Research
My research asks which parts of a factor model’s answer (its factors, its alphas, its portfolios) reflect information in the data, and which reflect only how the model measures that information.
A factor model explains many stocks’ returns with a few common drivers. To say how much of a stock’s expected return each driver explains, a model must first decide when two combinations of characteristics are really different, and how large each one is. The rule it uses for that is the ruler: technically, a metric (an inner product) on characteristic space.
Give each characteristic a weight and add them up: every stock gets a score. Such a weighted combination is called a direction (it is not a portfolio). The plain ruler (the Euclidean or identity metric) judges a direction by its weights alone. The characteristic ruler (the Gram metric) judges it by the scores it gives real stocks: how large they typically are, and whether two directions’ scores line up across stocks, high on the same stocks and low on the same stocks.
The ruler is the convention a model uses to measure how large a direction is and how much two directions overlap. Two value signals that pick out the same cheap stocks count as overlapping under the characteristic ruler but as unrelated under the plain ruler.
Each paper holds everything else fixed, changes one choice, and asks what moves. Read in the order below, they go from choosing the ruler, to reading the model’s alpha (the part of expected return its factors leave unexplained), to the holdings an investor ends up with.
- Which ruler? → Geometric Framework (earlier, broader version)
- What does the ruler decide, and how reliable is the answer? → Characteristic-Space Metrics
- What does a smaller alpha mean? → Interpreting Pricing Errors
- Same forecast, different portfolio? → Same Forecasts, Different Portfolios
- Same target, different stocks? → The Ruler Matters Job market paper
Short on time? Start with the job market paper.
Job market paper
Question 5 of 5: Same target, different stocks?
The Ruler Matters: Factor Exposures, Attribution, and Portfolio Risk
Same target, different stocks: a factor model that fits returns better need not represent the positions an investor cares about, and whether a hedge removes the model’s factor loadings depends on the ruler that measures them. Expanding the library from 36 to 153 characteristics raises in-sample fit, yet the model’s loadings then represent momentum and size positions less fully; hedging a value position with the plain ruler leaves, on average, 46.10% of the hedged position, by squared stock weight, still on the model’s factor loadings (pp. 6–7).
In more detail
The paper measures the characteristic ruler directly on the stocks, as the cross-sectional second moment of their characteristics (Section I.A, p. 4), and asks how much of a familiar position (book-to-market, momentum, larger-firm size) an estimated loading space from instrumented principal component analysis (IPCA) can reproduce. With three latent factors and six observed ones (market, size, value, profitability, investment and momentum) held fixed, expanding the Jensen–Kelly–Pedersen library from 36 to 153 characteristics raises total stock-return R² from 17.01% to 17.70% (Table II, p. 18). The share of the momentum position that the loading space represents falls from 82.27% to 62.33%, and that of size from 90.17% to 71.93% (p. 22). Reallocating exposure shared by observed and latent factors changes their reported contributions without changing the joint fit: for size, the observed factors’ contribution rises from 1.38 to 45.53 basis points a month while the total stays at 2.16 (Section VII.A, p. 38). With the coefficients fixed, portfolios that measure exposure on the current month’s stocks rather than on each March’s have 26.62% to 69.39% lower subsequent variance at the same characteristic tilt in twelve exploratory comparisons; nine of the twelve simultaneous intervals exclude zero, and mean returns and trading do not improve uniformly (Section VI.B, p. 34). The 46.10% measures how fully a hedge removes the loadings, not a reduction in return variance (p. 7).
Working papers
Question 1 of 5: Which ruler?
A Geometric Framework for Identification in Characteristic-Based Factor Models
Same data, different factors: which combinations of characteristics count as distinct depends on the ruler. The paper takes the ruler (the Gram metric) from how characteristics spread and overlap across stocks, and builds identification, no-arbitrage restrictions and inference on it.
Earlier, broader version of Characteristic-Space Metrics · grows out of the PhD thesis · listed on SSRN under the name Yang Liu.
In more detail
Take it from the data: the long-run average of how characteristics spread and overlap across stocks. The paper’s estimator, NA-IPCA, is instrumented principal component analysis (IPCA) rebuilt on that ruler. It caps how much expected return the intercept may carry; the cap is the “no-arbitrage” in the name. IPCA is the special case with the plain ruler and a cap too loose to matter. In simulations calibrated to U.S. stocks, where the true priced directions are placed by design on the combinations along which characteristics vary most, NA-IPCA recovers them (subspace distance about 0.01, on a scale from 0 for exact recovery to about 1.73 for entirely wrong directions). An IPCA benchmark with no intercept block does not (about 0.73). This design does not separate the ruler from the missing intercept block. In the paper’s efficiency and stress-test simulations, IPCA comes close as well, and the two estimators’ simulated out-of-sample fit is essentially the same.
Question 2 of 5: What does the ruler decide, and how reliable is the answer?
Characteristic-Space Metrics in Factor Models: Identification and Inference
Same fitted returns, different split: the ruler decides how returns are credited to latent factors (common drivers inferred from the data), observed factors such as the market, and an intercept (the part of expected return that no factor in the model explains). When the ruler is estimated from the data, its error belongs in the confidence intervals.
