minor formatting

README.md

@@ -10,7 +10,7 @@ An end-to-end quant research pipeline — from raw price data to a backtested lo
 
 ## Contents
 
-1. [Finance $\leftrightarrow$ Linear Algebra Dictionary](#finance--linear-algebra-dictionary)
+1. [Finance ↔ Linear Algebra Dictionary](#finance--linear-algebra-dictionary)
 2. [Setup and Reproducibility](#setup-and-reproducibility)
 3. [Part I — Data and Factor Analysis (Notebooks 01–03)](#part-i--data-and-factor-analysis-notebooks-0103)
 4. [Part II — Backtest and Risk Decomposition (Notebooks 04–05)](#part-ii--backtest-and-risk-decomposition-notebooks-0405)
@@ -256,7 +256,7 @@ The long-only portfolio beats the EW universe by **3.6% per year**, though that
 
 Portfolio risk is the quadratic form $w^\top \Sigma w$. This notebook decomposes it using the eigendecomposition of $\Sigma$.
 
-* **PCA via scikit-learn** (`PCA` from `sklearn.decomposition`), fit on the centered return matrix (a balanced panel of 394 stocks over 240 months). The linear-algebra content is the same eigendecomposition $\Sigma = V\Lambda V^\top$ — we now delegate the SVD-based solve to sklearn rather than rolling it by hand.
+* **PCA via scikit-learn** (`PCA` from `sklearn.decomposition`), fit on the centered return matrix (a balanced panel of 394 stocks over 240 months). The linear-algebra content is the same eigendecomposition $\Sigma = V\Lambda V^\top$.
 * **Marchenko–Pastur cutoff** (random matrix theory): eigenvalues of a random covariance matrix fall inside $[\lambda_-, \lambda_+]$; eigenvalues above $\lambda_+$ are statistically significant factors. Here **9 eigenvalues** exceed the cutoff. PC1 explains 34% of variance and is essentially the market factor (correlation 0.96 with MKT-RF); PC2 loads on value (HML, 0.65).
 * **Ledoit–Wolf shrinkage** ($\hat\Sigma = \delta F + (1-\delta)S$) regularizes the noisy sample covariance; out-of-sample it lowers the Frobenius estimation error by ~5%.
 * **Variance decomposition:** splitting $\Sigma$ into the top-$k$ factor subspace and its orthogonal complement,
@@ -273,15 +273,15 @@ The momentum long-only portfolio's risk is **~90.7% systematic** and **~9.3% idi
 
 | Metric | Long-Only (net) | EW Universe | Long-Short (net) |
 |--------|-----------------|-------------|------------------|
-| Annualized return | 19.6% | 15.9% | -0.7% |
+| Annualized return [mean of the inner product $w^\top r_{t+1}$] | 19.6% | 15.9% | -0.7% |
-| Sharpe ratio | 1.01 | 0.95 | -0.04 |
+| Sharpe ratio [$\bar r_p / \mathrm{std}(r_p)$ — a signal-to-noise ratio] | 1.01 | 0.95 | -0.04 |
-| Max drawdown | -57% | -47% | -70% |
+| Max drawdown [largest peak-to-trough drop of the compounded wealth curve] | -57% | -47% | -70% |
-| FF 4-factor alpha (annualized) | **+5.95% (t = 3.95)** | — | -2.95% (t = -1.41) |
+| FF 4-factor alpha (annualized) [orthogonal residual of the OLS projection onto the factor basis] | **+5.95% (t = 3.95)** | — | -2.95% (t = -1.41) |
-| Active return vs EW universe | +3.6% (IR 0.43, t = 1.92) | — | — |
+| Active return vs EW universe [$w^\top r$ minus its projection onto $\mathbf{1}$] | +3.6% (IR 0.43, t = 1.92) | — | — |
-| Walk-forward: Sharpe positive | 4 of 4 windows | — | — |
+| Walk-forward: Sharpe positive [positive signal-to-noise in each sub-window] | 4 of 4 windows | — | — |
-| Walk-forward: active positive | 3 of 4 windows | — | — |
+| Walk-forward: active positive [positive projection residual in each sub-window] | 3 of 4 windows | — | — |
-| Survivorship drag to lose alpha | ~3% per year | — | — |
+| Survivorship drag to lose alpha [bias from the non-random column set needed to cancel $\alpha$] | ~3% per year | — | — |
-| Systematic risk share | ~90.7% | — | — |
+| Systematic risk share [variance in the top-$k$ eigenspace, $w^\top B\Sigma_f B^\top w$, as a share of $w^\top \Sigma w$] | ~90.7% | — | — |
 
