Mathematics

The difference of a fast and a slow exponential average of the close, its own signal line and the histogram between them.

\text{MACD}_t = \EMA_{12}(P)_t - \EMA_{26}(P)_t, \qquad S_t = \EMA_{9}(\text{MACD})_t, \qquad H_t = \text{MACD}_t - S_t

(3.6)

Relative strength index (Wilder)

Wilder-smoothed average gains over average losses, seeded with the simple means of the first n changes.

G_t = \frac{(n-1)\,G_{t-1} + \max(\Delta P_t, 0)}{n}, \qquad L_t = \frac{(n-1)\,L_{t-1} + \max(-\Delta P_t, 0)}{n}

(3.7)

\text{RSI}_t = 100 - \frac{100}{1 + G_t/L_t} \quad (L_t = 0:\ 100,\ \text{or } 50 \text{ when } G_t = 0)

Parameters of Relative strength index (Wilder)
Symbol	Value	Meaning
$n$	14	sessions

— Wilder, J. W. (1978). New Concepts in Technical Trading Systems.

Stochastic oscillator

Where the close sits in the recent high–low range, and its three-session average.

\%K_t = 100\,\frac{P_t - \min_{14} L}{\max_{14} H - \min_{14} L}, \qquad \%D_t = \SMA_3(\%K)_t

(3.8)

Directional movement, ATR and ADX

Directional movements and the true range, Wilder-summed, give the directional indicators; the average directional index is the Wilder average of their normalised gap.

\text{TR}_t = \max\big(H_t - L_t,\ |H_t - P_{t-1}|,\ |L_t - P_{t-1}|\big), \qquad \text{ATR}_t = \text{Wilder}_{14}(\text{TR})_t

(3.9)

u_t = H_t - H_{t-1},\ \ d_t = L_{t-1} - L_t, \qquad +\text{DM}_t = u_t\,\mathbb{1}[u_t > d_t,\ u_t > 0], \quad -\text{DM}_t = d_t\,\mathbb{1}[d_t > u_t,\ d_t > 0]

S^{X}_t = S^{X}_{t-1} - \frac{S^{X}_{t-1}}{n} + X_t, \qquad \pm\text{DI}_t = 100\,\frac{S^{\pm\text{DM}}_t}{S^{\text{TR}}_t}, \qquad \text{DX}_t = 100\,\frac{|{+\text{DI}_t} - {-\text{DI}_t}|}{{+\text{DI}_t} + {-\text{DI}_t}}

\text{ADX}_t = \text{Wilder}_{n}(\text{DX})_t

(3.10)

Parameters of Directional movement, ATR and ADX
Symbol	Value	Meaning
$n$	14	sessions

— Wilder, J. W. (1978). New Concepts in Technical Trading Systems.

Bollinger bands

Coverage screener Company · Price & trend

A moving average with bands two population standard deviations away, the close's position within them, and their width.

m_t = \SMA_{20}(P)_t, \quad s_t = \sqrt{\tfrac{1}{20}\textstyle\sum_{i=0}^{19}(P_{t-i} - m_t)^2}, \quad U_t, L_t = m_t \pm 2 s_t

(3.11)

\%B_t = \frac{P_t - L_t}{U_t - L_t}, \qquad \text{BW}_t = \frac{U_t - L_t}{m_t}

Trend regression

Ordinary least squares of the log close on time over the last 63 sessions: the slope, annualised, its t-statistic and the fit.

\ln P_{t-k} = a + b\,(-k) + e_k, \quad k = 0,\dots,62; \qquad \text{slope}^{\text{ann}} = 252\,\hat b, \qquad t_b = \frac{\hat b}{\widehat{\text{se}}(\hat b)}

(3.12)

A close that did not change over the 63 sessions has no trend to test (t is 0/0): the slope, t-statistic and R² are shown as unavailable.

Donchian channels

Company · Levels & volume

The highest high and lowest low over a window.

\text{DC}^{\uparrow}_t = \max_{0\le i<n} H_{t-i}, \qquad \text{DC}^{\downarrow}_t = \min_{0\le i<n} L_{t-i}

RS

Relative strength

Performance against the benchmark and the coverage: differences of returns, the log price ratio's level against its own history, and its change.

Relative performance by horizon

Relative strength Coverage screener

The company's return less the benchmark's over the same sessions, at each horizon, ranked within the coverage.

\text{RS}^{(h)}_t = R_{t,h} - R^{B}_{t,h}, \qquad \text{RS20}_t = \text{Momentum20}_t - \text{Momentum20}^{B}_t\ \ \text{(pp)}

(4.1)

Rotation map: relative-strength level and momentum

Relative strength

The log price ratio's z-score against its own trailing 126 sessions (level), and its change over ten sessions (momentum); weekly trails show the path.

\ell_t = \ln\frac{P_t}{B_t}, \qquad z_t = \frac{\ell_t - \bar\ell_{t,126}}{\sd_{126}(\ell)_t}, \qquad \dot z_t = z_t - z_{t-10}

(4.2)

Parameters of Rotation map: relative-strength level and momentum
Symbol	Value	Meaning
$126$	sessions	level window
$10$	sessions	momentum lag
$8$	weeks	trail length

VOL

Volatility

Five realised estimators that use different parts of the daily bar, the RiskMetrics exponentially weighted estimator, GARCH(1,1) by maximum likelihood, and volatility cones. All annualised with √252.

Realised volatility estimators

Company · Volatility Coverage screener

Over a window of n sessions. Range estimators use the high and low; Rogers–Satchell allows drift; Yang–Zhang adds the overnight gap and is unbiased under both.

\hat\sigma^2_{\text{CC}} = \frac{1}{n-1}\sum_{t}(r_t - \bar r)^2

(5.1)

\hat\sigma^2_{\text{P}} = \frac{1}{4n\ln 2}\sum_t \ln^2\!\frac{H_t}{L_t}

(5.2)

\hat\sigma^2_{\text{GK}} = \frac1n\sum_t\left[\tfrac12\ln^2\!\frac{H_t}{L_t} - (2\ln2 - 1)\ln^2\!\frac{P_t}{O_t}\right]

(5.3)

\hat\sigma^2_{\text{RS}} = \frac1n\sum_t\left[\ln\frac{H_t}{P_t}\ln\frac{H_t}{O_t} + \ln\frac{L_t}{P_t}\ln\frac{L_t}{O_t}\right]

(5.4)

\hat\sigma^2_{\text{YZ}} = \hat\sigma^2_{o} + k\,\hat\sigma^2_{c} + (1-k)\,\hat\sigma^2_{\text{RS}}, \qquad k = \frac{0.34}{1.34 + \frac{n+1}{n-1}}, \quad \hat\sigma^2_o = \Var\!\left(\ln\tfrac{O_t}{P_{t-1}}\right),\ \hat\sigma^2_c = \Var\!\left(\ln\tfrac{P_t}{O_t}\right)

(5.5)

Parameters of Realised volatility estimators
Symbol	Value	Meaning
$n$	20	sessions (screener and company page)

— Parkinson, M. (1980). Journal of Business, 53(1).
— Garman, M. B. & Klass, M. J. (1980). Journal of Business, 53(1).
— Rogers, L. C. G. & Satchell, S. E. (1991). Annals of Applied Probability, 1(4).
— Yang, D. & Zhang, Q. (2000). Journal of Business, 73(3).

Exponentially weighted volatility (RiskMetrics)

Company · Volatility

Element t is the forecast for the next session from returns through t.

\sigma^2_{t+1\mid t} = \lambda\,\sigma^2_{t\mid t-1} + (1-\lambda)\,r_t^2, \qquad \sigma^2_{20\mid 19} = \frac{1}{20}\sum_{t<20} r_t^2

(5.6)

Parameters of Exponentially weighted volatility (RiskMetrics)
Symbol	Value	Meaning
$\lambda$	0.94	decay

— J.P. Morgan/Reuters (1996). RiskMetrics — Technical Document, 4th ed.

