Lesson 16: Portfolio Construction and Exposure Management

Multi-strategy does not equal low risk. Correlation is the essence of risk.

A Typical Scenario (Illustrative)

Note: The following is a synthetic example to illustrate common phenomena; numbers are illustrative and don't correspond to any specific individual/account.

In early 2022, an investor showed me their "diversified portfolio":

They asked me: "These four strategies are all excellent, combining them should be more stable, right?"

I did a simple correlation analysis:

Finding: All strategy correlations are above 0.75.

Result: In 2022, this "diversified portfolio" drew down 35% - worse than any single strategy.

Why?

These four strategies look different, but they're all doing the same thing: going long US tech stocks. When tech stocks declined as a group, they all lost simultaneously, correlations approached 1, diversification effect vanished.

This is the cost of lacking a portfolio layer - you think you're diversifying risk, but actually stacking the same bet.

16.1 The Necessity of a Portfolio Layer

16.1.1 Standard Quant Flow

A complete quant system should be:

Signals -> Portfolio Optimization -> Risk Control -> Execution
 |              |                      |              |
 +-- Each       +-- Unified            +-- Secondary  +-- Real
    Strategy       Allocation             Review         Trading
    Signal

Many systems are missing the "Portfolio Optimization" layer:

Signals -> [Missing] -> Risk Control -> Execution
 |                          |              |
 +-- Direct trading         +-- Passive    +-- May
    from signals               checking       explode

16.1.2 What Problems Does the Portfolio Layer Solve?

16.1.3 Signal Quality Does Not Equal Portfolio Quality

This is a key distinction many overlook:

Single Strategy View:
  Strong signal -> Big position -> High return

Portfolio View:
  Strong signal + High correlation -> Big position -> Concentrated risk -> May lose big
  Strong signal + Low correlation -> Big position -> Diversified risk -> More robust

16.2 Position Sizing Methods

16.2.1 Four Common Methods Compared

16.2.2 Equal Weight vs Equal Risk Contribution

Paper Exercise:

You have two strategies:

• Strategy A: 10% annualized volatility
• Strategy B: 30% annualized volatility

16.2.3 Kelly at the Portfolio Level

Single strategy Kelly:

f* = (p x b - q) / b

Multi-strategy Kelly (considering correlation):

f* = Sigma^(-1) x mu

Where:
- f* = Optimal weight vector
- Sigma = Covariance matrix
- mu = Expected return vector

Key:
- High correlation strategies get down-weighted
- Negative correlation strategies get up-weighted

Practical recommendation:

• Use 1/2 Kelly or 1/4 Kelly
• Use robust estimation for covariance matrix (see next section)
• Set single strategy weight caps

16.3 Covariance Estimation and Shrinkage

16.3.1 Problems with Sample Covariance

Mean-variance optimization needs covariance matrix Sigma. The natural approach is to estimate from historical data:

Sample Covariance Matrix:
Sigma_hat = (1/T) x Sum((r_t - mu_hat)(r_t - mu_hat)')

Where:
  r_t = Return vector at time t
  mu_hat = Sample mean vector
  T = Number of samples

Problem: When asset count N approaches sample count T, the sample covariance matrix becomes extremely unstable.

Paper Exercise:

Why small T/N causes problems:

1. High estimation noise
   - Covariance matrix has N x (N+1)/2 parameters
   - 100 stocks = 5,050 parameters
   - 252 samples estimating 5,050 parameters -> extremely unreliable

2. Matrix may be singular
   - When N > T, Sigma_hat is singular (determinant = 0)
   - Cannot invert -> cannot do portfolio optimization

3. Extreme weights
   - Estimation errors get amplified by optimizer
   - Produces unreasonable large weights or short positions

16.3.2 Ledoit-Wolf Shrinkage Estimation

Core idea: "Shrink" the unstable sample covariance matrix toward a stable target matrix.

Shrinkage estimate:
Sigma_shrunk = delta x F + (1-delta) x Sigma_hat

Where:
  F = Target matrix (structured, stable)
  Sigma_hat = Sample covariance matrix (unbiased but noisy)
  delta = Shrinkage intensity (0 <= delta <=1)

When delta -> 1: Approaches target matrix (stable but biased)
When delta -> 0: Approaches sample matrix (unbiased but noisy)

Common Target Matrices:

16.3.3 Code Implementation

import numpy as np
from sklearn.covariance import LedoitWolf, OAS

def get_shrunk_covariance(returns: np.ndarray, method: str = 'ledoit_wolf') -> dict:
    """
    Calculate shrinkage covariance matrix

