Part X · Edges, one by one · Chapter 10F

Equity factors

Quality, momentum, value, low volatility and size

A factor tilt modulates the equity sleeve to capture a documented, systematic excess return. The five factors here are quality, momentum, value, low volatility and size, each with its own mechanism, decay risk and drawdown profile. Most had their best decade in academic papers, while live retail experience is humbler. They are worth moderate allocations as tilts rather than concentrations, although harvesting them takes patience measured in decades.

1 · What a factor is

An equity factor is a trait by which stocks can be ranked, where the ranking has predicted returns. Researchers measure it with a long-short portfolio, which buys the stocks at the favored end of the ranking and sells short an equal dollar amount of those at the other end. The difference in return between the two sides is the factor's return in month $t$:

Here $R^{\text{top}}_t$ and $R^{\text{bottom}}_t$ are the average returns of the two groups. Suppose you hold $10,000 of the top group and short $10,000 of the bottom group, which return 1.4% and 0.2% in a month. You make $140 on one side and lose $20 on the other, so $F_t = 1.2\%$. Because one side offsets the other, the factor's return usually depends little on whether the market rose or fell, a rule that momentum breaks below. Each factor in this chapter is a version of (10F.1) with its own ranking.

The figure builds a factor from fifty made-up stocks and runs it for twenty years. At the default link, its true Sharpe ratio is 0.40, about what papers report for value. Drag the link down, and the longest stretch below an old peak grows. Press new stocks for another twenty years of the same factor.

trait link

0.15% per σ

stocks in each group 5 10 25

new stocks

true Sharpe ratio = 0.40

t = 2 takes 25 years

this run: Sharpe 0.40, t = 1.81

longest below a peak: 6.3 years

A tilt is the version you can hold. Instead of a long-short portfolio, you own a fund that holds more of the favored stocks than their share of the index, and nothing short. Its return is roughly the market's plus a fraction of the factor's.

How to read the evidence

Factor papers report monthly returns, Sharpe ratios and the t-statistic you know from trials, an estimated effect divided by its standard error. For a factor, the t-statistic has a simple link to the Sharpe ratio.

From a Sharpe ratio to a t-statistic

Step 1: the t-statistic of a mean. With $n$ monthly returns of mean $m$ and standard deviation $s$, the standard error of the mean is $s/\sqrt{n}$, so $t = m\sqrt{n}/s$. A long-short factor is already an excess return, so $m/s$ is its monthly Sharpe ratio and $t = \text{SR}_{\text{month}}\sqrt{n}$.

Step 2: annualize. The annual Sharpe ratio is $\text{SR}_{\text{month}}\sqrt{12}$ (Chapter 1D), and $Y$ years hold $n = 12Y$ months. Substituting both, the factors of $\sqrt{12}$ cancel and leave the result below.

Here SR is the annual Sharpe ratio and $Y$ is the number of years of data. A factor with a Sharpe ratio of 0.4 reaches $t = 2$ after 25 years, since $0.4 \times \sqrt{25} = 2$, but shows only $0.4 \times 3 = 1.2$ after 9 years. Published t-statistics come from particular samples and constructions, so use (10F.2) for orders of magnitude rather than to recompute them.

From paper to account

The academic Sharpe ratio is the start of a chain rather than what you earn. Chapter 13A sets out the retail haircut, the part of the Sharpe ratio lost at each step from a paper portfolio to a live fund and then to your own account. The Edge Book gives the chain for three of the factors, all as Sharpe ratios:

Value starts almost level with quality on paper, yet it can end in an account with as little as a third of quality's figure, for reasons the sections below give.

2 · Quality

The quality factor is multi-dimensional. In a 2019 paper in the Review of Accounting Studies (RAS), Asness, Frazzini and Pedersen build their quality minus junk (QMJ) factor on four pillars:

• Profitability: gross profits divided by assets, return on assets (ROA) and return on equity (ROE).
• Growth: the stability and the rate of growth of profits.
• Safety: low beta, low leverage and low volatility of earnings.
• Payout: stable dividends and share buybacks.

Holding high-quality stocks long and junky ones short gives the QMJ factor.

The evidence

• Novy-Marx (2013), "The Other Side of Value: The Gross Profitability Premium", in the Journal of Financial Economics (JFE), won the 2013 Fama-DFA Prize.
• Asness, Frazzini and Pedersen (2019) report a QMJ alpha, the return beyond what standard factor models explain (Chapter 1D), of 0.66% a month in the US and 0.45% globally, across about 24 countries. The US figure is roughly 8% a year.
• Quality was the single factor that did not crash through the value drawdown of 2017 to 2022, a key test in real markets rather than in a backtest.