GARCH(1,1) by maximum likelihood with variance targeting

Company · Volatility Coverage screener

Conditional variance driven by the last shock and the last variance; the constant is tied to the sample variance so the long-run level is the sample's. Fitted on the latest 504 returns (at least 250) by Nelder–Mead over a reparameterisation that keeps α, β ≥ 0 and α + β < 1, from three starts.

r_t = \mu + \varepsilon_t,\quad \varepsilon_t = \sqrt{h_t}\,z_t,\ z_t\sim\mathcal N(0,1), \qquad h_t = \omega + \alpha\,\varepsilon^2_{t-1} + \beta\,h_{t-1}

(5.7)

\bar\sigma^2 = \frac1n\sum_t\varepsilon_t^2, \qquad \omega = \bar\sigma^2(1-\alpha-\beta), \qquad h_1 = \bar\sigma^2

\mathcal{L}(\alpha,\beta) = -\frac12\sum_{t=1}^{n}\left[\ln 2\pi + \ln h_t + \frac{\varepsilon_t^2}{h_t}\right]

(5.8)

p = 0.9995\,\Lambda(x_1), \quad \alpha = p\,\Lambda(x_2), \quad \beta = p - \alpha, \qquad \Lambda(x) = \frac{1}{1+e^{-x}}

\E\big[h_{T+k}\big] = \bar\sigma^2 + p^{\,k-1}\big(h_{T+1} - \bar\sigma^2\big), \qquad \sigma^{\text{ann}}(H) = \sqrt{\frac{252}{H}\sum_{k=1}^{H}\E[h_{T+k}]}, \qquad t_{1/2} = \frac{\ln\frac12}{\ln p}

(5.9)

Parameters of GARCH(1,1) by maximum likelihood with variance targeting
Symbol	Value	Meaning
$n$	504 (≥ 250)	daily returns in the fit
$p = \alpha+\beta$	< 0.9995	persistence

Symmetric in the sign of the shock (no leverage term); Gaussian innovations understate the tails; re-estimated at each as-of date.

— Bollerslev, T. (1986). Journal of Econometrics, 31(3).
— Engle, R. F. & Mezrich, J. (1996). GARCH for Groups. Risk, 9(8) — variance targeting.
— Nelder, J. A. & Mead, R. (1965). The Computer Journal, 7(4).

Volatility cone

Company · Volatility

For each horizon, the distribution of every overlapping close-to-close volatility in the history, and where the current one sits in it.

\hat\sigma_h(t) = \sqrt{252}\,\sd\big(r_{t-h+1},\dots,r_t\big), \qquad Q_q(h) = \text{quantile}_q\{\hat\sigma_h(t)\}_t, \quad q\in\{0.1, 0.25, 0.5, 0.75, 0.9\}

(5.10)

Parameters of Volatility cone
Symbol	Value	Meaning
$h$	10, 21, 42, 63, 126	sessions

— Burghardt, G. & Lane, M. (1990). How to Tell if Options Are Cheap. Journal of Portfolio Management, 16(2).

RISK

Loss and drawdown

One-session loss measures reported as positive fractions, drawdowns and risk-adjusted ratios. The risk-free rate is taken as zero, and said so.

Value at risk and expected shortfall

Company · Risk Coverage screener

Historical (interpolated sample quantile), Gaussian and Cornish–Fisher value at risk; historical and Gaussian expected shortfall, at confidence c (95% by default).

\text{VaR}^{\text{hist}}_c = -\left[r_{(\lfloor h\rfloor+1)} + (h - \lfloor h\rfloor)\big(r_{(\lfloor h\rfloor+2)} - r_{(\lfloor h\rfloor+1)}\big)\right], \qquad h = (n-1)(1-c)

(6.1)

\text{VaR}^{\mathcal N}_c = -(\hat\mu + \hat\sigma z_{1-c}), \qquad \text{VaR}^{\text{CF}}_c = -(\hat\mu + \hat\sigma \tilde z), \quad \tilde z = z + \frac{(z^2-1)S}{6} + \frac{(z^3-3z)K}{24} - \frac{(2z^3 - 5z)S^2}{36}

(6.2)

\text{ES}^{\text{hist}}_c = -\frac{1}{m}\sum_{i=1}^{m} r_{(i)},\ \ m = \lceil n(1-c)\rceil, \qquad \text{ES}^{\mathcal N}_c = -\hat\mu + \hat\sigma\,\frac{\varphi(z_{1-c})}{1-c}

(6.3)

Parameters of Value at risk and expected shortfall
Symbol	Value	Meaning
$S, K$	sample	skewness and excess kurtosis
$c$	0.95, 0.99	confidence

— Cornish, E. A. & Fisher, R. A. (1938). Revue de l'Institut International de Statistique, 5(4).
— Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9(3).

Drawdowns and the Ulcer index

Company · Risk Monitoring rules

Distance below the running peak, its worst value, non-overlapping peak-to-recovery episodes, and the root-mean-square of percentage drawdowns.

\text{DD}_t = \frac{P_t}{\max_{s\le t} P_s} - 1, \qquad \text{MDD} = \min_t \text{DD}_t, \qquad \text{DD}^{252}_t = \frac{P_t}{\max_{t-251\le s\le t} P_s} - 1

(6.4)

\text{UI} = \sqrt{\frac1n\sum_t \big(100\,\text{DD}_t\big)^2}

(6.5)

— Martin, P. & McCann, B. (1989). The Investor's Guide to Fidelity Funds.

Downside deviation and risk-adjusted ratios

Company · Risk

Annualised, with a zero risk-free rate.

\sigma^{-} = \sqrt{\frac{252}{n}\sum_t \min(r_t - \tau, 0)^2}, \qquad \frac{252\,\bar r}{\sqrt{252}\,\hat\sigma}, \qquad \frac{252\,\bar r}{\sigma^{-}}

(6.6)

— Sortino, F. & van der Meer, R. (1991). Downside Risk. Journal of Portfolio Management, 17(4).

BETA

Market model and time-varying beta

Single-index regressions on the benchmark, rolling and state-space betas, and asymmetric sensitivity.

Market model by ordinary least squares

Company · Factor & beta

The company's daily return on the benchmark's, with classical standard errors.

r_t = \alpha + \beta\,r^B_t + \varepsilon_t, \qquad \hat{\boldsymbol\theta} = (X^\top X)^{-1}X^\top r, \qquad \widehat{\Var}(\hat{\boldsymbol\theta}) = \hat s^2 (X^\top X)^{-1},\ \ \hat s^2 = \frac{\sum\hat\varepsilon_t^2}{n-2}

(7.1)

R^2 = 1 - \frac{\sum\hat\varepsilon_t^2}{\sum(r_t - \bar r)^2}, \qquad \sigma_\varepsilon^{\text{ann}} = \sqrt{252}\,\hat s, \qquad \text{TE} = \sqrt{252}\,\sd(r_t - r^B_t)

(7.2)

\beta^{(w)}_t = \frac{\Cov_w(r, r^B)}{\Var_w(r^B)}

— Sharpe, W. F. (1963). A Simplified Model for Portfolio Analysis. Management Science, 9(2).

Kalman-filter beta

Company · Factor & beta

Alpha and beta follow random walks; the observation noise and the prior come from OLS on the first 60 sessions only, so no later observation leaks backwards.

\mathbf x_t = \begin{pmatrix}\alpha_t\\ \beta_t\end{pmatrix} = \mathbf x_{t-1} + \mathbf w_t,\ \ \mathbf w_t\sim\mathcal N(0, Q); \qquad r_t = \mathbf h_t^\top\mathbf x_t + v_t,\ \ \mathbf h_t = \begin{pmatrix}1\\ r^B_t\end{pmatrix},\ v_t\sim\mathcal N(0, R)

(7.3)

P^- = P + Q, \quad S = \mathbf h^\top P^- \mathbf h + R, \quad K = \frac{P^-\mathbf h}{S}, \quad \mathbf x \leftarrow \mathbf x + K\big(r_t - \mathbf h^\top\mathbf x\big), \quad P \leftarrow P^- - K\,\mathbf h^\top P^-

(7.4)

Parameters of Kalman-filter beta
Symbol	Value	Meaning
$Q$	diag(1e−9, 1e−4)	state drift (alpha, beta)
$R$	OLS residual variance	first 60 sessions
$\mathbf x_0, P_0$	OLS estimates, squared SEs	first 60 sessions

— Kalman, R. E. (1960). Journal of Basic Engineering, 82(1).