Listed on SSRN under the name Yang Liu.
In more detail
The ruler decides the split, not the fit. Models with identical fitted returns can credit them differently to latent factors, observed factors and an intercept, and R² cannot choose between them. A convention does. Because the ruler is estimated from the same data, its error belongs in the confidence intervals. In a stylised simulation built to isolate this mechanism (12 characteristics, 100 stocks, 960 time periods, 2,000 replications; not calibrated to U.S. data), nominal 95% confidence intervals (meant to contain the true value 95 times in 100) contain it only about 89–92 times in 100 when that error is left out, and about 94–95 times when it is included.
Question 3 of 5: What does a smaller alpha mean?
Interpreting Estimated Pricing Errors: Evidence from Characteristic-Based Return Forecasts
Same ruler, different target: an alpha can shrink because the factors explain more, or because the expected-return estimate they must explain (the target) has changed. Pairing each model’s estimate with each model’s factors on the same stocks, the paper finds that the large alpha gap between equal-weighted and value-weighted training mainly reflects different targets.
In more detail
In the paper’s comparison, a smaller alpha mainly reflects a change in the thing being explained, not factors that explain more. A characteristic-based model’s forecast splits into the part its factors can produce and a leftover, the alpha. The same model is trained twice, once weighting stocks equally and once by market capitalisation. Both are scored with the same ruler, on the same stocks and with the same scoring weights. The equal-weighted version’s alpha has a root-mean-square size of 117.37 basis points (hundredths of a percentage point) a month; the value-weighted version’s has 77.27. Giving the equal-weighted version only the other’s factors leaves its alpha at 116.67; giving it only the other’s expected-return estimate brings it to 75.02. Choosing the model with the smaller alpha each year adds little to forecast accuracy. Shrinking the alpha component toward zero gives more accurate forecasts than keeping it whole in 20 of 24 test settings, but more accurate than a simple average of the two models’ forecasts in only 8. The evidence is exploratory.
Question 4 of 5: Same forecast, different portfolio?
Same Forecasts, Different Portfolios: Factor Normalization and Regularization
Same forecasts, different portfolios: the small constant a portfolio rule adds to each factor’s risk is a penalty on factor bets, written in the ruler’s units, so one number gives two portfolios. Written in one set of units, or removed, the penalty gives one portfolio.
In more detail
Two versions of one IPCA fit, estimated separately under the plain and the characteristic ruler, give stock forecasts that differ by a median of 4.8 parts in a million (Table II, p. 24). With the same penalty number, their portfolios have Sharpe ratios (average excess return per unit of volatility) of 0.69 and 1.07 before trading costs over 594 months (Table I, p. 19). Written in one set of units, the penalty gives one portfolio; without it, both versions hold the same portfolio, at 1.62 (Table I, p. 19). Separately, using the full fitted mean, including the intercept the model already estimates (the part of expected return that no factor in the model explains), gives a Sharpe ratio of 2.51 before trading costs that barely moves with the penalty. Its gain over the best portfolio without an intercept is 0.89, 95% interval [0.59, 1.26], and 0.57 [0.29, 0.89] net of 25 basis points per dollar traded (Table III, p. 30). The gain weakens over time: since October 2007 its net advantage has an interval that includes zero (p. 7). This second result uses a different model, objective and sample from those in Interpreting Pricing Errors, so neither result confirms or contradicts the other.
Research in progress
Exposure or Intercept? Identifying Corporate Bond Premia with No-Arbitrage IPCA
With Paolo Zaffaroni.
Using No-Arbitrage IPCA (built on instrumented principal component analysis) on public TRACE data, the paper asks whether bond characteristics predict returns through factor exposures or through an intercept, and how much of that split depends on the choice of ruler (the metric) and on the cap on how much the intercept may carry.
Portfolio construction from linear out-of-sample return forecasts
An ongoing research collaboration with Prof. Semyon Malamud (EPFL and Swiss Finance Institute) on portfolio construction from linear out-of-sample return forecasts. The project complements his work on nonlinear return prediction and began after a seminar presented at EPFL in June 2026 at his invitation.
PhD thesis
No-Arbitrage IPCA: A Framework for Cross-Sectional Asset Pricing
The thesis develops NA-IPCA; the Geometric Framework paper sets out the theory behind it. Imperial College London, 2026. Available on request. The estimators are implemented in the NA-IPCA toolkit; see Data & Code.
Method showcases. Separate from the papers, the Research Toolkit shows reusable analysis tools at work on archived examples, each with its inputs, tests and limits stated:
Portfolio risk diagnostics: Return-State Behavior Profile, how a return stream behaved across market states (archived backtest, not actual trading and not audited; monthly data Apr 2007 – Nov 2025).
Macro forecast evaluation: Macro Forecast Evaluation & Regime Diagnostics, comparing forecast models on matched pseudo-out-of-sample windows, with the look-ahead documented (February 2026 research snapshot).