Skip to content

FRM Exam Part I · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Portfolio Risk and Return, and How Diversification Works

Updated 11 October 2026 · Fact-checked

Portfolio expected return is the weighted average of asset returns. Portfolio variance is not a weighted average: it also needs covariances. For two assets, σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. The lower the correlation, the more diversification reduces risk. Standard deviation is the square root of variance.

Understand Portfolio Risk and Return, Diversification

A portfolio is a set of assets held in weights that sum to 1. Its expected return is simple: multiply each asset's expected return by its weight and add. Risk does not work this way.

Portfolio variance depends on three things: each asset's variance, the weights, and how the assets move together. How they move together is measured by covariance or correlation. Covariance = ρ × σ1 × σ2. If two assets do not move in lockstep, some of their ups and downs cancel out. That cancelling is diversification.

Correlation ranges from -1 to +1. At ρ = +1, portfolio standard deviation is exactly the weighted average of the two standard deviations, so there is no benefit. At any ρ below 1, portfolio standard deviation is lower than that weighted average. At ρ = -1, risk can be driven to zero with the right weights.

Total risk splits into two parts. Unsystematic (specific, idiosyncratic) risk is unique to a firm or sector and can be diversified away by holding many assets. Systematic (market) risk affects all assets and cannot be diversified away. As you add more assets, portfolio variance falls toward the average covariance between assets, not toward zero. Because investors can remove specific risk for free, CAPM says only systematic risk earns a risk premium.

Key formulas to remember

Expected portfolio return
E(Rp) = Σ wi × E(Ri)
Weights sum to 1. Short positions have negative weights.
Covariance and correlation
Cov(1,2) = ρ12 × σ1 × σ2
Correlation = Cov ÷ (σ1 × σ2). It always lies between -1 and +1.
Two-asset portfolio variance
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2
Take the square root for standard deviation. Do not forget the cross term.
General portfolio variance
σp² = Σi Σj wi wj Cov(i,j)
Sum over all pairs, including i = j, where Cov(i,i) = σi².
Perfect correlation case
σp = w1σ1 + w2σ2 when ρ = +1 and weights are positive
This is the upper bound for a long-only two-asset portfolio.
Equally weighted n-asset portfolio
σp² = (1/n) × average variance + ((n − 1)/n) × average covariance
As n grows, variance approaches the average covariance.
Minimum variance weight (two assets)
w1 = (σ2² − ρσ1σ2) ÷ (σ1² + σ2² − 2ρσ1σ2)
w2 = 1 − w1. Gives the lowest-variance mix of the two assets.

How to solve Portfolio Risk and Return, Diversification questions

Use this order for any portfolio risk and return question.

  1. 1Write down the weights, expected returns, standard deviations and correlation (or covariance). Check that weights sum to 1.
  2. 2Compute expected return as the weighted sum of expected returns.
  3. 3If you are given correlation, convert to covariance: ρ × σ1 × σ2. If you are given covariance, use it directly.
  4. 4Compute each squared term: w1²σ1² and w2²σ2². Work in decimals, for example 0.20 not 20%.
  5. 5Compute the cross term: 2 × w1 × w2 × covariance. Keep the sign of the correlation.
  6. 6Add the three pieces to get variance, then take the square root for standard deviation.
  7. 7Sanity check: the result should lie between the ρ = -1 and ρ = +1 values. For long-only positive weights, it should not exceed the weighted average of the standard deviations.
  8. 8For conceptual questions, decide whether the risk in question is systematic or diversifiable and answer accordingly.

Quickest way: Compare to the ρ = +1 bound and eliminate options

When to use it: Use when the question gives four numeric options and time is tight.

  1. Compute the weighted average of the standard deviations. This is the maximum for long-only, ρ ≤ 1.
  2. Any option above that value is wrong when ρ < 1.
  3. If ρ = 0, variance is just w1²σ1² + w2²σ2². Skip the cross term.
  4. If ρ = -1 with positive weights, σp = |w1σ1 − w2σ2|.
  5. Only do the full calculation if two options remain.

Common mistakes in Portfolio Risk and Return, Diversification

  • Taking the weighted average of standard deviations as portfolio risk.

    Expected return is a weighted average, so students assume risk is too.

    Fix: This is true only when ρ = +1. Otherwise use the full variance formula.

  • Forgetting the factor 2 in the cross term.

