Last week I discussed absolute measures of portfolio performance and management, specifically between two portfolios that are composed of different assets (utilities and tech). I began with comparing the basics of return, standard deviation, and Sharpe ratio to some other possible portfolio in the Markowitz cloud. But, simply comparing the difference between these possible portfolios can be sensitive to the spread of stats within a specific Markowitz cloud. In other words, it’s not scale independent. A larger spread of possible stats can make a portfolio look bad due to the spread return/standard deviation/Sharpe ratio alone.
In this post I introduce quasi-relative measures. Again, I lean on the Markowitz cloud. They’re pasted below (Utilities on the left, tech on the right).

If we can somehow express the returns, volatilities, and Sharpe ratios on a common scale that is independent of the level values, then we can make the realized portfolios more comparable. One thing that we can do is to express a stat as a weighted linear average between the maximum and minimum possible values. Conditional on the realized standard deviation, there exists a maximum and minimum of possible return. Something like the below. Rho is the weight on the maximum return. It’s also the proportion of possible conditional returns that are lower than the realized return.

The unconditional version is the same, but would be relative to the global maximum and minimum stats. We can represent the weigh on the maximum return and the percentile among possible returns as gamma.
A final quasi-relative measure of performance is the dissimilarity index between the realized portfolio weights and some reference portfolio weights. This provides a measure of how much the asset weights would need to change in order to adjust the portfolio. If changing portfolio weights is costly, then it’s also a measure of the transaction cost of reallocation. It’s quasi-relative because it is independent of the spread of possible performance stats.
Below are the quasi-relative measures for each the utility and tech company portfolios.
stat | w_o | Max r|Same sd Min sd|Same r Min Var Max Return Max Sharpe
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rho_r | 0.864226 | 1.000000 +0.135774 | 1.000000 +0.135774 | 1.000000 +0.135774 | 1.000000 +0.135774 | 1.000000 +0.135774 |
rho_sd | 0.712544 | 1.000000 +0.287456 | 1.000000 +0.287456 | 1.000000 +0.287456 | 1.000000 +0.287456 | 1.000000 +0.287456 |
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gamma_r | 0.373136 | 0.426631 +0.053495 | 0.373136 +0.000000 | 0.199977 -0.173159 | 1.000000 +0.626864 | 1.000000 +0.626864 |
gamma_sd | 0.909618 | 0.909618 +0.000000 | 0.946991 +0.037373 | 1.000000 +0.090382 | 0.000000 -0.909618 | 0.000000 -0.909618 |
gamma_sharpe | 0.606674 | 0.693228 +0.086553 | 0.623060 +0.016386 | 0.350158 -0.256516 | 1.000000 +0.393326 | 1.000000 +0.393326 |
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dissimilarity | 0.000000 | 0.340000 | 0.369431 | 0.336911 | 0.670000 | 0.670000 |
stat | w_o | Max r|Same sd Min sd|Same r Min Var Max Return Max Sharpe
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rho_r | 0.949497 | 1.000000 +0.050503 | 1.000000 +0.050503 | 1.000000 +0.050503 | 1.000000 +0.050503 | 1.000000 +0.050503 |
rho_sd | 0.886174 | 1.000000 +0.113826 | 1.000000 +0.113826 | 1.000000 +0.113826 | 1.000000 +0.113826 | 1.000000 +0.113826 |
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gamma_r | 0.390961 | 0.402907 +0.011947 | 0.390961 +0.000000 | 0.044748 -0.346213 | 1.000000 +0.609039 | 1.000000 +0.609039 |
gamma_sd | 0.774443 | 0.774443 -0.000000 | 0.787130 +0.012687 | 1.000000 +0.225557 | 0.000000 -0.774443 | 0.000000 -0.774443 |
gamma_sharpe | 0.774467 | 0.797550 +0.023083 | 0.786451 +0.011985 | 0.118443 -0.656024 | 1.000000 +0.225533 | 1.000000 +0.225533 |
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dissimilarity | 0.000000 | 0.198673 | 0.195145 | 0.422477 | 0.670000 | 0.670000 |
The conditional ‘rhos’ are lower for the utility portfolio. That implies that even if the desired volatility was achieved in each portfolio, the tech portfolio did a better job getting the highest possible return conditional on that variance. For comparability, the rho for standard deviation is multiplied by the minimum value such that rhos closer to 1 are better and values closer to zero are worse.
What about the unconditional stat percentiles? These results are more of a mixed bag. Both realized portfolios are comparable as measured by the return percentile, at 37% & 39%. The utility portfolio scores much better on volatility. Importantly, this measure has *nothing* to do with the fact that utility companies are less volatile generally. The gamma stats measure the portfolio performance relative to what could have been achieved with the same assets. The tech portfolio comes out ahead in terms of the Sharpe ratio too.
Note that all of the rho values on the efficient frontier (EF) are unity. That’s because the definition of the EF is that it minimizes variance at each return and also often maximizes return at each variance. There are related patterns for the gamma values.
What’s the drawback of quasi-relative performance measures? Except for the dissimilarity index, all of the rho and gamma values describe a portfolio among possible returns, standard deviations, and Sharpe ratios – irrespective of how likely they are. But, as can be seen in a previous post, the density of possible portfolios is not uniform in (sigma, return) space. So, the quasi-relative performance measures identify percentile among possible performance stats, but do not measure the percentile among possible portfolios. Specifically, more possible portfolios (or weight combinations) tend to have lower variances and fewer tend to have higher variances. These different densities of portfolio weights are more pronounced as the constituent assets differ more by return and variance. I’ll address how to overcome the non-uniform distribution of possible portfolio weights across return and variance space in my next post on relative measures of performance.
Bartsch, Zachary. 2025. “Portfolio Efficient Frontiers & Diagnostics for Python.”
Ave Maria University. https://github.com/zacharybartsch/frontier_segments











