What is correlation between investments?

Correlation measures how closely the returns of two investments move together. It ranges from +1 (they always move in the same direction, in proportion) through 0 (no consistent relationship) to −1 (they always move in opposite directions). Correlation is what makes diversification work: combining investments that do not move in lockstep lowers the risk of the whole below the average risk of its parts.

The formula

The correlation between the returns of two investments $A$ and $B$ is their covariance divided by the product of their volatilities:

\rho_{A,B} = \frac{\operatorname{Cov}(r_A, r_B)}{\sigma_A \, \sigma_B}

It is usually estimated from daily, weekly or monthly returns over several years.

How correlation lowers risk

The volatility of a portfolio of two investments with weights $w_A$ and $w_B$ is:

\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 \, w_A w_B \, \rho_{A,B} \, \sigma_A \sigma_B}

The last term is where correlation enters: the lower it is, the lower the portfolio's volatility.

A simple illustration

Two investments each have a volatility of 20%, and a portfolio holds half of each:

Correlation	Portfolio volatility
+1.0	20.0%
+0.5	17.3%
0.0	14.1%
−0.5	10.0%

Two holdings of 20% volatility each. Neither becomes less risky - only their correlation changes.

With perfect correlation, combining the two does nothing for risk. Every step down in correlation lowers the volatility of the mix without changing its expected return.

Typical correlations

Stocks of different companies in the same market: often between 0.3 and 0.6.
Stock markets of different developed countries: often between 0.6 and 0.9, and higher than decades ago as markets have become more connected.
Stocks and high-quality government bonds: close to zero or slightly negative for much of the past two decades, but clearly positive in 2022, when both fell together as interest rates rose.

Limits

Correlations rise in crises. In a sell-off many investments fall together, exactly when diversification is needed most. Correlations measured in calm years overstate its benefit in bad ones.
It is not stable. A correlation describes the relationship over a particular period, and it can change when the economic environment changes.
It measures only linear co-movement. Two investments can be uncorrelated on average and still crash together in extreme events.

A worked example

Worked through on a sample portfolio. The figures below are that portfolio’s, not yours.

Which positions would likely fall together in a crisis?

These holdings show the strongest historical co-movement with your portfolio:

Asset	Weight (%)	Volatility p.a. (%)	Vol. contrib. (%)
Amazon.com Inc.	6,39%	27,76%	12,23%
Vanguard FTSE All-World UCITS ETF	13,96%	9,43%	10,92%
iShares Core S&P 500 UCITS ETF USD (Acc)	12,82%	9,40%	9,74%
Bitcoin	4,80%	36,07%	9,43%
Tesla Inc.	4,52%	35,08%	8,71%
iShares Core MSCI EM IMI UCITS ETF USD (Acc)	6,91%	17,40%	8,56%
ASML Holding N.V.	5,31%	31,74%	8,45%
NVIDIA Corp.	3,40%	30,56%	6,79%
Microsoft Corp.	6,16%	28,37%	6,29%
iShares Core MSCI World UCITS ETF	9,11%	9,31%	5,12%
Ethereum	1,57%	52,22%	4,21%
Allianz SE	6,36%	14,91%	2,74%
SAP SE	3,85%	29,18%	2,64%
LVMH S.A.	2,99%	26,66%	2,60%
Deutsche Lufthansa AG	2,80%	26,33%	1,31%
Coca Cola Co.	3,29%	17,30%	0,12%
Gold	0,00%	22,14%	0,00%

Interpretation: positions with high correlation to the portfolio and large contributions to portfolio volatility are likeliest to fall together in a crisis. That group in your table is mainly large ETFs and big-cap tech names, plus high-volatility exposures such as crypto and certain growth stocks; they therefore drive concurrent moves during market stress.

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