Have you ever pondered the nuanced process governing asset management firms when proffering investment recommendations? Is it an outcome of spontaneous ideation or a result of deliberate and methodical contemplation? Does it rely on qualitative analyses, or does it delve into the quantitative models? And if the latter, what is the origin of the data employed, and can its reliability withstand scrutiny? Intriguingly, the entire theory underpinning contemporary asset allocation practices can be traced back 72 years to a paper authored by a 25-year-old young man. Today, let us embark on a detailed exploration, elucidating this foundational model through a concrete case study.

Summary

• Harry Markowitz, renowned for his 1952 paper "Portfolio Selection," revolutionized the landscape of asset management and individual investment decisions by introducing the quantitative relationship between risk and return based on mathematical/statistical models.
• Modern Portfolio Theory (MPT) leverage mean and variance to delineate returns and risks. Mean denotes the weighted average return of portfolio assets, while variance gauges the dispersion of data. Calculating portfolio variance necessitates factoring in asset weights and correlations.
• Markowitz's insights reveal that combining multiple assets in a portfolio can mitigate certain risks without compromising the average expected return. Optimizing correlations between assets enables adjustments in portfolio risk, achieving an optimal return-risk equilibrium.
• The Efficient Frontier illustrates the highest expected returns at various risk levels, aiding investors in making optimal trade-offs between risk and return. Computational aspects involve advanced numerical optimization techniques like Lagrange multipliers or sequential least squares, progressively replaced by programming in practice nowadays.
• Although the Efficient Frontier offers a theoretically optimal solution for portfolio allocation, it has limitations such as reliance on historical data for future predictions, overlooking diverse risk types and transaction costs, and assuming a normal distribution of returns. In reality, a more detailed analysis and strategic adjustments are essential.

Modern Portfolio Theory

Harry Markowitz, acclaimed as the "father of Modern Portfolio Theory," left an indelible mark with his groundbreaking work on MPT, fundamentally reshaping perceptions and practices related to investment risk and portfolio management.  

In the 1950s, Markowitz, during his doctoral dissertation at the University of Chicago, astutely observed that traditional investment theories were predominantly fixated on individual asset selection and analysis. He contended that such an approach overlooked potential interactions between assets and the risk characteristics of the overall investment portfolio. Markowitz then began applying probability theory and statistics to the realm of finance. In 1952, he formally published his 14-page paper "Portfolio Selection" in The Journal of Finance. As technological advancements ensued, this theory gradually found validation in the securities market, ultimately leading to Markowitz being bestowed with the Nobel Prize in Economic Sciences in 1990.

Below are some of the Key Insights from "Portfolio Selection":

1. Risk and Return Relationship: Investors, when constructing portfolios, must not solely focus on anticipated returns but must also factor in risk, quantified through variance in statistical analysis.
2. Portfolio Diversification: Allocating funds across diverse asset categories enables investors to curtail overall portfolio risk—a contemporary interpretation of the adage "not putting all eggs in one basket."
3. Covariance and Correlation: Covariance elucidates the correlation between returns of different assets. Opting for assets with lower covariance further mitigates the overall portfolio risk.
4. Efficient Frontier: The Efficient Frontier manifests as a curve illustrating the highest expected returns achievable at various risk levels, assisting investors in optimizing their choices between risk and return.
5. Portfolio Optimization: Constructing portfolios to maximize expected returns at a given risk level is the ultimate objective. This involves intricate mathematical calculations and statistical analysis.

Mean and Variance of Portfolios

Modern portfolio investments leverage mean and variance to delineate returns and risks. Mean signifies the portfolio's expected return—a weighted average of individual asset expected returns over a specific period.

Variance gauges the deviation between actual returns and the mean, indicative of overall data dispersion. However, calculating portfolio variance isn't a straightforward weighted sum; it requires considering correlations between different assets, expressed mathematically as covariance.

Illustrative Example:

Consider Mr. A's investment portfolio comprising Apple, Boeing, McDonald's, and NVIDIA stocks. Analysts from Poseidon provide Mr. A with expected returns and daily closing prices from 2000 to 2018 for each asset. The portfolio's expected return is computed using the weighted average expected returns for each asset.

In this context, the portfolio's expected return is calculated as follows:

This computation involves multiplying the individual stocks' 18-year average annual return rates by their respective weights and summing them up. Therefore, the resulting portfolio expected return is 25.75%.  

In terms of the calculation of an investment portfolio's variance, Markowitz's groundbreaking insight emphasizes that portfolio risk isn't a mere summation of individual asset risks. It's a nuanced interplay that demands accounting for their correlations – which means that the arithmetic of 1+1 transcends a simplistic additive nature. Instead, optimizing asset correlations becomes the key to fine-tuning portfolio risk in pursuit of a targeted return-risk equilibrium.

