.3% 0.3% to 0.4% 0.4% to 0.5% 0.5% to 0.7% 0.7% to 0.8% 0.8% to 0.9% 0.9% to 1.1% 1.1% to 1.2% 1.2% to 1.3% 1.3% to 1.5% trailed the index fund ←
fig 1.2.9 — the lab's skill-less active managers over the whole sample, after costs: the distribution is shifted left by the cost gap, and the coral bars — managers who trailed the index fund — are the large majority

Real data agree. Fama & French (2010) find that the aggregate portfolio of US active equity mutual funds is close to the market portfolio, and that the high costs of active management show up almost intact as lower returns to investors. The arithmetic is contested at the edges: Pedersen (2018) points out that real index funds must trade whenever shares are issued, bought back or reshuffled in an index, so active managers who supply that trading can, in aggregate, earn something for it. That softens the arithmetic; it does not repeal it.

Who is on the other side of your trade?

Zero-sum turns §1.2.1's question into a test. Every time you buy, someone sells to you. If the honest answer to "who, and why are they wrong to?" is a pension fund rebalancing on a fixed date, a forced seller meeting redemptions or a buyer paying for insurance, you may have an edge. If the answer is another quant with the same data and a better model, you are the one paying. If you cannot answer, assume the second. Session 1.4 looks at the evidence on who actually wins.

💡 Looking ahead

The Picker's forecasts call the direction of a stock's specific return correctly only 50.3% of the time. That is enough because it makes the call on 300 stocks every month — a breadth of 3,600 bets a year. Grinold & Kahn's fundamental law of active management says the information ratio is roughly the information coefficient times the square root of breadth, IR≈ICBR\mathrm{IR} \approx \mathrm{IC}\sqrt{\mathrm{BR}}: here 0.01 × √3,600 = 0.60. The lab measured 0.57. Session 8.7 — Combining Signals builds it properly — including why real bets are never as independent as that square root assumes.

1.2.7 Try it

Open the companion lab and take the costumes off with your own hands.

🔧 Try it (15 min, in the lab)

1. Predict the Rocket's average return from its beta alone: 2.06 × the market's 9.5% ≈ 19.7%. Run the lab and compare with its section 4 track records (the first table in §1.2.4).
2. Set ROCKET_LEVERAGE = 2.0. Predict the new market beta and whether the alpha's t-statistic changes, then run. Why can't leverage make an alpha more significant?
3. In section 5, add a third model to MODELS: "mkt_style": ["MKT", "STYLE"]. Does the Bargain's alpha vanish with just these two factors? What does that tell you about the exposure it was hiding?
4. Success looks like this: you can say in one sentence each why the Rocket, the Insurer and the Bargain are not alpha, and which factor unmasks each one.

1.2.8 Glossary · Check yourself · Go deeper

Edge — a durable reason your trades earn more than they cost: a named counterparty who pays you, and a reason they keep paying.
Risk premium — the extra expected return for holding a risk others want to shed; it persists because its bad times are real.
Anomaly — a return pattern that known risks do not explain, usually traced to biases or constraints; trading can erode it.
Liquidity premium — the return for supplying immediacy to traders who must trade now, or for holding assets that are hard to sell.
Informational edge — profit from better or faster data, or better processing of it, than the price already reflects.
Latency edge — profit from acting on public information faster than others, e.g. trading against stale quotes.
Beta — a strategy's sensitivity to a factor: how much its return moves per unit move in the factor's return.
Alpha — the average excess return left after subtracting what a strategy's factor exposures earned; always relative to a chosen model.
Zero-sum game — a contest where winners' gains equal losers' losses; beating the market is one among active investors, before costs.
Equity premium — the expected return of stocks above the risk-free rate — the reward for bearing equity risk.
Index fund — a fund that holds an index's constituents at index weights, aiming to match the index rather than beat it.

Check yourself

1. A fund averaged 22% a year over a period when the market averaged 9.5%, with a Sharpe ratio of 0.66 against the market's 0.60. Its manager claims 12.7 percentage points of alpha. What do you check first?
Its market beta. The lab's Rocket had exactly these numbers and a beta of 2.06: beta times the market's average explains 19.7% a year out of its 22.2%. The leftover 2.5% is noise around a long-run value of 0.3%. The similar Sharpe ratios are the tell: high beta and leverage raise return and risk together (Session 2.3).
2. Over a long record, a strategy has made money in 74% of months, with a Sharpe ratio of 0.53, a market beta of 0.16 and a market-only alpha of 4.0% a year. Should you invest?
Not before looking at its shape and asking who pays it. Those are the Insurer's long-run numbers; its monthly skewness is −5.0. It is selling crash insurance — a risk premium, not skill. Regress it on a put-selling factor and the alpha disappears; size it as crash risk, and expect some long stretches, like the lab's sample, to show almost nothing.
3. Classify each by source, and name who pays: (a) quoting both sides of a thinly traded stock all day; (b) buying last year's biggest winners; (c) holding high-interest currencies funded in low-interest ones; (d) buying a retailer after satellite images show full car parks, before earnings.
(a) liquidity provision — traders who need immediacy pay the spread; (b) a behavioural anomaly (momentum) — investors who under-react, though some call it a risk premium; (c) a risk premium (carry) — investors avoiding currency-crash risk; (d) an informational edge — less-informed traders, before the news is in the price.
4. A colleague says: "Active management is zero-sum, so nobody can have an edge." What is right and what is wrong?
Right: relative to the market, active investors as a group earn the market return before costs and less after (Sharpe, 1991). Wrong: zero-sum means every winner is matched by a loser, not that nobody wins. An edge is possible if you can name the losing side — forced sellers, insurance buyers, the less informed; and Pedersen (2018) argues active managers can earn something for supplying the trading index funds must do.
5. A market-neutral fund shows a large alpha against the market. What regression would you run next, and what result would change your mind about paying its fees?
Regress it on the known style factors — value, momentum, carry, quality — and a volatility factor. If the alpha shrinks towards zero, as with the lab's Bargain (4.8% against the market alone, 0.1% against the full model), you are being sold factor beta that is available far more cheaply. Only an alpha that survives every priced factor, after costs, might be skill.
🔧 Homework (~1 h before Session 1.3)