Up- and down-market beta and capture

Company · Factor & beta

Betas and average-return ratios on sessions where the benchmark rose and where it fell (each needs 20 sessions).

\beta^{\pm} = \frac{\Cov(r_t, r^B_t \mid r^B_t \gtrless 0)}{\Var(r^B_t\mid r^B_t \gtrless 0)}, \qquad \text{Capture}^{\pm} = \frac{\E[r_t\mid r^B_t\gtrless0]}{\E[r^B_t\mid r^B_t\gtrless0]}

(7.5)

COV

Correlation structure and coverage risk

How the covered companies move together: shrinkage covariance, principal components, clustering, a risk decomposition of the equal-weight basket, breadth and dispersion.

Ledoit–Wolf shrinkage

Risk model

The sample covariance (denominator T) shrunk towards a scaled identity with the data-driven optimal intensity. Squared Frobenius norms are divided by n, as in the paper.

S = \frac1T X^\top X, \qquad m = \frac{\operatorname{tr} S}{n}, \qquad d^2 = \frac1n\big\lVert S - mI\big\rVert_F^2

(8.1)

\bar b^2 = \frac{1}{T^2}\sum_{k=1}^{T}\frac1n\big\lVert \mathbf x_k\mathbf x_k^\top - S\big\rVert_F^2, \qquad b^2 = \min(\bar b^2, d^2), \qquad \delta = \frac{b^2}{d^2}

\hat\Sigma^{*} = \delta\, m\, I + (1-\delta)\,S

(8.2)

Using ‖xxᵀ − S‖²_F = (xᵀx)² − 2xᵀSx + ‖S‖²_F, the estimator is computed without forming T outer products.

Correlations are ρᵢⱼ = Sᵢⱼ / √(SᵢᵢSⱼⱼ). A company whose price did not move over the window has no defined correlation and is treated as uncorrelated (ρ = 0) in averages, components and clustering.

— Ledoit, O. & Wolf, M. (2004). A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices. Journal of Multivariate Analysis, 88(2).

Principal components and the absorption ratio

Risk model Overview

Eigen-decomposition of the correlation matrix by cyclic Jacobi rotations (sign fixed so each vector's largest entry is positive); the absorption ratio is the share of variance in the first k components.

C = V\Lambda V^\top,\ \ \lambda_1\ge\dots\ge\lambda_n, \qquad \text{AR}_k = \frac{\sum_{i=1}^{k}\lambda_i}{\sum_{i=1}^{n}\lambda_i}, \qquad k = \left\lceil \tfrac{n}{5}\right\rceil

(8.3)

— Kritzman, M., Li, Y., Page, S. & Rigobon, R. (2011). Principal Components as a Measure of Systemic Risk. Journal of Portfolio Management, 37(4).

Correlation distance and average-linkage clustering

Risk model

A metric on correlations, merged agglomeratively by average distance; the leaf order places companies that move together next to each other.

d_{ij} = \sqrt{\tfrac12\,(1 - \rho_{ij})} \in [0,1], \qquad D(\mathcal A, \mathcal B) = \frac{1}{|\mathcal A|\,|\mathcal B|}\sum_{i\in\mathcal A}\sum_{j\in\mathcal B} d_{ij}

(8.4)

— Mantegna, R. N. (1999). Hierarchical Structure in Financial Markets. European Physical Journal B, 11(1).

Euler risk decomposition and diversification ratio

Risk model

Portfolio volatility is homogeneous of degree one in the weights, so it splits exactly into each holding's contribution.

\sigma_p = \sqrt{\mathbf w^\top\Sigma\mathbf w}, \qquad \text{MC}_i = \frac{\partial\sigma_p}{\partial w_i} = \frac{(\Sigma\mathbf w)_i}{\sigma_p}, \qquad \text{CC}_i = w_i\,\text{MC}_i, \qquad \sum_i \text{CC}_i = \sigma_p

(8.5)

\text{DR} = \frac{\sum_i w_i\sigma_i}{\sigma_p}

(8.6)

— Choueifaty, Y. & Coignard, Y. (2008). Toward Maximum Diversification. Journal of Portfolio Management, 35(1).

Breadth, average correlation and dispersion

Overview Regimes & breadth

How many companies are above their averages and at new highs or lows, how correlated they are on average, and how far apart their daily returns are.

\text{B}^{50}_t = \frac1n\sum_i \mathbb 1\big[P_{i,t} > \SMA_{50}(P_i)_t\big], \qquad \text{AD}_t = \text{AD}_{t-1} + \sum_i \sgn(r_{i,t})

(8.7)

\bar\rho = \frac{2}{n(n-1)}\sum_{i<j}\rho_{ij}, \qquad \text{Disp}_t = \sqrt{252}\;\frac{1}{21}\sum_{s=t-20}^{t}\sd_i\big(r_{i,s}\big)

(8.8)

REGM

Regimes: the hidden Markov model

A two-state Gaussian hidden Markov model for daily returns, fitted by expectation–maximisation with scaled forward–backward recursions. State 1 is always the higher-volatility state.

Two-state Gaussian HMM

Regimes & breadth Company · Regime Backtest lab

A Markov chain of states with Gaussian returns in each. Filtered probabilities use returns up to each session only; smoothed ones also use later returns inside the fitted window, never beyond the as-of date.

\Pr(s_t = j \mid s_{t-1} = i) = a_{ij}, \qquad r_t \mid s_t = k \sim \mathcal N(\mu_k, \sigma_k^2), \qquad b_k(r) = \frac{1}{\sigma_k}\varphi\!\left(\frac{r - \mu_k}{\sigma_k}\right)

(9.1)

\hat\alpha_t(j) = \frac{b_j(r_t)\sum_i\hat\alpha_{t-1}(i)\,a_{ij}}{c_t}, \qquad \hat\beta_t(i) = \frac{\sum_j a_{ij}\,b_j(r_{t+1})\,\hat\beta_{t+1}(j)}{c_{t+1}}, \qquad \ln L = \sum_t \ln c_t

(9.2)

\gamma_t(i) \propto \hat\alpha_t(i)\hat\beta_t(i), \qquad \xi_t(i,j) \propto \hat\alpha_t(i)\,a_{ij}\,b_j(r_{t+1})\,\hat\beta_{t+1}(j)

\hat a_{ij} = \frac{\sum_t\xi_t(i,j)}{\sum_t\gamma_t(i)}, \qquad \hat\mu_k = \frac{\sum_t\gamma_t(k)\,r_t}{\sum_t\gamma_t(k)}, \qquad \hat\sigma_k^2 = \max\!\left(\frac{\sum_t\gamma_t(k)(r_t - \hat\mu_k)^2}{\sum_t\gamma_t(k)},\ (0.05\,\hat\sigma)^2\right)

(9.3)

\Pr(s_t = 1 \mid r_{1:t}) = \hat\alpha_t(1), \qquad \Pr(s_t = 1\mid r_{1:T}) = \gamma_t(1), \qquad \E[\text{duration}_k] = \frac{1}{1 - a_{kk}}, \qquad \pi_1 = \frac{a_{01}}{a_{01}+a_{10}}

(9.4)

\delta_t(j) = \max_i\big[\delta_{t-1}(i) + \ln a_{ij}\big] + \ln b_j(r_t) \quad\text{(Viterbi path)}

Parameters of Two-state Gaussian HMM
Symbol	Value	Meaning
$A^{(0)}$	[[0.97, 0.03], [0.10, 0.90]]	initial transitions
$\text{split}$	80th percentile of \|r\|	initial calm / stressed returns
$\text{tolerance}$	1e−8 in ln L, ≤ 300 iterations	convergence

The benchmark regime refits on every benchmark return to the as-of date. The backtest lab's market-regime measure refits every 21 sessions on the latest 504 returns and filters daily, so no session uses a later return.

