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Optimizing Diversification And Hedge Funds - The Beryl Consulting Group

Rich Wenzel The Beryl Consulting Group Consultant 8 April 2010


In a series of articles, The Beryl Consulting Group looks at issues facing hedge funds and related investments. This is the first in a series being published here.


Given today’s investing environment, diversification is mandatory. This paper shows that even a simple volatility minimization optimization using historical data can significantly lower risk when investing in hedge funds.

Using data from 2002 to 2008, our results show that a naïvely diversified hedge fund portfolio (equal weighted across hedge fund strategy types) has on average 40 per cent more volatility than an optimized portfolio with the same return expectations. Our methodology is simple and practical in its approach and easily replicable.

Why Diversification

Our goal for this paper matches our investing advisory philosophy at The Beryl Consulting Group: It must provide practical, tangible, replicable benefits to investors. In this case, our investor is the reader.  To provide value to our investors we are providing historical research using a replicable and practical methodology that will show that diversifying a portfolio across hedge fund strategy types provides benefits that are real and tangible.

In addition, we are providing an introductory understanding of portfolio diversification and investment allocation optimization. While we expect the majority of interest will come from endowments, pension funds, RIAs, and family offices, we are sure that any investor considering investing in hedge funds will benefit from the research contained herein.

What this Paper is

Investors/managers are faced with daunting challenges in today’s investing environment.  We face all the traditional challenges: Will the market go up? Will the new regime increase taxes? What will inflation do over the next year? Investors are faced with more complicated investment vehicles and decisions than ever before. After hedge fund scandals, bank, credit, and corporate failures, and highly leveraged derivative structures belying true liquidity and exposure risk, diversification has reached beyond good investment mantra into absolutely mandatory investment policy.

We see Beryl as a solution for those seeking to move forward having learned the lessons of 2007-2009. We see a new normal coming into focus. If the last decade was defined by highly leveraged structures and reliance on mathematical models for securities that practitioners knew were underestimating exposure/capital risk, the future will be defined by common sense investing. Comprehensive detailed due diligence and risk management will be the most valuable services in the hedge fund investing sector.

This paper will show that from 2005 to 2008 a basic optimization process using historical data lowered volatility when compared to a naively diversified portfolio. Based on rolling twenty-four periods of returns for a naively diversified portfolio and a simply optimized portfolio using historical data we were able to achieve significant volatility reduction. The naively diversified portfolio (as defined by equally weighting the various hedge fund strategies in the Credit Suisse/Tremont HFI) on average had 40 per cent more volatility than the optimized portfolio with similar return expectations.

We will discuss the benefits of using diversification under a Multi-Manager/Fund of Funds approach. Using historical data and simple optimization techniques we build portfolios that provide better performance metrics; including lower risk. The results will show that diversification benefits not only help in stable markets but can also be beneficial in extreme markets.

For this analysis, we used Credit Suisse/Tremont HFI data and simple quadratic optimization to see if we could outperform from a risk perspective investing in single or naively diversified hedge fund portfolios.

What this Paper is not

While it is useful to discuss what this paper will provide to readers, it is also important to discuss what it does not provide.

The methodology here is simple. That is one of its benefits, it’s not fitted, contrived, contorted, or construed specifically for the most recent past markets. It is also not something that should be used blindly. We are not selling a purely objective model or methodology that can be used dogmatically.

What is shown here is an insight into the benefits of diversifying hedge fund investments across different investment strategy types. 

Structure of the Paper

Sections II to V explain diversification, optimization, and the data and analyses we used. Those familiar with these topics need only to glance over these sections to better understand how we set up our research.

Sections VI and beyond analyze and explain our results as well as mention the next steps to take this approach forward.


A Definition

While most readers will be versed with the benefits of diversification, we are going to provide a brief explanation and example of diversification.

The primary benefit from diversification is lowering volatility/risk for the same expected return. One of the most powerful concepts in finance is that the returns between assets are additive and that lack of correlation between assets is volatility decreasing. 

Said differently, if a person had an expectation for hedge funds A and B both to return 10 per cent and both to have volatilities of 10 per cent, by investing in both A and B instead of 100 per cent into either A or B, the portfolio of securities should have lower volatility than either A or B alone.


