Correlation and diversification · RiskAtoms Academy

By the end you will be able to

1Calculate the covariance and correlation of two return series.
2Calculate the volatility of a two-asset portfolio from the weights, the volatilities and the correlation.
3Explain how the benefit of diversification depends on correlation.
4Explain the limits of correlation: it changes over time and can rise in a crisis.

Why it matters

Our fund holds 60% in equities with a volatility of 17.4% and 40% in bonds at 5.1%. The weighted average is 12.5%. The fund's actual volatility is 10.1%. Where did the rest go?

Core explanation

Risks that partly cancel

Volatilities do not simply add up, because assets do not all move at the same time. On a day when equities fall, bonds may rise, and part of the loss is offset. That is diversification, and the number that measures how much offset to expect is the correlation.

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

Moving together. On each day, look at whether each fund was above or below its average. Multiply the two deviations: the product is positive if both were on the same side, negative if they were on opposite sides. Average the products and you have the covariance. Its size depends on how big the moves are, so we scale it by the two standard deviations to get the correlation, always between −1 and +1:

+1: they always move together, in proportion.
0: no straight-line relationship.
−1: they always move in opposite directions, in proportion.

Over the 500 days, the equity index fund and the bond fund have a correlation of −0.27. They moved in opposite directions on 320 days and fell together on 88.

ExploreEach day as a dot

Stretch of the data

Correlation

−0.27

500 days, 88 of them with both funds down.

Correlation is not constant. In the calm stretch the correlation was −0.32; during the turbulent spell it weakened to −0.16. On 3 of the 5 worst equity days, the bond fund fell too. The offset you count on can shrink exactly when you need it. Real markets show the same pattern: in 2022, rising inflation and interest rates pushed stocks and bonds down together.

The volatility of a portfolio

For two assets with weights $w_1$ and $w_2$ , the portfolio's variance has three parts: each asset on its own, plus a cross term for how they move together. The correlation sits in the cross term:

Correlation ρ	Volatility of the 60/40 fund
1.00	12.51%
0.50	11.63%
0.00	10.66%
−0.27 (the data)	10.10%
−1.00	8.41%

At +1 the fund's volatility is just the weighted average, 12.51%: nothing cancels. Every step down in correlation lowers it. At the data's −0.27 the diversification benefit is 2.41 percentage points. A negative correlation even makes a mix less volatile than its calmer part alone: holding about 13% in equities and the rest in bonds gives the lowest volatility, 4.4%, below the bond fund's 5.1%.

Quick checks

Your callWhich pair diversifies best?

0 of 2

You hold a US equity index fund. Which addition, at the same volatility, lowers your portfolio's volatility the most?
Two bank shares, both with a volatility of 30% and a correlation of 0.9. Does splitting money equally between them cut the volatility much?

The formal version

Take two return series $x_t$ and $y_t$ over $n$ days, with means $\bar x, \bar y$ and sample standard deviations $s_x, s_y$ (n − 1, as in L2.5.01: Measuring volatility). Covariance and correlation are new notation, so let's build them.

Now combine two assets. The rule for the variance of a sum of random variables (M-C11: Combining random variables) gives the portfolio's variance.

The formulas, for reference

\rho_{xy} = \frac{\operatorname{cov}(x,y)}{s_x s_y}, \qquad \operatorname{cov}(x,y) = \frac{1}{n-1}\sum_{t=1}^{n}(x_t-\bar x)(y_t-\bar y)

\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\,w_1 w_2\,\rho_{12}\,\sigma_1\sigma_2}

The weights add up to 1 and are held constant (the portfolio is rebalanced). Daily or annual volatilities both work, as long as the two are on the same basis. The maths is in M-C10: Covariance and correlation and M-C11: Combining random variables.

Try it

Start at the 60/40 fund. Drag the correlation up to +1 and watch the curve straighten into the dashed line: no diversification. Then drag it to −1, where the right mix brings volatility to zero. Finally, move the weight to find the lowest point of the curve at the data's correlation.

Try itTwo-asset portfolio

Many assets: covariance matrix toolkit →

Weight in asset 160%

Correlation ρ−0.27

Asset 1 volatility (a year)17.4%

Asset 2 volatility (a year)5.1%

Defaults: asset 1 is the equity index fund, asset 2 the bond fund, with the correlation of their 500 daily returns (illustrative data).

