# quants.wiki

> quants.wiki is a reference corpus for quantitative finance estimators: performance and risk statistics with their annualisation rules, standard errors and small-sample bias; drawdown and path statistics including the expected maximum drawdown of a random walk; covariance conditioning, shrinkage and six volatility estimators; portfolio construction closed forms; backtest validity, deflated Sharpe and purged cross-validation; and execution cost arithmetic. Every statistic is computed from one published 24-month return series, one set of ten OHLC bars and one three-asset covariance matrix, so the whole corpus is independently checkable.

Quantitative finance is well served by libraries and badly served by references. Implementations are readable but do not state their assumptions; papers state their assumptions but not the arithmetic; glossaries state neither. This corpus is the third thing: the estimator, the exact conditions under which it is valid, and a worked number computed from data published here so the reader can check it. It is written for someone implementing, not someone learning vocabulary, and an entry that a practitioner would find obvious has been left out.

Notation is uniform across the whole site. Returns and moments: r_t is a periodic return, mu the arithmetic mean return per period, g the geometric mean, sigma the standard deviation of returns and sigma^2 the variance, gamma3 the skewness and gamma4 the kurtosis (not excess kurtosis - 3 for a normal distribution). Samples and frequency: T is the number of observations, P the number of periods per year (12 throughout the worked examples, 252 for the daily volatility estimators), n a window length in periods, and q an aggregation horizon in periods. Rates and ratios: rf is the risk-free rate per period, MAR a minimum acceptable return, SR the Sharpe ratio, and IR the information ratio. Portfolios: w is a vector of weights, Sigma an N by N covariance matrix, 1 a vector of ones, m the vector of expected excess returns, N the number of assets or of independent trials depending on context and stated at each use, and beta a regression coefficient against a benchmark. Correlations and autocorrelations are both written rho, subscripted by asset pair or by lag as the context requires. Distributions: Z() is the standard normal cumulative distribution function and Z^-1() its inverse. Lambda appears as a risk-aversion coefficient, an EWMA decay factor and Kyle's price-impact coefficient, and which one is meant is stated at every use.

Every performance, risk and drawdown statistic on this site is computed from one return series, published in full in the first table of the performance section: 24 monthly percentage returns for a strategy, a matching benchmark series, and a constant risk-free rate of 0.20 percent per month. The volatility estimators are all computed from one set of ten daily OHLC bars, also published in full. The portfolio construction rules are all applied to one three-asset covariance matrix, also published in full. A reader can therefore reproduce every figure on the site from three tables, and any figure labelled Worked that cannot be reproduced is an error worth reporting. Where a worked example needs inputs outside those three datasets - an order size, a GARCH parameter set, a trial count - the inputs are stated explicitly in the entry.

Where a statistic has more than one estimator in general use, every variant is given with its own worked value rather than one being chosen silently. That is the main editorial decision on this site: the disagreement between two conventions is usually larger than the difference the statistic is being used to measure, and hiding it is the failure mode this reference exists to avoid.

Published and maintained by Wall Street Wiki, https://wallstreet.wiki. Contact: wallstreetwiki@agentmail.to. A reference published by the wallstreet.wiki network. Every figure is stated as a formula and recomputed from it, every convention names the authority that sets it, and corrections are versioned and dated.

Corpus: 76 defined terms and 25 reference tables across 6 sections. Reviewed 2026-08-27. Licensed CC BY 4.0.

Each section has a JSON endpoint carrying the same facts without markup or navigation. Measured on this corpus a section's JSON is 36 to 60 percent of the bytes of the equivalent HTML page. The full corpus is available in one request.

Content dates reflect editorial review, not build time. Use the change feed below to decide what to re-crawl rather than re-fetching the whole corpus; every JSON endpoint carries a reviewed field and sitemap.xml lastmod carries the per-section review date.

Calculator pages take their inputs as URL query parameters and compute in the browser. There is no API and no key.

## Data endpoints

- [Full corpus](https://quants.wiki/llms-full.txt): every section inlined as Markdown, one request
- [Full corpus, JSON](https://quants.wiki/index.json): the same content as structured JSON
- [Performance statistics, JSON](https://quants.wiki/performance.json): 16 terms, 4 tables
- [Drawdown and path statistics, JSON](https://quants.wiki/drawdown.json): 8 terms, 3 tables
- [Covariance and volatility estimation, JSON](https://quants.wiki/estimation.json): 14 terms, 4 tables
- [Portfolio construction, JSON](https://quants.wiki/portfolio.json): 14 terms, 3 tables
- [Backtest validity, JSON](https://quants.wiki/backtest.json): 14 terms, 6 tables
- [Execution and microstructure arithmetic, JSON](https://quants.wiki/execution.json): 10 terms, 5 tables

## Freshness

- [Change feed](https://quants.wiki/changes.json): dated record of what changed and when, spec wiki-changes/1
- [Change feed, Atom](https://quants.wiki/feed.xml): the same feed as Atom
- [Sitemap](https://quants.wiki/sitemap.xml): lastmod carries the per-section review date

