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Asset Allocation: Balancing Risk and Return | Part 2 - The Mechanics and Mathematics of Diversification

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Asset Allocation: Balancing Risk and Return | Part 2 - The Mechanics and Mathematics of Diversification

Alex Serman • April 11, 2024




Our first article in this series examined the fundamental principles of diversification, showing the benefits of increasing the number of complementary assets in our strategy. This allowed us to create a series of “efficient frontiers” where each increase in diversification produced a better set of results. We frequently hear the explanation of “efficiency” in the context of either: a) the highest possible return for a given level of volatility risk, or b) the lowest volatility for a given level of return. This is intuitive and easy to understand. But this still leaves this critical question unanswered: “How do I achieve this result?”

We need to understand how to identify these “complementary” asset, and how to combine them effectively, so that diversification is maximized.

Essentially, we need to understand the factors that are at work, and then manage these factors so that each asset in our strategy contributes equally to both return and risk. This is like performing an attribution analysis on the sources of return and risk – but doing this on a forward-looking basis. In this article, we explain the mechanics and the mathematics of creating a well-diversified and efficient asset allocation that meets a client’s required return while staying within their risk guidelines and limits.

We begin with a pair of diversification examples from the first article in this series. Our “2-Asset Portfolios” include allocations to the S&P 500 and the Aggregate Bond Index, with equity exposure ranging from 50% to 100 percent. The fully-diversified “12-Asset Portfolios” add alternative investments to the mix of broad asset classes, while also diversifying within the equity and bond segments. Diversification across major asset classes and within each asset class increased return substantially while increasing risk only slightly.

We examine our results in detail by comparing the traditional “60-40” mix of the S&P 500 and Aggregate bonds with its comparable allocation in the fully-diversified strategy. The effect of diversifying the strategy more fully added 88 bps of return while increasing risk by only 26 bps.

It is also worth noting that efficiency increases as we move to higher allocations of the growth assets of equity and alternatives. The “all growth” portfolio increases return by 151 bps while increasing risk by only 17 bps.


The diversified version of the 60-40 strategy splits the growth segment between 46% in global equity and 14% in alternative investments. The detailed allocation for the strategy and its segment details are illustrated in this table.


Creating efficient asset allocation strategies comes down to a simple concept: equalizing the contributions to both return and risk for each asset segment. Essentially, we want each part of the portfolio to “pull its own weight.”

To determine the right mix of assets and the appropriate weightings, we start with the return and risk of each asset segment, and then compare every pairing of segments to determine which of them are most complementary. Then we must find the ideal weightings of each segment, so that we meet our required return with the minimum of volatility risk.

Our solution is driven by the diversification offered by the various pairings of assets. The correlation statistic measures the degree to which assets are completely complementary (a value of -1.0) or identical (a value of +1.0.) This can be visualized easily: imagine the pattern of returns between two assets overlapping each other perfectly, so that there is no difference between them; we would describe that as a “1-to-1” relationship with a correlation of +1.0. If they were opposite (i.e. one moves above its average to the same degree that the other asset moves below its average) then these would have a reverse relationship; they would be completely complementary in terms of offsetting each other’s risk, with a correlation of -1.0.

We present a “correlation grid” that shows all pairings of assets. The diagonal is the interaction of each asset with itself. In isolation, these assets offer no diversification benefit. The other cells display all possible asset pairings. Each column (or each row) shows the complete set of pairings for every asset segment in the portfolio. The lowest correlations (from +0.27 to -0.42) are between bonds and all growth assets. These combinations reduce risk substantially, because the volatility patterns tend to be offsetting. The highest correlations (+0.96 to +0.68) are within global equity; these provide relatively little diversification amongst themselves. The alternatives provide reasonable diversification with global equity (from +0.58 to +0.81.) By color coding the correlation grid, we see the opportunities for diversification more easily, and it becomes apparent that allocating to the lower-correlation pairs will tend to create a more efficient strategy.


Ordinarily, we might expect contributions to both return and risk for each segment to be a simple calculation involving each statistic and its weighting. While this is true for the expected return (i.e., mean return) it is not true for risk. Why not?

We know that the risk of a diversified portfolio is always lower than the weighted average risk of its individual assets. If that were not the case, there would be no diversification benefit. We also know that low correlations between the portfolio’s assets are the source of this diversification benefit. To calculate the contribution to risk from each asset segment, we need the correlation between the asset and the overall portfolio. Once we have that correlation, contribution to portfolio risk is solved.

