You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.
Covariance and correlation
Covariance measures whether two variables move together, but its size depends on their units. Correlation scales it: r = covariance ÷ (sₓ × s_y), always between −1 and +1. It captures linear association only, can be distorted by outliers, and never proves causation.
Before this: Range, mean absolute deviation, variance and standard deviation
On the exam
- Interpret correlation between two variables to address an investment problem
Interpret the correlation between two variables. Expect r from a covariance and two standard deviations, a scatter plot to read, and a limitation to spot: nonlinearity, an outlier or a spurious correlation.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
How do you measure whether two returns move together?
Start with a picture. A scatter plot puts one variable on each axis, one dot per pair of observations. An upward drift means they tend to rise together, a downward drift means they move opposite ways, and a shapeless cloud means no clear pattern.
The sample covariance puts a number on that drift. For each pair, multiply how far X is above its mean by how far Y is above its mean, then average, using n − 1.
Covariance is positive when the variables tend to be above or below their means together, and negative when one is above while the other is below. Its size, though, depends on the units, so it cannot tell you how strong the link is.
How does correlation fix the size problem?
The sample correlation coefficient divides the covariance by the product of the two standard deviations. The units cancel, and the result lies between −1 and +1.
Correlation of +1 means the dots lie on an upward-sloping straight line, −1 on a downward-sloping one, and 0 means there is no straight-line pattern.
Often the exam hands you the covariance and the standard deviations instead.
Two assets have a covariance of returns of 0.0024 and standard deviations of 8.00% and 5.00%. Their correlation is closest to:
0.0024 ÷ (0.08 × 0.05) = 0.6
Correlation, r0.6
- 0.0024÷(0.08×0.05)=
- 0.0024ENTER0.08ENTER0.05×÷
ENTER enter (on the HP, puts the number on the stack)
Try this 0 of 3
- Load “One outlier” and drag the ringed point until r reaches 0.60 or more.
- Load the U-shape and show the straight line: r is about 0, yet Y clearly depends on X.
- Switch to returns as decimals and compare covariance and r with the percent version.
Worked example: 4 pairs, covariance 3.33 (%²), correlation r = 0.85, a strong positive straight-line relation. Correlation measures straight-line association only, and a single extreme point can change it.
Drag a point, or focus it and use the arrow keys.
- Pair of returns
Show working
Means: X̄ = 2.00, Ȳ = 2.00 (n = 4).
Deviations X: 0.00, 2.00, −3.00, 1.00; Y: −1.00, 1.00, −2.00, 2.00.
Cross-products: 0.00 + 2.00 + 6.00 + 2.00 = 10.00.
Covariance = 10.00 ÷ (4 − 1) = 3.33.
sₓ = √(14.00 ÷ 3) = 2.1602; s_y = √(10.00 ÷ 3) = 1.8257.
r = 3.33 ÷ (2.1602 × 1.8257) = 0.8452: a strong positive straight-line relation.
- 2NDDATA2NDCLR WORKclear old data
- 2ENTER↓1ENTER↓X01, Y01
- 4ENTER↓3ENTER↓X02, Y02
- 1+|-ENTER↓0ENTER↓X03, Y03
- 3ENTER↓4ENTER↓X04, Y04
- 2NDSTATpress 2ND SET until LIN shows
- ↓↓↓↓↓↓↓↓a = 0.5714 (intercept)
- ↓b = 0.7143 (slope)
- ↓r = 0.8452
2ND second function (the shifted label above a key) · DATA data worksheet · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · +|- change sign · STAT statistics worksheet
- fCLEAR Σ
- 1ENTER2Σ+Y first, then X
- display1
- 3ENTER4Σ+Y first, then X
- display2
- 0ENTER1CHSΣ+Y first, then X
- display3
- 4ENTER3Σ+Y first, then X
- display4
- 0gŷ,rSTO7a = 0.5714 (intercept: ŷ at x = 0
- storeitinR7,sinceΣ+keepsnandthesumsinR1toR6)
- x≷yr = 0.8452
- 1gŷ,rRCL7−b = 0.7143 (slope: ŷ at x = 1 minus ŷ at x = 0)
f gold shift key · CLEAR Σ clear the statistics registers · ENTER enter (on the HP, puts the number on the stack) · Σ+ add a data point · CHS change sign · g blue shift key · ŷ,r estimated y and correlation · STO store · x≷y swap the last two numbers · RCL recall a stored number
Display: b₀ 0.5714, b₁ 0.7143, r 0.8452
Drag a point and watch r and the covariance. Try the U-shape preset, then the One outlier preset.
Exam tip Rearranged, covariance = r × sₓ × s_y. Keep standard deviations as decimals (8% is 0.08) when the covariance is a decimal.
What are the limits of correlation?
A correlation is a summary of a straight-line pattern. Before relying on it, check four things.
- Reading r Memorize +1 upward line, −1 downward line, 0 no linear pattern
- Limitations Use it check these before trusting r
- Linear only a strong curve can give r near 0
- Outliers one point can create or hide correlation
- Spurious correlation chance, or a hidden third variable
- Not causation association says nothing about cause
-
| Limitation | What happens | Example |
|---|---|---|
| Nonlinear relation | r near 0 despite a strong relation | An option’s payoff against the share price around the strike |
| Outlier | One point creates or hides correlation | A single crash month |
| Spurious correlation | Chance, or a hidden third variable | Ice-cream sales and sunburn cases, both driven by summer |
| Not causation | Association only | A high r never shows which variable moves the other |
Go deeper Why covariance has such awkward units
Covariance multiplies two deviations, so for two return series its unit is percent times percent, or %². That is why a covariance of 0.0024 tells you nothing until you set it against the two standard deviations. It is also why you should never compare covariances across asset pairs with different volatilities: a bigger covariance may only mean the assets are more volatile.