This underscores our strong advocacy for investors to embrace diversified multi-asset allocations. Markowitz's empirical observations substantiate the notion that amalgamating various assets in a portfolio can act as a risk mitigator without compromising the average expected return. As highlighted in our House View on July 26, 2023, " From the figure, it can be seen that diversified asset class portfolios (triangles, diamonds and circles) are generally better than single asset class portfolios (squares), because they can concentrate in the upper left area (low volatility and high return), while single asset class portfolios are scattered in the lower right area (high volatility and low return). This means that diversified asset class portfolios can provide higher risk-adjusted returns, which is the Sharpe ratio. It is measured by dividing excess returns by volatility to measure the returns obtained per unit of risk. "

Based on trading data spanning 2000 to 2018, we delved into the calculation of correlation coefficients between assets as the below figure shows. Notably, the initial column data reveals a negative correlation between Apple's daily returns and those of Boeing and McDonald's. This dynamic implies that when Apple's stock ascends, there's an increased likelihood of Boeing and McDonald's stocks witnessing a downturn. Conversely, NVIDIA tends to surge when Apple's stock experiences an upswing. In Mr. A's investment portfolio, the counteracting price dynamics across the four assets contribute significantly to a reduced overall portfolio risk.

While the nitty-gritty details of the formula for calculating portfolio variance need not be expounded upon here, the essence lies in factoring in asset weights within the portfolio and the intricate dance of correlations between them. The final standard deviation for Mr. A's stock portfolio stands at 27.44%, significantly below the simplistic weighted sum of individual stock standard deviations, totaling 39%. The result we got here further echoes the efficacy of diversified stock investments in mitigating overall portfolio volatility mentioned earlier.

In the grander scheme, this in-depth analysis illuminates that within a portfolio equally weighted across Apple, Boeing, McDonald's, and NVIDIA – contingent on the assumption that stock price trends from 2000 to 2018 can mirror future trajectories – the portfolio not only outperforms Boeing's individual return of 18% but also does so with a commendable reduction in risk.  

‍Efficient Frontier and Mean-Variance Model

In the aforementioned equally weighted portfolio, the portfolio achieves a superior risk-adjusted return compared to holding individual assets. The ultimate goal for investment managers is to seek an asset allocation strategy that minimizes portfolio risk at a given level of expected return or maximizes the return at a specified risk level.

Markowitz introduced the Efficient Frontier in his paper to address this challenge. The theory involves using the formula for minimum variance (risk) of a portfolio, subject to constraints on expected return and investment weights. Further computations require numerical optimization techniques such as Lagrange multipliers or sequential least squares, with many practitioners nowadays employing programming languages like Python.

After solving Mr. A's portfolio, the resulting graph is depicted below, with risk on the horizontal axis and return on the vertical axis. The randomly-generated 25,000 blue dots represent individual portfolios, each corresponding to a unique combination of return and risk. The portfolios along the black border constitute the Efficient Frontier, indicating the minimum risk for each level of return.

The graph also highlights two special points – the green star representing the Minimum Volatility point and the red star denoting the Maximum Sharpe Ratio point – each holding unique significance:

• Minimum Volatility Point: This point signifies the portfolio with the lowest volatility along the Efficient Frontier. For risk-averse investors, this is a crucial point as it represents a portfolio that minimizes risk while still generating returns.
• Maximum Sharpe Ratio Point: The Sharpe Ratio compares a portfolio's excess return (above the risk-free rate) to its volatility, serving as a measure of investment performance. The Maximum Sharpe Ratio point represents the portfolio that provides the highest excess return per unit of risk undertaken. This is particularly important for investors aiming to maximize returns while accepting a certain level of risk.  

Therefore, in the final configuration, assuming Mr. A has an aversion to risk, a recommended allocation would be 11.57% in Apple, 26.16% in Boeing, 62.17% in McDonald's, and 0.10% in NVIDIA, targeting an annualized return of 16.30% with a volatility close to 20.5%.

However, considering that stocks themselves are the primary driver of capital appreciation in the overall investment portfolio, to achieve the maximum risk-adjusted return, an alternative allocation could be 33.21% in Apple, 24.66% in Boeing, 21.51% in McDonald's, and 20.62% in NVIDIA, targeting an annualized return of 25.86% with a volatility of 27.39%.

Limitations of Mean-Variance Model

The equity allocation strategy devised for Mr. A theoretically achieves the optimal risk-return balance for the investment portfolio. However, in practical terms, analysts need to factor in additional variables and recognize certain limitations.

Primary among these considerations is the model's reliance on historical returns and covariance as the basis for future expectations. Historical performance may not always serve as a reliable indicator of future outcomes, and if market behaviour diverges from historical data, real risks may be either underestimated or overestimated. Therefore, when collecting data, it's crucial to assess whether there have been fundamental changes in assets compared to the past. For instance, evaluating whether Intel can replicate past performance nowadays in an era marked by rapid developments in AI remains a key question to PMs.  

Secondly, the model predominantly employs price variance as a measure of risk but neglects other forms of risk such as liquidity risk, credit risk, and operational risk. Moreover, transaction costs, taxes, and other frictional costs can significantly impact portfolio returns as well.

Last but not least, the calculations often assume that returns follow a normal distribution, simplifying computations to some extent. However, financial returns often exhibit kurtosis, skewness, and other non-normally distributed anomalies. Black Swan events can disproportionately impact portfolio returns underly extreme conditions.

Nevertheless, all financial models entail fitting biases, resulting in disparities between backtesting and out-of-sample data for portfolios. We reckon that the Mean-Variance Model can provide investors with foundational data when constructing portfolios. However, effective portfolio performance will necessitate continuous optimization through quarterly rebalancing, timing considerations in asset allocation, and fundamental analysis of each position.