    Parameters:
    -----------
    returns : Returns matrix (T x N), each row is a period, each column is an asset
    method : 'ledoit_wolf' or 'oas' (Oracle Approximating Shrinkage)

    Returns:
    --------
    dict : Contains shrinkage covariance matrix and shrinkage intensity
    """
    if method == 'ledoit_wolf':
        estimator = LedoitWolf()
    elif method == 'oas':
        estimator = OAS()
    else:
        raise ValueError(f"Unknown method: {method}")

    estimator.fit(returns)

    return {
        'covariance': estimator.covariance_,
        'shrinkage': estimator.shrinkage_,
        'sample_cov': np.cov(returns, rowvar=False)
    }


def compare_covariance_stability(returns: np.ndarray, n_splits: int = 5) -> dict:
    """
    Compare stability of sample vs shrinkage covariance

    By splitting data, compare consistency of both methods across subsamples
    """
    T, N = returns.shape
    split_size = T // n_splits

    sample_covs = []
    shrunk_covs = []

    for i in range(n_splits):
        start = i * split_size
        end = start + split_size
        subset = returns[start:end]

        sample_covs.append(np.cov(subset, rowvar=False))

        lw = LedoitWolf()
        lw.fit(subset)
        shrunk_covs.append(lw.covariance_)

    # Calculate differences across subsamples (Frobenius norm)
    sample_diffs = []
    shrunk_diffs = []

    for i in range(n_splits):
        for j in range(i+1, n_splits):
            sample_diffs.append(np.linalg.norm(sample_covs[i] - sample_covs[j], 'fro'))
            shrunk_diffs.append(np.linalg.norm(shrunk_covs[i] - shrunk_covs[j], 'fro'))

    return {
        'sample_cov_variation': np.mean(sample_diffs),
        'shrunk_cov_variation': np.mean(shrunk_diffs),
        'stability_improvement': np.mean(sample_diffs) / np.mean(shrunk_diffs)
    }

16.3.4 Usage Example

import numpy as np
from sklearn.covariance import LedoitWolf

# Simulate 50 stocks, 1 year daily data
np.random.seed(42)
n_assets = 50
n_days = 252

# Generate returns with factor structure (more realistic)
factor_returns = np.random.normal(0.0005, 0.015, n_days)
betas = np.random.uniform(0.5, 1.5, n_assets)
idio_returns = np.random.normal(0, 0.02, (n_days, n_assets))
returns = np.outer(factor_returns, betas) + idio_returns

# Sample covariance
sample_cov = np.cov(returns, rowvar=False)

# Ledoit-Wolf shrinkage covariance
lw = LedoitWolf()
lw.fit(returns)
shrunk_cov = lw.covariance_

print(f"Asset count: {n_assets}")
print(f"Sample count: {n_days}")
print(f"T/N ratio: {n_days/n_assets:.2f}")
print(f"Shrinkage intensity delta: {lw.shrinkage_:.3f}")
print(f"Sample cov condition number: {np.linalg.cond(sample_cov):.0f}")
print(f"Shrunk cov condition number: {np.linalg.cond(shrunk_cov):.0f}")

# Example output:
# Asset count: 50
# Sample count: 252
# T/N ratio: 5.04
# Shrinkage intensity delta: 0.234
# Sample cov condition number: 847
# Shrunk cov condition number: 142

Interpretation:

• Shrinkage intensity delta = 0.234 means 23.4% from target matrix, 76.6% from sample matrix
• Condition number dropped from 847 to 142, meaning more stable matrix, more reliable inversion

16.3.5 When to Use Shrinkage Estimation

Practical recommendations:

1. Default to shrinkage estimation
   - Even with high T/N, shrinkage won't hurt
   - sklearn automatically computes optimal shrinkage intensity

2. Rolling windows need shrinkage even more
   - Rolling window = fewer samples
   - Shrinkage helps smooth covariance changes

3. Combine with factor models
   - Large portfolios (>100 assets) use factor models for dimensionality reduction
   - Apply shrinkage to residual covariance

16.4 Factor Exposure Management

16.4.1 What Is Factor Exposure?

Factors are common sources driving asset returns. Common factors:

16.4.2 Hidden Factor Exposure

Problem: You may not know what factors you're betting on.