— Hamilton, J. D. (1989). Econometrica, 57(2).
— Rabiner, L. R. (1989). Proceedings of the IEEE, 77(2).
— Baum, L. E., Petrie, T., Soules, G. & Weiss, N. (1970). Annals of Mathematical Statistics, 41(1).

STAT

Serial dependence and stationarity

Tests of whether returns are independent and whether a series wanders or reverts, with their small-sample corrections.

Autocorrelation and the Ljung–Box test

Company · Statistics Co-movement

\hat\rho_k = \frac{\sum_{t=k+1}^{n}(x_t-\bar x)(x_{t-k}-\bar x)}{\sum_{t=1}^{n}(x_t-\bar x)^2}, \qquad Q = n(n+2)\sum_{k=1}^{m}\frac{\hat\rho_k^2}{n-k} \;\overset{H_0}{\sim}\; \chi^2_m

(10.1)

— Ljung, G. M. & Box, G. E. P. (1978). Biometrika, 65(2).

Augmented Dickey–Fuller test with MacKinnon critical values

The t-ratio of the lagged level in a regression of differences; critical values from MacKinnon's response surfaces at the sample size.

\Delta y_t = a + b\,t + \gamma\,y_{t-1} + \sum_{i=1}^{p}\phi_i\,\Delta y_{t-i} + e_t, \qquad \tau = \frac{\hat\gamma}{\widehat{\text{se}}(\hat\gamma)}

(10.2)

\text{cv}_\alpha(T) = \tau_\infty + \frac{\beta_1}{T} + \frac{\beta_2}{T^2}, \qquad \text{e.g. constant, 5\%: } -2.86154 - \frac{2.8903}{T} - \frac{4.234}{T^2}

(10.3)

Parameters of Augmented Dickey–Fuller test with MacKinnon critical values
Symbol	Value	Meaning
$p$	1	augmenting lags
$\text{deterministic}$	constant (unit-root test); none (Engle–Granger residuals)

— Dickey, D. A. & Fuller, W. A. (1979). Journal of the American Statistical Association, 74(366).
— MacKinnon, J. G. (2010). Critical Values for Cointegration Tests. Queen's Economics Department Working Paper 1227.

Hurst exponent by rescaled range, Anis–Lloyd corrected

The rescaled range grows like n^H; the raw slope is biased upward in small samples, so it is measured against the expected R/S of independent data.

(R/S)_n = \frac{1}{c}\sum_{\text{chunks}}\frac{\max_k\sum_{i\le k}(x_i - \bar x) - \min_k\sum_{i\le k}(x_i-\bar x)}{s}, \qquad n = 8, 16, 32, \dots \le \tfrac{N}{4}

(10.4)

\E[(R/S)_n] = \frac{n-\frac12}{n}\,\frac{\Gamma\!\left(\frac{n-1}{2}\right)}{\sqrt\pi\;\Gamma\!\left(\frac n2\right)}\sum_{i=1}^{n-1}\sqrt{\frac{n-i}{i}}

(10.5)

H = \tfrac12 + \hat\kappa, \qquad \ln(R/S)_n - \ln\E[(R/S)_n] = c + \kappa\ln n + e_n

— Hurst, H. E. (1951). Transactions of the ASCE, 116.
— Anis, A. A. & Lloyd, E. H. (1976). Biometrika, 63(1).
— Peters, E. E. (1994). Fractal Market Analysis.

Lo–MacKinlay variance ratio

Company · Statistics Co-movement

Overlapping q-period variance against q times the one-period variance, with the heteroskedasticity-consistent statistic.

\hat\mu = \frac{\ln P_T - \ln P_0}{T},\quad \hat\sigma^2_a = \frac{1}{T-1}\sum_{t}(d_t)^2,\ d_t = \Delta\ln P_t - \hat\mu, \qquad \hat\sigma^2_c = \frac{1}{m}\sum_{t=q}^{T}\big(\ln P_t - \ln P_{t-q} - q\hat\mu\big)^2

(10.6)

m = q\,(T-q+1)\left(1 - \frac qT\right), \qquad \text{VR}(q) = \frac{\hat\sigma^2_c}{\hat\sigma^2_a}

z^* = \frac{\text{VR}(q) - 1}{\sqrt{\hat\theta/T}}, \qquad \hat\theta = \sum_{j=1}^{q-1}\left(\frac{2(q-j)}{q}\right)^2\hat\delta_j, \qquad \hat\delta_j = \frac{T\sum_t d_t^2\,d_{t-j}^2}{\big(\sum_t d_t^2\big)^2}

(10.7)

— Lo, A. W. & MacKinlay, A. C. (1988). Review of Financial Studies, 1(1).

AR(1) mean-reversion half-life

\Delta y_t = a + b\,y_{t-1} + e_t, \qquad \phi = 1 + \hat b, \qquad t_{1/2} = -\frac{\ln 2}{\ln\phi}\quad (0<\phi<1)

(10.8)

Moments and the Jarque–Bera test

Company · Levels & volume Technical flags

S = \frac{\frac1n\sum(x-\bar x)^3}{\left(\frac1n\sum(x-\bar x)^2\right)^{3/2}}, \qquad K = \frac{\frac1n\sum(x-\bar x)^4}{\left(\frac1n\sum(x-\bar x)^2\right)^{2}} - 3, \qquad \text{JB} = \frac n6\left(S^2 + \frac{K^2}{4}\right)\overset{H_0}{\sim}\chi^2_2

(10.9)

— Jarque, C. M. & Bera, A. K. (1980). Economics Letters, 6(3).

PAIR

Co-movement and cointegration

Same-sector pairs whose log prices share a stationary combination.

Engle–Granger two-step test

Co-movement

A levels regression of one log price on the other, then a unit-root test on its residual against the two-variable critical values. Each leg is also tested for a unit root, since a stationary leg makes the test meaningless.

\ln Y_t = c + h\,\ln X_t + u_t, \qquad \Delta\hat u_t = \gamma\,\hat u_{t-1} + \phi\,\Delta\hat u_{t-1} + e_t, \qquad \tau_{\text{EG}} \text{ vs } \text{cv}^{(2,c)}_\alpha(T)

(11.1)

z^{\text{spread}}_T = \frac{\hat u_T - \bar{\hat u}}{\sd(\hat u)}

(11.2)

Parameters of Engle–Granger two-step test
Symbol	Value	Meaning
$\text{cv}^{(2,c)}_{5\%}$	−3.33613 − 6.1101/T − 6.823/T²	MacKinnon (2010), two variables, constant

— Engle, R. F. & Granger, C. W. J. (1987). Econometrica, 55(2).

LVL

Volume, liquidity and levels

Participation, price impact and the levels price has traded at.

Volume measures

\text{OBV}_t = \text{OBV}_{t-1} + \sgn(P_t - P_{t-1})\,V_t, \qquad z^{V}_t = \frac{\ln V_t - \overline{\ln V}_{[t-60,\,t-1]}}{\sd_{[t-60,\,t-1]}(\ln V)}

(12.1)

\frac{V_t}{\frac1{20}\sum_{i=0}^{19}V_{t-i}} - 1, \qquad \text{UD}_{20} = \frac{\sum_{\Delta P>0}V}{\sum_{\Delta P<0}V}, \qquad \text{ADV}_{20} = \frac1{20}\sum P_tV_t

(12.2)

A session with no volume (a halt) has no logarithm: it is left out of the z-score's baseline, which then needs at least 54 of its 60 sessions, and has no z-score of its own.

— Granville, J. (1963). Granville's New Key to Stock Market Profits.

Amihud illiquidity

Coverage screener

Average absolute return per unit of dollar volume, in basis points per USD 1 million traded.

\text{ILLIQ} = \frac{1}{20}\sum_{t}\frac{10^4\,|r_t|}{P_tV_t/10^6}

(12.3)

— Amihud, Y. (2002). Illiquidity and Stock Returns. Journal of Financial Markets, 5(1).

VWAP, range position, volume profile and pivots

Company · Levels & volume

The volume profile spreads each session's volume evenly over its low–high range; the value area grows from the point of control towards the heavier neighbour until it holds 70% of the volume.