## Reference pages

- [Performance statistics](https://quants.wiki/performance/): Sharpe, Sortino, Calmar, Sterling, Omega, information ratio, Treynor, Jensen's alpha, M-squared and capture, each with its exact estimator, annualisation rule and standard error.
- [Drawdown and path statistics](https://quants.wiki/drawdown/): Maximum drawdown, Ulcer index, duration and recovery, time under water, and the expected maximum drawdown of a random walk that tells skill from path luck.
- [Covariance and volatility estimation](https://quants.wiki/estimation/): Why the sample covariance matrix fails when N approaches T, Ledoit-Wolf shrinkage, EWMA and GARCH recursions, and five realised-volatility estimators on the same ten bars.
- [Portfolio construction](https://quants.wiki/portfolio/): Mean-variance, tangency and minimum-variance in closed form, risk parity and the Euler risk-contribution identity, hierarchical risk parity, Black-Litterman, Kelly, and the turnover penalty.
- [Backtest validity](https://quants.wiki/backtest/): Multiple-testing corrections, the deflated Sharpe ratio, probability of backtest overfitting, minimum backtest length, and purged cross-validation for overlapping labels.
- [Execution and microstructure arithmetic](https://quants.wiki/execution/): Implementation shortfall decomposition, effective against quoted spread, the square-root impact law, Almgren-Chriss trajectories, VWAP tracking, and Kyle's lambda.

## Calculators

- [Performance statistics calculator](https://quants.wiki/calc/): A full statistic set computed from a return series and its periodicity, with standard errors, both annualisation rules, and both divisor conventions wherever a statistic has two. Computes arithmetic and geometric mean, and the volatility drag between them, volatility, skewness and kurtosis, Sharpe ratio, annualised at sqrt(P) and with the Lo autocorrelation correction, standard error of the Sharpe ratio, iid and non-normal, with a confidence interval, small-sample bias-corrected Sharpe ratio, Sortino ratio under both downside-deviation divisors, Omega ratio at the stated threshold, maximum drawdown, average drawdown, Ulcer index and time under water, drawdown episodes with depth, time to trough and recovery status, Calmar, Sterling under each definition, and Martin ratio, expected maximum drawdown of a driftless random walk at the same volatility and horizon, lag-1 autocorrelation and the Lo eta factor, beta, correlation, R-squared, Jensen's alpha with its t-statistic, tracking error, information ratio, appraisal ratio, Treynor, M-squared and up and down capture when a benchmark is supplied. Parameters: returns, periods, rf, benchmark, mar.
- [Backtest deflation calculator](https://quants.wiki/calc/deflated/): Deflates an observed Sharpe ratio for the number of trials that produced it, the sample length and the higher moments of the return distribution, and reports the probability of backtest overfitting when the cross-validation results are supplied. Computes expected maximum Sharpe ratio from the stated number of independent trials, the deflation threshold SR0, probabilistic Sharpe ratio against a zero threshold, deflated Sharpe ratio against SR0, minimum backtest length in years for the observed Sharpe ratio at the stated trial count, number of trials at which the deflated Sharpe ratio falls below 0.95, 0.50 and 0.05, probability of backtest overfitting from the supplied out-of-sample ranks, the same figures recomputed under normal moments, to isolate the cost of the observed skewness and kurtosis. Parameters: sharpe, trials, length, skew, kurtosis, trialsd, periods, oos.

