Alpha and signal research formulas

The research pipeline: from hypothesis to live capital

How precisely a Sharpe ratio is known

\operatorname{SE}(\hat S) \approx \sqrt{\frac{1 + \hat S^2/2}{T}}

For independent, normally distributed returns, with $\hat S$ annualised and $T$ in years (Lo, 2002). Fat tails and autocorrelation make it larger. The striking thing is how little $T$ is: every year of data is one unit of information, whatever the sampling frequency.

Remember

Version the code and the data together, so a result can be reproduced exactly.

Constructing a signal: standardisation, neutralisation and combination

Smoothing with an exponential average

\tilde s_t = (1 - \lambda)\,s_t + \lambda\,\tilde s_{t-1}, \qquad h = \frac{\ln 0.5}{\ln\lambda}

The smoothed signal $\tilde s_t$ forgets old values geometrically; $h$ is the half-life in periods. Larger $\lambda$ means slower trading and less turnover, at the cost of reacting later to new information.

Remember

Standardise within each date, after winsorising or ranking.

Measuring a signal: IC, breadth and the fundamental law

The fundamental law of active management

\mathrm{IR} \approx \mathrm{IC} \times \sqrt{\mathrm{BR}}

The information ratio you can achieve is the quality of each forecast times the square root of how many independent forecasts you make.

How noisy a single date’s IC is

\operatorname{SE}(\widehat{\mathrm{IC}}_t) \approx \frac{1}{\sqrt{N}}

For a cross-section of $N$ assets and a small true IC. One date tells you almost nothing; the evidence is in the average over many dates and in its $t$ -statistic.

Remember

IC is the cross-sectional correlation between forecast and forward return; $0.03$ is genuinely good.

Factor models: CAPM, Fama–French and statistical factors

CAPM

\mathbb{E}[r_i] - r_f = \beta_i\left(\mathbb{E}[r_m] - r_f\right), \qquad \beta_i = \frac{\mathrm{Cov}(r_i, r_m)}{\mathrm{Var}(r_m)}

Only market risk is compensated; idiosyncratic risk is diversifiable and earns nothing.

A linear factor model

r_i - r_f = \alpha_i + \sum_{k=1}^{K}\beta_{ik}\,f_k + \varepsilon_i

Returns in excess of cash are a sum of factor returns weighted by exposures, plus an idiosyncratic part. Arbitrage pricing theory says that in a well-diversified market the alphas must be close to zero; the factors carry all the priced risk.

Remember

CAPM: only undiversifiable risk is paid, and empirically the beta–return line is too flat.

Risk models: covariance estimation, VaR and expected shortfall

Expected shortfall

\mathrm{ES}_\alpha = \mathbb{E}\big[\text{loss} \mid \text{loss} > \mathrm{VaR}_\alpha\big]

The average loss in the tail beyond VaR. It is coherent — in particular subadditive — which is why regulation has moved toward it.

Remember

Sample covariance is singular and noisy when $n$ approaches $T$ ; shrink or use a factor model.

Portfolio construction: mean-variance, and why nobody uses it raw

The solution

w^* \propto \Sigma^{-1}\mu, \qquad \mathrm{SR}^2_{\max} = \mu^\top\Sigma^{-1}\mu

Maximise expected return per unit of variance and the weights come out proportional to inverse covariance times expected return.

The two-asset minimum-variance portfolio

w_A = \frac{\sigma_B^2 - \rho\,\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho\,\sigma_A\sigma_B}

It needs no expected returns at all, which is its appeal: covariances are estimated far more precisely than means.

The Sharpe ratio of many correlated strategies

S_p = \frac{S\sqrt N}{\sqrt{1 + (N - 1)\rho}} \;\xrightarrow{N\to\infty}\; \frac{S}{\sqrt\rho}

For $N$ strategies with equal Sharpe $S$ , equal risk and common pairwise correlation $\rho$ . The limit is the most diversification can deliver: correlation, not the number of strategies, sets the ceiling.

Remember

$w^* \propto \Sigma^{-1}\mu$ , and both inputs are badly estimated.

Execution: market impact, implementation shortfall and capacity

The square-root law

\text{impact} \approx Y\,\sigma\sqrt{\frac{Q}{V}}

Impact in price terms scales with volatility and with the square root of the order size as a fraction of daily volume.

The profit-maximising size

\Pi(Q) = \alpha Q - cQ^{3/2} \;\Rightarrow\; Q^* = \left(\frac{2\alpha}{3c}\right)^2 = \tfrac{4}{9}\,Q_{\text{break-even}}

Gross profit is linear in size and total impact cost grows as $Q^{3/2}$ . Profit peaks at four-ninths of the size at which costs consume all of it, $Q_{\text{break-even}} = (\alpha/c)^2$ .

Remember

Impact $\approx Y\sigma\sqrt{Q/V}$ — concave in size, and remarkably universal.

The overfitting problem: deflated Sharpe and what discipline looks like

What luck produces

\mathbb{E}\left[\max_{i \le N} \mathrm{SR}_i\right] \approx \sqrt{\frac{2\ln N}{T}}

The expected best Sharpe ratio among $N$ worthless strategies over $T$ years. Any reported Sharpe must clear this before it is evidence of anything.

Correlated trials count for less

N_{\text{eff}} \approx \bar\rho + (1 - \bar\rho)\,N

An approximation for $N$ trials with average pairwise correlation $\bar\rho$ between their returns: identical strategies count once, independent ones count fully. Clustering the trials and counting clusters is the more careful version.

Remember

Require an economic mechanism before testing, which raises the base rate.

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