\text{VWAP}_{20} = \frac{\sum \frac{H_t+L_t+P_t}{3}\,V_t}{\sum V_t}, \qquad \text{RP}_{252} = \frac{P_t - \min_{252}P}{\max_{252}P - \min_{252}P}

(12.4)

v(b) = \sum_t V_t\,\frac{\big|[L_t, H_t]\cap b\big|}{H_t - L_t}, \qquad \text{POC} = \argmax_b v(b)

\text{PP} = \frac{H+L+C}{3},\quad R_1 = 2\,\text{PP} - L,\quad S_1 = 2\,\text{PP} - H,\quad R_2 = \text{PP} + (H-L),\quad S_2 = \text{PP} - (H-L)

(12.5)

MON

Monitoring, attribution and ranking

How the research workflow turns measurements into changes for review: a severity for each rule, where a move sits, and a ranking that learns from the team's decisions without hiding large changes.

Rule severity

Changes Monitoring

Each monitoring rule measures how far past its threshold a change is, on a 1–3 scale.

\text{sev} = \clamp_{[1,3]}\!\left(\frac{|\text{magnitude}|}{\text{threshold}}\right) \quad\text{(rule-specific magnitude and threshold; see the rules below)}

(13.1)

Where a move sits

Changes

Over a change's window, the company's log return c is compared with its sector peers' average s and the benchmark's b. A description of where the move sits, not a claim about its cause.

\text{class} = \begin{cases} \text{mixed} & |c| < 0.015 \ \wedge\ |c-b| < 0.015\\ \text{market} & \sgn b = \sgn c \ \wedge\ |c-b| \le 0.35\,|c|\\ \text{company} & \omega \ge 0.65\\ \text{sector} & \omega \le 0.35\\ \text{mixed} & \text{otherwise}\end{cases} \qquad \omega = \frac{|c-s|}{|c-s| + |s-b|}

(13.2)

Learned weights and ranking

Overview · Needs attention Changes

Each decision counts for or against its kind of change, smoothed as if three neutral decisions had already been made. Learned weights reorder but never hide, and a change well past its rule keeps at least its own severity.

w_k = \frac{\sum_{d\in\mathcal D_k}\omega(d)}{|\mathcal D_k| + 3}

(13.3)

\text{score} = \text{sev}\,\big(1 + 0.35\,w_k\big)\cdot\big(1.25\ \text{if a stated priority}\big)\cdot\{1.3,\,1,\,0.75\}_{\text{Titan priority}}, \qquad \text{score} \ge \text{sev}\ \text{if } \text{sev}\ge 1.8

(13.4)

Parameters of Learned weights and ranking
Symbol	Value	Meaning
$\omega(\text{important})$	1	decision weight
$\omega(\text{investigate})$	0.75	decision weight
$\omega(\text{relevant})$	0.5	decision weight
$\omega(\text{watch})$	0.5	decision weight
$\omega(\text{dismissed})$	-0.25	decision weight
$\omega(\text{noise})$	-1	decision weight

An undecided change older than 13 weeks becomes history ("not reviewed") rather than a to-do.

LQ

Liquid Quant

Whether a company's recent relative strength remains positive after accounting for market and sector movements, across a declared grid of estimation choices.

Factor-adjusted relative strength over a grid

Liquid Quant Liquid Quant queue

For each evaluation day d and estimation length L, the company's daily log return is regressed on the factor returns over the L sessions ending at d − 1; the day's adjusted return subtracts the factor exposure only (the intercept carries the stock-specific drift the claim is about). The measure sums W of them.

r_s = a_d + \mathbf b_d^\top \mathbf f_s + e_s,\quad s \in [d-L,\ d-1], \qquad \mathbf f_s = \begin{cases} r^{M}_s & \text{market}\\ (r^M_s,\ r^{S}_s)^\top & \text{market and sector}\end{cases}

(14.1)

u_d = r_d - \hat{\mathbf b}_{d-1}^\top\mathbf f_d, \qquad \mathcal M(L, W) = 100\sum_{d = \tau - W + 1}^{\tau} u_d \quad\text{(log-pp)}

(14.2)

\tilde X = X D^{-1},\ D = \operatorname{diag}\lVert\mathbf x_j\rVert, \quad \tilde X = U\Sigma V^\top, \quad \hat{\boldsymbol\theta} = D^{-1}V\Sigma^{+}U^\top y, \qquad \kappa = \frac{\sigma_{\max}}{\sigma_{\min}} \le 30

(14.3)

\text{supports}(L,W) \iff \mathcal M(L,W) > 10^{-9}, \qquad \text{small} \iff |\mathcal M| \le 0.1

Parameters of Factor-adjusted relative strength over a grid
Symbol	Value	Meaning
$L$	60, 90, 120, 160, 200, 252	estimation sessions
$W$	20, 40, 60, 80, 100, 120	assessment sessions
$\kappa$	≤ 30	Belsley condition index; above it the cell is collinear and left empty

Every window sits on calendar positions: a missing price makes the cell unavailable with the series and date named; nothing is filled. Only the market-and-sector layer can establish the claim; the market layer is shown for comparison. Counts of supporting settings are never turned into a probability.

— Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression Diagnostics.
— Demmel, J. & Veselić, K. (1992). Jacobi's Method Is More Accurate than QR. SIAM J. Matrix Anal. Appl., 13(4).

LAB

Backtest lab

Event studies, measure studies, model correlations and rule backtests on up to ten years, with the statistics that say how much weight a result can bear.

Event studies

An occurrence is a session on which a condition becomes true (or a flag fires). Outcomes are measured from the occurrence close over each horizon, against ordinary sessions, with moving-block bootstrap intervals over calendar weeks.

y_{o,h} = R_{t_o\to t_o+h} - R^{B}_{t_o\to t_o+h}, \qquad \bar y_h = \frac{1}{|\mathcal O|}\sum_{o}y_{o,h}, \qquad \Delta_h = \bar y_h - \bar y^{\text{ordinary}}_h

(15.1)

\hat\theta^{*} = \frac{\sum_{g\in\mathcal G^*}S_g}{\sum_{g\in\mathcal G^*}N_g}, \quad \mathcal G^* = \text{blocks of } \lceil h/5\rceil \text{ consecutive weeks}, \qquad \text{CI}_{95} = \big[\hat\theta^{*}_{(2.5\%)},\ \hat\theta^{*}_{(97.5\%)}\big]

(15.2)

— Künsch, H. R. (1989). The Jackknife and the Bootstrap for General Stationary Observations. Annals of Statistics, 17(3).
— MacKinlay, A. C. (1997). Event Studies in Economics and Finance. Journal of Economic Literature, 35(1).

Measure studies: information coefficient

At each rebalance, companies are sorted by the measure into groups; the information coefficient is the rank correlation between the measure and what followed. Overlapping horizons use Newey–West with a matching lag.

\text{IC}_t = \rho_S\big(x_{i,t},\ R_{i,t\to t+h}\big), \qquad \rho_S = 1 - \frac{6\sum d_i^2}{n(n^2-1)}\ \ (\text{average ranks for ties})

(15.3)

\hat s^2 = \hat\gamma_0 + 2\sum_{k=1}^{\ell}\left(1 - \frac{k}{\ell+1}\right)\hat\gamma_k, \qquad t = \frac{\overline{\text{IC}}}{\sqrt{\hat s^2/n}}, \qquad \ell = \left\lceil\frac{h}{\Delta}\right\rceil - 1

(15.4)

— Newey, W. K. & West, K. D. (1987). Econometrica, 55(3).
— Grinold, R. C. & Kahn, R. N. (2000). Active Portfolio Management.

Rule backtests and performance

Decisions at a close, fills at the next open; long only, no leverage, units drift between trades, cash earns nothing, and every trade pays the stated cost on each side.

