minor formatting

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."
    ]