What is Optimization

In mathematics, logistics, operations, and finance, optimization can be defined as finding the best possible solution given a utility function and constraints. A utility function is an equation that is the goal of the optimization. It may be to maximize, minimize, or to equal a preset value. For example, let us say you wanted to get the lowest volatility for an expected return of 10 per cent. These Funds aren’t publicly traded and one cannot take a short position in them. In addition, let’s assume that investor does not want to take any leverage and that he wants to be fully invested, that is the weights in funds A and B are to equal 100 per cent and that each is capped at 100 per cent and floored at 0 per cent.

Ra, R = the expected returns in Funds A and B.

Types of Optimization in Investing

While it is possible to set up an optimization with almost any utility function and constraint, there are three primary types of optimizations used in investing. They can be defined by their utility functions. The example we used is a minimization utility function. In our example there was a desired return and volatility was to be minimized. Another example might be, an investor who wanted a certain annuity flow from a fixed-income portfolio. The investor might select the fixed-income securities by minimizing the cost to invest subject to the constraint that the minimum interest paid out each year was at least as high his desired annuity flow.

The next approach commonly seen involves a maximization type utility function. For example an investor may have a volatility constraint, such as a desire to match the volatility of a benchmark while outperforming the benchmark. Thus the investor’s goal will be to maximize return subject to volatility equaling S&P 500 volatility.

These two types of optimizations can be used to create one of the most common concepts in investing, the efficient frontier.

The third type of optimization, and probably the most common type of optimization performed in investing today, is what is called mean-variance optimization. That is a utility function that looks at a return expectation minus volatility. So the utility function will be an expected return minus variance subject to weighting and other constraints. The benefit of this is that the optimization is not constrained to either a specific volatility or return (though it can be) and is allowed to seek return to the point where increasing volatility actually lowers utility faster than the increased return is increasing utility.

While many investors look at return/risk metrics, Sharpe Ratios, Information Ratios, etc. the reason you don’t see optimizations specifically optimizing that utility function has to do with the difficulty in that optimization. While it was said earlier that anything could be optimized, practically it is not completely true. The less linear the utility function and/or constrains of an optimization becomes, the more difficult it is to solve. The mean-variance optimization is designed to bring similar results, at least in concept, as optimizing on a return/risk type of metric.


Basic Premise and Assumptions

Our investment analysis assumed that an investor would place his starting investment on January 1, 2005 and could not rebalance the portfolio more frequently than every six months. The reason we started on 2005 instead of 2002 is because we needed to generate a correlation matrix and wanted at least thirty periods to build our correlation matrix. More details on this part of the analysis are presented later.  We have a four-year investment period, that allowed for seven rebalances.

In order to make our optimization make sense, i.e. to have a reasonable diversification goal, we had to make assumptions about returns.

When an investor invests in something, he/she has to have an explicit or implicit expectation of returns. The most explicit would be ranking individual investment returns such as Fund ABC will return 9 per cent. The most implicit, i.e. the least descriptive expectation would be to say something such as; this portfolio of funds will return the same return as an aggregate hedge fund index. This means the investor wouldn’t be focusing on returns, but rather replicating a benchmark while minimizing volatility. The investor doesn’t have to specify actual returns to run his/her optimization.

A practice some investors use is a ranking methodology similar to that used by many research analysts. What the investor is saying is that securities ranked say 5 (on a scale of 1 to 5 with 5 being highest) would outperform, but not give specifics of how much outperformance. This is also called ordinal ranking.

Any model that predicts future returns that does not have a test phase of ex ante returns will be limited by design. For this analysis, we had to make some necessary assumptions.

As there are thirteen strategy types and we were investing over four years with six-month reallocation ability, we ranked each strategy’s return every six months. We decided that the bottom three would be the underperformers and the top three would be the outperformers for that period.

The only hedge fund strategy that was not among the top or bottom three during the time span, was the Multi-Strategy (MS). This is intuitive because a good MS strategy manager can lever and de-lever strategies dynamically to avoid strategies they believed would underperform going forward.

This was very good for the diversification study for two reasons. The first is that some investors might say that a MS manager will be able to diversify on their own and therefore investors can invest in them and be safe.  We are able to test this belief.

