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Correlation: how many strategies do you really have?

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Start with the idea

Covariance measures whether deviations tend to share a sign; correlation removes their measurement scale. It describes linear co-movement in a sample. Diversification reduces variance through the cross terms, but can disappear when dependence changes during stress.

Symbols, units & horizon

xᵢ, yᵢ: paired observations
x̄, ȳ: sample means
n: pair count
Cov: sample covariance
sₓ, sᵧ: sample standard deviations
ρ (rho): dimensionless correlation between −1 and 1
Sxy: sum of cross-products of centred observations
Sxx, Syy: sums of squared centred observations
σ: common asset volatility in return units
σ_b: equal-weight blend volatility
Var: variance
X,Y: paired random return variables, with σ_X and σ_Y denoting their SDs (estimated by sₓ,sᵧ in sample calculations)
a,b,ε in y=a+bx+ε: intercept, slope and residual
N: number of equally weighted, equal-volatility streams in the diversification example

When and why to use this

Use correlation for an initial redundancy check, then use covariance with actual weights for allocation. Compare normal periods with stressed co-movement before calling a second strategy a hedge.

Correlation $ρ$ runs from $- 1$ to $+ 1$ and measures how much two return streams move together, after removing their scale.

ρ_{X Y} = \frac{Cov (X, Y)}{σ_{X} σ_{Y}} = \frac{\sum (x_{i} - \overline{x}) (y_{i} - \overline{y})}{\sqrt{\sum (x_{i} - \overline{x})^{2} \sum (y_{i} - \overline{y})^{2}}}

Algebra and arithmetic

Cancel the sample denominators in correlation

Write $Cov = \frac{S_{x y}}{(n - 1)}$ , $s_{x} = \sqrt{\frac{S_{x x}}{(n - 1)}}$ , $s_{y} = \sqrt{\frac{S_{y y}}{(n - 1)}}$ .
Dividing covariance by $s_{x} s_{y}$ cancels n−1 and gives $\frac{S_{x y}}{\sqrt{S_{x x} S_{y y}}}$ . Zero variance makes correlation undefined.

Work it by hand

x=(1,2,3), y=(2,4,6) give Sxy=4, Sxx=2, Syy=8; ρ=4/√16=1. Reversing y gives −1.

When you run several strategies, their combined volatility is not the average of their volatilities. For two equal-weight streams with the same $σ$ :

σ_{blend} = σ \sqrt{\frac{1 + ρ}{2}}

Algebra and arithmetic

Expand the equal-weight blend

For equal vol σ and weights ½, $Var [\frac{(X + Y)}{2}] = \frac{1}{4} [σ^{2} + σ^{2} + 2 ρ σ^{2}]$ .
Factor out $σ^{2}$ : variance is $\frac{σ^{2} (1 + ρ)}{2}$ . Take the square root. To solve for correlation from blend vol, rearrange to $ρ = 2 (\frac{σ_{b}}{σ})^{2} - 1$ .

Work it by hand

σ=20%, ρ=.5 gives blend vol = .2√.75=17.32%, not 10%.

ρ	blend vol / single vol	what it means
+1.0	100%	one strategy, two names
+0.9	97%	one strategy, two names, plus paperwork
+0.5	87%	some diversification
0.0	71%	uncorrelated — volatility falls by √2
−0.5	50%	a hedge that still earns

You add a second, independent (

ρ = 0

) strategy with identical EV and

σ

, 50/50 weight. Compared with running one, EV per unit of risk goes…

EV of the blend is the same; $σ$ drops to $\frac{σ}{\sqrt{2}} \approx 0.71 σ$ . EV/σ rises by $\frac{1}{0.71} = 1.41$ . The Sharpe ratio scales with $\sqrt{N}$ independent bets — this is the whole argument for diversification, and why it fails when the bets are not independent.

Regression: correlation with a slope

Linear regression fits $y = a + b x + ε$ . In trading the slope $b$ shows up as a hedge ratio (how much of X to hold against Y), as beta (sensitivity to the market), and as the core of mean-reversion signals (regress spread on time or on its own lag). Logistic regression does the same for a yes/no target — will the next bar close up? — producing a probability instead of a level, which plugs straight into the EV formula.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from statistics import mean
from math import sqrt

def correlation(x, y):
    if len(x) != len(y) or len(x) < 2:
        raise ValueError("Matching samples of at least two required")
    dx, dy = [v-mean(x) for v in x], [v-mean(y) for v in y]
    denominator = sqrt(sum(v*v for v in dx)*sum(v*v for v in dy))
    if denominator == 0:
        raise ValueError("Correlation undefined for constant inputs")
    return sum(a*b for a,b in zip(dx,dy))/denominator

def equal_weight_volatility(sigma, rho):
    return sigma*sqrt((1+rho)/2)

print(correlation([1,2,3], [2,4,6]), equal_weight_volatility(.2, .5))
from statistics import linear_regression

def hedge_regression(x_returns, y_returns):
    slope, intercept = linear_regression(x_returns, y_returns)
    return intercept, slope

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