Why it exists

The mechanism is mixed. As a risk premium, quality firms carry lower distress risk, for which investors might rationally pay. As a behavioral effect, investors overpay for a narrative of growth and underpay for boring profitability. The mixed source explains why quality has held up better than the purely behavioral factors: the risk-based part of the premium does not decay, while the behavioral part might (Chapter 1G).

3 · Momentum

This is cross-sectional momentum, which compares stocks with one another. Its cousin, the time-series momentum of Chapter 10D, trades each asset on its own past.

The factor is called UMD, for up minus down. It is (10F.1) with past returns as the ranking:

The evidence

• Jegadeesh and Titman (1993, Journal of Finance, JF) is the canonical study. UMD earned about 1% a month, with a t-statistic of about 3 and a Sharpe ratio of about 0.5 unlevered, that is, without borrowing to scale the portfolio up.
• Asness, Moskowitz and Pedersen (2013, JF), "Value and Momentum Everywhere", found momentum pervasive across 8 asset classes globally.
• Daniel and Moskowitz (2016, JFE) add the critical caveat that momentum lost 73% from March to May 2009, in the V-bottom that ended a bear market.
• Jensen, Kelly and Pedersen (2023, JF) document a decline of about 47% after publication across hundreds of factors.

Why it exists

Momentum is mostly behavioral, Source 2 of Chapter 1G. Investors under-react to news because they anchor on old information, then over-react at trend extremes by chasing what has already risen. Momentum is possibly also compensation for crash risk in V-shaped reversals, whose mechanics a toy example shows.

4 · Value

Modern variations may control for sector exposure and rank on a composite: E/P (earnings to price), B/M, FCF/P (free cash flow, FCF, to price) and EV/EBITDA. The last divides enterprise value (EV), the shares plus debt less cash, by earnings before interest, taxes, depreciation and amortization (EBITDA).

The evidence

• Fama and French (1993, JFE) is the canonical study. HML earned 0.40% to 0.50% a month, with a t-statistic of about 3 and a long-run Sharpe ratio of about 0.4.
• The years 2010 to 2020 brought the worst documented value drawdown in modern history. AQR Equity Market Neutral lost about 38% from peak to trough by December 2020.
• Arnott, Harvey, Kalesnik and Linnainmaa (2021) wrote "Reports of Value's Death May Be Greatly Exaggerated" in the Journal of Portfolio Management (JPM). They attribute the drawdown to a widening of the valuation spread, which is cyclical, not to fundamentals.
• The rebound followed. AQR Absolute Return gained 43.5% in 2022 and 18.5% in 2023, while the Russell 1000 Value index gained 24.9% in 2024.

The valuation spread is how many times more expensive the expensive stocks are than the cheap ones. Write each group's price as a multiple $m$ of its book value. The spread is then $V = m_e/m_c$, where $e$ marks the expensive group and $c$ the cheap one. From the start of a period, 0, to its end, 1, regrouping the multiples and the book values gives

where $R$ is each group's return and $g$ the growth of its book value. Value's return against expensive stocks is the change in the spread times the ratio of their fundamental growth.

Why it exists

The source of the value premium is disputed among three views:

• The risk-premium view of Fama and French: value stocks bear the risk of financial distress.
• The behavioral view of Lakonishok, Shleifer and Vishny: investors over-extrapolate growth and underpay for boring value.
• The intangibles view of Lev and Srivastava: book-to-market is mismeasured as intangible assets have grown. An HML adjusted for intangibles works better after 1993.

Accounting records most spending on research, software and brands as an expense rather than an asset. A company built that way reports a small book value, so the sort files it as expensive even when it is not. Probably all three views hold: the source is mixed, mostly behavioral and partly risk-based.

5 · Low volatility and betting against beta

The textbook model is the capital asset pricing model (CAPM). It says a stock's expected excess return is its beta times the market's:

Here $R_i$ is the stock's return, $R_m$ the market's, $r$ the risk-free rate and $\beta_i$ the stock's beta.

The factor scales each side to a beta of 1 before taking the difference:

Here $L$ marks the low-beta portfolio and $H$ the high-beta one, and the portfolio's beta is $\beta_L/\beta_L - \beta_H/\beta_H = 0$. With the example's numbers and $r = 0$, the long side earns $8\%/0.6 = 13.3\%$ and the short side costs $11\%/1.5 = 7.3\%$. BAB earns 6.0% a year with no market exposure, though only after borrowing 67 cents for every dollar in the low-beta side, since $1/0.6 = 1.67$.

The evidence

• Frazzini and Pedersen (2014, JFE), "Betting Against Beta", report a US BAB Sharpe ratio of 0.78 over 1926 to 2012.
• Novy-Marx and Velikov (2022, JFE), "Betting Against Betting Against Beta", add a critical caveat. The Frazzini-Pedersen Sharpe ratio is largely an artifact of two choices: a non-standard rank weighting that overweights micro- and nano-caps, the smallest companies, and a specific beta shrinkage. In a value-weighted version the BAB Sharpe ratio falls to about 0.49, still positive at a t-statistic of 3.5 and 76 bps a month, but materially smaller.
• Bali and colleagues (2017) find that lottery demand for high-beta, lottery-like stocks creates the same effect.

