The Effects of the Introduction of Volume-Based Liquidity Constraints in Portfolio Optimization with Alternative Investments
<p>Correlation matrix of the ETFs returns. Reference period: 2016–2023.</p> "> Figure 2
<p>Correlations between the liquidity measures, and correlations of the liquidity measures with the average and standard deviation of the asset returns. By Amihud, KO and CVVol we denote the reciprocal of the original illiquidity measures (<a href="#FD3-mathematics-12-02424" class="html-disp-formula">3</a>)–(<a href="#FD5-mathematics-12-02424" class="html-disp-formula">5</a>). Reference period: 2016–2023.</p> "> Figure 3
<p>Mean–variance–liquidity efficient frontiers <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi>M</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>V</mi> <mo>}</mo> </mrow> </semantics></math> obtained with the MV-L model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) using the liquidity measures AveVol (<b>a</b>), Amihud (<b>b</b>), KO (<b>c</b>), and CVVol (<b>d</b>) as the target return <span class="html-italic">M</span> and the target liquidity <span class="html-italic">L</span> vary. The red dashed line represents the mean variance frontier obtained with the MV model (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>). Reference period: 2016–2023.</p> "> Figure 3 Cont.
<p>Mean–variance–liquidity efficient frontiers <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi>M</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>V</mi> <mo>}</mo> </mrow> </semantics></math> obtained with the MV-L model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) using the liquidity measures AveVol (<b>a</b>), Amihud (<b>b</b>), KO (<b>c</b>), and CVVol (<b>d</b>) as the target return <span class="html-italic">M</span> and the target liquidity <span class="html-italic">L</span> vary. The red dashed line represents the mean variance frontier obtained with the MV model (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>). Reference period: 2016–2023.</p> "> Figure 4
<p>Level curves of the mean–variance–liquidity efficient frontiers obtained with the MV-L model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) using the liquidity measures AveVol (<b>a</b>), Amihud (<b>b</b>), KO (<b>c</b>), and CVVol (<b>d</b>) as the target return <span class="html-italic">M</span> and the target liquidity <span class="html-italic">L</span> vary. Reference period: 2016–2023.</p> "> Figure 5
<p>Optimal portfolio weights of the MV optimization model (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>). Reference period: 2016–2023.</p> "> Figure 6
<p>Optimal portfolio weights of the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) for the different liquidity measures considered, with a low target value for the liquidity constraint. Reference period: 2016–2023.</p> "> Figure 7
<p>Optimal portfolio weights of the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) for the different liquidity measures considered, with a medium target value for the liquidity constraint. Reference period: 2016–2023.</p> "> Figure 7 Cont.
<p>Optimal portfolio weights of the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) for the different liquidity measures considered, with a medium target value for the liquidity constraint. Reference period: 2016–2023.</p> "> Figure 8
<p>Optimal portfolio weights of the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) for the different liquidity measures considered, with a high target value for the liquidity constraint. Reference period: 2016–2023.</p> "> Figure 9
<p>Optimal portfolio weights for the backtesting investigation, computed with the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) using the CVVol liquidity measure for a low, medium, and high target liquidity value; the weights are compared to those obtained with the MV model (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>). Reference period: 2016–2022.</p> "> Figure 10
<p>Dynamics of the values of the optimal portfolios selected in the backtesting investigation; the portfolios have been computed with the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) using the CVVol liquidity measure for a low, medium, and high target liquidity value; for comparison, the MV portfolio (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>) is also displayed. Reference period: 2023.</p> "> Figure 11
<p>Mean–variance–liquidity efficient frontiers <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi>M</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>V</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> obtained with the MV-L model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>), as the target return <span class="html-italic">M</span> and the target liquidity <span class="html-italic">L</span> vary, for the extended asset set including Bitcoin; the liquidity measures used are AveVol (<b>a</b>), Amihud (<b>b</b>), KO (<b>c</b>), and CVVol (<b>d</b>). The red dashed line represents the mean–variance frontier obtained with the MV model (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>). Reference period: 2016–2023. The red dashed line represents the mean–variance frontier obtained with the MV model.</p> "> Figure 12
<p>Optimal portfolio weights of the MV optimization model (<a href="#FD7-mathematics-12-02424" class="html-disp-formula">7</a>) for the extended asset set including Bitcoin. Reference period: 2016–2023.</p> "> Figure 13
<p>Optimal portfolio weights of the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) for the extended asset set including Bitcoin; the liquidity measures used are AveVol (<b>a</b>), Amihud (<b>b</b>), KO (<b>c</b>), and CVVol (<b>d</b>) and the target value for the liquidity constraint is set to the medium level. Reference period: 2016–2023.</p> "> Figure 13 Cont.
