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Asset Allocation: Optimization and Constraints | Part 3 - Balancing Opportunity and Acceptability

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Asset Allocation: Optimization and Constraints | Part 3 - Balancing Opportunity and Acceptability

Alex Serman • May 02, 2024




Our first two articles explained the principles of diversification and its role in producing an adequate portfolio return while keeping volatility risk to a minimum. This condition is essential to meeting our monetary goals while reducing the likelihood of mission failure resulting from making withdrawals when markets are weak. We observed the value of spreading our investments across various asset classes, while also investing broadly within each major asset group (bonds, stocks and alternatives.)

We focused on finding complementary assets with offsetting return patterns, so that each pair of assets typically reflects strong returns in one while the other experiences weaker-than-expected returns. The asset allocation task was to identify a complete set of complementary pairs with the correct weightings for each asset. To accomplish this, we examined the mathematics at work within the “black box” of asset allocation, and we showed that the correlation of an asset to its portfolio was the key to equalizing the contribution each asset makes to total return and total risk.

This third installment demonstrates the difference between theoretically-correct portfolios and investable portfolios. Not surprisingly, the “best” answer (from the optimizer) may not be accepted as reasonable or even prudent. To bridge the gap between mathematics and practicality, we need to provide guidance that lets us balance return opportunity and reasonable standards of practice; we must provide solutions with the necessary familiarity and comfort that lead to adoption of the investment plan. We will optimize a familiar-but-simplistic strategy and demonstrate a process of introducing realistic limits to the optimization process. This balances mathematics, investment theory, client goals and investor judgement. 


Our starting point is the familiar strategy of investing 60% of our assets in the S&P 500 equity index and 40% in the Aggregate bond index. This approach continues to be used as a baseline for evaluating both strategies and performance, since it provides a minimum level of market exposure and diversification, it is simple and easy to understand, and it is a low-cost/low-maintenance approach. For our purposes, it is an ideal place to begin our analysis. Using our forecast of the capital markets (return, risk and correlations) we developed a visual of our set of investment segments. At first glance, it appears that the 60-40 strategy provides a fair compromise between the safety of bonds and the return opportunity from stocks. Plus, it provides the expectation of outperforming cash by over 3.5% annually, while earning about 1% over expected inflation.

But as we saw in our last article, the opportunity to improve the return-vs-risk result is even more significant, provided we expand our set of investment alternatives.


In our prior article, we demonstrated how we could create a series of superior strategies that lie above the simplistic opportunity set of portfolios in our “2-Asset” starting approach. We used an intuitive process to increase our exposure to the other 53 asset segments that are displayed on this chart of market opportunities. Our solutions allocated the portfolio’s funds across a global set of stocks, bonds and alternative investments. We were guided by our familiarity with common practice and our sense of comfort with exposures across the set of investments in our portfolio. It was encouraging (and perhaps surprising) to see how much our results improved by simply estimating our set of allocations, provided we included an “assortment of flavors” within all three of the major asset classes.

This raises a few questions:

  • How much more could we improve our results by using optimization?
  • How much should we constrain the asset selection process?
  • How do we create a reasonable approach?


Our starting point is the 60-40 asset mix, and we took two approaches to improving our results. First, we sought the highest return for the level of risk we bear with the 60-40 strategy. This is the “highest return” set of options. Then we sought the lowest risk for the return offered by the 60-40 mix.

Our initial optimization was unconstrained in terms of the exposure to the three asset classes (bonds, stocks and alternatives) and it was free to select any (or all) of the segments within each asset class. It is critical to note that this unconstrained optimization produces the maximum benefit possible: either the highest return or the lowest risk. (The only practical limits we introduced were to avoid short selling and leverage, since our context is a “long-only” strategy.)

We are reminded that financial theory is based on our indifference to anything other than the return and volatility of each asset and the correlations between all asset pairs. This process is blind to any of the ways that we group assets; a true optimization gives no consideration to the number of assets in the portfolio or what they are called or how they are classified. The solution ignores practical limitations such as:

  • Concentrations of money
  • Narrow focus of assets
  • Liquidity


We created a “solutions triangle” that starts with our basic allocation, and moves upward on the return axis to its theoretical maximum return; it also moves to the left on the risk axis to the minimum risk solution. We see about 350 bps of return enhancement and almost 700 bps of risk reduction, relative to our starting point. We can also see the “cost” of our constraints, which introduce limitations to the exposures in the portfolios, making the strategies more “investable.”


The practice of constraining the optimization places limits on the assets that may be used. Within this context, the optimizer finds the set of asset weightings that produces the desired result (either a high return or a low level of risk.) Effectively, some assets are excluded by the process, producing a subset of assets to select from in creating our efficient asset allocation. Our starting set of 55 asset segments was reduced by our set of constraints to 14 final segments used in our final optimized solutions. These are the segments with return, volatility and correlation characteristics that produce the most desirable risk and return results. In isolation, some of these seem to be less efficient than others, but in the context of the overall portfolio, all of them contribute to its efficiency. Our optimization process then assigns weights to the set of allowable asset segments.


