Portfolio Optimization

What is Portfolio Optimization?

Portfolio optimization is the process of selecting and combining different assets with the aim to achieve the best possible outcome in terms of risk and return. It involves using quantitative techniques and the analysis of metrics related to the portfolio’s expected returns, correlation and volatility. The objective is to create an “efficient” portfolio  that is diversified and is in-line with the investor’s risk profile.

An efficient portfolio offers the highest expected return for a given level of risk. On the efficient frontier, it is positioned where no other portfolio provides better risk-adjusted return.

Strategies for Portfolio Optimization

There are various strategies that have been developed to optimize a portfolio and picking what might work best depends on the characteristics of the investment strategy such as specific goals and constraints. Below we discuss some of the most popular methods.

Modern Portfolio Theory (MPT)

Pioneered by Harry Markowitz, the MPT provides a framework on how to evaluate risk and reward, assuming that all investors are rational (i.e. given an acceptable level of risk, all investors attempt to achieve the highest level of return for that level of risk). It involves combining assets with low correlation to each other to avoid significant losses should one asset underperforms.

Mean-Variance Optimization (MVO)

MVO is the process of allocating assets based on their risk-reward trade-off. The MVO involves varying the weightings of various assets in a portfolio determine which provide the best risk-adjusted returns. Such analysis results in what is called an “efficient frontier”. Below is an example that observes three different portfolios and their respective weightings.

Black-Litterman Model

Black-Litterman Model is an asset allocation tool which uses market equilibrium as a starting point and considers the investors’ subjective market views to calculate how the optimal asset weights should differ from the initial portfolio allocation. The model aims to create more efficient portfolios based on investors’ insight.

Monte Carlo Simulation

Monte Carlo Simulation also known as “stochastic modelling” or “probability analysis”, this is a statistical method for analysing random portfolio returns. It uses a computer program that selects annual returns based on all assumptions, including the distribution of those returns, their volatility and the correlation between assets. This process is the repeated thousands of times and shows a range of outcomes rather than just an estimated value.

Risk Parity

Risk Parity as the name suggests, this approach aims to achieve a level of risk that is equal for all portfolio holdings. After evaluating the risk of all assets (returns are not considered) they get allocated accordingly. Risk parity would generally aim to create a more stable portfolio.

How to Optimize a Portfolio? Steps to Portfolio Optimization

The process of optimizing a portfolio could be divided into several steps, each of which looks into different aspects and characteristics of the investment strategy. Below we show some examples of these characteristics:

• Time Horizon – depending on investor’s time horizon, different approaches can be applied.
• Return Expectations – should the investor has specific return targets, the Black-Litterman model can be useful.
• Risk Tolerance – different approaches account for risk in different ways. For example, the MVO focuses on minimizing variance .
• Historical Data – approaches such as the MVO rely heavily on historical return data. If the data is unreliable, more robust methods like Robust Optimization may be

After considering the characteristics of the portfolio and the strategy, there are gradual steps that investors can take to optimize their portfolios. Below is a practical list:

1. Clearly define the investment objectives (such as time horizon and risk tolerance) and any specific constraints like liquidity levels or transaction costs.
2. Evaluate the quality and quantity of historical data and other relevant inputs.
3. Compare different methods by matching the characteristics of each to the investment strategy. This may include running simulations or back testing.
4. Determine the available resources and level of expertise in order to implement and maintain the chosen method.
5. Implement the chosen method on a small scale to assess its effectiveness before full-scale implementation.

Advantages and Disadvantages

Portfolio optimization offers a number of advantages that can enhance the risk and return characteristics of a portfolio, but also has some limitations.

Advantages

• Maximising risk-adjusted returns – portfolio optimization helps to build portfolios that provide the highest expected return for a given level of risk. These can be analysed through various risk-adjusted measures.
• Risk management – methods like the MVO promote diversification by selecting a mix of assets that reduces overall portfolio risk. That allows investors to remain within their preferred risk parameters.
• Improved idea generation – during the whole process, investors use a lot of data, which may help them identify potential market opportunities.

Disadvantages

• Reliance on historical data – this could lead to estimation errors, resulting in suboptimal portfolios. In addition, structural changes in the market or unprecedented events can render historical data less relevant and lead to misguided results.
• Model assumptions – some methods assume that asset returns are normally distributed, but asset returns often exhibit skewness and kurtosis (they can have more extreme values than the normal distribution). This may overlook the potential impact of extreme events.
• Implementation challenges – some models may become too complex to run due to either the large number data points and could require significant resource and expertise to implement and maintain.

Portfolio Optimization Formula

As already mentioned, the Black-Litterman model is a common method for portfolio optimization, which combines the Capital Asset Pricing Model (CAPM) equilibrium market returns with investor’s unique views to create an optimal portfolio. Below is the formula:

Where:

E(R) = The expected return

τ = A scalar number indicating the uncertainty of the CAPM distribution

P = A matrix with investors views; each row a specific view of the market and each entry of the row represents the weights of each assets

Q = The expected returns of the portfolios from the views described in matrix P

Ω = A diagonal covariance matrix with entries of the uncertainty within each view

Σ = Covariance matrix of returns

Π = Vector of implied equilibrium expected returns

Portfolio Optimization Excel Example

In this example, an investor is considering to purchase a fund and has four options. Given this information, we are asked to determine which fund would be the optimal solutions in terms of risk and reward. Access the full version in the free download section to check your answer.

Conclusion

Portfolio optimization is fundamental process in investment management. It aims to achieve the best possible trade-off between the risk taken and potential returns over the long-term, which can be executed through a number of strategies such as the MPT or the Black-Litterman Model. However, this is a complex exercise that requires expert knowledge and usually the usage of specialist software.

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Additional Resources

Modern Portfolio Theory

Capital Asset Pricing Model (CAPM)

Risk Adjusted Measures

Portfolio Management Certification