A value-weighted portfolio weights each stock by its market capitalization, while a rank weighting gives tiny companies as much weight as giants. Shrinkage pulls each noisy beta estimate part of the way toward 1. Such choices are ordinary, which makes a result that depends on them fragile.

Why it exists

Low volatility is mainly an institutional friction, Source 3 of Chapter 1G. Leverage-constrained investors such as mutual funds and pensions cannot borrow, so they buy high-beta stocks to get more equity exposure, and high beta becomes overpriced. Lottery demand for high-beta names adds a behavioral overlay.

6 · Size, once the junk is removed

Size is usually measured as SMB, for small minus big. Conditioned on quality, it compares small, high-quality companies with large ones:

Asness, Frazzini, Israel, Moskowitz and Pedersen (2018, JFE), "Size Matters, If You Control Your Junk", find that the premium nearly doubles after controlling for quality. The source is mixed: a risk premium for the illiquidity and distress risk of small companies, plus behavioral underpricing of obscure ones. Conditioning on quality removes the "junk" drag.

7 · Combining the factors

Within the equity allocation, the Edge Book suggests this set of tilts:

The factor tilts together take 25% to 40% of the equity sleeve. Do not go higher, because the tracking error against broad equity becomes large while the diversification benefit between factors saturates. Read the table as a range for each tilt rather than as a sum. A total held to 25% to 40% leaves 60% to 75% for the index, more than the remainder column says, and AVUV appears in two rows, so count it only once. The Edge Book's own summary of the category, in Chapter 10A, puts the tilts lower, at 15% to 30% of the sleeve.

Factors do not all underperform at the same time:

• 2020 to 2021: momentum strong, value weak, quality held up.
• 2022: value strong, momentum mixed, quality held up.
• 2008: low volatility best, momentum and value badly hit.
• The 1990s: value and momentum strong, low volatility weaker.

Diversifying across 4 to 5 factors smooths the path materially, while a single-factor concentration produces multi-year drawdowns that test patience. The algebra of Chapter 1F shows how much combining helps, and where the help stops.

The Sharpe ratio of an equal mix of factors

Take $N$ factors, each with volatility $\sigma$ and Sharpe ratio SR, with the same correlation $\rho$ between every pair. An equal-weight mix holds $N$ variance terms of $\sigma^2/N^2$ and $N(N-1)$ covariance terms of $\rho\sigma^2/N^2$, so its variance is

$$\frac{\sigma^2}{N}\big(1 + (N-1)\rho\big)$$ (10F.9)

The mix earns the same excess return per dollar as each factor. Dividing that return by the square root of the variance gives the result below.

With a hypothetical Sharpe ratio of 0.4 for each factor and a correlation of 0.2 between them, four factors give $0.4 \times \sqrt{4/1.6} = 0.63$. As $N$ grows, the fraction under the root approaches $1/\rho$, so no number of such factors beats $0.4/\sqrt{0.2} = 0.89$. That ceiling is what "the diversification benefit saturates" means: the fifth factor adds less than the second, and a twentieth adds almost nothing.

What the whole set adds

An academic combined factor portfolio earns about 2% to 3% a year of excess return over cap weighting, with a Sharpe ratio of about 0.6. A realistic retail implementation earns about 0.5% to 1.5% a year, with a Sharpe ratio of about 0.2 to 0.4 above cap weighting, which is the information ratio of Chapter 1D. The retail haircut is large for three reasons: the ETFs implement the factors only partially, tax friction matters and behavioral exits during multi-year drawdowns are common.

The Edge Book also says that combining quality, momentum, value and low volatility can add 0.2 to 0.4 of Sharpe ratio to the equity sleeve. Read that as the tilts' information ratio, not as a rise in the whole sleeve's ratio.

That is why the tilts are worth doing only with a multi-decade commitment and an explicit understanding of the path.

8 · Why not time the factors

Factor performance is highly cyclical and difficult to predict. Most of the academic literature finds that factor timing, shifting weight toward whichever factor looks set to do well, adds no value relative to a static, diversified factor allocation. The exception is timing value by its valuation spread when the spread is extreme, and even that is contentious. Default to static, diversified tilts.

Chapter 10G turns to calendar and flow effects: the FOMC cycle, end-of-quarter (EOQ) rebalancing, options expiration (OPEX) and the seasons. They are smaller than the factor tilts and modulate existing positions rather than holding a sleeve of their own.

9 · Check yourself