\text{NAV}_{t} = \text{cash}_t + \sum_i q_{i,t}P_{i,t}, \qquad \text{cash} \leftarrow \text{cash} - \sum_i \Delta q_i\,O_{i,t+1} - c\sum_i|\Delta q_i|\,O_{i,t+1}

(15.5)

\text{CAGR} = \left(\frac{\text{NAV}_T}{\text{NAV}_0}\right)^{252/n} - 1, \qquad \widehat{\text{SR}} = \sqrt{252}\,\frac{\bar r}{\hat\sigma}, \qquad \text{Calmar} = \frac{\text{CAGR}}{|\text{MDD}|}

(15.6)

Parameters of Rule backtests and performance
Symbol	Value	Meaning
$c$	10 bp	default cost per side

Sharpe ratio inference: standard error, PSR and deflated Sharpe

Backtest lab Analyst ratings

The Sharpe ratio's sampling error under non-normal returns, the probability that the true ratio exceeds a benchmark, and that probability against the best a set of variants would show by luck.

\widehat{\text{se}}(\widehat{\text{SR}}) = \sqrt{\frac{1 + \frac12\widehat{\text{SR}}^2 - \hat\gamma_3\widehat{\text{SR}} + \frac{\hat\kappa}{4}\widehat{\text{SR}}^2}{n}}

(15.7)

\text{PSR}(\text{SR}^*) = \Phi\!\left(\frac{(\widehat{\text{SR}} - \text{SR}^*)\sqrt{n-1}}{\sqrt{1 - \hat\gamma_3\widehat{\text{SR}} + \frac{\hat\gamma_4 - 1}{4}\widehat{\text{SR}}^2}}\right), \qquad \hat\gamma_4 = \hat\kappa + 3

(15.8)

\E\big[\max_N \widehat{\text{SR}}\big] \approx \sqrt{\Var[\widehat{\text{SR}}_n]}\left[(1-\gamma)\,\Phi^{-1}\!\left(1 - \frac1N\right) + \gamma\,\Phi^{-1}\!\left(1 - \frac{1}{Ne}\right)\right], \qquad \text{DSR} = \text{PSR}\big(\E[\max_N\widehat{\text{SR}}]\big)

(15.9)

Parameters of Sharpe ratio inference: standard error, PSR and deflated Sharpe
Symbol	Value	Meaning
$\hat\gamma_3, \hat\kappa$	sample	skewness and excess kurtosis of daily returns
$\gamma$	0.5772…	Euler–Mascheroni constant
$N$	variants tried	in the conversation

— Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4).
— Mertens, E. (2002). Comments on Variance of the IID Estimator in Lo (2002).
— Bailey, D. H. & López de Prado, M. (2012). The Sharpe Ratio Efficient Frontier. Journal of Risk, 15(2).
— Bailey, D. H. & López de Prado, M. (2014). The Deflated Sharpe Ratio. Journal of Portfolio Management, 40(5).

Graded assessment

Every study and record is graded by named checks with stated thresholds: the core check (an interval, or the information coefficient's t-statistic) gates any evidence; the rest decide strong, moderate or weak.

\text{grade} = \begin{cases}\text{none} & \text{core fails}\\ \text{weak} & \text{core cautions, or any other check fails}\\ \text{moderate} & \text{others only caution}\\ \text{strong} & \text{all pass}\end{cases}

(15.10)

ANR

Analyst ratings

Third-party ratings, measured: when a rating counts, the unit a record measures, what followed in the rated direction, and rankings that do not reward short lucky records.

Stance, timing and consensus

Analyst ratings

Each firm's words map to a stance; a rating counts from the first close after it was published; the consensus averages current ratings on a 1–5 scale.

s(\text{rating}) \in \{+1, 0, -1\}, \qquad t^{\text{eff}} = \begin{cases} t_{\text{pub}} & \text{published on a session before 16:00 ET}\\ \text{next session} & \text{otherwise}\end{cases}

(16.1)

\bar c = \frac{5N_{+} + 3N_{0} + N_{-}}{N}, \qquad \text{balance} = \frac{N_+ - N_-}{N}, \qquad \text{dispersion} = \frac{\sd(T_j)}{\overline T}, \qquad \text{gap} = \frac{\overline T}{P_\tau} - 1

(16.2)

Parameters of Stance, timing and consensus
Symbol	Value	Meaning
$\text{current}$	365 days	a rating counts for this long after the analyst's latest action
$\text{recent}$	28 days	window for rating changes

Spells and directional excess

Analyst ratings

A spell is one analyst's continuous stance on one company, ended by a change, the end of coverage, a firm move or a year. Its outcome is the company's return less a benchmark, signed by the stance.

\text{spell} = [t_e,\ t_x],\quad t_x = \min\big(t_{\text{change}},\ t_{\text{end}},\ t_{\text{move}},\ t_e + 252\big)

(16.3)

x = s\cdot\left[\frac{P_{t_x}}{P_{t_e}} - \frac{Y_{t_x}}{Y_{t_e}}\right], \qquad Y \in \{\text{market},\ \text{sector},\ \text{coverage average}\}, \qquad \text{hit} = \mathbb 1[x > 0]

(16.4)

C_t = C_{t-1}\left(1 + \frac1{n_t}\sum_{i}\left(\frac{P_{i,t}}{P_{i,t-1}} - 1\right)\right) \quad\text{(coverage average, rebalanced daily)}

\text{1Y signed} = \prod_{t=\tau-252+1}^{\tau}\Big(1 + s_{t-1}\big(R_{t} - R^{Y}_{t}\big)\Big) - 1

(16.5)

Records, intervals and empirical-Bayes rankings

Analyst ratings · Rankings

A record averages directional excess with a moving-block bootstrap interval over the weeks spells took effect. Rankings shrink each record towards the average of everyone ranked by the one-way random-effects estimate of how much records truly differ.

\bar x_i = \frac{1}{n_i}\sum_{k}x_{ik}, \qquad \mu = \frac{\sum_i n_i\bar x_i}{\sum_i n_i}, \qquad \hat\sigma^2_w = \frac{\sum_i (n_i - 1)s_i^2}{\sum_i(n_i - 1)}

(16.6)

n_0 = \frac{N - \sum_i n_i^2/N}{k-1}, \qquad \hat\tau^2 = \max\!\left(0,\ \frac{\frac{1}{k-1}\sum_i n_i(\bar x_i - \mu)^2 - \hat\sigma^2_w}{n_0}\right)

(16.7)

\tilde x_i = \mu + B_i\,(\bar x_i - \mu), \qquad B_i = \frac{n_i\hat\tau^2}{n_i\hat\tau^2 + \hat\sigma^2_w}

(16.8)

Parameters of Records, intervals and empirical-Bayes rankings
Symbol	Value	Meaning
$n_i \ge 10$	completed spells	to be ranked
$\text{blocks}$	4 weeks	bootstrap block length, 1,000 draws

With k records ranked, about 0.05·k would show an interval clear of zero by chance alone; the ranking says so.

— Efron, B. & Morris, C. (1975). Data Analysis Using Stein's Estimator and Its Generalizations. JASA, 70(350).
— Searle, S. R., Casella, G. & McCulloch, C. E. (1992). Variance Components.

Price targets and what followed rating actions

Analyst ratings

A target is reached when a close within a year is at or beyond it; actions are studied as events from the close before they became known.

\text{reached}(T_j) = \mathbb 1\Big[\exists\, t\in(t_e,\, t_e + 252] : \sgn(T_j - P_{t_e})\,(P_t - T_j) \ge 0\Big]

(16.9)

\text{CAR}_k = \frac{P_{t_e+k}}{P_{t_e-1}} - \frac{Y_{t_e+k}}{Y_{t_e-1}}, \qquad k = -10,\dots,60

(16.10)

Parameters of Price targets and what followed rating actions
Symbol	Value	Meaning
$h$	5, 20, 60 (graded at 20)	sessions after day 0

SYN

The synthetic universe

The labelled demonstration data: 26 fictional companies, a generated benchmark and ten years (2610 sessions) from one seed (20260918). Its structure plants answers the models can be checked against.

Regimes, factors, GARCH shocks, jumps and trends

Every sample page

A two-state Markov regime drives the market factor; each sector has its own factor; each company adds GARCH(1,1) idiosyncratic shocks, rare jumps and a slow trend wave. Each component draws from its own seeded stream.

