6. Independence and dependence between events

Level 1: Probability: the language of uncertainty

Two positions, each with an 8% standard deviation. If they are independent, together they come to roughly 11.3%.

If they move almost in step, they come to 15.6%. That is to say, practically the same as not having diversified at all.

The two portfolios look alike on paper. In real risk they look nothing alike, and the difference has a precise name.

What independent means, exactly

Two events, or two random variables, are independent when knowing the outcome of one provides absolutely no information that changes the probability of the other.

Formally: the conditional probability of A given B is exactly equal to the unconditional probability of A, with no shift at all.

When this does not hold — when knowing B does change the estimated probability of A — the two are said to be dependent. That is exactly the situation studied in detail in the module on conditional probability.

Why diversification works

This distinction is the precise mathematical basis of something the market theory course explained intuitively: that diversification reduces risk when the assets combined are not perfectly correlated.

The exact reason is this. The variance of a sum of random variables equals the sum of their individual variances, plus an extra term for each pair of variables, called the covariance, which measures how much they tend to move together.

Two scenarios follow from that:

• When the variables are independent, or simply uncorrelated, those covariance terms are zero, and the total variance is just the sum of the parts.
• When dependence exists, those terms are not zero. They add to the combined variance if the relationship is positive, or subtract from it if it is negative.

The 8% arithmetic

A numerical example clarifies the size of the effect. Two positions each have a standard deviation of 8%, that is, a variance of 64 in units of percent squared.

First case, perfectly independent, with zero correlation:

• The variance of their sum would be 64 + 64 = 128.
• Their standard deviation, roughly 11.3%.
• That is clearly below the simple sum of the two standard deviations, which would be 16%. That reduction is the mathematical essence of diversification.

Second case, highly correlated, with a correlation of 0.9:

• The covariance term adds an extra 2 × 0.9 × 8 × 8 = 115.2.
• The combined variance rises to 243.2.
• The standard deviation ends up around 15.6%, very close to the 16% you would get with no diversification at all.

Ten tickers are not ten bets

This carries a very practical warning.

Many retail portfolios are built from stocks that feel like separate, independent bets: different tickers, different companies, different narratives.

In practice they are strongly correlated because they share a common driver:

• The same sector.
• The same macro exposure: interest rates, dollar strength.
• The same style factor: momentum, growth.

Ten large-cap technology stocks are not ten independent bets. Taken together they behave far more like a single concentrated bet with some added noise than like ten distinct sources of risk.

Correlations spike at the worst possible moment

A second warning, just as important: correlations between assets are not fixed constants over time.

A repeatedly documented fact is that correlations between risk assets tend to rise sharply during periods of stress and broad market declines.

This means something very uncomfortable. The diversification benefit that looked solid in calm periods, measured with a historically low correlation, can shrink dramatically at precisely the moment it is needed most. And the cause is exactly that the dependence between assets is not stable over time.

Zero correlation is not independence

It is also worth distinguishing precisely between independence and zero correlation. They are not the same thing, and conflating them is a common mistake.

Independence is a stronger condition than the absence of linear correlation. Two variables can have a correlation of zero, with no detectable straight-line relationship, and still be dependent through a non-linear relationship.

Correlation, as usually defined and computed, is incapable of detecting that relationship.

A zero correlation reading is therefore not proof of real independence. It is only proof that there is no measurable linear relationship.

The same applies to combining strategies

The same logic applies not only to combining assets but also to combining trading strategies within a single account.

Think of two apparently distinct systematic strategies that ultimately react to the same underlying driver. For example, if both are momentum variants. Or if both are secretly exposed to the same volatility regime.

Running them together in the name of diversification delivers far less genuine risk reduction than combining two truly independent sources of edge.

Correctly assessing the real dependence between positions, instead of assuming independence just because the positions look different on the surface, is exactly the mathematical foundation that separates real diversification from a mere illusion of it.