The second reason is for the optimization premise. We wanted the optimization to have a minimal amount of assumptions. However, when we did this optimization, we needed to have something to outperform. What we want to show is that for the same expectation of returns we could outperform naïve investing from a return/risk ratio.

As mentioned previously, there is always at least an implicit assumption of returns. We wanted our analysis to have the most “believable” assumptions possible. So we designed our optimization to have the same expectations as the MS strategy, i.e. the middle of the road. That does not mean the fourth highest and the tenth highest hedge fund strategy for a six month period couldn’t have significant differences, but that the expectations were for “average” performance.

How did MS help us in this regard? The MS index was able to avoid severe downturns, which lends credence to the belief that managers can predict with some confidence which strategies will be underperforming in the near future.


Return Expectations

For every six month period, we looked forward six months, and ranked each hedge fund strategy. The top three performers were then given a rank of two, the bottom three a rank of zero, and the rest a rank of one. The MS strategy always had a rank of one as it never out/underperformed. This means we did not try to predict specific returns for each hedge-fund strategy type, but rather made the assumption that an investment manager would be able to predict which strategy types would underperform.

Details of the Optimization

We used a variance minimizing optimization for our analysis. Our optimization was very simple and has few constraints for the simple reason that the more complicated the utility function/constraints, the more fitted the answer.

Minimize Volatility

Subject to:

Wf  ≥ 0;

Wf  ≤ 15%;

∑ Wf  = 1;

∑ Wf  R = 1;


Wf  = the weights in fund strategy type;

R = the return rank of a fund strategy;

Because of the nature of our optimization, the total weight of investment in funds with a rank of two must equal the total weight invested in rank zero. It is possible to not invest in any rank zero/two strategies in a period.



For our analysis, we constructed two naïve strategies. The first was to fully invest in MS (Multi Strategy) and the second was to invest equally amongst all hedge fund strategies (equal weighted or naïvely diversified portfolio). 

The optimized portfolio had the lowest volatility for the total period and the best return/risk ratio. It significantly outperformed prior to the last two months of 2008. Unfortunately, the most diversifying strategy types at the June/July reallocation were also the most significantly underperforming strategies. The optimizer chose them and it significantly hurt the statistics.

The astute observer will note immediately that the optimized portfolio had the highest return and thus inflated its Return/Risk ratio which was not the goal of the optimization. While this is true, the main goal for the optimization was to reduce risk, and the optimization did that by having the lowest volatility. The similar returning equal-weighted (Eq. Wgt.) portfolio has a higher volatility than the optimized portfolio. Until very recently the optimized portfolio highly outperformed from a risk metric both strategies as seen in figures 7, 8, & 9.

At Beryl, our optimization strategy is much more vested and dynamic in nature and will not be discussed in depth in this paper. Using our ordinal rankings that incorporate risk management and allocation parameters for each fund should allows us to construct portfolios that can outperform this simple methodology.

A Closer Look at the Results

While it is valid to say that the return outperformance of the diversified portfolio was fortunate, as seen in Figure 6, the smoothness of the equity curve compared to the equal weighted portfolio is evident and is not luck.


Return Calculations

Return calculations can be a sensitive subject among practitioners, academics, researchers, and managers. For this paper, we are going to state how we calculated returns and why. What is important is that our analysis is replicable in real-life investing decisions.

For our correlation matrix, we wanted to base the correlations on actual returns that an investor would use. Instead of using monthly return percentages to take correlation numbers and volatility numbers, we calculated six-month return numbers. We used these numbers for our optimization.


The first and most important benefit of this methodology is that it actually reflects what the investor faces. An investor in hedge funds cannot rebalance his portfolio daily like a stock portfolio and is subject to six-month returns.

The second benefit is evident as the longer the holding period, the more mean reversion is captured in the volatility. Imagine a high intra-day volatility stock that almost always is up or down one percent per week. There is a mean reversion to a plus /minus one percent. If the investor only rebalances weekly, using shorter period returns, the measure will overstate the risk. 


The most contentious aspect of this methodology is that the returns generated by definition have serial correlation (just like a moving average does with itself). We think that this is not really detracting but rather benefitting for this type of investor as the returns are what he/she experiences.

