Vector-First Matrix Construction

Step 1: Raw Input Data

📈 data_bench (SPY benchmark)

Date	spytr
2024-01-01	0.012
2024-01-02	-0.008
...	...
2024-01-08	0.007
2024-01-09	0.021
2024-01-10	-0.005
...	...
2024-01-22	0.011

📅 15 total dates, need last 10 for lookback

📊 data_tr (Stock returns - 5 stocks)

Date	gvkey	tr
2024-01-08	001001	0.015
2024-01-08	001002	0.009
2024-01-08	001003	0.021
2024-01-08	001004	0.031
2024-01-08	001005	0.041
...	...	...

🔍 Thousands of rows, 5 stocks in example

🎯 baseline (Universe filter)

gvkey	wgt
001001	0.40
001002	0.35
001003	0.25

⚡ Only 3 stocks selected from data_tr

Step 2: Build Benchmark Vector FIRST (Establishes Master Calendar)

🎯 Benchmark Vector Construction

Input: data_bench, rebal_date=2024-01-22, lookback_days=10

lookback_dates = last 10 trading days before 2024-01-22

↓

Index	Date	SPY Return
[0]	2024-01-08	0.007
[1]	2024-01-09	0.021
[2]	2024-01-10	-0.005
[3]	2024-01-11	0.009
[4]	2024-01-12	0.018
[5]	2024-01-16	-0.003
[6]	2024-01-17	0.014
[7]	2024-01-18	-0.009
[8]	2024-01-19	0.006
[9]	2024-01-22	0.011

✨ Output:

RetBM_vector = [0.007, 0.021, -0.005, 0.009, 0.018, -0.003, 0.014, -0.009, 0.006, 0.011]

lookback_dates = ['2024-01-08', '2024-01-09', ..., '2024-01-22']

Step 3: Build Stock Matrix Using Same Dates (Forced Alignment)

📊 Stock Return Matrix Construction

Input: data_tr, baseline, required_dates (from vector!)

1. Filter universe: baseline selects only 3 stocks from 5 available

2. Initialize matrix: (10 dates × 3 stocks) filled with zeros

3. Fill column by column using EXACT same dates:

Date	001001 (AAPL)	001002 (MSFT)	001003 (GOOGL)
2024-01-08	0.015	0.009	0.021
2024-01-09	-0.012	0.018	0.005
2024-01-10	0.008	-0.003	0.000
2024-01-11	0.022	0.014	0.017
2024-01-12	-0.007	0.025	-0.011
2024-01-16	0.011	-0.008	0.009
2024-01-17	0.019	0.012	0.023
2024-01-18	-0.004	0.007	0.000
2024-01-19	0.013	-0.002	0.004
2024-01-22	0.006	0.016	0.012

🔴 Key Observations:

TSLA (001004) & NVDA (001005) completely ignored despite having data
GOOGL missing on 2024-01-10 & 2024-01-18 → filled with 0.000
Matrix shape: (10, 3) not (10, 5)

Step 4: Perfect Date Alignment for Optimization

🎯 Ready for Optimization Problem

Benchmark Vector (10,)

[0] → 0.007

[1] → 0.021

[2] → -0.005

...

[9] → 0.011

×

Stock Matrix (10, 3)

Row [0] → [0.015, 0.009, 0.021]

Row [1] → [-0.012, 0.018, 0.005]

Row [2] → [0.008, -0.003, 0.000]

...

Row [9] → [0.006, 0.016, 0.012]

@

Weights (3,)

w₁

w₂

w₃

Optimization:
minimize ||Matrix @ weights - benchmark_vector||²

✅ Perfect alignment: both have same 10 dates in same order

💡 Why "Vector-First" is Critical

🚨 Without Vector-First Approach:

Matrix might have dates: [Mon, Tue, Thu, Fri] (missing Wed)
Vector might have dates: [Mon, Tue, Wed, Thu, Fri]
❌ MISALIGNED OPTIMIZATION PROBLEM!

✅ With Vector-First Approach:

Vector establishes "master calendar": [Mon, Tue, Wed, Thu, Fri]
Matrix FORCED to use same dates: [Mon, Tue, Wed, Thu, Fri]
Missing stock data → fill with zeros
✅ PERFECT ALIGNMENT GUARANTEED!

🎯 The Result: Every position in the benchmark vector corresponds exactly to the same date in the matrix rows, enabling robust optimization across changing stock universes and missing data.

📊 Vector-First Matrix Construction