We seek the minimum number of constraints in the asset allocation process. This provides the greatest opportunity for improved results. After all, if we were to constrain every asset class, we would effectively supply the answer that the optimizer is supposed to find. The process of assigning constraints is a balancing act, and finding the right balance is a matter of testing and judgement. Our approach is to begin with the unconstrained solution and examine it for anything objectionable. That objectional situation becomes our first constraint, and the optimization is run once again. The new solution will satisfy the first constraint, but will likely produce other objectionable asset holdings. Once again, we constrain the new problem and repeat the process until we find an acceptable overall solution.

We outlined the constraint process in this table of emerging allocations. In each case, we see the allocation problem and its resulting constraint, followed by the next solution with its own problem(s.) After five iterations, we found our ideal solution. By comparing these solutions, we can evaluate the “cost” in efficiency that eventually produced the acceptable solution.


The initial solution allocated almost 2/3rds of the portfolio to securitized bonds, with the remaining assets concentrated in illiquid alternative investments and Asian equity. We chose to address only the concentration in securitized bonds in our first level of constraints. This first constraint eliminated the initial problem at a cost of almost 35 bps of increased risk, while introducing a problem of no exposure to global equity. This became our second constraint, with a solution that increased volatility risk by 75 bps, and introduced a concentration in risky, leveraged loans, while continuing the concentration in Asian equity. This pair of problems formed our third constraint, and its solution increased risk by another 25 bps, leaving only a concentration of China within the Asian equity segment. The 4th constraint diversified the equity segment with a minimal increase in risk. The final problem was a concentration in illiquid alternatives, which was eliminated by the final constraint. Devoting the alternatives segment to liquid real estate was considered an acceptable option, with less than 7.5% devoted to this segment. This is an example of exercising judgement regarding acceptable concentrations.

We created an acceptable low-risk solution with a cost of about 150 bps of higher volatility. It is not surprising that most of its assets are in the bond segment, with an emphasis on high quality, along with a few opportunistic/diversifying segments to maintain the strategy’s target return at such a low level of risk.



Not surprisingly, our unconstrained “highest-return” solution is concentrated almost exclusively in equity. We did not constrain this solution regarding its exposure to bonds; we simply sought to eliminate unacceptable concentrations within the equity segment – we want a well-diversified asset class, especially when it is our dominant one.

The initial solution excluded developed markets equity; that became our first constraint. The solution reduced return by almost 100 bps, and it left a “barbell” of mid-sized US equity and Asian equity. We corrected the lack of diversification within US equity, with a very small effect on return. The concentrations in and within Asian equity became the focus of the next two levels of constraints: introducing European equity and eliminating the concentration in China. Our final constraint was the same as with our “lowest-risk” optimization: the elimination of illiquid alternatives in favor of liquid US real estate.



We see that our optimization process produced a balance between objective results and subjective limits that ensures conformity to norms of prudent investment strategy. Our strategies are familiar enough to be comfortable, while producing significant benefits in terms of risk and return, relative to less-diversified approaches. We measured these improvements objectively, demonstrating the tradeoff between theoretical efficiency and investable reality.

Returning to our baseline 60-40 portfolio, we see the benefits of optimization using our subjective constraints through a new efficient frontier of investable opportunities. We also illustrate the benefits of a naïve diversification approach, where we invested equally across all 55 asset segments available to us. That portfolio was relatively equally-weighted across bonds, equity and alternatives (at 33%, 31% and 36% respectively) and it outperformed the 60-40 strategy by 60 bps with about 140 bps less risk. Of course, that strategy would have been quite undesirable, having so many tiny positions and requiring so much work to monitor its set of relatively small and similar investments. We see that the same-risk portfolio on the optimized frontier outperforms that naïve strategy by 90 bps – this quantifies the value of optimization relative to naïve diversification. These results are consistent with our earlier paper illustrating our concept of “tiers of diversification.”

Lastly, we show the in-depth results from our optimization process, with its effects on both return and risk. Our constraints do lower the return of the theoretical solution, but that “excess return” is not achievable, since the concentrated solutions are not truly investable. The “cost of constraints” is the difference between textbook solutions and real-life solutions.

The improvement in the Sharpe Ratio is far greater in the lowest risk solution. This may surprise some, but we note that the Sharpe Ratio naturally declines along any efficient frontier, since risk is increasing faster than return.


In our final installment, we look at the portfolio in terms of broad benefits that we want it to provide. This approach is highly intuitive and gaining in popularity, because it examines the “functionality” that we want from the portfolio – quite a divergence from the traditional approach of thinking of the portfolio as a set of asset classes in rigid groupings! We want to focus on benefits resulting from the functions of the portfolio and its assets.

What are these functions? We suggest the following:

  • Cash flow matching – to provide certainty to the portfolio’s outflows
  • Inflation protection – for “intergenerational equity”
  • Growth – the source of both withdrawals and preserving real portfolio value
  • Illiquid alpha – additional sources of diversifying market return and excess return

Written in partnership with Stephen Campisi