<p>Optimal portfolio weights of the MV-L optimization model (<a href="#FD9-mathematics-12-02424" class="html-disp-formula">9</a>) for the extended asset set including Bitcoin; the liquidity measures used are AveVol (<b>a</b>), Amihud (<b>b</b>), KO (<b>c</b>), and CVVol (<b>d</b>) and the target value for the liquidity constraint is set to the medium level. Reference period: 2016–2023.</p> ">
Abstract
:1. Introduction
- We analyze some volume-based liquidity measures already proposed in the literature; in addition, we devise a novel measure of liquidity, based on the coefficient of variation of the market volume;
- Following one of Lo et al. [9]’s suggestions, we devise an extension of the classical mean variance optimization problem which includes an additional constraint that guarantees the portfolio a desired liquidity level; furthermore, we derive, through an in-depth sensitivity analysis, the trade-offs among the target return, the target liquidity, and the variance of the optimal portfolios;
- Making use of the liquidity measures and the extended portfolio optimization problem discussed in the first part of this paper, in the second part we analyze the effects of the introduction of a liquidity constraint in portfolio selection problems including a wide range of alternative investments, in the form of thematic ETFs. Moreover, we undertake a back-testing procedure;
- Finally, we investigate the effects of the introduction of an extremely high expected return and risk asset class, represented by a Bitcoin ETF.
2. Methodology
2.1. Liquidity Measures
2.2. Portfolio Optimization with Liquidity Constraints
2.3. Sensitivity Analysis and Liquidity–Return Trade-Off
3. Empirical Analysis: The Data
4. Empirical Analysis: Portfolio Optimization Results
4.1. Comparison of the Portfolio Optimization Results
4.2. Liquidity–Return Trade-Off
4.3. Portfolio Backtesting
4.4. Introducing an Asset with Extremely High Return and High Volatility: The Case of Bitcoin
ETF | Mean | SD | Maximum | Minimum | Skewness | Kurtosis |
Bitcoin | 0.0034 | 0.0035 | 0.412 | −0.256 | 0.525 | 6.928 |
ETF | AveVol | Amihud | KO | CVVol |
Bitcoin | 110.013 | 86.249 | 422.098 | 0.618 |
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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ETF Sector | ETF Name | Class |
---|---|---|
Energy | iShares US Energy ETF | S |
Materials | iShares US Basic Materials ETF | S |
Industrials | iShares US Industrials ETF | S |
Consumer discretionary | iShares US Consumer discretionary ETF | S |
Consumer staples | iShares US Consumer staples ETF | S |
Healthcare | iShares US Healthcare ETF | S |
Financials | iShares US Financials ETF | S |
Information technology (IT) | iShares US Technology ETF | S |
Communication services | iShares US Telecommunications ETF | S |
Utilities | iShares US Utilities ETF | S |
Real estate | iShares US Real estate ETF | S |
Treasuries | iShares US Treasury Bond ETF | S |
Corporate | iShares iBoxx $ Investment Grade Corporate Bond ETF | S |
Gold | SPDR Gold Shares | A |
Silver | iShares Silver Trust | A |
