In modern portfolio theory, the efficient frontier (or portfolio frontier) is an investment portfolio which occupies the “efficient” parts of the risk–return spectrum. Formally, it is the set of portfolios which satisfy the condition that no other portfolio exists with a higher expected return but with the same standard deviation of return (i.e., the risk). The efficient frontier was first formulated by Harry Markowitz in 1952.
A combination of assets, i.e. a portfolio, is referred to as “efficient” if it has the best possible expected level of return for its level of risk (which is represented by the standard deviation of the portfolio’s return). Here, every possible combination of risky assets can be plotted in risk–expected return space, and the collection of all such possible portfolios defines a region in this space. In the absence of the opportunity to hold a risk-free asset, this region is the opportunity set (the feasible set). The positively sloped (upward-sloped) top boundary of this region is a portion of a hyperbola and is called the “efficient frontier”.
If a risk-free asset is also available, the opportunity set is larger, and its upper boundary, the efficient frontier, is a straight line segment emanating from the vertical axis at the value of the risk-free asset’s return and tangent to the risky-assets-only opportunity set. All portfolios between the risk-free asset and the tangency portfolio are portfolios composed of risk-free assets and the tangency portfolio, while all portfolios on the linear frontier above and to the right of the tangency portfolio are generated by borrowing at the risk-free rate and investing the proceeds into the tangency portfolio.
Work
It is represented by plotting the expected returns of a portfolio and the standard deviation of returns. The y-axis is made up of the expected returns of the portfolio. The x-axis is labeled as the standard deviation of returns, which is a measure of risk.
A portfolio is then plotted onto the graph according to its expected returns and standard deviation of returns. The portfolio is compared to the efficient frontier. If a portfolio is plotted on the right side of the chart, it indicates that there is a higher level of risk for the given portfolio. If it is plotted low on the graph, the portfolio offers low returns.
Significance of an Efficient Frontier
The efficient frontier is the foundation for modern portfolio theory, which is the idea of how investors aim to create a portfolio that maximizes expected returns based on a specific level of risk. It helps investors understand the potential risks and returns in their portfolios and analyze how they compare to the optimal set of portfolios that are considered to be efficient. Doing so helps investors to accordingly change their investing strategies by understanding the level of risk that pertains to each portfolio.
It should be noted that there is no single efficient frontier for everyone. Each one is different for every investor because it depends on multiple factors – such as the number of assets in the portfolio, the industry of the assets, and the degree of the investor’s risk tolerance.
Creation of Efficient Frontier
Creation of the efficient frontier is a fundamental concept in Modern Portfolio Theory (MPT), developed by Harry Markowitz. It represents a set of portfolios that offers the highest expected return for a given level of risk or the lowest risk for a given level of return.
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Collect Data
First, gather historical return data for the assets you are considering for your portfolio. This data is used to calculate the expected returns, variances, and covariances for each asset.
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Calculate Expected Returns and Risk
- Expected Return: Calculate the average returns for each asset, which serve as an estimate of future returns.
- Variance and Standard Deviation: Measure the volatility of each asset, which represents the risk.
- Covariance: Determine how returns on different assets move in relation to each other. This helps in understanding how asset combinations can diversify risk.
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Construct a Portfolio
Combine assets in various proportions. For each combination, you need to calculate:
- Portfolio Expected Return: This is the weighted average of the expected returns of the individual assets, where the weights are the proportions of each asset in the portfolio.
- Portfolio Variance: This is more complex as it not only involves the variances of individual assets but also their covariances. The formula for the variance of a two-asset portfolio, for example, is:
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Simulate Different Combinations
Systematically vary the asset mix to calculate the expected returns and risk (standard deviation of returns) for a wide range of potential portfolios. This step often involves using optimization software or a financial model because the number of potential combinations can be very large.
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Plot the Portfolios
On a graph, plot each portfolio’s risk (standard deviation) on the X-axis and the expected return on the Y-axis. Each point on this graph represents a possible portfolio.
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Identify the Efficient Frontier
Among all plotted points, the efficient frontier is the portion of the curve that forms the upper boundary of the set of possible portfolios. This frontier starts from the portfolio with the minimum variance (lowest risk) and extends to the portfolio offering the highest return at the maximum acceptable risk.
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Optimization for Maximum Utility
The final step is selecting a portfolio along the efficient frontier that aligns with the investor’s specific risk tolerance and return expectations. This is typically done using utility functions or by imposing additional constraints such as requiring certain assets or asset classes to be included or excluded from the portfolio.
