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Glossary, notation & formulas
Check a term or symbol without making optional mathematics a prerequisite. Formulas describe stated models and assumptions, not evidence that a strategy works.
Read the notation
- Subscripts such as xᵢ identify a component or observation. A superscript such as x² denotes a power; T on a matrix denotes transpose in the stated convention.
- Σ means add terms; Π means multiply them. Bounds say which indices participate. Do not confuse the summation symbol with a covariance matrix named Σ.
- ln is the natural logarithm and exp its inverse. √ is a square root. A vertical bar in P(A|B) means conditioning on B, not division.
- f′ is a single-variable derivative; ∂ denotes a partial derivative. ∇f collects partials, H is a Hessian, and ∫ accumulates over the specified domain.
- One percent is 0.01 as a fraction; one basis point is 0.0001. State whether variance uses decimal-return units squared or percentage-point units squared.
- A vector has ordered coordinates, a matrix has rows and columns, and multiplication needs compatible dimensions and aligned economic labels.
Equations below use native browser MathML with a plain-language reading and explicit assumptions. Long equations scroll within their own focusable panel, not the whole page.
Concept glossary
- Autocorrelation
Association of a series with its lagged values; overlapping targets can create it without independent new evidence.
Revisit the concept in its section →- Base rate
The event frequency before considering the additional signal; rare events can produce many false alerts even with a sensitive detector.
Revisit the concept in its section →- Basis point
0.01 percentage points, or 0.0001 as a fractional return. A rate moving 2% to 3% changes by 100 basis points.
Revisit the concept in its section →- Benchmark
A feasible comparator chosen to challenge the actual claim, using compatible dates, risks, costs, and information.
Revisit the concept in its section →- Brier score
Mean squared binary probability error; lower is better on the same eligible outcome sample.
Revisit the concept in its section →- Brownian motion
A continuous-path stochastic model with independent Gaussian increments and variance equal to elapsed time; not an empirical claim about every market.
Revisit the concept in its section →- CAGR
Constant annual compound growth rate matching positive starting and ending wealth over stated years without external flows.
Revisit the concept in its section →- Calibration
Agreement between probabilities and repeated event frequencies in comparable groups, with counts and uncertainty.
Revisit the concept in its section →- Capacity
How feasible net opportunity changes with size, trading, liquidity, and constraints; a curve rather than a fixed return label.
Revisit the concept in its section →- Collider
A shared consequence of two causes; conditioning on it can create association between those causes.
Revisit the concept in its section →- Concept drift
A changed conditional input-to-outcome relationship; stable input histograms alone do not rule it out.
Revisit the concept in its section →- Conditional probability
Probability within a specified information or event group; reversing the condition generally changes the denominator.
Revisit the concept in its section →- Conditioning
Sensitivity of a mathematical problem to perturbed inputs, distinct from an algorithm's numerical stability.
Revisit the concept in its section →- Confidence interval
An interval from a procedure with stated repeated-sampling coverage under assumptions, not a future-loss bound.
Revisit the concept in its section →- Covariance
Paired co-movement about the respective means, retaining the product of the variables' units.
Revisit the concept in its section →- Difference-in-differences
Treated-group change minus comparison-group change; causal interpretation needs a justified identification design.
Revisit the concept in its section →- Discrimination
Ability to rank cases by outcome risk; useful ordering need not imply accurate probability levels.
Revisit the concept in its section →- Drawdown
Loss from a running wealth peak to a later observation; historical maximum is not a future maximum-loss guarantee.
Revisit the concept in its section →- Embargo
A justified exclusion of nearby later training observations in certain blocked designs; not a universal arbitrary gap.
Revisit the concept in its section →- Excess kurtosis
Ordinary kurtosis minus three; distinguish convention and noisy tail estimation before interpreting a reported number.
Revisit the concept in its section →- Exchangeability
A joint-distribution symmetry assumption supporting standard conformal arguments; dependence and regime change can violate it.
Revisit the concept in its section →- Expected shortfall
Average upper-tail loss quantiles under a stated convention; probability masses can require fractional weighting at the threshold.