Go deeper Where this goes next
Correlation is the input that makes diversification work: the lower the correlation between assets, the more portfolio risk falls (portfolio mathematics). Whether a sample correlation is statistically different from zero is a hypothesis test (tests of independence), and the square of the correlation between two variables is the R² of a simple regression.
Memorize this
MemorizeCorrelation at a glance
| r | Scatter plot | Meaning |
|---|---|---|
| +1 | points on an upward line | perfect positive linear link |
| 0 | no straight-line pattern | no linear link (curves possible) |
| −1 | points on a downward line | perfect negative linear link |
Memory hook Correlation is covariance with the units divided out: that is why it is stuck between −1 and +1.
Know it cold , so . Sample covariance divides by .
- Reading r Memorize +1 upward line, −1 downward line, 0 no linear pattern
- Limitations Use it check these before trusting r
- Linear only a strong curve can give r near 0
- Outliers one point can create or hide correlation
- Spurious correlation chance, or a hidden third variable
- Not causation association says nothing about cause
-
Quick example
Cross-products of the deviations add to 10 over n = 4 pairs. 10 ÷ (4 − 1) = 3.33 (%²).
- 10÷(4−1)= 3.33
- 10ENTER4ENTER1−÷ 3.33
ENTER enter (on the HP, puts the number on the stack)
Quick example
Covariance 0.0045, standard deviations 15% and 10%. r = 0.0045 ÷ (0.15 × 0.10) = 0.30.
- 0.0045÷(0.15×0.1)= 0.3
- 0.0045ENTER0.15ENTER0.1×÷ 0.3
ENTER enter (on the HP, puts the number on the stack)
How it is asked
-
Correlation from covariance
Covariance 0.0024, standard deviations 8% and 5%. Correlation?
0.0024 ÷ (0.08 × 0.05) = 0.60. Traps: 0.38 (÷ variance of X), 0.02 (÷ sum of the standard deviations).
-
U-shaped scatter with r near 0
A clear U-shape in the scatter plot, correlation 0.02. Conclusion?
Any relationship is not linear. Not "unrelated".
-
Two series that rise together for no reason
Umbrella sales and taxi rides are highly correlated.
Spurious correlation, driven by a third variable (rain). Not causation.
Exam traps
- Reading r = 0 as 'no relationship' rather than 'no linear relationship'.
- Inferring causation from a high correlation.
- Comparing covariances across pairs of assets with different volatilities. Compare correlations instead.
- Forgetting that one outlier can swing r from near 0 to high.
Checkpoint
5 exam-style questions. About 90 seconds each.
1Two assets have a covariance of returns of 0.0024 and standard deviations of 8% and 5%. Their correlation is closest to:
2A scatter plot shows a clear U-shaped pattern between two variables and a correlation of 0.02. The analyst should most appropriately conclude that:
3Monthly sales of umbrellas and of taxi rides are highly correlated. This is most likely an example of:
4Two assets have a correlation of −0.40, and their standard deviations are 5% and 10%. The covariance of their returns is closest to:
5An analyst finds that adding a single extreme month to a data set raises the correlation between two return series from 0.1 to 0.7. The analyst should most appropriately conclude that:
Key takeaways
- r = covariance ÷ (sₓ × s_y), between −1 and +1, with no units.
- Correlation measures linear association only.
- Outliers, nonlinearity and hidden variables can mislead.
- Correlation is not causation.
Quick example
Cross-products of the deviations add to 10 over n = 4 pairs. 10 ÷ (4 − 1) = 3.33 (%²).
- 10÷(4−1)= 3.33
- 10ENTER4ENTER1−÷ 3.33
ENTER enter (on the HP, puts the number on the stack)
Quick example
Covariance 0.0045, standard deviations 15% and 10%. r = 0.0045 ÷ (0.15 × 0.10) = 0.30.
- 0.0045÷(0.15×0.1)= 0.3
- 0.0045ENTER0.15ENTER0.1×÷ 0.3
ENTER enter (on the HP, puts the number on the stack)
Questions about this lesson?
Where this comes back
- Portfolio Mathematics Quantitative Methods
- Hypothesis Testing Quantitative Methods
- Simple Linear Regression Quantitative Methods
Questions candidates ask
What is the difference between covariance and correlation?
Both show whether two variables move together. Covariance is in the product of their units (%² for two returns), so its size is hard to interpret. Correlation divides covariance by the two standard deviations, so it is unit-free and always between −1 and +1.
Does zero correlation mean no relationship?
No. It means no linear relationship. Two variables can have a strong curved relationship, such as a U-shape, and still show a correlation near zero. Look at a scatter plot as well as the number.
What is spurious correlation?
A correlation that appears by chance or because both variables respond to a hidden third variable. Ice-cream sales and sunburn cases both rise in summer. They are correlated, but neither causes the other. A high correlation alone never proves a causal link.
Spotted an error? Tell us and we will fix it.