Limitations of an Efficient Frontier
The efficient frontier is built on assumptions that may not accurately portray realistic situations. For example, it assumes that all investors think rationally and avoid risks. It also assumes that fluctuations in market prices do not depend on the number of investors, and all investors enjoy equal access to borrowing money at a risk-free interest rate.
Such assumptions are not always true, as some investors may not make rational decisions, and some investors are high risk-takers. Not all investors obtain equal access to borrowing money as well.
Additionally, it assumes that asset returns result in a normal distribution. However, in reality, asset returns often do not follow a normal distribution, as it often varies three standard deviations away from the mean.
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)]
Assumptions of the Efficient Frontier Model
- All investors have a common goal: avoid the risk because they are risk-averse and maximize the return as far as possible and practicable.
- Investors are rational and know all the facts about the markets. This assumption implies that all the investors are vigilant enough to understand the stock movements, predict returns, and invest accordingly. That also means that this model assumes all investors are on the same page regarding knowledge of the markets.
- There are not many investors who would affect the market price.
- Investors lend and borrow money at a risk-free interest rate.
- The markets are efficient.
- Investors have unlimited borrowing power.
- The assets follow a normal distribution.
- Markets absorb information quickly and accordingly base the actions.
- The investors’ decisions are always based on expected return and standard deviation as a measure of risk.
Merits
- This efficient frontier graph helps investors choose the portfolio combinations with the highest and least possible returns.
- This theory portrayed the importance of diversification.
- It represents all the dominant portfolios in the risk-return space.
Drawbacks/Demerits
- The theory can be applied, or the frontier can be constructed only when a concept of diversification is involved. If there is no diversification, the theory would certainly fail.
- The assumption that all investors are rational and make sound investment decisions may not always be true because not all investors would have enough knowledge about the markets.
- Also, the assumption that investors have unlimited borrowing and lending capacity is faulty.
- The real costs, like taxes, brokerage, fees, etc., are not considered while constructing the frontier.
- The assumption that the assets follow a normal distribution pattern might not always stand true. In reality, securities may have to experience returns far from the respective standard deviations, sometimes like three standard deviations away from the mean.
Efficient frontier and Investor Utility
Efficient frontier consists of the set of all efficient portfolios that yield the highest return for each level of risk. The efficient frontier can be combined with an investor’s utility function to find the investor’s optimal portfolio, the portfolio with the greatest return for the risk that the investor is willing to accept.
- Efficient Frontier
The efficient frontier is a graphical representation of optimal portfolios that maximizes returns at each level of risk or minimizes risk at each level of return. This boundary is crucial because it helps investors make decisions based on their risk tolerance and return expectations.
- Investor Utility
Investor utility is a measure of the satisfaction or benefit derived from a particular portfolio, considering both the expected return and the risk associated with it. Utility functions typically incorporate an investor’s risk aversion, balancing the desire for high returns against the discomfort of increased risk.
Connection between Efficient Frontier and Investor Utility
The connection between the efficient frontier and investor utility is fundamental in portfolio selection:
- Utility Maximization:
Investors aim to select a portfolio that maximizes their utility. Utility functions vary among individuals because risk tolerance can differ significantly. A common utility function format in financial modeling might be:
- Optimal Portfolio Selection:
On the efficient frontier, each point represents a portfolio that is optimal for a given level of risk. The investor’s utility function helps in choosing which point on this frontier is most desirable. Essentially, the optimal portfolio is where the investor’s highest utility (maximum satisfaction) intersects with the efficient frontier.
- Risk Aversion and Portfolio Choice:
Highly risk-averse investors will likely choose portfolios near the left end of the efficient frontier, where there is lower risk (and lower return). Conversely, less risk-averse investors may opt for portfolios toward the right end of the frontier, accepting higher risk for the chance of higher returns.
- Dynamics in Utility and Efficient Frontier:
Changes in an investor’s financial situation, goals, or market conditions can shift their utility functions, thereby altering their optimal choice on the efficient frontier. For instance, as an investor approaches retirement, they may become more risk-averse, shifting their utility maximum towards less risky portfolios.
Practical Application
In practice, financial advisors use these concepts to help clients build portfolios that align with their risk tolerance, financial goals, and market outlook. By understanding where a client’s utility function intersects the efficient frontier, advisors can tailor investment strategies that optimize client satisfaction with risk-return trade-offs.