Revisit the concept in its section →- Expected value
Probability-weighted average outcome under a distribution; not a loss bound or a guarantee for one realization.
Revisit the concept in its section →- FDR
Expected false-rejection proportion among rejections under a procedure's assumptions, not the probability one discovery is false.
Revisit the concept in its section →- Gradient
Vector of partial derivatives; Euclidean directional derivatives use a dot product with a unit direction.
Revisit the concept in its section →- Hessian
Matrix of second partial derivatives describing local quadratic curvature and cross-sensitivity for a smooth function.
Revisit the concept in its section →- Holdout
An evaluation sample protected from fitting and selection; its information role changes once used to redesign a procedure.
Revisit the concept in its section →- Hypothesis
A measurable claim about a defined population, outcome, horizon, and comparison that evidence can contradict.
Revisit the concept in its section →- Implementation shortfall
Execution outcome relative to a defined decision benchmark, accounting for relevant delay and unfilled opportunity without double counting.
Revisit the concept in its section →- Itô correction
A second-derivative contribution from Brownian quadratic variation that modifies the ordinary smooth-path chain rule.
Revisit the concept in its section →- Jacobian
Output-by-input derivative matrix mapping a small input shock to a first-order output change.
Revisit the concept in its section →- Log return
Natural logarithm of a positive wealth ratio; additive through time for a consistent series.
Revisit the concept in its section →- Look-ahead bias
Historical decisions use information not yet available at their stated decision time.
Revisit the concept in its section →- Marginal risk
Local risk sensitivity under fixed estimated inputs; funded substitutions must include the leg reduced elsewhere.
Revisit the concept in its section →- Multiple testing
Repeated research choices create selection opportunities; the discarded experiments matter when interpreting a selected winner.
Revisit the concept in its section →- Nested evaluation
Outer evaluation of a full inner selection-and-fitting procedure; each learned inner transformation respects its own training boundary.
Revisit the concept in its section →- Null space
Input vectors mapped to zero by a linear transformation; its dimension plus rank equals the number of input coordinates.
Revisit the concept in its section →- P-value
Extremeness probability for the statistic under the null and sampling model; not a probability that the null is true.
Revisit the concept in its section →- Pearson correlation
Covariance normalized by positive standard deviations; describes linear association, not general independence or causality.
Revisit the concept in its section →- Percentage point
The direct difference between percentage rates. Moving 4% to 6% is two percentage points and a 50% relative increase.
Revisit the concept in its section →- Physical versus risk-neutral measure
Real-world forecasting probabilities and model-based pricing probabilities answer different questions and need their respective assumptions.
Revisit the concept in its section →- Pinball loss
Asymmetric residual loss targeting a requested quantile under population conditions; not a maximum-loss guarantee.
Revisit the concept in its section →- Point-in-time data
Values and identities representing what was actually available at each historical decision, including relevant vintages and latency.
Revisit the concept in its section →- Precision
True positive calls divided by all positive calls; undefined zero-denominator cases need a convention.
Revisit the concept in its section →- Purging
Removing ineligible training labels whose information intervals overlap a held-out interval under a defined design.
Revisit the concept in its section →- Recall
Detected positive events divided by actual positive events; a different conditioning group from precision.
Revisit the concept in its section →- Regularization
Changing a fitting objective or constraint to control sensitivity/complexity; its strength is a selected model choice, not merely display cleanup.
Revisit the concept in its section →- Sharpe ratio
Same-frequency mean excess return divided by excess-return standard deviation, with explicit annualization and cost conventions.
Revisit the concept in its section →- Shrinkage
Moving an estimate toward a structured target to trade bias for lower variance or improved stability under suitable conditions.
Revisit the concept in its section →- Simple return
Gain relative to beginning wealth over one period; compound consecutive periods by multiplying growth factors.
Revisit the concept in its section →- Skewness
A measure of distributional asymmetry. Negative skew can coexist with positive average return.
Revisit the concept in its section →- Sortino ratio
Return above a stated target relative to downside deviation; target and denominator conventions must be declared.
Revisit the concept in its section →- Span
All linear combinations of a collection of vectors; coefficients need not sum to one or be positive.
