side_notes

My side-notes while reading

Core Financial Terms

graph TD
    A[Total PnL]
    A --> B[Factor PnL]
    A --> C[Idiosyncratic PnL]

    B --> D[Market + Industry PnL]
    B --> E[Style PnL]

    E --> F[Momentum]
    E --> G[Value]
    E --> H[Liquidity]
    E --> I[Other Styles]

Term	Definition
PnL (Profit and Loss)	A measure of the portfolio's financial gain or loss over time. Cumulative PnL sums daily or periodic returns.
Total PnL	The overall profit or loss from the strategy, including contributions from all sources (market, factors, idiosyncratic).
Idiosyncratic PnL	Profit or loss from stock-specific movements that cannot be explained by known risk factors (i.e., alpha).
Factor PnL	PnL attributable to exposure to systematic risk factors, such as market, industry, or style factors.
Style PnL	The component of factor PnL that comes from style-based factors like value, momentum, size, volatility, etc.
Sharpe Ratio	A risk-adjusted return metric defined as the mean return divided by the return volatility. Higher Sharpe = more return per unit of risk.
Alpha	The excess return of a portfolio that cannot be explained by exposure to known risk factors. Often interpreted as "manager skill."
Beta	A measure of a portfolio's sensitivity to a particular risk factor, most commonly the market. A beta of 1 means the portfolio moves with the market.

Factor Investing Terminology

Term	Definition
Market Factor	Exposure to broad market returns, often captured by a benchmark index like the S&P 500.
Industry Factor	Exposure to sector or industry-specific movements (e.g., tech, energy).
Style Factors	Quantitative attributes that drive return patterns across stocks. Common ones include:
→ Value	Stocks that are "cheap" relative to fundamentals (e.g., low P/E ratio). Historically tend to outperform.
→ Momentum	Stocks that have performed well recently tend to continue performing well (trend-following behavior).
→ Liquidity	Measures sensitivity to trading volume or ease of execution. Illiquid stocks may offer higher returns but more risk.
→ Other	Could include size (small vs. large cap), volatility, quality, or other non-standard styles depending on the model used.

Risk and Attribution Concepts

Term	Definition
Performance Attribution	A breakdown of portfolio returns to understand what contributed to performance (e.g., market, factor exposures, stock picks).
Variance Contribution	A measure of how much each factor or component contributes to the portfolio's overall return volatility.
Exposure	The degree to which a portfolio is influenced by a specific factor or risk — often represented as a portfolio beta to that factor.
Net of Market / Total ex Market	A portfolio return measure that excludes market returns, isolating factor or alpha components.
Drawdown	A peak-to-trough decline in PnL or returns. Used to assess risk and performance slumps.

Meta Concepts

Term	Definition
"Not Destiny" (re: Style PnL)	Style PnL can be managed — through hedging, optimization, or factor neutrality — unlike some external shocks.
Time-Series Attribution	Attribution over time, breaking PnL into components like market, factor, and idiosyncratic contributions period-by-period, not just on average.
Alpha Unlocked	Refers to identifying and retaining genuine skill-based returns by removing unintentional risks like style drags.

What is "Cross-Section" 橫截面 in Investing?

A cross-section in finance means:

The set of all assets (e.g., stocks, bonds) being observed or analyzed at a specific point in time.

For example:
- On July 12, 2025, the prices of 500 S&P stocks — that's a cross-section of 500 data points.
- The return of all stocks in a universe on the same day/week/month is a cross-section of returns.

Think of it as a "snapshot" across many items, at one point in time.

This is different from a time-series, which is:

One asset's performance over time.
E.g., Apple's stock price over 5 years is a time-series.

What is "Cross-Sectional Equalization"?

When we say cross-sectional equalization, it means:

We are adjusting numbers across all assets at a single time point, to standardize or normalize them.

For example:

If I compute the momentum score for each stock in the S&P 500 today, the raw scores might have:
- Some huge
- Some tiny
- Some negative

To make them comparable and balanced, I can:

Rescale them to have a mean of zero and standard deviation of 1 (Z-score normalization).
Or rank them and assign weights proportionally.

What Is Being "Distributed" and Why?

When we say:

"Raw factor scores across stocks can have uneven distributions."

We mean:

Factor scores (like momentum, value) across all stocks can be:
- Skewed
- Concentrated
- Unevenly spread

For instance:

A momentum score could range wildly, causing the portfolio to concentrate heavily in a few stocks with high scores.

This is a problem because:

You may inadvertently take on other risks (like overweighting certain industries, sizes, or market sensitivities).
You don't want one or two stocks dominating your portfolio due to extreme scores.

So we equalize to ensure:

No single stock or subset dominates.
The factor exposure is balanced across the whole stock universe.

Why Does This Affect "Unintended Market Beta"?

➡️ Market Beta:

Measures how sensitive a stock/portfolio is to market returns.
E.g., a beta of 1.2 means if the market goes up 1%, your stock is expected to go up 1.2%.

➡️ Problem:

If your raw factor scores (e.g., momentum) give higher scores to high-beta stocks, and you don't equalize, then:

Your portfolio will unintentionally be exposed to the market (beta) — you're not just capturing momentum but also extra market sensitivity.
So, even if you think you're trading "momentum", you're also accidentally loading on "market up/down movement".

➡️ Equalization Fix:

By cross-sectionally equalizing:

You standardize factor exposures, avoiding hidden correlations with the market beta.
You isolate the pure effect of the factor (e.g., momentum).

Core Idea: Trading events like earnings announcements requires balancing expected returns with transaction costs and risk constraints, using quantitative heuristics for optimal position sizing and timing.

Context and importance:
- Although fundamental investing isn't event-driven by nature, earnings trades can contribute 25–50% of a PM’s PnL.
- Event trading demands a systematic, cost-aware approach distinct from general position updates.
Heuristics for event trading:
1. Position size should scale with expected return.
2. Higher transaction costs → smaller positions.
3. Less time to the event → smaller positions, due to limited build-up time.
4. Liquidate post-event to free capital and risk budget.
Market impact modeling:
- Trading moves prices—impact costs grow faster than linearly with trade size (~size² relationship).
- Optimal trade size formula:
  
  $$ \text{Optimal Size} = C \times \frac{\alpha \times V \times T}{2\sigma} $$
  - α: expected return
  - V: daily dollar volume
  - T: time to event
  - σ: daily volatility
  - C: market-specific constant (requires calibration)
Further refinements:
- Constant volume participation (VWAP) is generally optimal for small trades.
- For large, concentrated positions:
  - Risk constraints become critical.
  - Trading should accelerate into the event and decelerate (faster liquidation) afterward.
  - May involve temporary risk limit breaches, requiring judgment.
Practical considerations:
- PMs should monitor position concentration and risk closely when trading events.
- For complex or large trades, custom simulations and specialized models are advisable to optimize both PnL and risk.
Key takeaway:
- A simple quantitative model provides a solid starting point, but real-world trading requires adjustments for risk, liquidity, and timing dynamics, especially in concentrated portfolios.