1. In the lab, set INSURE_MARKUP = 1.0 (insurance sold at exactly its expected payout). Predict the Insurer's long-run average return and market-only alpha, then read them in section 3. What is left to pay you for the crash risk?
2. Set IC = 0.02. Use the fundamental law to predict the Picker's information ratio, then check it. Restore it, set N_ASSETS = 75 and predict again; explain any gap using the standard error of 20 years of data.
3. Take one strategy you have seen advertised. Write five lines: which source, who pays, why it persists, its capacity, the shape you expect. Keep it: it becomes the "who pays us" line of your first pre-registration in Session 1.3.
4. Read Sharpe (1991) — three pages — then the introduction of Pedersen (2018) for the counter-argument.

📚 Go deeper

  1. Sharpe, W. F. (1991). "The Arithmetic of Active Management." Financial Analysts Journal 47(1), 7–9. doi:10.2469/faj.v47.n1.7
  2. Ilmanen, A. (2011). Expected Returns: An Investor's Guide to Harvesting Market Rewards. Wiley. ISBN 978-1-119-99072-7
  3. Grinold, R. C. & Kahn, R. N. (2000). Active Portfolio Management, 2nd ed. McGraw-Hill. ISBN 978-0-07-024882-3
  4. Pedersen, L. H. (2018). "Sharpening the Arithmetic of Active Management." Financial Analysts Journal 74(1), 21–36. doi:10.2469/faj.v74.n1.4
  5. Asness, C. S., Moskowitz, T. J. & Pedersen, L. H. (2013). "Value and Momentum Everywhere." Journal of Finance 68(3), 929–985. doi:10.1111/jofi.12021

Sources for the numbers

  1. Mehra, R. & Prescott, E. C. (1985). "The Equity Premium: A Puzzle." Journal of Monetary Economics 15(2), 145–161. doi:10.1016/0304-3932(85)90061-3
  2. Jegadeesh, N. & Titman, S. (1993). "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency." Journal of Finance 48(1), 65–91. doi:10.1111/j.1540-6261.1993.tb04702.x
  3. Nagel, S. (2012). "Evaporating Liquidity." Review of Financial Studies 25(7), 2005–2039. doi:10.1093/rfs/hhs066
  4. Katona, Z., Painter, M. O., Patatoukas, P. N. & Zeng, J. (2025). "On the Capital Market Consequences of Big Data: Evidence from Outer Space." Journal of Financial and Quantitative Analysis 60(2), 551–579. doi:10.1017/S0022109023001448
  5. Aquilina, M., Budish, E. & O'Neill, P. (2022). "Quantifying the High-Frequency Trading 'Arms Race'." Quarterly Journal of Economics 137(1), 493–564. doi:10.1093/qje/qjab032
  6. Brunnermeier, M. K., Nagel, S. & Pedersen, L. H. (2008). "Carry Trades and Currency Crashes." NBER Macroeconomics Annual 23, 313–347. doi:10.1086/593088
  7. Daniel, K. & Moskowitz, T. J. (2016). "Momentum Crashes." Journal of Financial Economics 122(2), 221–247. doi:10.1016/j.jfineco.2015.12.002
  8. Carr, P. & Wu, L. (2009). "Variance Risk Premiums." Review of Financial Studies 22(3), 1311–1341. doi:10.1093/rfs/hhn038
  9. Credit Suisse AG (2018). "Credit Suisse AG Announces Event Acceleration of its Daily Inverse VIX Short Term ETNs" (XIV), press release, 6 February 2018. https://mondovisione.com/news/credit-suisse-ag-announces-event-acceleration-of-its-daily-inverse-vix-short-ter-201826/
  10. Fama, E. F. & French, K. R. (2010). "Luck versus Skill in the Cross-Section of Mutual Fund Returns." Journal of Finance 65(5), 1915–1947. doi:10.1111/j.1540-6261.2010.01598.x
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