Introduction

Modern Portfolio Theory (MPT) offers valuable insights for financial advisors seeking to optimize portfolio construction and enhance risk-adjusted returns. One of the key aspects of MPT involves the use of optimizers, such as the minimum variance, maximum return, and minimum Conditional Value-at-Risk (CVaR) approaches. In this write-up, we will explore these optimizers, provide a simple description of CVaR, and offer examples of how they can be used to construct portfolios.

Optimizers Used 

When applying MPT, financial advisors often employ different optimizers to create well-balanced portfolios tailored to clients’ risk tolerance and objectives. The following optimizers are commonly utilized:

1) Minimum Variance: The minimum variance optimizer seeks to minimize portfolio volatility by allocating investments in a manner that reduces overall risk. It aims to find the optimal asset allocation that achieves the lowest possible variance of returns.

Suppose you are a risk-averse investor and require stable and steady returns. By utilizing the minimum variance optimizer, your advisor can identify an asset allocation that emphasizes low volatility assets such as bonds and stable dividend-paying stocks. The resulting portfolio may have lower overall risk while aiming to provide steady, consistent returns.

2) Maximum Return: The maximum return optimizer focuses on maximizing portfolio returns while considering the given risk constraints. It aims to identify the asset allocation that provides the highest potential returns given the specified level of risk tolerance.

For an investor seeking higher returns, a financial advisor may use the maximum return optimizer. By identifying asset classes with historically higher returns, such as growth stocks or emerging markets, the advisor can construct a portfolio that focuses on maximizing potential returns while maintaining an acceptable level of risk.

3) Minimum CVaR: Conditional Value-at-Risk (CVaR) is a risk measure that quantifies the expected losses beyond a certain confidence level. The minimum CVaR optimizer seeks to minimize the potential downside risk by constructing a portfolio with the lowest expected losses at a given confidence level.

Minimizing CVaR protects you against the worst 5% or worst 1% of possible outcomes, and hence protects you against downside risk.

CVaR provides a measure of the potential loss magnitude beyond a specified confidence level. It offers insights into the tail risk associated with an investment or portfolio. CVaR represents the expected loss in the event that losses exceed a predetermined confidence level, such as 95% or 99%. It provides a more comprehensive risk measure compared to traditional metrics like standard deviation or Value-at-Risk (VaR). CVaR considers the severity of potential losses beyond the confidence level, providing a clearer picture of the downside risk.

By constructing a portfolio with assets that exhibit lower downside risk and incorporating diversification strategies, your advisor can minimize potential losses beyond a specified confidence level.

Examples 

We now illustrate the use of these optimizers for constructing portfolios that reflect investor preferences. For the purposes of illustration, we limit our universe of assets to the following ETFs. The methodology can of course be extended to single stocks, individual bonds, and mutual funds.

Table 1: List of Assets Used as Asset Universe 

Our asset universe is defined over 15 assets spanning equities and fixed income along with Gold. Equities and Fixed income assets are in turn split into many different asset classes. These different asset classes are specified within Fabric’s internal taxonomy of assets. Assets that share similar characteristics are pooled together into a single level of the taxonomy. A taxonomy provides a structured framework for classifying assets into various categories and assists in identifying asset classes or subclasses that complement each other in terms of risk and return characteristics.

Maximum Return Portfolio 

As a first example, we will create a portfolio with a maximum possible return using all the assets in our asset universe defined in Table 1. We also want to limit the amount of volatility in the portfolio. Hence, we need to specify a maximum level of volatility to fully define the problem. Running Fabric’s optimizer and specifying a maximum volatility of 10.5% annually, results in the following portfolio with an annualized return of 6.53%, (we omit taxonomy details to keep the presentation succinct):

Table 2: Asset allocations for Maximum return portfolio with no constraints

We observe that of the 15 assets in our asset universe, the final portfolio is highly concentrated in just a few positions. This is a known problem with mean-variance optimizers. Without guardrails, maximum return optimizers will suggest portfolios with weights that are concentrated in positions which have the highest expected return. In this preliminary case the allocations are somewhat diversified into fixed income assets because we added a limit on the maximum possible volatility at 10.5%. Thus, the optimizer converged to a final portfolio with a balanced allocation spread between multiple fixed income assets and a low-volatility equity asset.

Optimizers that converge to highly concentrated portfolios have limited usefulness. This undesirable behavior can be tempered by regularization. Regularization is a mathematical technique that introduces a penalty term into the optimization objective function. This penalty encourages simpler and more stable portfolio solutions by discouraging extreme allocations or excessive reliance on a few assets. In simpler terms, regularization helps us find a balance between diversification and concentration, ensuring that our portfolio is not too scattered or too focused on a small number of assets.

We add the regularization and re-run the optimizer, and we retrieve the following portfolio:

Table 3: Asset allocations for Maximum return portfolio with regularization

With regularization added to the portfolio optimizer, we retrieve a portfolio that is well-balanced across the different assets. The portfolio is not concentrated in any one single position. The expected return for this portfolio is now 5.44% as compared to the earlier 6.5%. Fabric’s optimization engine allows you to control the strength of the regularization to allow for complete customization. In Figure 1 below, we show the relationship between the strength of the regularization and the expected return for the universe of assets defined in Table 1.

