Portfolio Standard Deviation

What Is Portfolio Standard Deviation?

Portfolio standard deviation measures the overall risk or volatility of a portfolio by considering both the risk of individual investments and how those investments move relative to one another. Unlike the standard deviation of a single asset, portfolio standard deviation incorporates diversification by accounting for the correlation between asset returns.

For CFA Level I candidates, understanding portfolio standard deviation is essential because it explains why combining assets can reduce overall portfolio risk even when individual investments remain volatile.

In this study note, you’ll learn:

• What portfolio standard deviation measures.
• How to calculate portfolio variance and standard deviation.
• Why correlation affects portfolio risk.
• How diversification lowers volatility.
• How portfolio standard deviation is tested in the CFA exam.

AnalystPrep Summary

Portfolio standard deviation measures the total risk of a portfolio rather than simply averaging the risks of individual investments.

Its calculation depends on three variables:

• The weight of each asset.
• Each asset’s standard deviation.
• The correlation (or covariance) between asset returns.

As correlation decreases, diversification benefits increase and portfolio risk generally falls.

Key Takeaways

• Portfolio standard deviation measures total portfolio volatility.
• Portfolio variance must be calculated before finding standard deviation.
• Diversification reduces portfolio risk when assets are not perfectly positively correlated.
• Correlation is one of the most important determinants of portfolio risk.
• Portfolio standard deviation is one of the core quantitative concepts tested in CFA Level I Portfolio Management.

The standard deviation of a portfolio of assets, or portfolio risk, is simply not the sum of the risk of the underlying securities. Due to the correlation between securities, the computation of portfolio risk must incorporate this correlation relationship.

Why Is Portfolio Standard Deviation Important?

Investors rarely hold just one investment. Instead, they combine multiple assets to build diversified portfolios.

Portfolio standard deviation provides a measure of the overall volatility of the entire portfolio by considering both the risk of each investment and the relationship between investments.

Because assets do not always move together, diversification can reduce total portfolio risk. Understanding this relationship forms the foundation of Modern Portfolio Theory and is heavily tested throughout the CFA curriculum. (CFA Institute)

How Do You Calculate Portfolio Standard Deviation?

The portfolio standard deviation and variance are important. They involve the variance of the assets and the covariance between asset pairs. For a portfolio with assets X and Y, the portfolio variance can be calculated as follows:

$$ \text{Portfolio variance} = w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2 w_{X} w_{Y} \sigma_{X} \sigma_{Y} \rho_{XY} $$

Therefore,

$$ \text{Portfolio standard deviaton} = \sqrt{w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2 w_{X} w_{Y} \sigma_{X} \sigma_{Y} \rho_{XY}} $$

Where:

w = Weight of the asset within the portfolio.

\(\sigma\) = Standard deviation.

\( \rho \) = Correlation coefficient.

Note that \( \sigma_{X} \sigma_{Y} \rho_{XY} = \text{Covariance}_{XY}\)

How Does Correlation Affect Portfolio Standard Deviation?

Correlation has a significant impact on portfolio risk.

• Correlation = +1: Assets move perfectly together and diversification provides no benefit.
• Correlation between 0 and +1: Diversification reduces overall risk.
• Correlation = 0: Asset returns are unrelated, creating greater diversification opportunities.
• Correlation below 0: Assets often move in opposite directions, reducing portfolio volatility even further.
• Correlation = –1: Under certain weight combinations, portfolio risk can theoretically be eliminated.

This explains why professional portfolio managers seek investments that are less than perfectly correlated. (CFA Institute)

Portfolio Variance vs. Portfolio Standard Deviation

Portfolio variance is primarily used during calculations, while portfolio standard deviation is easier to interpret because it is expressed in the same units as returns.

Real-World Example of Portfolio Standard Deviation

Suppose an investor owns shares in a technology company and a utility company.

Although the technology stock is more volatile, the utility stock tends to perform differently during changing market conditions.

Because the two investments are not perfectly correlated, combining them reduces the overall portfolio standard deviation compared to investing entirely in the technology stock.

This illustrates why diversification lowers portfolio risk without necessarily reducing expected return.

CFA Exam Tip

Portfolio standard deviation questions frequently test your understanding of diversification rather than simply your ability to apply formulas.

When solving exam questions:

• Identify the correct asset weights.
• Use the appropriate standard deviations.
• Carefully include the correlation or covariance term.
• Remember that lower correlation generally reduces portfolio risk.

Many incorrect answers result from omitting the covariance component.

Question

Consider two assets in a portfolio. Asset A has an allocation of 80% and a standard deviation of 16%. Asset B has an allocation of 20% and a standard deviation of 25%. The correlation coefficient between asset A and asset B is 0.6. In this case, the portfolio standard deviation is closest to:

A. 16.3%.

B. 2.7%.

C. 22%.

Solution

The correct answer is A.

We determine the portfolio variance as follows:

Portfolio variance = (0.8)^2 \times (0.16)^2 + (0.2)^2 \times (0.25)^2 + 2(0.8)(0.2)(0.16)(0.25)(0.6)

Then, we use the square root of the variance to get the standard deviation:

\( \text{Portfolio standard deviation} =\sqrt{2.66\%} = 16.3\% \)

Glossary

Portfolio Standard Deviation — A measure of the total volatility of a portfolio.

Portfolio Variance — The weighted variance of portfolio returns before taking the square root.

Diversification — Reducing portfolio risk by combining assets that are not perfectly correlated.

Correlation — A measure ranging from -1 to +1 showing how two assets move relative to each other.

Covariance — A measure of how two asset returns move together.

Volatility — The degree to which investment returns fluctuate over time.

Why Portfolio Standard Deviation Matters in CFA Level I

Portfolio standard deviation is a core CFA Level I Portfolio Management concept. Candidates may be tested on diversification benefits, correlation effects, portfolio risk measurement, and how combining assets can change overall return volatility.

Build confidence with the Level I CFA study program featuring guided lessons, practice questions, and full mock exams.

Summary of Portfolio Standard Deviation

Portfolio standard deviation demonstrates that portfolio risk depends not only on individual investments but also on how those investments interact.

Frequently Asked Questions

What is portfolio standard deviation?

Portfolio standard deviation measures the overall risk or volatility of a portfolio by considering both the risk of individual assets and their relationships with one another.

Why isn’t portfolio risk simply the average of individual risks?

Because assets are correlated. Their relationships determine how much diversification reduces total portfolio risk.

What is the difference between portfolio variance and portfolio standard deviation?

Portfolio variance is the squared measure of portfolio risk, while portfolio standard deviation is the square root of variance and is easier to interpret.

Why does diversification reduce portfolio standard deviation?

Diversification combines assets that do not move perfectly together, reducing the portfolio’s overall volatility.

What happens if two assets have a correlation of +1?

They move perfectly together, so diversification provides no reduction in portfolio risk.

Is portfolio standard deviation tested in CFA Level I?

Yes. Candidates are expected to calculate, interpret, and understand how correlation and diversification affect portfolio standard deviation. (CFA Institute)