Formula Library

This page collects the formulas used throughout the school, grouped by topic. Each one links to the lesson that explains it with worked examples and, where available, to a calculator that does the maths for you. Formulas are written in plain text so they are easy to copy into a spreadsheet or code. Remember that formulas describe models; real markets add costs, slippage, gaps and fat tails that no formula fully captures.

Returns and growth#

Simple return = End value / Start value - 1
Log return = ln(End value / Start value)
CAGR = (End value / Start value) ^ (1 / Years) - 1
Gain needed to recover = 1 / (1 - Drawdown) - 1

Lessons: Measuring Returns and CAGR, Maximum Drawdown. Calculators: Compound Growth and CAGR Calculator, Drawdown Recovery Calculator.

Position sizing and trade maths#

Position size = (Account × Risk %) / |Entry - Stop|
Reward to risk = |Target - Entry| / |Entry - Stop|
Break even win rate = 1 / (1 + Reward to risk)
Expectancy = Win rate × Average win - Loss rate × Average loss
Profit factor = Gross profit / Gross loss
Kelly fraction = p - (1 - p) / b

Lessons: Position Sizing, Risk/Reward Ratio, Expectancy, Profit Factor, Kelly Criterion. Calculators: Position Size Calculator, Risk/Reward Calculator, Expectancy and Profit Factor Calculator, Kelly Criterion Calculator.

Risk and performance ratios#

Volatility (annual) = SD of periodic returns × √(Periods per year)
Sharpe ratio = (Return - Risk free rate) / Volatility
Sortino ratio = (Return - Target) / Downside deviation
Calmar ratio = Annual return / Maximum drawdown
Information ratio = Active return / Tracking error
Treynor ratio = (Return - Risk free rate) / Beta
Beta = Cov(asset, market) / Var(market)
Parametric VaR = z × Volatility × √(Days) × Portfolio value

Lessons: Variance and Standard Deviation, Sharpe Ratio, Sortino Ratio, Calmar and MAR Ratio, Information Ratio and Tracking Error, Treynor Ratio, Alpha and Beta, Value at Risk (VaR). Calculators: Sharpe and Sortino Calculator, Correlation and Beta Calculator, Portfolio Volatility and VaR Calculator.

Portfolio maths#

Two asset variance = w1² σ1² + w2² σ2² + 2 w1 w2 ρ σ1 σ2
Diversified volatility = σ × √(ρ + (1 - ρ) / N)
Risk contribution = w_i × (Σw)_i / Portfolio volatility

Lessons: Modern Portfolio Theory and the Efficient Frontier, Diversification, Risk Contribution and Risk Decomposition.

Options#

Call payoff at expiry = max(S - K, 0)
Put payoff at expiry = max(K - S, 0)
Put call parity: C - P = S × e^(-qT) - K × e^(-rT)
Black Scholes call = S e^(-qT) N(d1) - K e^(-rT) N(d2)
d1 = [ln(S/K) + (r - q + σ²/2) T] / (σ √T),  d2 = d1 - σ √T
Expected 1 SD move = S × Implied volatility × √T

Lessons: Option Payoff Diagrams, Put-Call Parity, Black-Scholes Model, Implied Volatility (IV). Calculators: Option Payoff Calculator, Black-Scholes and Greeks Calculator, Implied Volatility Calculator, Binomial Option Pricing Calculator.

Bonds and rates#

Bond price = Σ C / (1 + y)^t + Face / (1 + y)^n
Modified duration = Macaulay duration / (1 + y)
Price change ≈ - Modified duration × Δy × Price + ½ × Convexity × Δy² × Price
DV01 = Modified duration × Price × 0.0001

Lessons: How Bonds Work, Yield to Maturity, Duration, Convexity, DV01. Calculator: Bond Price, Duration and DV01 Calculator.

Futures and forex#

Futures fair value = Spot × e^((r - y) × T)
Futures P&L = Ticks × Tick value × Contracts
FX forward = Spot × (1 + r_quote × t) / (1 + r_base × t)
Pip value (USD quote) = Units × Pip size

Lessons: Basis and Basis Trading, Tick Size and Tick Value, Covered and Uncovered Interest Parity, Pips and Pipettes. Calculators: Futures Basis and Forward Price Calculator, Futures Tick Value Calculator, FX Forward Points Calculator, Pip Value Calculator.

Prediction markets#

Implied probability = Contract price / Payout
Expected value per contract = Your probability × Payout - Price

A Polymarket share priced at $0.62 implies a 62% probability. If you believe the true chance is 70%, the expected value is 0.70 times $1 minus $0.62, or $0.08 per share, before fees. Lessons: Reading Odds as Probabilities, Expected Value, Prediction Market Strategies and Risks.

Using formulas safely#

A formula is only as good as its inputs. Volatility, correlation and win rates estimated from short or calm periods can make risk look smaller than it is. When a formula feeds a real decision, such as position size or leverage, use conservative inputs, round sizes down and check the result against common sense. If a calculation says a tiny stop allows a huge position, the formula is working but the trade may still be a bad idea because of gaps and slippage. See Slippage and Stress Testing and Scenario Analysis.

Frequently asked questions#

What is the most important trading formula?#

Position size, because it controls how much each trade can lose; it is the foundation of risk management.

Can I copy these formulas into a spreadsheet?#

Yes. They are written in plain text so they translate directly into spreadsheet or code formulas.

Do formulas guarantee trading results?#

No. They describe models and risk; real results depend on your edge, costs and market conditions.

Next, browse the key diagrams in the Visual Library.