Figure 1: Impact of regularization on maximum possible return and diversification. Left: Variation of the maximum possible return for a given target volatility as the strength of regularization is varied. Larger values (close to 1.0) implies a stronger penalty term. Right: Variation of the number of non-zero positions as strength of regularization is increased.

We observe that as the strength of regularization is increased, there is an (almost) linear increase in the number of non-zero positions in the optimized portfolio. However, while there is a sharp increase in the number of non-zero positions, we dont necessarily have to take the regularization strength to its maximum possible value (equal to 1.0). We observe that beyond a certain critical value of the regularization there is no gain to be accrued from increasing the regularization.

We observe a similar behavior with the maximum possible return as well. Initially with no regularization, we achieve the maximum possible return, but with no diversification. As we increase the regularization strength, the maximum achievable return slowly decreases. Beyond a critical value of the regularization strength, the maximum achievable return stays constant for a given level of the volatility. We also observe that as we increase the level of volatility, the maximum possible return also increases. This is reassuring since regularization is not impacting the risk-return tradeoff: for greater levels of risk, we can achieve greater levels of return.

Adding Asset Constraints 

Let us reconsider the optimized portfolio in Table 3. This portfolio is not overly concentrated, and there is no single position that dominates the others. An investor might not be comfortable with some of these allocations. For instance, an allocation of greater than 5% to Emerging Markets or greater than 3% to MBS might not be within the mandate for a given investor. In such a case, we can explicitly limit the optimizer to produce optimal portfolios that obey constraints for specific groups of assets.

Within Fabric’s optimizer we can apply constraints to any level of the taxonomy. Let us apply the following set of constraints:

1) Maximum Allocation to Equities: 40%

2) Maximum Allocation to Emerging Market equities: 5%

3) Maximum Allocation to MBS/ABS: 3%

4) Maximum Allocation to U.S. High Yield: 5%

5) Maximum Allocation to U.S. High Yield: 10% 

Note that the constraints are being specified at multiple levels even within a single asset- class: we can have constraints on the overall equity allocation, and then constraints on specific parts of our equity allocation (Emerging markets here). We run the optimizer twice: once without the regularization and once with the regularization. We keep the same target volatility of 10.5%.

Table 4: Asset allocations for Maximum return portfolio with regularization and constraints 

In Table 4 we show the optimized portfolios with and without regularization. The impact of regularization is clear: it results in less concentrated portfolios. We also note that the constraints are satisfied in each case, but differently. Because the unregularized portfolio concentrates the portfolio in a few positions, it can satisfy the constraints trivial: by setting the emerging market ETF position to 0%, we have automatically satisfied the condition the maximum possible allocation to this position shouldn’t exceed 5%. For the U.S. High Yield constraint, things are a bit different: within the regularized portfolio the weights are less disparate between High Yield Corps and High Yield Munis as compared to the unregularized portfolio. Overall, the regularized portfolio is a more stable one with the tradeoff of a lower expected return: 5.3% for the regularized portfolio vs 6.21% for the unregularized portfolio.

Sector Constraints 

A final consideration while constructing portfolios is accurately tracking exposures to various sectors. Within the Fabric optimizer, you can specify constraints that you would like to impose on the eleven sectors that form the highest level of the industry classification taxonomy of the GICS. We run the optimizer again but limit the exposure to the information technology sector to be less than 20% and exposure to the Consumer Discretionary sector to be at least 10%. Such a preference could be expressed due to prevailing market conditions where the investor would prefer higher allocation to defensive sectors. We maintain the constraints on the overall Emerging Markets exposure as well as limiting the allocation to MBS/ABS to 3%. The portfolio that results is presented in Table 5, and the sector exposures presented in Figure 2. The annualized return and volatility for this portfolio are 5.5% and 10.75% respectively.

Table 5: Asset allocations with constraints on sectors

Figure 2: Exposures to different sectors of the GICS taxonomy for the portfolio in Table 5

Combining Model Portfolios 

A final use case that is accommodated within the Fabric optimizer is the ability to combine different model portfolios. Model portfolios are pre-constructed investment portfolios designed to meet specific investment objectives and risk profiles. These portfolios can consist either of a diversified mix of asset classes or could be focused on a single asset class. One can create different model portfolios based on risk tolerance of the investors, income needs, or high growth. These model portfolios can then form the building blocks for creating a customized portfolio for a given investor.

Our prior discussion has focused on creating a portfolio from the ground-up; starting from individual assets, we have applied constraints based on our preferences to construct a portfolio out of individual assets. We can perform the same set of operations using model portfolios as well. A useful mental model when thinking about model portfolios is that they are in many ways analogous to ETFs. An ETF can also be considered as a portfolio of assets but whose underlying positions can not be traded outside of the wrapper of the ETF. On the flip-side, a model portfolio is also a collection of assets but whose underlying positions can be changed/traded. The extra constraint on these positions is that the allocation within a portfolio should conform to its allocation within its parent model. We make it clear with an example of two models. Consider the two model portfolios presented in Table 6 and Table 7. The first is an Equity model and the second is a Fixed Income model.

