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Financial Management and Business Data Analytics · Risk and Return

Portfolio Return and Risk: Correlation and Diversification

Updated 10 October 2026 · Fact-checked

Portfolio return is the weighted average of the expected returns of the assets. Portfolio risk is not a weighted average of standard deviations. For two assets, variance = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. The lower the correlation, the more diversification reduces risk.

Understand Portfolio Return and Risk, Correlation and Diversification

A portfolio is a combination of two or more assets held together. An investor cares about the return and the risk of the whole portfolio, not of each asset alone.

The expected return of a portfolio is simple. It is the weighted average of the expected returns of the assets, using the proportion of money invested in each as the weight. The weights add up to 1.

Risk behaves differently. Risk is measured by standard deviation, and it depends on how the assets move relative to each other. This is measured by covariance (the direction and joint size of movement) and correlation (covariance scaled to lie between -1 and +1). Covariance is hard to read, so correlation is used to judge the relationship.

Diversification works because assets do not move in perfect step. When one falls, another may rise or fall less, and the ups and downs partly cancel. If correlation is +1, there is no risk reduction: portfolio standard deviation is just the weighted average of the two standard deviations. For any correlation below +1, portfolio standard deviation is less than that weighted average. At -1, risk can be reduced to zero by choosing suitable weights.

Diversification removes only the risk specific to individual assets (unsystematic risk). Market-wide (systematic) risk stays.

Key rules to remember

Portfolio expected return (two assets)
Rp = w1R1 + w2R2, where w1 + w2 = 1
Weights are proportions of the total amount invested. Works for any correlation.
Covariance from probabilities
Cov(1,2) = Σ p × (R1 − E(R1)) × (R2 − E(R2))
Take deviations of each asset from its own expected return, multiply, then weight by probability.
Correlation coefficient
ρ12 = Cov(1,2) ÷ (σ1 × σ2)
Always between -1 and +1. So Cov(1,2) = ρ12 × σ1 × σ2.
Portfolio variance (two assets)
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2
The last term can also be written 2 w1 w2 Cov(1,2).
Portfolio standard deviation
σp = √σp²
Take the square root only at the end.
Special case: ρ = +1
σp = w1σ1 + w2σ2
No diversification benefit.
Special case: ρ = -1
σp = |w1σ1 − w2σ2|; zero when w1 = σ2 ÷ (σ1 + σ2)
Risk-free portfolio possible with these weights.

How to solve Portfolio Return and Risk, Correlation and Diversification questions

Use this order for any two-asset portfolio question. It keeps the marks even if you slip on arithmetic.

  1. 1Write the data: weights, expected returns, standard deviations, and correlation or covariance. Convert percentages to decimals or work consistently in percent.
  2. 2Check that the weights add to 1. If amounts are given in rupees, divide each by the total.
  3. 3Compute portfolio return as the weighted average of expected returns.
  4. 4If the question gives only correlation, find the covariance term as ρ × σ1 × σ2. If it gives covariance, use it directly. If it gives correlation but asks for covariance, compute it.
  5. 5Substitute into the variance formula. Show each of the three terms separately.
  6. 6Add the three terms to get variance, then take the square root for standard deviation.
  7. 7Compare with the weighted average of the individual standard deviations to show the diversification benefit, and state the conclusion in a line.

Quickest way: Three-term shortcut

When to use it: Use for MCQs and for the calculation part of written answers when the data is clean.

  1. Compute the three terms: w1²σ1², w2²σ2² and 2w1w2ρσ1σ2.
  2. Work in percent units (variance in %²) to avoid decimals, then take the square root at the end.
  3. Test the extremes mentally: ρ = +1 gives the weighted average of the standard deviations; the answer must be at or below that for any ρ below +1.
  4. If an option equals the weighted average of the standard deviations and ρ < 1, reject it.

Common mistakes in Portfolio Return and Risk, Correlation and Diversification

  • Taking portfolio standard deviation as the weighted average of individual standard deviations.

    Return is a weighted average, so students assume risk is too.

    Fix: Use the variance formula with the correlation term. The weighted average is valid only when ρ = +1.

  • Forgetting the factor 2 in the covariance term.

    The formula looks like a square of a sum but is written from memory.

    Fix: Remember it as (a + b)² = a² + b² + 2ab, with a = w1σ1 and b = w2σ2 and the 2ab scaled by ρ.