The second biggest detraction is the opposite of the second benefit. Across longer-term return periods this method can understate short-term volatility. This is a valid concern and when doing risk calculations, an investor should understand his short-term and long-term risk.



The first improvement to the process would be to have robust return predictions for individual hedge funds. Quite clearly, the less generalized we make the optimization (i.e. instead of using indices, we use actual hedge funds), the better the optimization. When we guide clients, this is our method. For clarity of exposition we made a host of simplifying assumptions.

The returns could be in the form of actual returns for a hedge fund. A simpler method would be to line up hedge funds within a strategy type and rank them, similar to research style rankings. Then one would either rank or assign returns to the strategy types. The optimizer could then incorporate a maximized return based on these ranking while restricting volatility or using mean-variance optimization.


One of the first improvements for the optimization would be to include non-linear risk. Correlation measures the linear relationship between two assets, i.e. how well a line can explain each asset’s returns. Including skewness and kurtosis into the optimization would capture some of the non-linear risk.

Moving to a mean-variance optimization would also mean giving the optimization more flexibility. More flexibility allows the optimizer to achieve better expected results.

Finally optimizations that use multiple correlation tables or running optimizations on two different correlation tables will give the investor/manager an idea of how sensitive his/her portfolio is to correlation changes.


Incorporating VaR analysis can benefit any investor. Understanding better historical or conjectured (either analytically or through Monte-Carlo simulation) portfolio VaR allows the investor to have a better understanding of how likely he is to lose a certain value of money. Like the optimization itself, it is only as good as the assumptions made. However, if one understands the assumptions, he/she also understands their limitations.


Greater Effort Required

To actually invest in this manner it takes more effort than just placing money with a few managers. Aside from the technical work, one has to conduct drill down due diligence on managers. For the average investor, this is by far the biggest constraint. It takes a lot of work to fully investigate a hedge fund manager and to maintain that knowledge. This is the greatest benefit of using a well equipped M-M consultant or FoF. For example at Beryl Consulting we have detailed information on over 1,000 hedge funds including the most sought top-tier managers.

M-M consultants and FoF place more money than individual investors do and this gives them more leverage in getting good information.

Quantitative Analysis is Based on Assumptions

Just ask anyone who managed mortgage portfolios, CDO, or CDS in any form, mathematical assumptions are just that, assumptions and they don’t always hold.

However, we have shown that a simple optimization can hedge risk even during very stressful market considerations.

Frankly, between the due diligence and quantitative abilities available to a FoF/Consultants, they should outperform the Credit/Suisse Tremont HFI.

Fully investigating a fund manager on investment, allocation, and risk management approaches gives excellent information about a manager. Placing money in hedge funds without this information is an act of faith. Some large hedge funds have the traders as their own risk managers. These are red flags, and while a fund that operates in this style may outperform, the bulk will not especially in stressful times.

At Beryl Consulting for each of the managers we recommend we strive to understand not only their investing decision criteria but also investigate these managers on their strategy allocation criteria and risk management. Ruling out managers that do not have strong investment process from entry to risk management from the investment process will provide for more stable results.


We at Beryl Consulting hope that we created value for our readers with this paper. What is evident is that developing a diversification methodology is not only easy but returns very tangible benefits. Investing in hedge funds has more emptor caveat than any other security type except private equity. Limited information, illiquid investments, higher fees, and more intensive research requirements makes hedge fund investing daunting for investors with limited resources and experience.

However, hedge fund investing can be very rewarding for a variety of reasons. Unlike mutual fund managers that have a hard time outperforming the S&P500 or other indices over time, M-M consultants and FoF should be able to outperform hedge fund indices. In addition, hedge funds tend to have low correlation with other asset classes making them an excellent diversifying investment. Hedge fund managers have less constraints than traditional asset managers allowing them to exploit advantages more quickly and fully.

In a future paper we will also show that hedge fund investing can have a very beneficial diversification benefit for a multiple asset class portfolio.

Prior to working at Beryl, Rich Wenzel was most recently at Société Générale where he created new trading strategies and helped design and develop a satellite algorithmic FX on the NY trading desk. Previous experience includes investment banking corporate advisory at Salomon Smith Barney, FX risk client advisory at Bank of America, and structured products internal due diligence/compliance at Citigroup.

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