For much of the twentieth century, investors looked at risk through a narrow lens. They judged each security on its own, asking whether a stock seemed volatile or a bond appeared secure. The common belief was that combining individually stable investments would naturally create a prudent portfolio.
But this view missed an important part of the picture. Investments don't work in isolation. Their returns interact, sometimes amplifying or offsetting each other in unexpected ways. It was this insight that American economist and Nobel laureate Harry Markowitz introduced in the 1950s. His work shifted the way we understand risk: not as a feature of individual assets, but as something that only becomes clear when looking at the portfolio as a whole.¹
What is mean-variance analysis?
The mean-variance framework is a cornerstone of financial economics and a systematic approach to understanding how risk and return interact within an investment portfolio. Developed by Harry Markowitz, it focuses on two core inputs: expected return, which reflects what an investor anticipates earning over time, and risk, measured by variance or standard deviation, which captures how widely returns may fluctuate.¹
The framework's defining insight is that assets should not be judged on their own. A single investment may look risky, but when combined with others, it can actually make the portfolio more stable. Therefore, the mean-variance analysis shifts attention from individual securities to how assets behave together, particularly how their returns move relative to one another.¹
Using this approach, investors can identify portfolios that offer the highest expected return for a given level of risk, a process known as portfolio optimization. While similar ideas had appeared earlier in other fields, Markowitz was the first to formalize risk and return into a coherent, general framework for portfolio selection, a contribution that became the foundation of Modern Portfolio Theory and remains central to asset allocation today.¹,²
How to calculate portfolio risk using the mean-variance model
In Markowitz's mean-variance framework, portfolio risk is calculated by examining how the risks of individual assets combine within a portfolio. Rather than focusing on securities in isolation, the framework uses several key inputs to assess how assets interact and to identify portfolios that deliver the highest expected return for a given level of risk.¹,³
1. Estimate the expected return
The starting point of the mean-variance framework is estimating the expected return of each asset, representing the average outcome an investor anticipates over time. At the portfolio level, expected return is the weighted average of the returns of its individual components.
2. Understand risk through variance and standard deviation
The next step is to understand risk and how much an investment's returns can rise and fall over time. Instead of focusing only on potential losses, Markowitz viewed risk as the overall variability around the average return. Simply put, the bigger the swings, the riskier the asset. Measuring this variability lets investors compare different investments, like a volatile tech stock and a more stable government bond, using a consistent method.
3. Adjust the portfolio weights
Portfolio weight refers to how much of each asset an investor holds, which influences the portfolio's overall risk and return. Changing these weights adjusts the balance between risk and reward even if the underlying assets stay the same. This can also include short selling, where an investor takes a negative position on an asset to potentially fund other opportunities.
4. Calculate the correlation and covariance
Correlation describes the degree to which asset returns move together and is a central component of the mean-variance framework. When two assets tend to move in the same direction, such as shares of companies within the same industry, they exhibit positive correlation. When their returns move in opposite directions, as is often observed between assets such as gold and technology stocks, the correlation is negative. It is this negative or low correlation that helps reduce overall portfolio risk through diversification.
5. Apply diversification to reduce risk
Within the context of Markowitz's mean-variance analysis, diversification is the process of combining assets with different return patterns to reduce overall portfolio risk. By spreading investments across various industries, companies, or asset types that behave differently, investors can lower specific risks unique to individual holdings. This approach helps the portfolio maintain expected returns while significantly reducing uncertainty.
6. Review the efficient frontier
Markowitz explains that the efficient frontier is an analytical curve that visually shows the best portfolios that offer the maximum possible return for a given level of risk. If an investor's portfolio lies below this curve, it is considered inefficient, meaning it takes on unnecessary risk without providing additional return.
What are the limitations of the mean-variance model?
While the Markowitz mean-variance model remains foundational in portfolio management, it has important limitations. It relies on historical data to estimate returns, volatility, and correlations, yet markets continuously evolve, and past patterns may not predict future outcomes. The model also assumes returns are normally distributed, often underestimating extreme events and sharp market movements. During periods of market stress, asset correlations tend to rise, which reduces diversification benefits and increases portfolio risk.⁴
In particular, markets like the Hang Seng Index face challenges such as frequent stock suspensions, which require excluding certain stocks from mean-variance optimization. The model depends on estimated returns, variances, and covariances, and uncertainty in these parameters can reduce its effectiveness, especially in markets with less stable data. Additionally, variance alone does not fully capture the risks investors face, such as extreme losses, because it does not reflect the full distribution of returns. As a result, the mean-variance model can serve as a useful reference in investment management but should be applied cautiously when parameters are uncertain.⁵
Final thoughts: a structured guide to understanding risk and return
Understanding the mean-variance framework is a crucial step toward smarter investing. While it’s not a perfect solution, it provides a structured way to think about risk and return that has stood the test of time. Investors seeking to optimize their portfolios should consider this approach alongside other strategies and market insights, always mindful that successful investing is as much about adapting to change as it is about applying timeless principles.
To discover more about financial concepts related to Markowitz’s mean-variance framework, visit UBS Nobel Perspectives & Economic Views. Gain insights and explore engaging ideas from Harry Markowitz and other Nobel laureates shaping the future of modern finance.
References
- Markowitz H.Portfolio Selection. Journal of Finance, 1952.
- Nobel Prize Outreach.Press release. NobelPrize.org, 1990.
- Markowitz HM.Harry M. Markowitz – Prize Lecture. NobelPrize.org, 1990.
- Kolm PN, Tütüncü R, Fabozzi FJ.60 Years of Portfolio Optimization: Practical Challenges and Current Trends. European Journal of Operational Research, 2014.
- Zhang H.Effectiveness and Limitation of Markowitz Mean-variance Model: Evidence from Hang Seng Index. Highlights in Business, Economics and Management, 2023.