SemiMetric Portfolio Optimization: A New Algorithm Reducing Simultaneous Asset Shocks
Abstract
:1. Introduction
1.1. Overview of Portfolio Optimization
1.2. Overview of Change Point Detection Methods
1.3. Overview of SemiMetrics
1.4. Motivation and Structure of This Paper
2. Proposed SemiMetric Change Point Optimization Framework
 Covariance is computed as an expectation $\mathrm{Cov}(X,Y)=\mathbb{E}(X\mathbb{E}X)(Y\mathbb{E}Y)$, which is an average (integral) over an entire probability space. In a financial context, this computes an average over time; in modern financial markets, especially since the global financial crisis, most time periods are bull markets, with most assets performing quite well together. As such, assets that rise together in a bull market but actually exhibit distinct dynamics may be erroneously identified as similar.
 Covariance fails to capture dissimilarity between time series during periods of market crisis and erratic behavior. Investors are often particularly concerned with the robustness of their portfolio during such times. Portfolios that are optimized using covariance as a risk measure fail to determine the impact of various asset combinations during times of market crisis. For instance, if two assets are simultaneously acting erratically, they may actually be negatively correlated during this time. If they are both included in a portfolio, this would increase rather than reduce erratic behavior. Structural breaks herald erratic behavior, so using distances between breaks in the objective function may better separate out erratic behavior in a portfolio.
 Investors are also interested in peaktotrough measures of asset performance, that is, the size of a drop in returns from a local maximum to a local minimum. Optimization algorithms using covariance measures fail to identify and minimize peaktotrough behavior. However, distances between sets of structural breaks (in the mean, variance, and other stochastic quantities) are better equipped to identify how similar two time series are with respect to peaktotrough measures. Thus, they may suitably allocate weights to minimize these precipitous drops.
 While various methods of portfolio optimization target downside risk directly, we believe that structural breaks may be a kind of “root cause” of the greatest erratic behavior and simultaneous downside risk, and thus are of the greatest priority to diversify away from.
3. Theoretical Properties
4. Simulation Study
4.1. Synthetic Data Simulation
4.2. Synthetic Data: Portfolio Optimization Experiments
5. Real Data Results
 We train our algorithm over a relatively long period to estimate the true dynamics between various assets’ structural breaks as precisely as possible. Training the algorithm on longer periods provides a more accurate assessment of similarity in varying market dynamics.
 However, there is a balance between going back far enough to learn appropriate dynamics between asset classes and using too much history that relationships between assets no longer behave the way in which they were estimated. The behavior of individual asset classes and their relationships may change over time.
 The period from January 2018–June 2019 is a suitable outofsample period to test the algorithm, due to the varied market conditions. Most of 2018 provided relatively buoyant equity market returns, with a sharp drop in December 2018, followed by a prolonged recovery until June 2019. We wish to examine how candidate portfolios will perform in various market conditions, particularly in the presence of large drawdowns. In addition, we do not wish to test our algorithm during a period that is too similar to the training interval, as performance could be artificially strong. Thus, this is a suitable period to compare the optimization algorithms’ performance.
 We did not include the COVID19 market crisis in our test data to ensure that our training data have broadly similar dynamics to the outofsample data set. We include a targeted analysis of the COVID19 crisis in Section 5.3.
 The role of asset allocation is often guided by an investment policy statement that provides upper and lower bounds for capital allocation decisions. This is captured in the candidate weights’ constraints. During pronounced bull and bear markets, institutional asset allocators may not have the flexibility to implement global optimization solutions. For example, if two asset classes had significantly higher returns and lower volatility than the remainder of candidate investments, the unconstrained solution would allocate all portfolio weight into these two assets. Investment weighting constraints prevent these contrived scenarios from occurring. For our constraints, we place a minimum 5% and maximum 25% of portfolio assets in any candidate investment. This is one of several typical constraints imposed in realworld policy statements—indeed, investment policy statements may include this as their only constraint (Coffey 2016). As mentioned in Section 3, we may impose additional constraints by combining with other optimization methods cited in Section 1.
 Our method provides an advantage over the simple correlation measure by addressing all three limitations in Section 2. One possible drawback to our proposed method, however, is that to learn meaningful relationships between assets’ structural breaks, a long time series history is needed, preferably with many structural breaks observed.
 When considering portfolio risk in an optimization framework, investors have a variety of measures they may choose to optimize over. Standard deviation, $\beta $, downside deviation, and tracking error are just several of these. Our CPO model introduces a mathematical framework that addresses peaktotrough (drawdown) losses and erratic behavior as a measure of risk. Specifically, the model captures simultaneous asset shocks and aims to minimize the size of drawdowns by creating a uniform spread of change points across all portfolio holdings. We are unaware of any existing measure with these properties.
5.1. Training and Validation Procedure
5.2. OutofSample Performance and Distributional Properties
5.3. Performance during COVID19
5.4. Sampling Study of Structural Breaks between Countries’ Financial Indices
6. Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
CPO  Change point optimization method 
MVO  Mean–variance optimization 
MSV  Mean–semivariance 
MAD  Mean absolute deviation 
FLPM  First lower partial moment 
SLPM  Second lower partial moment 
CVAR  Conditional value at risk 
EVAR  Entropic value at risk 
CDAR  Conditional drawdown at risk 
UCI  Ulcer index 
CPM  Change point model 
Appendix A. Change Point Detection Algorithm
Appendix A.1. Batch Detection (Phase I)
Appendix A.2. Sequential Detection (Phase II)
Appendix B. Overview and Properties of Distances between Sets
Appendix B.1. Overview of Metrics
 $d(x,y)\ge 0$, with equality if and only if $x=y$;
 $d(x,y)=d(y,x)$;
 $d(x,z)\le d(x,y)+d(y,z)$.
Appendix B.2. Distances between Sets
Appendix B.3. Illustration Study of Different (Semi)Metrics
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$$D=\left[\begin{array}{cccc}0& 100& 1000& 900\\ 100& 0& 900& 800\\ 1000& 900& 0& 900\\ 900& 800& 900& 0\end{array}\right];A=\left[\begin{array}{cccc}1& 0.9& 0& 0.1\\ 0.9& 1& 0.1& 0.2\\ 0& 0.1& 1& 0.1\\ 0.1& 0.2& 0.1& 1\end{array}\right]$$

