NISM-Series-X-A: Investment Adviser (Level 1) · Introduction to Modern Portfolio Theory

Covariance, Correlation and Portfolio Risk for NISM X-A

Updated 11 October 2026 · Fact-checked

Covariance measures how two assets' returns move together. Correlation is covariance scaled to a range of -1 to +1. Portfolio risk depends on weights, each asset's standard deviation and correlation. Use σp = √(w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2). Lower correlation means more diversification benefit.

Understand Covariance, Correlation and Portfolio Risk

Portfolio risk is not just the weighted average of the risks of the assets you hold. It also depends on how the assets move relative to each other. That relationship is what covariance and correlation capture.

Covariance tells you the direction of co-movement. A positive covariance means the two assets tend to be above or below their average returns at the same time. A negative covariance means one tends to be up when the other is down. Its size is hard to read, because it depends on the units and volatility of the assets.

Correlation fixes that problem. It divides covariance by the product of the two standard deviations. The result always lies between -1 and +1. A value of +1 means perfect positive co-movement, -1 means perfect opposite movement, and 0 means no linear relationship.

Why does this matter for risk? When two assets are not perfectly correlated, a fall in one is partly offset by the other. So portfolio standard deviation is lower than the weighted average of the individual standard deviations. This is the diversification benefit. It is largest when correlation is -1 and disappears only when correlation is +1.

Return is different. Expected portfolio return is always the simple weighted average of the asset returns. Correlation does not change it. Only risk changes.

Key formulas to remember

Correlation: ρ12 = Cov(1,2) ÷ (σ1 × σ2); Always between -1 and +1. Same sign as covariance.
Covariance from correlation: Cov(1,2) = ρ12 × σ1 × σ2; Use this when the question gives correlation and standard deviations.
Portfolio expected return: Rp = w1R1 + w2R2; Weighted average. Correlation plays no role.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2w1w2Cov(1,2); Equivalent to using ρσ1σ2 in place of Cov(1,2).
Two-asset portfolio standard deviation: σp = √(w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2); Take the square root of variance as the last step.
Perfect positive correlation case: σp = w1σ1 + w2σ2 (when ρ = +1); The only case where risk is the weighted average of standard deviations.

How to solve Covariance, Correlation and Portfolio Risk questions

Use the same sequence for any two-asset risk question. Keep the numbers in decimals, such as 20% as 0.20, until the end.

1Write down the weights, standard deviations and either the correlation or the covariance.
2Check that weights add up to 1. Convert percentages to decimals.
3If the question gives correlation, you may use it directly. If it gives covariance, use it directly in the variance formula.
4Compute each term separately: w1²σ1², w2²σ2², and 2w1w2 times the covariance term.
5Add the three terms to get portfolio variance.
6Take the square root to get standard deviation. Do not stop at variance unless asked.
7Sanity check: the answer must lie between the lower and higher of the weighted extremes. It cannot exceed w1σ1 + w2σ2.

Quickest way: Bounds and special-case shortcut

When to use it: Use when the options are far apart or the correlation is +1, 0 or -1.

If ρ = +1, answer = w1σ1 + w2σ2. No square root needed.
If ρ = 0, drop the third term and compute √(w1²σ1² + w2²σ2²).
If ρ = -1, answer = |w1σ1 - w2σ2|.
For other values, first find the ρ = +1 value as the upper limit. Eliminate any option above it.
Then compute fully only if two or more options remain.

Common mistakes in Covariance, Correlation and Portfolio Risk

Reporting the weighted average of standard deviations as portfolio risk.
It works for returns, so students assume it works for risk.
Fix: Use the full formula with correlation. The weighted average is right only when ρ = +1.
Forgetting the factor 2 in the cross term.
The cross term looks like one product, but it appears twice in the expansion.
Fix: Always write 2 × w1 × w2 × ρ × σ1 × σ2.
Stopping at variance and choosing it as the answer.
Time pressure makes students skip the square root.
Fix: Standard deviation is the square root of variance. Check what the question asks.
Squaring the weights but not the standard deviations, or the reverse.
The terms w²σ² look alike and are written quickly.
Fix: Compute (wσ)² as a single quantity: first wσ, then square it.
Treating covariance as bounded between -1 and +1.
Covariance and correlation are confused.
Fix: Only correlation is bounded. Covariance can be any size and depends on the units.
Believing negative correlation changes expected return.
Students link diversification to both risk and return.
Fix: Expected return is the weighted average regardless of correlation. Only risk falls.

Worked examples

Example 1

A portfolio has 60% in Asset A (standard deviation 10%) and 40% in Asset B (standard deviation 20%). The correlation between A and B is 0.5. Find the portfolio standard deviation.

Show the solution

Weights: w1 = 0.6, w2 = 0.4. Standard deviations: σ1 = 0.10, σ2 = 0.20. ρ = 0.5.
w1σ1 = 0.6 × 0.10 = 0.06. Square: 0.0036.
w2σ2 = 0.4 × 0.20 = 0.08. Square: 0.0064.
Cross term = 2 × 0.06 × 0.08 × 0.5 = 0.0048.
Variance = 0.0036 + 0.0064 + 0.0048 = 0.0148.
Standard deviation = √0.0148 ≈ 0.1217, or about 12.17%.

Answer: About 12.17%. This is below the weighted average of 14%, showing diversification benefit.

Example 2

Two assets have standard deviations of 15% and 25%. Their covariance is 0.0150. Find the correlation, and say what it means.

Show the solution

Correlation = Cov ÷ (σ1 × σ2).
σ1 × σ2 = 0.15 × 0.25 = 0.0375.
ρ = 0.0150 ÷ 0.0375 = 0.4.
A value of 0.4 is positive but well below +1.

Answer: Correlation is 0.4. The assets tend to move in the same direction, but not perfectly, so combining them gives some diversification benefit.

Exam tips

Memorise the three special cases (ρ = +1, 0, -1). Many options can be eliminated or solved in seconds.
Check whether the question gives covariance or correlation. Using one in place of the other is a favourite trap.
Expect conceptual questions: which correlation gives the greatest risk reduction? The answer is -1.
Portfolio standard deviation can never exceed the weighted average of the individual standard deviations when correlation is below +1.
On 2-mark caselet questions a wrong answer costs more, so compute fully before you guess.

Practice questions from Introduction to Modern Portfolio Theory

Covariance, Correlation and Portfolio Risk in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Covariance, Correlation and Portfolio Risk: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows the direction in which two assets move together, but its size depends on units and volatility. Correlation divides covariance by the product of the two standard deviations, so it lies between -1 and +1 and is easy to compare.

How do I calculate portfolio standard deviation for two assets?

Find w1²σ1², w2²σ2² and 2w1w2ρσ1σ2. Add them to get variance, then take the square root. Use decimals for weights and standard deviations.

Does correlation affect the expected return of a portfolio?

No. Expected return is the weighted average of the asset returns. Correlation affects only portfolio risk.

When is there no diversification benefit?

When correlation is exactly +1. In that case portfolio standard deviation equals the weighted average of the individual standard deviations.