# The Univariate Collapsing Method for Portfolio Optimization

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## Abstract

**:**

## 1. Introduction

## 2. The Univariate Collapsing Method

#### 2.1. Motivation

#### 2.2. Model

#### 2.3. Model Estimation and ES Computation

## 3. Sampling Portfolio Weights

#### 3.1. Uniform, Corner, and Near-Equally-Weighted

#### 3.2. Objective, and First Illustration

#### 3.3. Sample Size Calibration via Markowitz

#### 3.4. Data-Driven Sampling

#### 3.5. Methodological Assessment

#### 3.5.1. Performance Variation: Use of Hair Plots

#### 3.5.2. Varying the Values of $\tau $ and s

#### 3.6. The DDS-DONT Sampling Method

## 4. Enhancing Performance with PROFITS

#### 4.1. PROFITS-Weighted Approach

#### 4.2. Increasing $\tau $ Amid Favorable Conditions

- the number of random portfolios satisfying the mean constraint exceeds ${k}_{C}$, and
- the ES corresponding to ${\mathbf{a}}^{\u2605}$ is less than a particular cutoff value, say ${k}_{\mathrm{ES}}$,

## 5. Performance Comparisons across Models

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mean Signal Improvement

## Appendix B. Model Diagnostics and Alternative APARCH Specifications

**Figure A1.**Power plots of two normality tests, based on actual NCT(6,0) data, having applied the cdf/inverse-cdf transforma, assuming different values of $\nu $ (

**Left**) and $\gamma $ (

**Right**), and using sample size $T=250$.

**Figure A2.**

**Left**: Quantiles of p-values computed on the 30 constituent series of the DJIA, based on the MSP (top) and JB (bottom) normality tests applied to cdf/inverse-cdf transformed NCT-APARCH filtered innovations based on the KP-method with fixed APARCH parameters (7), applied to 4787 rolling windows of size $T=250$ over 20 years of returns data.

**Right**: Number of violations, out of the 30 assets, at the 5% level, with expected number of violations under the null being 30/20 = 1.5.

**Figure A4.**Quantiles, computed over 30 series, for the APARCH parameter estimates ${\widehat{c}}_{0}$, ${\widehat{c}}_{1}$, ${\widehat{d}}_{1}$, over time, according to (6), based on rolling windows of size 250, with fixed value ${g}_{1}=0.4$.

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**Figure 1.**Scatterplot of the first two out of three portfolio weights, for different sampling schemes.

**Figure 2.**

**Left**: Cumulative returns over the entire DJIA data, from 1 June 1999 to 31 December 2014, using the $1/N$ portfolio, and the indicated sampling methods, based on $s=1000$ and window length of $w=250$, so that the first trade occurs around June 2000.

**Right**: Same, but with corner sampling, using the indicated values of exponent q.

**Figure 3.**

**Left**: Cumulative return sequences of the DJIA data from 2009 to 2015 using the Markowitz iid long-only framework (denoted Mark-NS), based on moving windows of $w=250$ returns. The green line corresponds to the numerically optimized portfolio, while blue corresponds to use of the UCM method with uniform sampling, and $s=\mathrm{10,000}$ replications. The result based on the equally weighted portfolio is also shown, in red.

**Right**: Blue circles indicate the average, over all the windows, of $\parallel {\mathbf{w}}^{A}-{\mathbf{w}}^{U}{\parallel}_{2}$, where ${\mathbf{w}}^{A}$ and ${\mathbf{w}}^{U}$ refer to the analytic (optimized) and UCM-based portfolio vectors, respectively. This was conducted $h=8$ times per sample size s for $s\le 1000$, and otherwise $h=2$ times. The red cross indicates the average over the h values.

**Figure 4.**

**Left**: Circles show $\widehat{\nu}$ for each of the 29 assets, over 30 windows of length 250, for the 29 DJIA stocks, from 1 June 1999 to 31 December 2014. The blue (black) line shows their medians (IQR).