\Pr(\text{calm}\to\text{stressed}) = 0.012, \quad \Pr(\text{stressed}\to\text{calm}) = 0.055, \qquad m_t = \mu_{s_t} + \sigma_{s_t}z_t

(17.1)

r_{i,t} = \mu_i + \underbrace{w_i\sin\!\left(\phi_i + \frac{2\pi c_i t}{n}\right)}_{\text{trend wave}} + \beta_i m_t + \lambda_i f^{S(i)}_t + \kappa_{s_t}\,\varepsilon_{i,t} + J_{i,t}

(17.2)

h_{i,t} = \omega_i + \alpha_i\varepsilon^2_{i,t-1} + \beta_i h_{i,t-1},\ \ \omega_i = \sigma^2_i(1-\alpha_i-\beta_i), \qquad J_{i,t} = \mathbb 1[U_t < \lambda^J_i/252]\,\sigma^J_i\,\eta_t

(17.3)

Parameters of Regimes, factors, GARCH shocks, jumps and trends
Symbol	Value	Meaning
$(\mu, \sigma)_{\text{calm}}$	(0.00052, 0.0066)	market factor, daily
$(\mu, \sigma)_{\text{stressed}}$	(−0.0011, 0.0158)	market factor, daily
$\sigma^{S}$	0.0042 / 0.0072	sector factor, calm / stressed
$\kappa_{\text{stressed}}$	1.35	idiosyncratic multiplier

Cointegrated pairs, the backward regime and the Gaussian bridge

Co-movement

A follower's log price is the leader's times a hedge ratio plus a stationary AR(1) spread. The ten-year history runs the regime backwards (a two-state chain is reversible) and bridges each spread exactly to the value the workspace starts from.

\ln F_t = h\ln L_t + u_t, \qquad u_t = \phi\,u_{t-1} + \sigma_u\eta_t,\ |\phi|<1

(17.4)

\pi_0\,p_{01} = \pi_1\,p_{10} \;\Rightarrow\; \Pr(s_{t-1}=j\mid s_t=i) = p_{ij}

(17.5)

\tilde u_t = u_t + (u^{\star} - u_N)\,\phi^{\,N-t}, \qquad \frac{\Cov(u_t, u_N)}{\Var(u_N)} = \phi^{\,N-t}

(17.6)

Synthetic analysts with planted skill

Analyst ratings (sample)

Each fictional analyst's view of a company mixes a noisy version of the company's next six months against the benchmark (the planted skill ρ) with persistent noise, past-move chasing and the firm's house leaning. Thresholds give about 55% positive, 35% neutral and 10% negative ratings before the leaning.

v_t = \rho\,z^{\text{fut}}_{126}(t) + \sqrt{1-\rho^2}\,e_t + c\,z^{\text{past}}_{63}(t) + \ell_{\text{firm}}, \qquad e_t = 0.75\,e_{t-1} + \sqrt{1-0.75^2}\,\eta_t

(17.7)

\text{stance} = \begin{cases}+ & v_t > \Phi^{-1}(0.45)\\ - & v_t < \Phi^{-1}(0.10)\\ 0 & \text{otherwise}\end{cases}, \qquad z^{\text{fut}}_{h}(t) = \frac{\ell_{t+h} - \ell_t}{\hat\sigma_\ell\sqrt h},\ \ \ell = \ln\frac{P}{B}

(17.8)

The future is used only here, to plant the answer. Tests require the engine to recover it: skilled analysts rank high, reliably wrong ones low, specialists in their home sector.

NUM

Numerical methods

The algorithms underneath, all implemented in the terminal and tested against closed forms and known answers.

Least squares, inverses and decompositions

Risk model Liquid Quant

Ordinary regressions solve the normal equations with a Gauss–Jordan inverse (partial pivoting); collinearity-sensitive regressions go through a one-sided Jacobi SVD so XᵀX is never formed; symmetric eigenproblems use cyclic Jacobi rotations.

\hat{\boldsymbol\beta} = (X^\top X)^{-1}X^\top y, \qquad \kappa(X^\top X) = \kappa(X)^2 \;\Rightarrow\; \text{SVD for ill-conditioned designs}

(18.1)

A^{(k+1)} = J_{pq}^\top A^{(k)} J_{pq},\quad \tan 2\theta = \frac{2a_{pq}}{a_{qq} - a_{pp}} \quad\text{(cyclic Jacobi)}

— Golub, G. H. & Van Loan, C. F. (2013). Matrix Computations, 4th ed.

Nelder–Mead simplex

Company · Volatility

Derivative-free minimisation for the GARCH likelihood, with the standard coefficients.

\mathbf x_r = \bar{\mathbf x} + 1\cdot(\bar{\mathbf x} - \mathbf x_h),\ \ \mathbf x_e = \bar{\mathbf x} + 2(\mathbf x_r - \bar{\mathbf x}),\ \ \mathbf x_c = \bar{\mathbf x} + \tfrac12(\mathbf x_h - \bar{\mathbf x}),\ \ \mathbf x_i \leftarrow \mathbf x_\ell + \tfrac12(\mathbf x_i - \mathbf x_\ell)

(18.2)

— Nelder, J. A. & Mead, R. (1965). The Computer Journal, 7(4).

Distribution functions

Every test statistic

The normal CDF by a Chebyshev fit of the complementary error function; its inverse by Acklam's rational approximation with one Halley step; chi-square and Student-t tails by regularised incomplete gamma and beta functions.

\Phi(x) = \tfrac12\operatorname{erfc}\!\left(-\tfrac{x}{\sqrt2}\right), \qquad x_{n+1} = x_n - \frac{e}{1 + \frac{x_n e}{2}},\ \ e = \frac{\Phi(x_n) - p}{\varphi(x_n)}

(18.3)

\Pr(\chi^2_k > q) = Q\!\left(\tfrac k2, \tfrac q2\right), \qquad \Pr(|t_\nu| > t) = I_{\nu/(\nu + t^2)}\!\left(\tfrac\nu2, \tfrac12\right)

(18.4)

— Press, W. H. et al. (2007). Numerical Recipes, 3rd ed.
— Acklam, P. J. (2003). An algorithm for computing the inverse normal cumulative distribution function.

OUT

What followed

Each rule's record of what followed its occurrences on the coverage, by a protocol fixed on 2026-10-26 (outcomes-protocol-1.1.0) after the history it scores, so the historical records are in-sample: when an occurrence could first be acted on, what it is compared with, the one quantity its record averages, how repeats and overlap are handled, and how a record is graded among the others.

Entry and the same-date comparison

What followed Changes

An occurrence at session t is entered at the close of the session after the one at which the weekly report containing it existed: the week-end, or the next session for a week cut short by a holiday. Its return over h sessions is compared with the average return of the other covered companies over exactly the same sessions.

e = k\big(w(t)\big) + 1, \qquad w(t) = \min\{w \in \mathcal W : w \ge t\}, \qquad k(w) = \begin{cases} w & w \text{ is a Friday} \\ w + 1 & \text{otherwise (a week cut short)}\end{cases}

(19.1)

R_{i,e,h} = \frac{P_{i,e+h}}{P_{i,e}} - 1, \qquad \bar R_{-i,e,h} = \frac{1}{|J|-1}\sum_{j \in J,\, j \ne i} R_{j,e,h}, \qquad a_{i,e,h} = R_{i,e,h} - \bar R_{-i,e,h}

(19.2)

\text{complete} \iff e + h \le \tau \qquad (\text{pending otherwise; never in a record})

Parameters of Entry and the same-date comparison
Symbol	Value	Meaning
$h$	5, 20, 60 sessions (records graded at 20)	horizons
$\|J\| - 1$	≥ 3	other companies with prices over the window
$\mathcal W$	complete week-ends	report dates; a monitoring change fires at one
$k(w)$	the session the report exists at	a week ending before Friday is known complete only once the next session arrives

The return from the flag's own close (e = t) and the return between t and e (the cost of the weekly delivery) are shown beside each record. The market-benchmark version replaces the others' average with B.

Sizes and falls in expected standard deviations

Size and depth records divide by the company's volatility expected beforehand (RiskMetrics EWMA at the entry close, scaled to the horizon), subtract the other companies' same quantity, and subtract what companies in the same volatility state showed — an EWMA forecast does not revert, so it overstates what follows a spike and understates what follows a lull, and only the state comparison removes that.

