Appendix A · Quick-reference formula sheet

How to use this formula sheet. This appendix condenses every operational formula in the course into two dense tables — one per area — using the exact notation of modules 7 through 9. Each row gives the concept, the formula, the symbols to fix before computing, and the usage caution; the most frequently consulted formulas also appear as display blocks, outside the table, so they can be copied at a glance. Global conventions: losses are positive unless stated otherwise; \(r_t\) = log return; \(\alpha\) = tail level (0.01 for 99% VaR); \(z_\alpha = \Phi^{-1}(1-\alpha)\); \(\varphi, \Phi\) = standard normal density and CDF; \(w\) = weights, \(\Sigma\) = covariances, \(\mathbf{1}\) = vector of ones. And the golden rule of the course: no formula is executed by the LLM — the LangChain agent orchestrates, explains, and alerts; the number is always produced by a deterministic tool.

13.1 Risk and performance

13.1.1 Formula table: delta-normal/historical/Monte Carlo VaR, ES/CVaR (Rockafellar-Uryasev), realized volatility, EWMA λ=0.94, GARCH(1,1), MDD, Sharpe/Sortino/Omega/Calmar/Sterling/Ulcer, beta, TE/IR + Grinold's law, skew/kurtosis, Cornish-Fisher, Component/Marginal VaR (Euler), Kupiec/Christoffersen LR, PSR/DSR

Key formulas for quick reference:

Table A.1 — Risk and performance formula sheet: formula, key symbols, and usage cautions

Usage notes — risk and performance. When each formula fails: delta-normal VaR systematically underestimates risk under fat tails and negative skewness (\(S<0\), \(K_{\mathrm{exceso}}>0\) in equity) and does not capture derivative non-linearities; historical VaR relies on a single path and at 99% with 250 days has only ~2.5 tail observations; Monte Carlo shifts the risk to the stochastic generator — the comparative evidence shows that Gaussian MC underestimates tails relative to filtered historical simulation. The \(\sqrt{h}\) rule and Sharpe annualization with \(\sqrt{252}\) are valid only under i.i.d. returns: with serial autocorrelation, Lo (2002) documents annualized Sharpe ratios overstated by up to 65% (Alpha Vantage) . Fixed course conventions: \(\lambda=0{,}94\) daily (0.97 monthly; effective memory ≈ 17 days), per the RiskMetrics Technical Document (Godel Discount) ; EWMA does not revert to the mean — if reversion is needed, use GARCH(1,1) with \(\alpha+\beta<1\). Sterling is the least standardized formula in the literature: the original version adds 10% to the denominator, but variants with \(-10\%\) or with \(R_f\) circulate; always state the convention used (±10% tolerance) (Alpha Vantage) . Cornish-Fisher stops being monotonic for \(\lvert S\rvert\gtrsim 2\) or \(K\gtrsim 6\): verify the monotonicity of \(z_{CF}\) in \(\alpha\) before reporting. Kupiec has low power with small \(T\) and does not detect time clustering — which is why it is complemented with Christoffersen. Calmar and MDD depend on the window: a crisis-free history inflates them. And no backtest Sharpe is evidence without DSR > 0.95 after declaring the \(N\) trials (PyPI) .

13.2 Portfolios and execution

13.2.1 Formula table: Markowitz MVO + GMV + tangency, CAPM, FF3/FF5, Black-Litterman (π=δΣw, posterior, Idzorek ω), Kelly (discrete/continuous/fractional), ERC (Spinu), Ledoit-Wolf δ*, HRP, Almgren-Chriss (trajectory, κ, E(X), V(X)), square-root law, signal cross-sectional z-score

Key formulas for quick reference:

Table A.2 — Portfolio and execution formula sheet: formula, key symbols, and usage cautions

Usage notes — portfolios and execution. The weak link of MVO is not the QP but \(\mu\): errors in means weigh ~11× those of variances (Chopra–Ziemba) and the optimizer systematically maximizes them (Michaud); that is why GMV, ERC, and HRP — which do not use \(\mu\) or do not invert \(\Sigma\) — tend to win out of sample, and long-only constraints with bounds act as implicit shrinkage (Jagannathan–Ma). In production, never feed the raw sample \(\Sigma\): Ledoit–Wolf with closed-form \(\delta^{*}\) is the standard before any \(\Sigma^{-1}\). In Black-Litterman, \(\tau\) scales the confidence in the equilibrium prior: institutional practice is \(\tau\in[0{,}025,\ 0{,}05]\) or \(\tau = 1/T\), with \(\Omega\) calibrated per Idzorek (Github) ; the views \((P,\mathbf{q},\Omega)\) are the auditable entry point for signals, including LLM-generated ones. Full Kelly maximizes growth but with intolerable drawdowns: in practice it is traded fractionally at 1/4–1/2 — half-Kelly retains ≈ 75% of the growth with ~50% of the volatility (Github) . In execution: \(\kappa T\ll1\) reproduces TWAP; \(\kappa T\gg1\) liquidates front-loaded with half-life \(1/\kappa\), and the naive strategy (\(\lambda=0\)) is never optimal for a risk-averse trader (Blockchain News) . The square-root law is concave in \(Q\) — "it costs ~1 day of volatility to move 1 day of volume" (🦜️🔗 LangChain) : liquidating \(X = 1{,}000{,}000\) shares with \(Q/V_D = 0{,}25\), \(\sigma_D = 2\%\) and \(Y=0{,}6\) costs \(\approx 0{,}6\times2\%\times0{,}5 = 60\) bp. The cross-sectional z-score requires winsorizing the signal by date before ranking; with LLM signals, explicitly handle the point mass in the neutral class.

Appendix A of 3 — LangChain for Quantitative Trading. Data and versions verified as of July 2026; prices and market figures subject to change. LangChain 1.3.14 / langchain-core 1.5.2.