Revisit the concept in its section →- Spearman correlation
Correlation of ranks, measuring monotonic ordering rather than linear spacing of raw values.
Revisit the concept in its section →- Standard error
Sampling spread of an estimator, distinct from the spread of individual observations. Independence-based formulas can fail under dependence.
Revisit the concept in its section →- Statistical power
Probability of rejecting a false null for a specified alternative under the test's assumptions.
Revisit the concept in its section →- Survivorship bias
Selection based on surviving to a later date can change the historical population and omit failures.
Revisit the concept in its section →- SVD
Singular-value decomposition into input/output orthogonal directions and nonnegative singular values; exposes rank and near-redundancy.
Revisit the concept in its section →- Turnover
Trading activity under a declared convention; per-executed-dollar costs need total bought-plus-sold notional when that is what is charged.
Revisit the concept in its section →- Variance
Average squared deviation under the chosen population or sample convention; standard deviation takes its square root.
Revisit the concept in its section →
Formula reference
Read the formula in words and check its assumptions
Price return equals ending price minus beginning price, divided by beginning price.
- P₀: beginning price
- P₁: ending price
- r: fractional return
Nonzero beginning price; this price-only formula excludes distributions and external cash flows.
Read the formula in words and check its assumptions
Ending wealth equals initial wealth multiplied by the product of one plus each period's simple return.
- W₀, Wₜ: initial and ending wealth
- rₜ: each period's simple return
- Π: multiply the terms
Consistent wealth/return conventions and no external contributions or withdrawals; leverage and losses require appropriate accounting.
Read the formula in words and check its assumptions
Log return equals the natural logarithm of one plus simple return. Simple return equals the exponential of log return minus one.
- ℓ: log return
- r: simple return
- ln and exp: inverse functions
Positive wealth ratio. Time aggregation of log returns differs from cross-asset portfolio weighting.
Read the formula in words and check its assumptions
Probability of A given B equals probability of B given A times probability of A, divided by probability of B.
- P(A): prior event probability
- P(B|A): conditional likelihood
- P(B): total probability of evidence
P(B)>0; include event and non-event states in the evidence denominator.
Read the formula in words and check its assumptions
Expected net equals event probability times gain, minus non-event probability times loss magnitude, minus cost.
- p: event probability
- g, l: positive gain and loss magnitudes
- c: cost per action
A simplified two-state fixed-payoff model. Positive expectation alone does not justify unrestricted risk.
Read the formula in words and check its assumptions
Sample variance equals the sum of squared deviations from the sample mean, divided by n minus one.
- s²: sample variance
- x̄: sample mean
- n: observation count
n>1. Squared units differ from standard deviation. Unbiasedness needs an appropriate sampling model.
Read the formula in words and check its assumptions
Correlation is covariance divided by the product of the two positive standard deviations.
- Cov(X,Y): covariance
- σₓ, σᵧ: standard deviations
- ρ: linear correlation
Finite nonzero variances and matched observations. Zero linear correlation does not generally mean independence.
Read the formula in words and check its assumptions
Estimated standard error of the mean equals sample standard deviation divided by the square root of n.
- s: sample standard deviation
- n: independent observation count
Independent identical finite-variance observations. Serial or common-shock dependence requires a suitable alternative.
Read the formula in words and check its assumptions
Sharpe equals mean excess return divided by standard deviation of excess return.
- r−r_f: excess return
- Mean and standard deviation use the same frequency
Nonzero denominator; state risk-free, cost, horizon and annualization conventions. CAGR is not this numerator.
Read the formula in words and check its assumptions
Net return per capital equals gross return per capital minus executed notional divided by capital times cost per executed dollar.
- G, N: gross and net returns per capital
- T: total executed notional/capital
- c: cost per executed dollar
Same horizon and compatible units; do not double count all-in charges. Additional financing may remain.
Read the formula in words and check its assumptions
Average the squared difference between each predicted event probability and its zero-or-one outcome.
- pᵢ: event probability
- yᵢ: 0 or 1 outcome
- n: scored eligible outcomes
Lower is better. A score and calibration diagnose different properties; use matched later observations.
