Equation Library

The mathematical foundations of quant finance and machine learning — each concept from equation to Python.

The mathematical foundations behind everything else on this site. This is my proof of work: rather than assume the standard formulas, I re-derive and re-implement them so I understand exactly what each one assumes and where it breaks.

Every entry follows the same template:

the equation itself,
what each symbol means,
a plain-English explanation,
why it matters in markets,
a simple worked numerical example,
a Python implementation,
a manual / Excel calculation,
a real financial-market example,
and the common mistakes people make with it.

Published entries

Published one at a time, to the standard above. The roadmap below is the plan.

ARIMA

time series

quant finance

ARIMA(p,d,q): difference d times to reach stationarity, then fit p AR and q MA terms. The Box-Jenkins workflow, the special cases, and an honest Nasdaq forecast that collapses to a random walk with drift.

Annualisation

quant finance

risk

Turning per-period figures into annual ones: returns scale with time (×N), volatility with the square root of time (×√N), because variance adds. The √252 rule, why the Sharpe ratio scales with √time, and where the assumption breaks.

Autocorrelation

time series

core statistics

Autocorrelation: the correlation of a time series with a lagged version of itself. The ACF, the significance band, and the two great stylised facts — returns are barely predictable, but volatility clusters.

Backpropagation

machine learning

neural networks

Backpropagation applies the chain rule backward through a network to compute the gradient of the loss w.r.t. every weight in one pass. The forward/backward passes, δ error signals, vanishing gradients, and an honest Nasdaq net.

Beta, CAPM & Alpha

quant finance

factor models

Beta as sensitivity to the market, CAPM’s prediction of expected return from beta, and alpha as the miss. Systematic vs idiosyncratic risk, the Security Market Line, and a Nasdaq-100 basket including a negative-beta name.

CAGR

quant finance

returns

Compound Annual Growth Rate: the constant yearly rate that connects a start value to an end value. Why it is the geometric mean, why it sits below the arithmetic average (volatility drag), and why it says nothing about the path.

Cointegration

time series

quant finance

Cointegration: two non-stationary series whose linear combination is stationary. The hedge ratio, the Engle-Granger test, why it isn’t the same as correlation, and why genuinely cointegrated pairs are rare.

Covariance & Correlation

core statistics

risk

Covariance and correlation: whether two return series move together, correlation as normalised covariance in [−1,1], the diversification link, and a Nasdaq-100 basket example where the correlations aren’t what you’d guess.

Cross-Entropy Loss

machine learning

classification

Cross-entropy: −Σ y·log p, the classification loss. Why it punishes confident-wrong predictions, equals maximum likelihood, has a clean (p−y) gradient with sigmoid/softmax, and an honest Nasdaq log-loss scorecard.

Entropy

machine learning

information theory

Shannon entropy H = −Σ p log p: the average surprise, or uncertainty, in a distribution. Bits vs nats, why it maxes at uniform, the link to cross-entropy and information gain, and the Nasdaq’s near-maximal daily uncertainty.

GARCH

time series

quant finance

GARCH(1,1): σ²_t = ω + α ε²_{t-1} + β σ²_{t-1}. Conditional variance, volatility clustering, the reaction/persistence split, why α+β matters, fat tails from normal shocks, and a Nasdaq fit.

Gini Impurity

machine learning

classification

Gini impurity = 1 − Σp²: the chance of mislabelling a random sample by a node’s class mix. Binary 2p(1−p), why CART/scikit-learn default to it, how it compares to entropy, and an honest Nasdaq split.

Gradient Descent

machine learning

optimisation

Gradient descent: θ ← θ − η∇J(θ). Step against the gradient to minimise a cost. The learning rate, convergence and divergence, batch vs stochastic, and why it scales where formulas can’t.

Information Gain

machine learning

information theory

Information gain = parent entropy − weighted child entropy: the uncertainty a split removes. Why decision trees maximise it, its identity with mutual information, the high-cardinality bias, and a near-zero Nasdaq split.

Kelly Criterion

quant finance

performance

The Kelly criterion: the fraction of capital that maximises the growth rate of wealth. The edge/odds and mu/sigma-squared forms, the link to the Sharpe ratio, over-betting, and why practitioners use fractional Kelly.

L1 & L2 Penalties (Lasso & Ridge)

machine learning

optimisation

L2 (Ridge, Σθ²) shrinks all weights smoothly; L1 (Lasso, Σ|θ|) drives weak weights to exactly zero for feature selection. The diamond-vs-circle geometry, soft-thresholding, Elastic Net, and a sparse Nasdaq fit.

Maximum Drawdown

quant finance

risk

Maximum drawdown: the largest decline from a running peak in an equity curve. Drawdown, recovery, the Calmar ratio, and a Nasdaq-100 example where the worst loss was −36% and NVDA fell two-thirds.

Mean Squared Error (MSE)

machine learning

optimisation

Mean squared error: MSE = mean of squared residuals, RMSE its root. Why squaring makes it smooth and outlier-sensitive, MSE vs MAE, the link to variance and R², and a fat-tailed Nasdaq example.

Moving Averages (SMA & EMA)

time series

quant finance

Simple and exponential moving averages: averaging the last k prices to reveal trend. The smoothing factor α = 2/(k+1), the lag-versus-responsiveness trade-off, crossovers, and an honest Nasdaq backtest.