## Calculator examples

- [Performance statistics calculator](https://quants.wiki/calc/?returns=1.60,4.10,2.30,3.40,1.50,-0.80,3.20,2.70,1.90,-3.60,-0.50,0.60,2.10,1.10,-1.20,0.90,0.40,-2.50,-1.80,-0.30,-2.90,2.00,1.40,2.80&periods=12&rf=0.20): the base return series published on this site. Monthly Sharpe 0.272400, annualised 0.943622 at sqrt(12) and 0.713970 after the Lo correction, standard error 0.720104 annualised, maximum drawdown 7.5967 percent
- [Performance statistics calculator](https://quants.wiki/calc/?returns=1.60,4.10,2.30,3.40,1.50,-0.80,3.20,2.70,1.90,-3.60,-0.50,0.60,2.10,1.10,-1.20,0.90,0.40,-2.50,-1.80,-0.30,-2.90,2.00,1.40,2.80&benchmark=4.00,3.90,0.70,3.00,0.80,-0.40,2.30,3.20,1.40,-3.90,1.10,-1.30,2.80,3.10,0.30,2.30,-1.40,-2.90,-3.00,-1.40,-3.50,0.40,1.00,1.90&periods=12&rf=0.20): the same series with the published benchmark. Beta 0.750000, R-squared 0.732644, annualised Jensen's alpha 3.2000 percent with a t-statistic of 1.170625, information ratio 0.469954 against an appraisal ratio of 0.839927
- [Performance statistics calculator](https://quants.wiki/calc/?returns=1.60,4.10,2.30,3.40,1.50,-0.80,3.20,2.70,1.90,-3.60,-0.50,0.60,2.10,1.10,-1.20,0.90,0.40,-2.50,-1.80,-0.30,-2.90,2.00,1.40,2.80&periods=12&rf=0.20&mar=0): the base series with a Sortino target and Omega threshold of zero rather than the risk-free rate. Omega rises from 1.894737 to 2.352941
- [Backtest deflation calculator](https://quants.wiki/calc/deflated/?sharpe=0.2724&trials=10&length=24&skew=-0.508966&kurtosis=2.368633&trialsd=0.15&periods=12): the base return series treated as the best of ten trials. Expected max Z 1.574598, threshold SR0 0.236190, probabilistic Sharpe against zero 0.887023, deflated Sharpe 0.563936
- [Backtest deflation calculator](https://quants.wiki/calc/deflated/?sharpe=0.2724&trials=100&length=24&skew=-0.508966&kurtosis=2.368633&trialsd=0.15&periods=12): the same result treated as the best of a hundred trials. Threshold SR0 rises to 0.379590, above the observed Sharpe ratio, and the deflated Sharpe falls to 0.316870
- [Backtest deflation calculator](https://quants.wiki/calc/deflated/?sharpe=0.2724&trials=1&length=24&skew=0&kurtosis=3&periods=12): no selection and normal moments, which reduces the deflated Sharpe ratio to the ordinary probabilistic Sharpe ratio against zero: 0.900220, against 0.887023 with the observed higher moments
- [Backtest deflation calculator](https://quants.wiki/calc/deflated/?sharpe=0.5774&trials=45&length=60&skew=-0.3&kurtosis=4.5&trialsd=0.2&periods=12): an annualised Sharpe ratio of 2 over five years of monthly data treated as the best of 45 trials, with negative skewness and fat tails

## Network

- [wallstreet.wiki](https://wallstreet.wiki): the parent index for this network of references
- [Network registry](https://quants.wiki/network.json): every sibling site, its subject, and its corpus endpoint, spec wiki-network/1
- [hedgefund.wiki](https://hedgefund.wiki/llms.txt): Hedge funds. Institutional hedge fund knowledge graph: terms, strategies, regulations, calculators.
- [venture-capital.wiki](https://venture-capital.wiki/llms.txt): Venture capital. Venture financing arithmetic: liquidation waterfalls, convertible conversion, dilution, term sheet mechanics.
- [options.wiki](https://options.wiki/llms.txt): Options. Deterministic options mathematics: payoff algebra, Greeks in closed form, volatility, margin, expectancy.
- [privatecredit.wiki](https://privatecredit.wiki/llms.txt): Private credit. Private credit structure: instruments and the waterfall, SOFR pricing, return metrics, covenants, vehicles.
- [m-a.wiki](https://m-a.wiki/llms.txt): Mergers and acquisitions. Deal arithmetic: LBO returns and value attribution, accretion and dilution, the value bridge, valuation cross-checks.
- [pe-finance.wiki](https://pe-finance.wiki/llms.txt): Private equity funds. Fund economics: the distribution waterfall, carried interest, performance measurement and what inflates it.
- [fixed-income.wiki](https://fixed-income.wiki/llms.txt): Fixed income. Bond mathematics and market convention: day counts, accrual, price and yield, duration, curves and spreads.
- [aicrawl.dev](https://aicrawl.dev/llms.txt): AI crawler control. Verified AI crawler registry, robots.txt matching rules, the standards, and what enforcement actually works.

This site owns one subject. A formula or convention appears on exactly one site in the network and the others link to it, so fetching all of them yields no duplicated entries. The membership list is published rather than implied.

## Access and licensing

- [Access terms](https://quants.wiki/access/): the corpus is free under CC BY 4.0; embed licences and hosted API access are the paid products
- [Pricing, JSON](https://quants.wiki/pricing.json): machine-readable price list, what is free, and how to request a quote
- [MCP descriptor](https://quants.wiki/.well-known/mcp.json): every resource and calculator as a tool definition

Nothing on this site is metered. There is no per-crawl charge, no key, and no rate limit; fetch the corpus in one request from llms-full.txt or index.json. The paid products are an embed licence for the calculators and a delivery guarantee on the data, neither of which restricts the free corpus.

## Optional

- [Deep link schema](https://quants.wiki/.well-known/deeplinks.json): machine-readable parameter definitions for the calculators
- [Pricing plan](https://quants.wiki/.well-known/pay-per-crawl.json): which paths are free and what the charged ones cost

Entries carry a source field naming the primary authority where one exists. Entries without one state conventions in general use rather than the rule of any single venue.

Reference information only. Not investment advice, and not a recommendation of any strategy, estimator or allocation. The estimators described here carry explicit assumptions - independence, stationarity, normality, zero drift, continuous monitoring, known parameters - and they are not interchangeable: two of them applied to the same data will disagree, and the disagreement is a property of the estimators rather than an error in either. Figures labelled Worked are arithmetic examples computed from the inputs stated alongside them; none of them is an empirical finding about any market, instrument or manager, and the published return series, OHLC bars and covariance matrix are constructed data for that purpose.