Read the formula in words and check its assumptions
The break-even probability is loss magnitude plus per-action cost, divided by gain plus loss magnitude.
- g: gain on event
- l: loss magnitude otherwise
- c: cost paid per action
g+l>0; fixed payoffs/costs. Positive net requires p above this threshold, subject to risk and implementation constraints.
Read the formula in words and check its assumptions
Variance equals squared weight times each own variance, plus twice the product of the weights, correlation, and volatilities.
- w and 1−w: sleeve weights
- σ_A, σ_B: compatible volatilities
- ρ: correlation
Fixed weights, finite moments, common horizon. Take a square root for volatility; stress changing dependence.
Read the formula in words and check its assumptions
The derivative at a is the limit as h approaches zero of f of a plus h minus f of a, all divided by h.
- h: nonzero input increment tending to zero
- f′(a): instantaneous slope
The finite limit must exist. Continuity alone does not imply differentiability.
Read the formula in words and check its assumptions
The definite integral of f from a to b equals an antiderivative F at b minus F at a.
- F′=f: antiderivative relationship
- a, b: lower and upper bounds
Use the theorem's integrability/antiderivative conditions. Improper boundaries need separate limits.
Read the formula in words and check its assumptions
The total derivative of f along x of t and y of t is partial f over x times x prime plus partial f over y times y prime.
- f_x, f_y: partial sensitivities
- x′, y′: rates along the actual input path
Suitable differentiability. Holding other inputs fixed is a different question from total path response.
Read the formula in words and check its assumptions
Multiply matching coordinates and add: the dot product is the sum of a_i times b_i.
- a, b: equal-dimension vectors
- i: matched component index
Coordinate order and units matter; the output is a scalar, not a vector or a claim of statistical independence.
Read the formula in words and check its assumptions
The projection of b onto the line spanned by u is u multiplied by u dot b divided by u dot u.
- u: nonzero direction vector
- b: target vector
- b−projection: perpendicular residual
Euclidean inner product and u≠0. Adding separate projections onto nonorthogonal directions generally double counts.
Read the formula in words and check its assumptions
For a differentiable function, the derivative in unit direction u equals the gradient dot u.
- ∇f: vector of partial derivatives
- u: Euclidean unit direction
Normalize u for change per unit distance. Scaling coordinates or changing the metric changes steepest direction.
Read the formula in words and check its assumptions
Function at a plus delta is approximately its value at a, plus gradient transpose times delta, plus one half delta transpose Hessian delta.
- δ: simultaneous input shock
- ∇f(a): gradient at the base point
- H(a): Hessian at that same point
Sufficient local smoothness and a suitably small shock; generic functions have remainder error.
Read the formula in words and check its assumptions
For the stated minimization convention, each multiplier is nonnegative, each constraint g_i is nonpositive, and their product is zero.
- g_i≤0: inequality convention
- λ_i≥0: corresponding multiplier
These are part of KKT, not the complete system. Stationarity and any equalities also apply; necessity and sufficiency need assumptions.
Read the formula in words and check its assumptions
A equals U times Sigma times V transpose.
- U: orthogonal output directions
- Σ: nonnegative singular values in a diagonal-shaped matrix
- V: orthogonal input directions
For full SVD of m×n A, dimensions are m×m, m×n, n×n. Numerical truncation changes the effective estimator.
Read the formula in words and check its assumptions
Under geometric Brownian motion, log price change has drift mu minus half sigma squared, plus sigma times the Brownian increment.
- μ: level drift under the specified measure
- σ: diffusion volatility
- W: standard Brownian motion
Positive S and the specified constant-coefficient diffusion; this is a model identity, not a claim that market returns are Gaussian.
Read the formula in words and check its assumptions
Estimated simulation mean standard error is sample payoff standard deviation divided by the square root of independent path count.
- s: payoff standard deviation
- N: independent finite-variance simulated payoffs
Distinguish sampling error from time discretization and model bias; paired or dependent paths need the appropriate variance calculation.
These are companion definitions, not a replacement for each section's worked examples and limitations. Read prerequisite refreshers in the optional resources whenever an algebra or calculus step is unfamiliar.