Appendix: Cross-Cutting Reference

The stuff that silently corrupts backtests if you get it wrong: annualization, return math, performance metrics, and the standard list of biases.

Annualization Factors

Multiply (or scale √ for volatility) by the periods-per-year for your bar frequency.

Rules.

• Returns scale linearly; volatility scales with √time. ${\sigma }_{\text{annual}}={\sigma }_{\text{period}}\times \sqrt{\text{periods/year}}$.
• $\text{return\_annual}=(1+\text{return\_period}{)}^{\text{periods/year}}-1$ (geometric) — not simple multiplication for compounded series.
• Match the factor to the data: annualizing daily-crypto vol with √252 understates it.

Return & Growth Math

Simple vs Log Returns

• Simple: $r=P_t/P_{t-1}-1$. Aggregates across assets (portfolio return = weighted sum).
• Log: $r=\ln (P_t/P_{t-1})$. Aggregates across time (sum of log returns = total log return). Symmetric, better for stats/vol.
• Algo: Use log returns for time-series modeling & vol; convert to simple for portfolio aggregation. Never mix.

CAGR

$\text{CAGR}=(P_{\text{end}}/P_{\text{start}}{)}^{1/\text{years}}-1$. Smooths the path — hides drawdowns and volatility.

Compounding

$\text{APY}=(1+\text{APR}/n{)}^n-1$, continuous: $e^{\text{APR}}-1$.

Performance & Risk Metrics

Sharpe Ratio

Def. Excess return per unit of total volatility. Formula. $\text{Sharpe}=(\text{mean}(r)-r_f)/\sigma (r)$, then $\times \sqrt{\text{periods/year}}$ to annualize. Algo. Use excess returns (subtract risk-free). Sensitive to non-normal returns; high Sharpe on short samples is often overfit.

Sortino Ratio

Def. Like Sharpe but penalizes only downside volatility. Formula. $\text{Sortino}=(\text{mean}(r)-r_f)/{\sigma }_{\text{downside}}$, where ${\sigma }_{\text{downside}}$ uses only returns below target (usually 0). Algo. Fairer for asymmetric/positively-skewed strategies; report alongside Sharpe.

Max Drawdown (MDD)

Def. Largest peak-to-trough equity decline. Formula. $\text{MDD}={\min }_t\left(\frac{{\text{Equity}}_t-{\text{running\_max}}_t}{{\text{running\_max}}_t}\right)$. Algo. The number that determines if you can survive the strategy psychologically & financially. Track duration of drawdown too.

Calmar Ratio

Def. Return relative to max drawdown. Formula. $\text{Calmar}=\frac{\text{annualized return}}{|\text{max drawdown}|}$. Algo. Good single number for trend/futures strategies where drawdown is the binding risk.

Volatility (realized)

$\sigma =\text{std}(\text{returns})$, annualized via $\sqrt{\text{periods/year}}$. Use sample std (ddof=1) for estimates; be explicit about ddof to match libraries.

Beta

Def. Sensitivity of an asset's returns to the market. Formula. $\beta =\frac{\text{Cov}(r_{\text{asset}},r_{\text{market}})}{\text{Var}(r_{\text{market}})}$. Algo. Estimate over a rolling window; β is unstable across regimes.

Value at Risk (VaR) / CVaR

Def. VaR = loss not exceeded with confidence X over horizon; CVaR = average loss beyond VaR. Formula. Historical: the Xth percentile of the return distribution. CVaR = mean of the tail beyond it. Algo. VaR ignores tail shape; CVaR (expected shortfall) is the coherent risk measure — prefer it for sizing.

Win Rate vs Expectancy

Def. Win rate alone is meaningless without payoff size. Formula. $\text{Expectancy}=(\text{WinRate}\times \text{AvgWin})-(\text{LossRate}\times \text{AvgLoss})$. Algo. A 40%-win strategy with 3:1 winners beats a 70%-win strategy with 1:3 winners. Optimize expectancy, not win rate.

Position Sizing

Kelly Criterion

Def. Growth-optimal bet fraction. Formula. $f^{\ast }=(bp-q)/b$, where b = win/loss payoff ratio, p = win prob, q = 1−p. For continuous: $f^{\ast }=\mu /{\sigma }^2$. Algo. Full Kelly is too aggressive (estimation error → ruin); use fractional Kelly (¼–½). Overestimating edge is catastrophic.

Volatility Targeting

Def. Scale position so portfolio hits a target volatility. Formula. $\text{position}=(\text{target\_vol}/\text{realized\_vol})\times \text{capital}$. Algo. Standard in CTA/managed futures; rebalance on a schedule, use lagged vol to avoid lookahead.

ATR-based Sizing

Def. Size by the asset's recent range so each trade risks a fixed $ amount. Formula. $\text{contracts}=\frac{\text{risk\_per\_trade (\$)}}{\text{stop\_distance}\times \text{\$/point}}$, stop_distance often $k\times \text{ATR}$. Algo. Normalizes risk across instruments of different volatility/price.

Backtesting Biases (the canonical failure list)

Slippage & Cost Model (minimum viable)

The fill price crosses the spread and adds impact/latency slippage; the total cost then sums commission, that slippage, and any borrow/funding on shorts/perps.

$$\text{fill\_price}=\text{quote}\pm \left(\frac{1}{2}\times \text{bid\_ask\_spread}\right)\pm (\text{slippage\_bps}\times \text{price})$$

$$\text{total\_cost}=\text{commission\_per\_trade}+\text{slippage}+\text{borrow/funding}$$

• Rule of thumb: if a strategy only works with zero costs, it doesn't work. Always include a cost model before trusting a backtest.
• Crypto perps: add funding cash flows per interval. Futures: add roll costs. Options: per-leg spread + commission.

Data Hygiene Checklist (before any backtest)

• Prices split/dividend adjusted (equities) — or you'll trade phantom gaps.
• Continuous futures: know the stitch method; recompute returns per contract if unsure.
• Fundamentals/macro keyed to release date, not period date.
• Timestamps in a single, explicit timezone.
• NaN/warm-up windows dropped, not forward-filled into signals.
• Annualization factor matches bar frequency (252 vs 365 vs intraday).
• Signals shifted ≥1 bar (no same-bar lookahead).
• Costs, slippage, and (where relevant) funding/borrow modeled.
• Out-of-sample period reserved and untouched until the end.