Markowitz model & efficient frontier explained

Markowitz model, modern portfolio theory, and efficient frontier are some of the most popular concepts in finance. In simple terms, the model helps you select the right assets to maximise returns for a given level of risk. Read on to find out what the Markowitz model is and how you can implement it.

Let’s start by breaking it down step by step.

What is an efficient portfolio?

An efficient portfolio is one that satisfies two conditions:

• It offers the highest expected return for a given level of risk.
• It has the lowest possible risk for a desired level of expected return.

In other words, it’s a portfolio that strikes the best balance between risk and return. To create an efficient portfolio, consider these steps:

1. Know your risk tolerance: Determine your risk appetite, which could be conservative, moderate, or aggressive, depending on how much risk you’re willing to accept.
2. Set realistic goals: Clearly define your financial objectives, taking into account your risk tolerance. Be cautious not to set overly ambitious goals that might discourage you or too conservative ones that could hinder your long-term gains due to excessive caution.
3. Diversify your assets: Spread your investments across different types of assets such as stocks, ETFs, CFDs, and index funds. Diversification helps to mitigate risk by not putting all your eggs in one basket.
4. Seek tax advice: After crafting your investment plan, you’ll need to address tax implications. It’s crucial to be informed about how to report your income for tax purposes, so you may want to consult with a tax expert to avoid penalties from tax authorities.
5. Invest in assets based on your risk profile: While we’ve mainly discussed stock market investments so far, remember that there are other avenues for profitable and potentially less risky investments, such as debt instruments (i.e., bonds) or cash and cash equivalents.

Who is Harry Markowitz?

Harry Markowitz, an economist born in 1927 in Chicago, Illinois, is a prominent figure in the world of finance. He belonged to the esteemed Chicago School of Economics and became a renowned investment analyst. In 1952, he published an article titled “Portfolio Selection”, a paper that would define his career and pioneer a new field in the study of constructing optimal investment portfolios.

In recognition of his groundbreaking contributions to portfolio management theory, he received the Nobel Prize in Economics in 1990, sharing it with Miller and Sharpe.

What is the Markowitz model?

The Markowitz model, developed by Harry Markowitz in the 1950s, is a mathematical framework used to assess the performance of investment portfolios. This model plays a crucial role in maximising returns while minimising risk by creating diversified portfolios composed of various asset classes. Diversification serves to reduce overall risk while offering the potential for high returns.

Today, the Markowitz model remains highly influential in the world of finance, utilized by investors and money managers worldwide.

What are the assumptions underlying the Markowitz model?

The Markowitz model is built on specific assumptions about investor behaviour and financial markets:

Investor behaviour

1. Rational behaviour: The model assumes that all individuals act rationally and aim to maximise their expected utility, considering both expected return (as a measure of profitability) and variance or standard deviation (as a measure of risk).
2. Utility: When comparing portfolios with the same variance (risk), the one with the highest expected return is preferred.
3. Risk aversion: Investors exhibit risk aversion. Thus, when comparing portfolios with the same expected return, the one with the lowest variance is preferred.
4. Indifference curves: The model incorporates indifference curves, which depict combinations of return and risk that provide the same utility to investors. Curves located further to the left and upward represent higher expected utility levels.

Assets and financial markets

1. Perfect information: Financial markets are considered to be perfect, where all information is equally accessible and free to all participants.
2. No transaction costs: There are no transaction costs involved in buying and selling financial assets.
3. Infinitely divisible assets: Financial assets are infinitely divisible, allowing investors to allocate any proportion of their budget to them.
4. No inflation or taxes: The model assumes an environment with no inflation or taxes.
5. Price-accepting investors: Investors accept prices as given and do not influence market prices.
6. Single-period planning horizon: All investors have the same planning horizon, which is one period. They buy a portfolio of securities at the start of this period and sell it at the end.
7. No risk-free asset: The model does not consider the existence of a risk-free financial asset for investment.
8. Immediate liquidity: Assets are immediately liquid at the end of the period.
9. No short sales or margin accounts: Short sales and margin accounts are not permitted.

How does the Markowitz model work?

The Markowitz model aims to identify the optimal portfolio that maximises an investor’s expected utility. This process involves four steps:

1. Identifying investment possibilities: The model examines N risky assets and potential portfolios available in the market. It estimates their expected returns, variances, and covariances between their returns. Portfolios with intermediate correlation are plotted on a hyperbola, and the curvature of this hyperbola depends on the correlation between the assets’ returns.
2. Determining the efficient frontier: An efficient portfolio satisfies two conditions: it has the lowest risk for a given expected return and the highest expected return for a given level of risk. The efficient frontier, represented as a graph with risk on the horizontal axis and return on the vertical axis, illustrates the optimal balance between risk and return through diversification across asset classes.
3. Selecting the appropriate portfolio: Depending on their risk tolerance, investors can choose a portfolio along the efficient frontier. Risk-averse investors select portfolios closer to the low-risk end, while those seeking higher returns opt for portfolios closer to the high-return end.
4. Minimum variance portfolio: The portfolio with the lowest risk on the efficient frontier is known as the minimum variance portfolio, marking the beginning of the efficient frontier.

In summary, the Markowitz model guides investors in constructing portfolios that achieve the best possible trade-off between risk and return by diversifying across asset classes. It’s a foundational framework in modern finance, helping investors make informed decisions to optimise their investment strategies.

How is the efficient portfolio calculated?

The formula for calculating the set of efficient portfolios is often expressed through the mean-variance optimisation model. The key elements are:

• Expected Portfolio Return
• Portfolio Risk (Standard Deviation)
• Weights of Assets in Portfolio

Here is a simplified formula for a two-asset portfolio:

Expected Return = (Weight of A1 * Return of A1) + (Weight of A2 * Return of A2)

Portfolio Risk = √ [(Weight of A1² * Standard Deviation of A1²) + (Weight of A2² * Standard Deviation of A2²) + (2 X Correlation Coefficient * Standard Deviation of A1 * Standard Deviation of A2)]

The formulas can become more complex when there are more than two assets. In that case, matrices and optimisation techniques are used to find the set of efficient portfolios.

What disadvantages does the Markowitz model have?

The Markowitz model has several drawbacks:

1. Mathematical complexity: Initially, the model was criticised for its complex resolution algorithm and the extensive number of expected returns, variances, and covariances that needed to be calculated. However, advancements in hardware and software have simplified the resolution process significantly.
2. Hypotheses of the model: The Markowitz model does not consider transaction costs or taxes, and is based on the perfect divisibility of selected assets. These limitations can be addressed by introducing additional constraints to the model.
3. Risk assessment: The model lacks a mechanism for investors to assess their risk tolerance and derive their utility function, which is crucial for determining their optimal portfolio.
4. Data used: The use of historical return series in estimating expected parameters can introduce biases. As a result, efficient portfolios in the model may consist of high-yield assets with low variance and low correlation with other assets, leading to highly concentrated portfolios with limited diversification and high risk. However, this issue can be mitigated by imposing restrictions on the maximum allocation to each asset.

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Markowitz model: summary

Markowitz’s work underscores that portfolio diversification isn’t solely about the number of assets but also the correlation of their returns. High correlation limits diversification potential, while low correlation allows for effective risk reduction.

In summary, the Markowitz model serves as the foundation of modern portfolio theory and the determination of the efficient frontier. While it has theoretical significance, its practical adoption by investment analysts and managers has been somewhat limited compared to its theoretical relevance.

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