$$D=\left[\begin{array}{cccc}0& 100& 111& 933\\ 100& 0& 189& 833\\ 111& 189& 0& 844\\ 933& 833& 844& 0\end{array}\right];A=\left[\begin{array}{cccc}1& 0.89& 0.88& 0\\ 0.89& 1& 0.78& 0.11\\ 0.88& 0.78& 1& 0.10\\ 0& 0.11& 0.10& 1\end{array}\right]$$

$$D=\left[\begin{array}{cccc}0& 1& 5& 461\\ 1& 0& 13& 306\\ 5& 13& 0& 327\\ 461& 306& 327& 0\end{array}\right];A=\left[\begin{array}{cccc}1& 0.998& 0.989& 0\\ 0.998& 1& 0.972& 0.336\\ 0.989& 0.972& 1& 0.291\\ 0& 0.336& 0.291& 1\end{array}\right]$$

$$D=\left[\begin{array}{cccc}0& 11& 61& 517\\ 11& 0& 72& 417\\ 61& 72& 0& 367\\ 517& 417& 367& 0\end{array}\right];A=\left[\begin{array}{cccc}1& 0.98& 0.88& 0\\ 0.98& 1& 0.86& 0.19\\ 0.88& 0.86& 1& 0.29\\ 0& 0.19& 0.29& 1\end{array}\right]$$

Asset  Number of Change Points  Weights 

Asset${}_{1}$  8  6.9% 
Asset${}_{2}$  8  6.9% 
Asset${}_{3}$  8  6.9% 
Asset${}_{4}$  3  5% 
Asset${}_{5}$  3  5% 
Asset${}_{6}$  3  5% 
Asset${}_{7}$  1  33.49% 
Asset${}_{8}$  1  30.7% 
Method  Cumulative Returns  Standard Deviation  Sharpe Ratio  Drawdown  Kurtosis 

CPO  107.04  0.0045  0.99  8.83  1.06 
MVO  98.64  0.0060    17.08  1.54 
MSV  105.76  0.0055  0.66  6.61  1.61 
MAD  102.28  0.0068  0.21  13.03  2.38 
FLPM  101.82  0.0063  0.18  11.05  0.90 
SLPM  102.26  0.0062  0.23  10.59  0.90 
CVAR  72.32  0.0061    29.23  0.90 
EVAR  148.55  0.0053  5.77  27.17  1.57 
CDAR  100.66  0.0066  0.063  14.22  2.25 
UCI  100.60  0.0055  0.069  12.47  1.61 
Method  Cumulative Returns  Standard Deviation  Drawdown  Kurtosis 

CPO  99.76  0.018  33.55  8.55 
MVO  88.82  0.036  65.13  15.48 
MSV  88.69  0.036  65.54  15.60 
MAD  88.06  0.038  67.71  15.99 
FLPM  88.08  0.038  68.39  16.11 
SLPM  88.69  0.036  65.51  15.60 
CVAR  88.45  0.035  64.41  15.54 
EVAR  89.12  0.033  60.99  14.90 
CDAR  101.43  0.025  38.38  9.49 
UCI  88.40  0.045  75.58  17.15 
Sample Size  Lower Limit  Upper Limit 

4  72.99  213.17 
6  100.70  210.92 
8  113.35  208.32 
10  127.02  201.08 
12  135.86  196.36 
14  143.81  191.59 
16  151.42  186.31 
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James, N.; Menzies, M.; Chan, J. SemiMetric Portfolio Optimization: A New Algorithm Reducing Simultaneous Asset Shocks. Econometrics 2023, 11, 8. https://doi.org/10.3390/econometrics11010008
James N, Menzies M, Chan J. SemiMetric Portfolio Optimization: A New Algorithm Reducing Simultaneous Asset Shocks. Econometrics. 2023; 11(1):8. https://doi.org/10.3390/econometrics11010008
Chicago/Turabian StyleJames, Nick, Max Menzies, and Jennifer Chan. 2023. "SemiMetric Portfolio Optimization: A New Algorithm Reducing Simultaneous Asset Shocks" Econometrics 11, no. 1: 8. https://doi.org/10.3390/econometrics11010008