Case: Four "Different" Strategies

Strategy A: Buy high ROE stocks
Strategy B: Buy low PE stocks
Strategy C: Buy financially solid stocks
Strategy D: Buy high dividend stocks

Looks like: Four different stock selection methods
Actually: All going long "Quality + Value" factor

Result: When Value factor fails (e.g., 2019-2020), all four strategies lose simultaneously

16.4.3 Factor Exposure Calculation

Paper Exercise:

Your portfolio holds:

Calculate portfolio factor exposures:

16.4.4 Factor Neutralization

If you want a factor-neutral portfolio:

Goal: Factor exposure ~ 0

Method 1: Constrained Optimization
  max  Expected return
  s.t. Factor exposure = 0
       Sum of weights = 1
       Weights >= 0

Method 2: Hedging
  If portfolio has 0.5 Value exposure
  Short 0.5 units of Value factor ETF

Method 3: Pair Trading
  For each Value stock bought
  Also buy a Growth stock (with equal Value Beta)

16.5 Hidden Leverage

16.5.1 What Is Hidden Leverage?

Explicit leverage: Borrow money to buy stocks, $1M capital + $1M borrowed = 2x leverage

Hidden leverage: No borrowing, but portfolio risk exposure exceeds capital

Case: Futures Portfolio

Capital: $1M
Holdings:
  - Stock index futures long: $2M notional (20% margin)
  - Bond futures long: $3M notional (15% margin)
  - Commodity futures long: $1.5M notional (15% margin)

Margin used: $500K
Idle cash: $500K

Appears to be: Only using 50% of capital
Actually: $6.5M notional exposure = 6.5x leverage!

16.5.2 Calculating True Leverage

True leverage = Sum(|Notional exposure|) / Capital

Or from risk perspective:

Risk leverage = Portfolio volatility / Benchmark volatility

Example:
  Portfolio annualized volatility: 30%
  S&P 500 volatility: 15%
  Risk leverage = 30% / 15% = 2x

16.5.3 Leverage Traps

16.5.4 Leverage Control Framework

16.6 Multi-Strategy Portfolio Pitfalls

16.6.1 The Two Faces of Correlation

Normal period: Strategy correlation = 0.3 (diversification works)

Crisis period: Strategy correlation -> 0.9 (diversification fails)

Why does correlation spike in crises?

1. Liquidity squeeze
   Everyone is selling -> All assets drop

2. Risk preference reversal
   "Risk-off" -> Only buy treasuries, sell everything else

3. Leverage liquidation
   Margin calls -> Forced selling -> Prices drop -> More margin calls

4. Panic contagion
   One market crashes -> Investors panic -> Sell all risk assets

16.6.2 Drawdown Synchronization Problem

Paper Exercise:

You have three strategies, each with 15% historical max drawdown.

16.6.3 Strategy Capacity Constraints

Problem: When strategy capacity is insufficient, continuing to add capital causes:

• Increased slippage
• Alpha decay
• Diminishing marginal returns

16.7 Multi-Agent Perspective

16.7.1 Portfolio Agent Responsibilities

16.7.2 Division with Risk Agent

Collaboration Flow:

Signal Agents --> Portfolio Agent --> Risk Agent --> Execution Agent
                       |                   |
                       v                   v
                  Optimal weights    Review/Reduce/Reject
                  Factor exposures       Leverage check
                  Correlations          Drawdown check

16.7.3 Portfolio Optimization Frequency

Recommendation:

• Only rebalance when weight change exceeds threshold (e.g., 5%)
• Avoid over-trading costs

Acceptance Criteria

After completing this lesson, use these standards to verify learning:

Lesson Deliverables

After completing this lesson, you will have:

1. Position Sizing method comparison - Equal weight, equal volatility, equal risk contribution
2. Factor exposure calculation framework - Identify hidden risk exposures in portfolio
3. Leverage control rules - Notional and risk leverage constraints
4. Portfolio Agent design template - Responsibilities and workflow for portfolio optimization

Lesson Summary

• Multi-strategy does not equal diversification, correlation determines diversification effect
• Equal weight allocation lets high volatility strategies dominate portfolio risk
• Factor exposure may be hidden, needs explicit monitoring
• Hidden leverage comes from notional exposure exceeding capital
• Crisis correlations spike, diversification effect fails

Further Reading

• Lesson 08: Beta, Hedging, and Market Neutrality - Basics of Beta and factors
• Lesson 15: Risk Control and Money Management - Risk control and portfolio layer collaboration
• Background: Statistical Traps of Sharpe Ratio - Statistical issues in portfolio evaluation
• Background: Famous Quant Disasters - Cases of leverage out of control

Next Lesson Preview

Lesson 17: Online Learning and Strategy Evolution

After the portfolio is built, markets change, strategies must evolve. How do you enable the system to continuously learn and self-update, rather than waiting until after a big loss to discover problems? Next lesson we explore online learning methods.