\sigma^2_{i,t} = \lambda\,\sigma^2_{i,t-1} + (1-\lambda)\,r_{i,t}^2,\ \ \lambda = 0.94, \qquad s_{i,e,h} = \sigma_{i,e}\sqrt h

(19.3)

q_{i} = \frac{\lvert a_{i,e,h} \rvert}{s_{i,e,h}}, \qquad d_{i} = \min\!\Big(0,\ \min_{1 \le k \le h} \frac{P_{i,e+k}}{P_{i,e}} - 1\Big), \qquad \delta_i = \frac{d_i}{s_{i,e,h}}

(19.4)

m_i = (q_i - \bar q_{-i}) - \mu^{q}_{s(i,e)}, \qquad f_i = (\delta_i - \bar\delta_{-i}) - \mu^{\delta}_{s(i,e)}, \qquad v_i = \frac{\operatorname{sd}(r_{i,e+1},\dots,r_{i,e+h})}{\sigma_{i,e}}

(19.5)

s(i,e) = \Big\lfloor 10\,p\big(\sigma_{i,e} \,\big|\, \{\sigma_{i,e-251},\dots,\sigma_{i,e}\}\big)\Big\rfloor, \qquad \mu^{x}_{k} = \operatorname{mean}\big\{x_{j,e'} - \bar x_{-j,e'} : s(j,e') = k,\ e' + h \le \tau\big\}

(19.5a)

— J.P. Morgan / Reuters (1996). RiskMetrics — Technical Document, 4th ed.

Records by question, and episodes

Each rule's record averages one fixed quantity: the signed same-date return for a one-sided direction rule, the balanced contrast for a two-sided one, the unsigned return for drift, the state-compared size or depth for magnitude and downside. Occurrences on a company entered at the same session count as one, in the direction of the latest (the state the weekly report showed); it joins the current episode when it is in the same direction and within a horizon of that episode's start, and otherwise starts a new one.

\text{one-sided: } \frac1n\sum_k s_k\,a_k, \qquad \text{two-sided: } \tfrac12\big(\bar a^{\uparrow} - \bar a^{\downarrow}\big), \qquad \text{drift: } \bar a, \qquad \text{magnitude: } \bar m, \qquad \text{downside: } \bar f

(19.6)

\text{same entry} \Rightarrow \text{one occurrence, sign of the latest}; \qquad \text{a new episode} \iff e_k - e_{\text{start}} \ge h \ \lor\ s_k \ne s_{\text{start}}, \ \text{otherwise it joins}

(19.7)

\text{side} = \frac1n\sum_k \mathbb 1\big[s_k R_k > s_k\,\operatorname{med}_{j\ne i}R_{j}\big] \quad(\text{one half when the occurrences carry no information})

(19.8)

The sign s comes from the occurrence's own direction (up +1, down −1), never from the outcome. Every test is two-sided: a direction record can show continuation (positive) or reversal (negative).

For a two-sided rule the side share is the average of the two sides' shares. Size and depth records carry no side share.

Overlap-aware intervals and the family

Episodes are summed by the calendar week they were entered, over every week of the period including empty ones; a circular block bootstrap draws runs of three times the horizon in weeks. The standard error is the replicates' spread; the interval, z, p and the smallest detectable effect all come from it, so the interval clears zero exactly when p < 0.05; q-values hold across the family under any dependence.

\hat\theta^{*}_b = g\Big(\sum_{w \in \mathcal B^{*}_b} S_w\Big), \qquad \ell = \max\big(2,\ 3\lceil h/5 \rceil\big) \text{ weeks}, \qquad \widehat{\mathrm{se}} = \operatorname{sd}_b\,\hat\theta^{*}_b

(19.9)

\text{CI} = \hat\theta \pm 1.96\,\widehat{\mathrm{se}}, \qquad z = \frac{\hat\theta}{\widehat{\mathrm{se}}}, \qquad p = 2\big(1 - \Phi(|z|)\big) = Q\!\left(\tfrac12, \tfrac{z^2}{2}\right), \qquad \text{MDE} = 2.8\,\widehat{\mathrm{se}}

(19.10)

q_{(k)} = \min_{j \ge k}\ \min\!\Big(1,\ \frac{m\,c(m)}{j}\,p_{(j)}\Big), \qquad c(m) = \sum_{i=1}^{m}\frac1i \quad\text{(Benjamini–Yekutieli)}

(19.11)

Parameters of Overlap-aware intervals and the family
Symbol	Value	Meaning
$B$	1000 at 20 sessions, 400 otherwise	bootstrap replications
$\text{weeks}$	≥ 8	for an interval
$m$	up to 26	company rules with an interval at 20 sessions: at least 10 episodes over at least 8 weeks, with at least 95% of resampled histories measurable (the two rules about the whole coverage are described, not tested)
$\|z\|$	≥ 3	the multiple-testing hurdle marked beside a record

— Politis, D. N. & Romano, J. P. (1992). A Circular Block-Resampling Procedure for Stationary Data. In Exploring the Limits of Bootstrap.
— Benjamini, Y. & Yekutieli, D. (2001). The Control of the False Discovery Rate in Multiple Testing under Dependency. Annals of Statistics, 29(4).
— Harvey, C. R., Liu, Y. & Zhu, H. (2016). … and the Cross-Section of Expected Returns. Review of Financial Studies, 29(1).

Grading a record

On licensed data a company rule's record at the primary horizon is graded by the Backtest lab's rule from named checks; nothing is graded on synthetic demonstration data.

\text{core} = \begin{cases}\text{pass} & 0 \notin \text{CI} \ \land\ q \le 0.1 \\ \text{fail} & \text{a CI, otherwise (no evidence)} \\ \text{not graded} & \text{no CI (value only, outside the family)}\end{cases}

(19.12)

\text{sample: } n \ge 30,\ \text{weeks} \ge 20\ (\text{caution } n \ge 10); \qquad \text{concentration: } \max_i \frac{n_i}{n} \le 0.25\ (\text{caution} \le 0.5)

(19.13)

\operatorname{sgn}\hat\theta_{\text{earlier half}} = \operatorname{sgn}\hat\theta_{\text{later half}} = \operatorname{sgn}\hat\theta, \qquad \operatorname{sgn}\hat\theta_{(0,5]} = \operatorname{sgn}\hat\theta_{(5,20]} = \operatorname{sgn}\hat\theta \ \ (\text{return records})

The halves split a record's episodes in two by entry date. The parts of the horizon do not overlap, unlike neighbouring horizons, so their agreement is informative; that check applies to return records (direction and drift) only, since the depth of a fall over sessions 6–20 is not part of the fall over 1–20. The out-of-sample record (occurrences on or after the protocol's date) is counted separately and never joined to the history.

Replay: a rule's record as returns over time

Each session, every company the rule flagged within the holding period is held at an equal share, in the rule's direction for a direction rule, against the rest of the coverage over that session; the sessions compound into the curve. Each session uses only closes up to itself, so the curve up to any date is what could have been drawn on that date. The funnel is how far a curve with no effect could have wandered by then.

H_s = \{\, i : e_i < s \le e_i + \ell \,\}, \qquad p_s = \frac{1}{|H_s|}\sum_{i \in H_s} \sigma_i \big(r_{i,s} - \bar r_{-i,s}\big), \qquad C_t = \prod_{s \le t}(1 + p_s) - 1

(19.14)

\text{funnel}_t = \exp\!\Big(\pm 1.96\, \operatorname{sd}\big(\log(1+p_s)\big)_{s \le t} \sqrt{n_t}\Big) - 1

(19.15)

Parameters of Replay: a rule's record as returns over time
Symbol	Value	Meaning
$\ell$	5, 20, 60 sessions	holding period, chosen by the reader
$\sigma_i$	+1 or −1	the flag's direction for a direction rule; +1 otherwise
$n_t$	sessions with a holding	up to t

A later flag on a company replaces the earlier one. Before costs; a description of this history, not a return anyone earned.

Strategy studio: a hypothetical simulation