  • Squaring the weights incorrectly or not squaring the standard deviations.

    Mixing up variance and standard deviation.

    Fix: The first two terms use w² and σ² (variance). Write them out before substituting.

  • Stopping at variance and reporting it as risk.

    The square root is the last step and is easily forgotten under time pressure.

    Fix: Circle the question wording. If it asks for standard deviation, take the square root.

  • Using covariance where correlation is given, or vice versa.

    The two terms sound alike.

    Fix: Correlation is a pure number between -1 and +1. Covariance carries units of %². Convert using Cov = ρσ1σ2.

  • Claiming diversification removes all risk.

    Overgeneralising from the ρ = -1 case.

    Fix: Say that diversification removes unsystematic risk only. Systematic risk remains, and zero risk needs ρ = -1 with specific weights.

Worked examples

Example 1

Asset A has expected return 12% and standard deviation 20%. Asset B has expected return 18% and standard deviation 30%. A portfolio has 60% in A and 40% in B. The correlation between A and B is 0.5. Find the portfolio expected return and standard deviation.

Show the solution
  1. Return: 0.6 × 12 + 0.4 × 18 = 7.2 + 7.2 = 14.4%.
  2. Term 1: w1²σ1² = 0.36 × 400 = 144.
  3. Term 2: w2²σ2² = 0.16 × 900 = 144.
  4. Term 3: 2 × 0.6 × 0.4 × 0.5 × 20 × 30 = 2 × 0.6 × 0.4 × 0.5 × 600 = 144.
  5. Variance = 144 + 144 + 144 = 432 (%²).
  6. Standard deviation = √432 = 20.78% (approx.).
  7. Check: weighted average of standard deviations = 0.6 × 20 + 0.4 × 30 = 24%. The portfolio risk 20.78% is lower, showing diversification.

Answer: Expected return = 14.4%; standard deviation ≈ 20.78%, below the 24% weighted average.

Example 2

Two securities X and Y have standard deviations of 10% and 20%. Their covariance is -100 (%²). Find the correlation coefficient. Then find the portfolio standard deviation if equal amounts are invested in each.

Show the solution
  1. Correlation = Cov ÷ (σX σY) = -100 ÷ (10 × 20) = -100 ÷ 200 = -0.5.
  2. Weights: wX = wY = 0.5.
  3. Term 1: 0.25 × 100 = 25.
  4. Term 2: 0.25 × 400 = 100.
  5. Term 3: 2 × 0.5 × 0.5 × (-100) = -50.
  6. Variance = 25 + 100 − 50 = 75 (%²).
  7. Standard deviation = √75 = 8.66% (approx.).
  8. Check: the weighted average of standard deviations is 0.5 × 10 + 0.5 × 20 = 15%, so the portfolio is much less risky than either the average or Y alone.

Answer: Correlation = -0.5; portfolio standard deviation ≈ 8.66%.

Exam tips

  • Show the three variance terms on separate lines. Step marks are given for the correct formula and substitution even if the final figure is slightly off.
  • Read whether the question gives correlation or covariance. Many papers give one and test whether you can use it correctly.
  • In MCQs, check the extreme cases: with ρ = +1 the answer equals the weighted average of the standard deviations, and anything above that is impossible.
  • Add a one-line interpretation: lower correlation means greater diversification benefit, and only unsystematic risk is reduced.
  • Round the square root to two decimals and state the unit (%).

Practice questions from Risk and Return

Portfolio Return and Risk, Correlation and Diversification in other exams

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

Portfolio Return and Risk, Correlation and Diversification: frequently asked questions

What is the formula for portfolio risk of two assets?

Portfolio variance = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Portfolio standard deviation is the square root of this value. The correlation term is what captures the diversification effect.

Why does a lower correlation reduce portfolio risk more?

When assets do not move together, losses on one are partly offset by gains or smaller losses on the other. The covariance term in the variance formula becomes smaller, so total risk falls. At ρ = +1 there is no offset at all.

What is the difference between covariance and correlation?

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

Can diversification eliminate all risk?

No. It removes unsystematic (asset-specific) risk. Systematic risk that affects the whole market remains. In the two-asset case, risk can reach zero only when correlation is -1 and the weights are chosen suitably.