Table 6: Example of an equity/growth model 

Table 7: Example of fixed income model 

Given these two model portfolios, we would like to construct a combination of these two model portfolios that optimizes for either maximum return or minimum variance. So, if we were to consider these two models as “synthetic ETFs” which have specific weightings to underlying assets, our task is to find the optimal combination of the Equity ETF (in Table 6) and Fixed Income ETF (in Table 7) that achieves a specific risk/return objective.

As an example, we consider the case where we want to compute the optimal combination of these two model portfolios that has the maximum possible return but that keeps the volatility to less than 12%. Using Fabric’s optimizer, we find that the optimal combination allocates 52.6% of the combined portfolio to the Fixed Income Model, and a 47.4% to the equity model. The final return achieved is 5.5% with a volatility of 11.5%. We summarize the portfolio allocations in the table below:

Table 8: Allocations of Equity and Fixed Income Model and the underlying assets 

Conclusion 

We have shown how using the Fabric optimizer, an investor can very easily construct portfolios tailored to their needs. The optimizer supports traditional mean-variance optimizers — maximum return and minimum variance — as well as the ability to minimize Conditional Value at Risk (CVaR). Using these optimizers as a foundation, the investor can express their preferences which are in turn expressed as constraints within the Fabric optimizers. Finally, an investment advisor can combine model portfolios, obtained through a TAMP or the CIO office, to produce optimal portfolios that suit their client’s specific risk and return preferences. In the course of this presentation, we have focused on examples and skipped the technical implementation details. These can be furnished upon request.

Appendix: Regularization for Mean-Variance Portfolios 

Mean-variance optimization is a popular technique for constructing portfolios that balance risk and return. However, naive mean-variance optimization can suffer from a number of problems, including:

• Sensitivity to the assumptions on expected returns and the covariance matrix. The
results of naive mean-variance optimization are extremely sensitive to the assumptions
on expected returns and the covariance matrix used for estimating risk. This can lead to
portfolios that are not robust to changes in these assumptions.

• Lack of diversification: Naive mean-variance optimization can often lead to portfolios
that are too concentrated in a small number of assets. This can increase the risk of the
portfolio and make it more volatile.

• Inability to handle constraints: Naive mean-variance optimization cannot handle constraints, such as limits on the number of assets in the portfolio or the maximum weight that can be allocated to any single asset.

To address these problems, we can add regularization to the mean-variance optimization
problem. Regularization is a technique that adds additional terms to the objective function in order to penalize certain behaviors. In the case of mean-variance optimization, we can use regularization to:

1) Prevent the portfolio from becoming too concentrated: We can add a penalty term to
the objective function that penalizes portfolios with high weights in any single asset. This
will help to ensure that the portfolio is diversified.

2) Make the portfolio more robust to changes in the assumptions on expected returns
and the covariance matrix: We can add a penalty term to the objective function that
penalizes portfolios that are sensitive to changes in these assumptions. This will help to
ensure that the portfolio is not too heavily influenced by any one set of assumptions.

3) Handle constraints: We can add constraints directly to the mean-variance optimization
problem. This will ensure that the portfolio meets the desired constraints.

There are a number of different ways to add regularization to the mean-variance optimization problem. The specific approach that you choose will depend on the specific needs of your portfolio.

Adding Asset-Weight Constraints 

One way to regularize the mean-variance optimization problem is to add asset-weight constraints. Asset-weight constraints limit the amount of weight that can be allocated to any single asset or group of assets. This can help to prevent the portfolio from becoming too concentrated and can also make the portfolio more robust to changes in the assumptions on expected returns and the covariance matrix.

There are a number of different ways to add asset-weight constraints. One common approach is to use a box constraint, which limits the weight of each asset to a specified range. Another common approach is to use an upper bound constraint, which limits the weight of each asset to a specified maximum value.

Another method is to add weight constraints on a collection of assets. These constraints can be applied at the attribute level, such as Asset class. Or they can be applied at the level of specific exposures, such as sector and regional exposures.

Within our implementation of the optimizer, we allow the user to implement constraints at the individual asset level, the attribute level, or the exposure level. In addition, the user can also set minimum and maximum weights for all assets in order to further ensure that the portfolio is not too concentrated.

Adding Penalty Terms 

Another way to regularize the mean-variance optimization problem is to add penalty terms to the objective function. Penalty terms penalize certain behaviors, such as high portfolio weights or sensitivity to changes in the assumptions on expected returns and the covariance matrix.

There are a number of different types of penalty terms that can be used. One common penalty term is the quadratic penalty term, which penalizes the square of the portfolio weights. Another common penalty term is the absolute penalty term, which penalizes the absolute value of the portfolio weights.

In our implementation of the mean-variance optimizer, we use a quadratic penalty term.

This penalty term penalizes the square of the portfolio weights. This helps to ensure that the portfolio is not too heavily influenced by any one asset. It also helps to correct for the instability of the covariance matrix.