    Students remember the term w1w2ρσ1σ2 but miss that it appears twice.

    Fix: Write 2 × w1 × w2 × ρ × σ1 × σ2 every time.

  • Stopping at variance and reporting it as standard deviation.

    Time pressure and similar-looking numbers in the options.

    Fix: Always take the square root. Check whether the question asks for variance or standard deviation.

  • Squaring the weights incorrectly, such as using w² × σ instead of w² × σ².

    Mixing up standard deviation and variance.

    Fix: Square the weight and the standard deviation separately: (wσ)².

  • Saying that diversification can remove all risk.

    Students remember that correlation below 1 helps and overgeneralize.

    Fix: Only unsystematic risk can be diversified away. Systematic risk remains, and perfect elimination needs ρ = -1 with specific weights.

  • Mixing percentages and decimals.

    Using 20 instead of 0.20 leaves variance in %² units while other terms are decimals.

    Fix: Convert everything to decimals first, then convert back at the end.

Worked examples

Example 1

Asset A has expected return 8% and standard deviation 12%. Asset B has expected return 12% and standard deviation 20%. The correlation is 0.30. A portfolio is 60% in A and 40% in B. Find the expected return and standard deviation.

Show the solution
  1. Expected return = 0.60 × 8% + 0.40 × 12% = 4.8% + 4.8% = 9.6%.
  2. w1²σ1² = 0.36 × 0.0144 = 0.005184.
  3. w2²σ2² = 0.16 × 0.04 = 0.0064.
  4. Cross term = 2 × 0.60 × 0.40 × 0.30 × 0.12 × 0.20 = 0.144 × 0.024 = 0.003456.
  5. Variance = 0.005184 + 0.0064 + 0.003456 = 0.01504.
  6. Standard deviation = √0.01504 ≈ 0.12263, or about 12.26%.
  7. Check: weighted average of standard deviations = 0.6 × 12% + 0.4 × 20% = 15.2%. Our answer is lower, as expected for ρ < 1.

Answer: Expected return 9.6%; standard deviation about 12.26%.

Example 2

Two assets each have a standard deviation of 10%. You hold them 50/50. What is the portfolio standard deviation if the correlation is (a) +1, (b) 0, (c) -1?

Show the solution
  1. Each squared term: 0.25 × 0.01 = 0.0025, so the two own-variance terms sum to 0.005.
  2. Cross term = 2 × 0.5 × 0.5 × ρ × 0.10 × 0.10 = 0.005ρ.
  3. (a) ρ = +1: variance = 0.005 + 0.005 = 0.01, so σp = 10%.
  4. (b) ρ = 0: variance = 0.005, so σp = √0.005 ≈ 7.07%.
  5. (c) ρ = -1: variance = 0.005 − 0.005 = 0, so σp = 0%.
  6. Lower correlation means lower risk, with no benefit at +1 and full elimination at -1.

Answer: (a) 10%; (b) about 7.07%; (c) 0%.

Exam tips

  • Questions often give correlation and ask for standard deviation. Practise the full calculation until you can do it in about two minutes.
  • Expect conceptual questions on which risk is diversifiable. Specific risk is diversifiable and not rewarded; systematic risk is not diversifiable and is rewarded.
  • If the question gives covariance, do not multiply by σ1σ2 again.
  • Watch for negative weights (short positions). The cross term sign changes with the product of the weights.
  • Use the memory of the extremes ρ = +1, 0, -1 to check your answer quickly.

Practice questions from Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

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

What is the portfolio variance formula for two assets?

σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. The last term uses the correlation, or you can replace ρσ1σ2 with the covariance. Take the square root to get standard deviation.

How does correlation affect diversification?

Lower correlation gives more risk reduction for the same weights. At ρ = +1 there is no benefit, and at ρ = -1 risk can be eliminated with the right weights. Most real assets sit between 0 and 1.

What is the difference between systematic and unsystematic risk?

Systematic risk comes from market-wide factors such as interest rates or recessions and cannot be diversified away. Unsystematic risk is specific to a firm or industry and falls as you add more assets. Under CAPM, only systematic risk earns a premium.

Why does portfolio risk not fall to zero as I add more assets?

Portfolio variance approaches the average covariance between assets as the number of assets grows. Since most assets are positively correlated, that average is above zero. This remaining part is systematic risk.