Correlated Pooling

432. Correlated Pooling

The generalization of the square-root law to demand sources with non-zero pairwise correlation. The crucial formula:

𝜎_{pool}^{2} = \sum_{𝑖 = 1}^{𝑁} 𝜎_{𝑖}^{2} + 2 \sum_{𝑖 < 𝑗} 𝜌_{𝑖 𝑗} 𝜎_{𝑖} 𝜎_{𝑗}

where $𝜌_{𝑖 𝑗}$ is the correlation between demands $𝐷_{𝑖}$ and $𝐷_{𝑗}$ .

432.1. Derivation

For random variables $𝐷_{1}, \dots, 𝐷_{𝑁}$ with $Var (𝐷_{𝑖}) = 𝜎_{𝑖}^{2}$ and $Cov (𝐷_{𝑖}, 𝐷_{𝑗}) = 𝜌_{𝑖 𝑗} 𝜎_{𝑖} 𝜎_{𝑗}$ :

Var (\sum 𝐷_{𝑖}) = \sum_{𝑖} Var (𝐷_{𝑖}) + 2 \sum_{𝑖 < 𝑗} Cov (𝐷_{𝑖}, 𝐷_{𝑗})

= \sum_{𝑖} 𝜎_{𝑖}^{2} + 2 \sum_{𝑖 < 𝑗} 𝜌_{𝑖 𝑗} 𝜎_{𝑖} 𝜎_{𝑗}

432.2. Three regimes

For identical $𝜎_{𝑖} = 𝜎$ and uniform $𝜌_{𝑖 𝑗} = 𝜌$ :

𝜎_{pool}^{2} = 𝑁 𝜎^{2} + 𝑁 (𝑁 - 1) 𝜌 𝜎^{2} = 𝜎^{2} [𝑁 + 𝑁 (𝑁 - 1) 𝜌]

𝜎_{pool} = 𝜎 \sqrt{𝑁 + 𝑁 (𝑁 - 1) 𝜌}

$𝜌$	Pooled $𝜎$ as $𝑁 \to \infty$	Benefit
$= 0$	$𝜎 \sqrt{𝑁}$	classic square-root law (full pooling benefit)
$> 0$	$𝜎 \sqrt{𝑁 + 𝑁 (𝑁 - 1) 𝜌} \to 𝜎 𝑁 \sqrt{𝜌}$ for large $𝑁$	reduced benefit — pool variance grows linearly in $𝑁$
$< 0$	$𝜎 \sqrt{𝑁 + 𝑁 (𝑁 - 1) 𝜌}$ can be smaller	enhanced benefit — pool variance grows sub-linearly
$= 1$	$𝜎 𝑁$	no benefit — perfectly correlated demand sums proportionally
$= - \frac{1}{𝑁 - 1}$	$0$	perfect cancellation — sum is deterministic

432.3. Numerical example

$𝑁 = 4$ stores, each $𝜎 = 30$ . Compare pooling under different correlations:

$𝜌$	$𝜎_{pool}$	SS at $𝑧 = 1.65$
$0$ (independent)	$𝜎 \sqrt{4} = 60$	$99$
$0.3$ (somewhat positive)	$\sqrt{4 + 12 \cdot 0.3} \cdot 30 \approx 86$	$142$
$0.5$	$\sqrt{4 + 12 \cdot 0.5} \cdot 30 \approx 95$	$156$
$1$ (perfect positive)	$\sqrt{4 + 12} \cdot 30 = 120$	$198$
$- 0.2$ (slight negative)	$\sqrt{4 - 12 \cdot 0.2} \cdot 30 \approx 51$	$84$

Compare: separate-stock total SS = $4 \cdot 1.65 \cdot 30 = 198$ .

Independent pooling cuts that in half. Highly correlated pooling provides no benefit. Negatively correlated pooling beats independent.

432.4. 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

432.5. Sources of correlation

Type	Typical $𝜌$
Same product, different regions, no shared shocks	near $0$
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 $-$

432.6. 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 $𝜌$ before claiming pooling savings — independence is rarely a safe assumption

432.7. See also

Risk Pooling — overview
Square-Root Law — independent case
Pearson Correlation