**Right**: The resulting ${r}_{u}$, ${r}_{c}$ and ${r}_{e}$ proportions (out of $s=1000$), based on (14) and (15).

**Figure 5.**Hair plots of eight cumulative return sequences, using the indicated method of sampling, of the DJIA data from 2009 to 2015 (so that the first trade occurs around January 2010), again with $s=1000$, $w=250$, and $\tau =10\%$. The thicker red line shows the exact $1/N$ performance. The bottom right panel corresponds to the data-driven sampling scheme (14) and (15) in Section 3.4.

**Figure 6.**Hair plots, based on window size $w=250$, of eight cumulative return sequences of the DJIA data (from June 1999 to December 2014), using data-driven sampling. The yearly percentage return, $\tau $, and number of samples, s, are indicated in the titles, along with the attained annualized Sharpe ratio as the average over the eight sequences. The thicker red line shows the cumulative return performance of the equally weighted portfolio.

**Figure 7.**

**Left**: Similar exercise as shown in Figure 6, but only for $\tau =10\%$, and using the cutoff strategy, such that, if out of the s samples, less than ${k}_{C}$ satisfy the mean constraint (as as indicated in the titles and legends), then trading is not conducted.

**Right**: Smoothed Sharpe ratios for all runs, as a function of ${k}_{C}$.

**Figure 8.**For $s=900$,

**left**panels show the number of samples satisfying the mean constraint, and the predicted expected shortfall of the $1/N$ and optimal under DDS portfolios. The

**right**panels plot the expected mean, sorted over the $s=900$ sampled portfolios, while the bottom panel shows a scatterplot of the predictive mean versus the predictive ES for the $s=900$ sampled portfolios, for time point t corresponding to the last observation in the data set.

**Figure 9.**

**Left**: Hair plot for $\tau =10$, for the three indicated sampling sizes s, based on the same data as in the comparable plots in Figure 7, using the PROFITS technique and the optimal values of parameters ${k}_{S}$ and ${k}_{CS}$.

**Right**: Obtained Sharpe ratios as a function of ${k}_{S}$ and ${k}_{CS}$, averaged over the eight runs, and smoothed, using a moving window of length two, for both dimensions, and for each of the three values of s.

**Figure 10.**

**Left**: Hair plots for $\tau =10\%$ using the ${\tau}^{*}$ strategy, for ${k}_{\mathrm{ES}}=2$. The same y-axis limits are used as throughout the paper for comparison purposes, and thus the upper values in the graphs corresponding to $s=300$ and $s=900$ have been truncated: For $s=900$, the terminal wealth of the best of the eight runs reaches 201, while the average over the eight runs is 182.

**Right**: Obtained Sharpe ratios as a function of ${k}_{\mathrm{ES}}$, averaged over the eight runs, and smoothed.

**Figure 11.**Illustration, using $s=300$, of the results from the ${\tau}^{*}$ method. It is based on the first of the eight conducted runs (they all result in very similar graphs). The graphs show the expected return, the predicted ES, the PROFITS measure, and the realized returns of DDS+DONT+${\tau}^{*}$ method minus those of DDS+DONT. The plot of ES divides by 100 simply to ensure that the top and bottom panels line up graphically via the spacing of the y-axis coordinates.

**Figure 12.**Comparison of cumulative returns. Methods are: The UCM method based on 900 replications (same as in the lower left panel of Figure 10, black lines); Markowitz (no short selling) based on the iid assumption (green line) and using the Gaussian DCC-GARCH model (blue line) for computing the expected returns and their covariance matrix; the equally weighted method (red line); and based on an iid two-component mixed normal distribution with parameters estimated via the MCD methodology, from Gambacciani and Paolella (2017).

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**MDPI and ACS Style**

Paolella, M.S.
The Univariate Collapsing Method for Portfolio Optimization. *Econometrics* **2017**, *5*, 18.
https://doi.org/10.3390/econometrics5020018

**AMA Style**

Paolella MS.
The Univariate Collapsing Method for Portfolio Optimization. *Econometrics*. 2017; 5(2):18.
https://doi.org/10.3390/econometrics5020018

**Chicago/Turabian Style**

Paolella, Marc S.
2017. "The Univariate Collapsing Method for Portfolio Optimization" *Econometrics* 5, no. 2: 18.
https://doi.org/10.3390/econometrics5020018