Regularisation

machine learning

optimisation

Regularisation adds a complexity penalty λR(θ) to the loss, so a model fits the data while staying simple. The bias-variance dial, weight shrinkage, why markets demand it, and an honest Nasdaq overfitting experiment.

Sharpe Ratio

quant finance

performance

The Sharpe ratio: excess return over volatility, the field’s standard risk-adjusted performance measure. √252 annualisation, the risk-free rate, and a Nasdaq-100 basket where the highest return is not the highest Sharpe.

Simple, Log & Cumulative Returns

quant finance

returns

The three ways to express a return: simple (percentage change), log (continuously compounded), and cumulative (total over a period). Why log returns add across time and simple returns average across assets, and the classic compounding trap.

Skewness & Kurtosis

core statistics

risk

Skewness and kurtosis: the shape of a return distribution beyond mean and variance — asymmetry, fat tails, excess kurtosis, and why a single σ understates crash risk. A Nasdaq-100 example against the normal.

Sortino Ratio

quant finance

performance

The Sortino ratio: excess return over downside deviation instead of total volatility. Why it stops penalising upside, the divide-by-n convention, and a Nasdaq-100 basket where it reranks Sharpe.

Stationarity & the ADF Test

time series

core statistics

Stationarity: constant mean and variance over time. Why prices aren’t and returns are, the unit-root idea, the Augmented Dickey-Fuller test, and the spurious-regression trap of ignoring it.

The AR Model (Autoregression)

time series

core statistics

The autoregressive model: y_t = c + φ y_{t-1} + ε. How the AR coefficient φ controls memory, why φ = 1 is the unit-root boundary, the link to the ACF and the ADF test, and an honest Nasdaq fit.

The Cost Function

machine learning

optimisation

The cost function J(θ): one number scoring how wrong a model is, as a function of its parameters. Loss vs cost, convex vs non-convex landscapes, why the choice of cost defines the model, and Nasdaq examples.

The Hurst Exponent

time series

quant finance

The Hurst exponent H: below 0.5 mean-reverting, 0.5 a random walk, above 0.5 trending. Variance-scaling and R/S estimation, why H = 0.5 is the √time rule, and an honest Nasdaq estimate.

The MA Model (Moving Average)

time series

core statistics

The moving-average model: y_t = μ + ε_t + θ ε_{t-1}. Regression on past shocks (not values), why its ACF cuts off after lag q, invertibility, the AR(∞) duality, and why it isn’t the SMA/EMA.

The Sigmoid Function

machine learning

classification

The sigmoid σ(z)=1/(1+e⁻ᶻ): squashing any real score into (0,1). The logit inverse, the σ(1−σ) derivative and vanishing gradients, logistic regression, and an honest Nasdaq up/down fit.

The Softmax Function

machine learning

classification

Softmax: exp-then-normalise a vector of scores into probabilities that sum to 1. Why it reduces to the sigmoid, shift-invariance and numerical stability, temperature, and a 3-class Nasdaq regime example.

Value at Risk & Expected Shortfall

quant finance

risk

VaR and Expected Shortfall (CVaR): the two standard tail-risk numbers. Historical vs Gaussian VaR, why ES is the coherent measure, and a Nasdaq-100 example where the normal model understates the tail.

Variance & Standard Deviation

core statistics

risk

Variance and its square root, standard deviation (volatility): the spread of a return series, sample vs population (n−1), √252 annualisation, and a worked Nasdaq-100 example — including the numpy/pandas ddof trap.

Win Rate, Profit Factor & Expectancy

quant finance

performance

The three numbers that judge a trading system: win rate, profit factor, and expectancy. Why win rate alone misleads, the break-even win rate, and a real Nasdaq mean-reversion example.

Z-score & the Normal Distribution

core statistics

distributions

The z-score (how many standard deviations from the mean) and the normal distribution it maps onto: the 68-95-99.7 rule, the central limit theorem, and the fat-tailed reality that breaks the Gaussian assumption in markets.

t-statistic, p-value & Confidence Interval

core statistics

inference

Statistical significance for a mean or a strategy: the standard error, the t-statistic, the p-value, and the confidence interval. Why the t-stat of a year’s return equals its Sharpe, and why proving an edge takes so long.

Mean Return

core statistics

returns

The arithmetic mean return: estimator of E[r], its gap to compound growth (the volatility drag), and a worked Nasdaq-100 example — by hand, in Python, and in Excel.

The roadmap

Core statistics

Mean return · variance · standard deviation · covariance · correlation · skewness · kurtosis · Z-score · normal distribution · t-statistic · p-value · confidence interval.

Quant finance

Simple return · log return · cumulative return · volatility · annualised return · annualised volatility · Sharpe ratio · Sortino ratio · maximum drawdown · Value at Risk · Expected Shortfall · beta · alpha · CAPM · Kelly criterion · CAGR · profit factor · win rate · expectancy.

Time series

Autocorrelation · moving average · exponential moving average · AR model · MA model · ARIMA · GARCH · Hurst exponent · cointegration · stationarity · Augmented Dickey–Fuller test.

Machine-learning mathematics

Cost function · gradient descent · mean squared error · cross-entropy loss · sigmoid · softmax · entropy · information gain · Gini impurity · regularisation · L1 and L2 penalties · backpropagation.