Portfolio volatility, a year

10.08%

Weighted average of the two12.48%

Diversification benefit2.40 pts

On $10m, one SD a year$1,008,239

your correlation ρ = +1 and ρ = −1

Worked example

The 60/40 fund's volatility from its parts

First a four-day warm-up by hand, then the fund itself. Try each step before revealing it.

Correlation by hand: four days

Step 1 of 3

1
Returns and means
x: 1%, -1%, 2%, 0%. y: 2%, -1%, 3%, 0%. Means $\bar x = 0.5\%$ and $\bar y = 1.0\%$ .

The 60/40 fund

Step 1 of 6

1
Inputs from 500 days
Annualised volatilities $\sigma_1 = 17.44\%$ (equity index fund) and $\sigma_2 = 5.13\%$ (bond fund), correlation $\rho = -0.2717$ , weights 0.6 and 0.4.

For more than two assets, the covariance matrix toolkit does the same calculation with a full matrix.

Regulatory lens

Regulators are wary of diversification benefits, because correlations measured in calm times can break down in a crisis. The Basel market risk standard (the Fundamental Review of the Trading Book, FRTB) shows this caution in its standardised approach: banks aggregate their risks with prescribed correlations, and must work out the capital under a high, a medium and a low correlation scenario, then hold the largest of the three. The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting. How firms use diversification to manage risk is the subject of L2.6.05: Diversification and its limits.

Sparks

SparkThe only free lunch

Harry Markowitz's paper “Portfolio Selection”, published in the Journal of Finance in 1952, showed that a portfolio's risk depends on how its holdings move together, not just on each one's own risk. He shared the 1990 Nobel prize in economics for it. Diversification is often called the only free lunch in finance: lower risk without giving up expected return.

Spark2022: when both legs fell

A mix of stocks and high-quality bonds relies on bonds rising when stocks fall. In calendar year 2022, as inflation and interest rates jumped, the S&P 500 lost about 18% (total return) and the Bloomberg US Aggregate bond index about 13%. A typical 60/40 portfolio had one of its worst years in decades.

Key takeaways

Covariance averages the products of two series' deviations; correlation scales it by the two standard deviations, so −1 ≤ ρ ≤ +1.
Two-asset portfolio volatility: σp = √(w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂).
At ρ = +1 there is no diversification; the lower the correlation, the larger the benefit. At ρ = −1 the right mix removes volatility altogether.
Our 60/40 fund: 10.1% volatility against a 12.5% weighted average, thanks to a correlation of −0.27.
Correlations change over time and can rise in a crisis. Correlation sees only straight-line co-movement; matrices and dependence beyond correlation come later.

Reaction

Check what stuck

Question 1 · objective 2

A portfolio is 50% in asset A (volatility 20%) and 50% in asset B (volatility 10%). Their correlation is 0.5. What is the portfolio's volatility, in per cent?

%

Question 2 · objective 1

Two daily return series have a covariance of 0.0006 (in decimals squared) and volatilities of 3% and 4%. What is their correlation?

Question 3 · objective 3

When does combining two risky assets bring no reduction in volatility at all?

AWhen their correlation is 0BWhen their correlation is +1CWhen their correlation is −1DWhen their volatilities are different

Question 4 · objective 3

Asset A has a volatility of 20%, asset B of 10%, and their correlation is exactly −1. What weight in A, in per cent, makes the portfolio's volatility zero?

%

Question 5 · objective 4

Which statement about correlation is right?

AA correlation estimated from calm years holds in a crisisBZero correlation means the two assets are unrelated in every wayCCorrelation measures average co-movement; in a crisis it can change sharply, and it can miss dependence in the tailsDCorrelation tells you which asset causes the other to move

Glossary

Covariance: The average product of two series' deviations from their means: positive if they tend to move together, negative if they tend to move apart.
Correlation: Covariance scaled by the two standard deviations, so it always lies between −1 and +1.
Diversification: Spreading money across assets that do not move in lockstep, so that their swings partly cancel.
Diversification benefit: How much lower a portfolio's volatility is than the weighted average of its parts' volatilities.
Portfolio weight: The share of a portfolio's value held in one asset.

Deep Dives & Origins

Deep Dives

Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill, the chapter on optimal risky portfolios. Intermediate.
J. C. Hull, Risk Management and Financial Institutions, Wiley, the chapter on correlations and copulas. Intermediate.

Origins

H. Markowitz, “Portfolio Selection”, Journal of Finance, 7(1), 1952, pp. 77–91.
Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2019 (the sensitivities-based method and its correlation scenarios).
S&P Dow Jones Indices and Bloomberg index factsheets, calendar-year 2022 total returns.

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