428. Correlated Pooling
The generalization of the square-root law to demand sources with non-zero pairwise correlation. The crucial formula:
where is the correlation between demands and .
428.1. Derivation
For random variables with and :
428.2. Three regimes
For identical and uniform :
| Pooled as | Benefit | |
|---|---|---|
| classic square-root law (full pooling benefit) | ||
| for large | reduced benefit — pool variance grows linearly in | |
| can be smaller | enhanced benefit — pool variance grows sub-linearly | |
| no benefit — perfectly correlated demand sums proportionally | ||
| perfect cancellation — sum is deterministic |
428.3. Numerical example
stores, each . Compare pooling under different correlations:
| SS at | ||
|---|---|---|
| (independent) | ||
| (somewhat positive) | ||
| (perfect positive) | ||
| (slight negative) |
Compare: separate-stock total SS = .
Independent pooling cuts that in half. Highly correlated pooling provides no benefit. Negatively correlated pooling beats independent.
428.4. Intuition
When demands move together (positive correlation):
- Peaks and troughs reinforce — combined demand is more variable per unit of mean
- Pooling does little; you’d need to cover the joint peak anyway
When demands move opposite (negative correlation):
- One source up while another down — combined demand is smoother
- Pooling is more beneficial than for independent demands
428.5. Sources of correlation
| Type | Typical |
|---|---|
| Same product, different regions, no shared shocks | near |
| Same product, similar markets / season | moderate |
| Same product nationwide (TV ads, viral) | high |
| Substitute products in same store | moderate |
| Complementary products (winter / summer items) | strong |
428.6. Practical consequences
- Diversification works like in finance: bundle products with low / negative correlation to reduce inventory volatility
- Common cause events (weather, holidays, economic shocks) drive correlations up — risk pooling benefit shrinks during them
- Always estimate empirical before claiming pooling savings — independence is rarely a safe assumption
428.7. See also
- Risk Pooling — overview
- Square-Root Law — independent case
- Pearson Correlation