 **Bottom line:** a sector-neutralized momentum signal, traded long-only, generates a Fama–French 4-factor alpha of **5.95% annualized (t = 3.95)**, with a positive Sharpe in all four walk-forward windows and robustness to plausible survivorship bias. The long-short variant does not work — the edge is on the long side.
 

notebooks/02_factor_analysis_and_diagnostics.ipynb

@@ -141,7 +141,7 @@
    "metadata": {},
    "source": [
     "## Factor definitions\n",
-    "We'll now compute our four standard factors. All are cross-sectional ranks (permutations) at each month-end, so they live on the same scale and can be combined later without further normalization.\n",
+    "We'll now compute our four standard factors. All are cross-sectional ranks (permutations) at each month-end, so they live on the same scale and can be combined later without further normalization.\n\nEach factor below is an economic hypothesis about a vector $f_t \\in \\mathbb{R}^{N_t}$; the rest of this notebook measures the angle between that vector and $r_{t+1}$ (their cosine similarity is the IC).\n",
     "\n",
     "- **Momentum**: Trailing 12-month return skipping the most recent month ($t - 12$ to $t - 2$). The intuition behind it is that stocks that went up over the past year tend to keep going up for another month or two\n",
     "- **Values**: we use price-based inverse momentum as a value proxy: 60-month trailing return, inverted. The intuition is that stocks that went down over 5 years are \"cheap\" and may mean-revert.\n",
@@ -791,7 +791,7 @@
     "\n",
     "IC at 1-month tells you the signal exists. IC at longer horizons tells you how fast it dies. This sets your rebalance frequency. A factor with positive IC out to 12 months is slow-moving — you can trade it quarterly and capture most of what you need it to. A factor that's dead after 2 months is fast — you need monthly rebalancing, and transaction costs can play a role. \n",
     "\n",
-    "Momentum is typically slow. Short-term reversal is fast. Low-vol is very slow. The decay curve tells you which is which."
+    "Momentum is typically slow. Short-term reversal is fast. Low-vol is very slow. The decay curve tells you which is which. In vector terms, 'slow' means the angle between $\\mathrm{rank}(f_t)$ and $\\mathrm{rank}(r_{t+h})$ stays small as the horizon $h$ grows; 'fast' means it opens quickly."
    ]
   },
   {

notebooks/04_backtest_and_performance.ipynb

@@ -143,7 +143,7 @@
    "source": [
     "## Benchmark Factors and Alpha\n",
     "\n",
-    "A portfolio that returns 20% sounds great — but if the market also returned 18%, most of that performance is just the market: the portfolio went up because everything went up. To claim skill, we need to show returns *above and beyond* what standard risk factors explain.\n",
+    "A portfolio that returns 20% sounds great — but if the market also returned 18%, most of that performance is just the market: the portfolio went up because everything went up. To claim skill, we need to show returns *above and beyond* what standard risk factors explain — that is, a component lying in the orthogonal complement of the factor span.\n",
     "\n",
     "### What are the Fama–French factors?\n",
     "\n",
@@ -1089,7 +1089,7 @@
    "source": [
     "## Walk-Forward Analysis\n",
     "\n",
-    "A single full-sample Sharpe ratio doesn't quite tell us whether we have a robust strategy. Walk-forward analysis splits the sample into non-overlapping 5-year windows and computes performance metrics in each. A strategy that's positive in *every window* is far more convincing than one that earned all its returns in one lucky period.\n",
+    "A single full-sample Sharpe ratio doesn't quite tell us whether we have a robust strategy. Walk-forward analysis splits the sample into non-overlapping 5-year windows and computes performance metrics in each. A strategy that's positive in *every window* is far more convincing than one that earned all its returns in one lucky period. Stability of the signal-to-noise ratio across sub-windows is evidence that the angle between $f_t$ and $r_{t+1}$ is a persistent feature of the data, not a single-period artifact.\n",
     "\n",
     "This is pure consistency checking; we're not optimizing any parameters or anything."
    ]