Contribution to Portfolio Risk (for a segment of the portfolio) = Weighting x Segment Risk x Correlation to Portfolio


Correlation of an asset to the portfolio = Sum of Correlations x Weights x Standard Deviations
Portfolio Standard Deviation

We demonstrate this calculation for the Treasury segment:

Armed with these correlations, we can develop an attribution of portfolio return and risk:

Correlation to the portfolio is the measure of the percentage of individual risk contribution (i.e., weighting x risk) that remains in the portfolio after diversification effects. We demonstrate this concept using Aggregate Bond (“Agg bonds.”)

Agg bonds have an individual volatility of 4.3% and a weighting of 20 percent. Their undiversified contribution to risk would be 0.2 x 4.3% = 0.86% or 86 bps. However, their correlation to the portfolio is 0.4, indicating that only 40% of their individual risk contribution remains in the portfolio. As a result, the contribution to risk from Agg bonds is 0.86% x 0.4 = 0.34% or only 34 bps.

Here is an interesting comparison of risk contributions:

Long Treasury bonds have an individual risk of 12.4% and an 8% weighting, but their low correlation to the portfolio indicates that only 6% of their individual risk contribution remains in the portfolio. This compares favorably to EM bonds, which have comparable volatility (12.2%) and only ¼ of the Treasury weight (2%) but they have a much higher correlation to the portfolio (0.75) so that most of their individual risk contribution remains in the portfolio. As a result, EM bonds contribute 3x the risk as our larger position in equally-volatile Treasury bonds (18 bps vs 6 bps.)


The key to evaluating a portfolio’s efficiency is to compare the contributions to return and to risk on a percentage basis.

We developed an efficiency measure, which is the difference between the contributions to return and to risk. In a perfect world, these contributions would be equal, with an efficiency metric of zero. This could be visualized as a set of assets sitting on the diagonal “Efficient Line” in our chart showing contributions to portfolio return vs risk. In the real world, we face limitations that prevent us from delivering these equal contributions. These limitations include prohibitions on using leverage and short selling, and client constraints such as maximum exposures to certain market segments or to illiquid asset classes. Within these limits, we seek to equalize the contributions to return and risk as much as possible.

Risk concentrations in our strategy are within reasonable maximums of about 10 - 20 percent, with most in single digits. We see that our high-quality bonds are “super-efficient” because they deliver positive efficiency values that completely offset the inefficiency of the growth sectors, especially within developed global equity. It is also noteworthy that these inefficiencies are relatively small, single-digit values. Most of our assets sit reasonably close to the Efficient Line.

“This is what efficiency looks like.”


Is our strategy truly superior to the simpler 60-40 strategy that is so often cited as a performance standard?
We prepared visuals to evaluate these two approaches.

The more diversified strategy reflects a relatively-even distribution of return and risk, with noted inefficiencies in the developed equity markets, and super-efficiencies in high-quality bonds.

By comparison, the 60-40 portfolio concentrates 94% of its risk in a single equity segment with only a 70% contribution to return. Some mistakenly compare the 60% equity weighting with a 70% contribution to return as a favorable outcome, while looking at the bond segment’s 40% weight and 30% contribution to return as inferior. That is a common viewpoint, but it is seriously flawed. The proper evaluation is between risk and return, rather than between weighting and return. The proper conclusion is that the bond segment is super-efficient, contributing 30% of the return and only 6% of the risk.

The real question for the client is this: “Are you comfortable with nearly 95% of your portfolio’s risk coming from a single segment of the strategy?” It is a safe bet to say that most clients would prefer that you took action to lower this risk concentration. That is the proper approach to encouraging clients to increase diversification.

We wonder why so many investors consider this 60-40 strategy to be a prudent approach or a reasonable benchmark!


In our last paper, we mentioned the benefits of “Double-Barreled Diversification” where return is increased as risk is decreased. We demonstrate this with our diversified strategy, comparing the portfolio’s actual return and risk to the average values of its assets.

The strategy’s compound return is 42 bps higher than the average of its assets, and its risk is 286 bps lower than the asset average. Let us examine how these benefits are related:

In a prior paper, we noted that the long-term compound return is separated from the 1-year mean return by half of the variance (the square of the standard deviation.) Given the low correlation between the assets in the portfolio, its standard deviation is reduced, and this lowers this “volatility drag” on the portfolio’s return. Since the portfolio’s mean is the average of its assets, and we subtract less than the average volatility drag, the portfolio’s compound return is higher than the average of its asset returns. (The underlying assumption of the average compound return is that the average asset volatility is subtracted from the portfolio mean return.)

We see that the portfolio’s compound return was increased on a relative basis by 6% while its risk was decreased by nearly 20 percent. This shows that increasing diversification lowers risk and increases long-term portfolio return. That is “double-barreled diversification” in action!

  • What is the value of optimization?
  • What about including some of the more esoteric and opportunistic asset segments?
  • What if we created a portfolio with no asset constraints?

Written in partnership with Stephen Campisi