Commodities | iShares S&P GSCI Commodity-Indexed Trust | A |
Rare earth metals | VanEck Rare Earth and Strategic Metals UCITS ETF | A |
Luxury | Amundi S&P Global Luxury UCITS ETF | A |
Hedge funds | IQ Hedge Multi-Strategy Tracker ETF | A |
Forestry | iShares Global Timber & Forestry UCITS ETF | A |
Sustainability | Invesco MSCI Sustainable future ETF | A |
ETF | Mean | SD | Maximum | Minimum | Skewness | Kurtosis | J-B | ADF |
---|---|---|---|---|---|---|---|---|
Energy | 0.00046 | 0.0195 | 0.161 | −0.207 | −0.464 | 16.580 | 16,096.41 * | −15.64 * |
Materials | 0.00049 | 0.0140 | 0.112 | −0.104 | −0.357 | 10.663 | 5145.19 * | −14.79 * |
Industrials | 0.00051 | 0.0127 | 0.123 | −0.125 | −0.385 | 17.279 | 17,764.13 * | −13.79 * |
Consumer discretionary | 0.00046 | 0.0125 | 0.091 | −0.112 | −0.574 | 12.849 | 8542.52 * | −14.30 * |
Consumer staples | 0.00041 | 0.0100 | 0.081 | −0.101 | −0.913 | 19.199 | 23,086.62 * | −9.93 * |
Healthcare | 0.00042 | 0.0107 | 0.076 | −0.099 | −0.244 | 12.644 | 8100.82 * | −14.15 * |
Financials | 0.00048 | 0.0135 | 0.117 | −0.135 | −0.407 | 18.365 | 20,568.62 * | −13.88 * |
Information technology (IT) | 0.00088 | 0.0154 | 0.113 | −0.136 | −0.285 | 10.721 | 5206.65 * | −11.98 * |
Communication services | 0.00006 | 0.0123 | 0.079 | −0.088 | −0.185 | 7.948 | 2139.10 * | −13.96 * |
Utilities | 0.00037 | 0.0121 | 0.121 | −0.109 | −0.029 | 20.330 | 26,092.63 * | −11.69 * |
Real estate | 0.00031 | 0.0132 | 0.085 | −0.169 | −1.124 | 23.647 | 37,472.71 * | −9.18 * |
Treasuries | 0.00004 | 0.0032 | 0.023 | −0.022 | 0.182 | 7.905 | 2101.56 * | −14.81 * |
Corporate | 0.00012 | 0.0056 | 0.074 | −0.050 | 0.639 | 34.751 | 87,724.23 * | −10.81 * |
Gold | 0.00034 | 0.0087 | 0.049 | −0.054 | −0.065 | 6.560 | 1102.23 * | −44.81 * |
Silver | 0.00038 | 0.0164 | 0.091 | −0.136 | −0.196 | 10.386 | 4752.76 * | −43.72 * |
Commodities | 0.00027 | 0.0142 | 0.068 | −0.121 | −0.866 | 10.478 | 5119.24 * | −44.13 * |
Rare earth metals | 0.00057 | 0.0217 | 0.147 | −0.160 | −0.018 | 6.716 | 1199.64 * | −45.68 * |
Luxury | 0.00050 | 0.0142 | 0.116 | −0.120 | −0.128 | 10.409 | 4773.97 * | −43.86 * |
Hedge funds | 0.00010 | 0.0037 | 0.028 | −0.036 | −1.041 | 17.260 | 18,043.36 * | −16.83 * |
Forestry | 0.00042 | 0.0141 | 0.120 | −0.136 | −0.689 | 17.361 | 18,081.47 * | −11.93 * |
Sustainability | 0.00043 | 0.0144 | 0.080 | −0.109 | −0.466 | 9.460 | 3700.85 * | −10.93 * |
ETF | ADF | ADF Drift | PP | PP Drift |
---|---|---|---|---|
Energy | −2.768 * | −4.231 * | −18.423 * | −26.530 * |
Materials | −2.691 * | −5.030 * | −30.552 * | −34.950 * |
Industrials | −2.223 * | −6.011 * | −33.764 * | −40.174 * |
Consumer discretionary | −3.365 * | −6.510 * | −34.334 * | −38.472 * |
Consumer staples | −2.043 * | −3.336 * | −28.680 * | −35.055 * |
Healthcare | −2.308 * | −8.405 * | −28.078 * | −38.799 * |
Financials | −2.541 * | −5.332 * | −17.127 * | −31.347 * |
Information technology (IT) | −2.989 * | −9.837 * | −31.691 * | −38.221 * |
Communication services | −2.225 * | −8.505 * | −26.680 * | −35.788 * |
Utilities | −2.852 * | −4.580 * | −26.949 * | −34.604 * |
Real estate | −1.268 | −4.962 * | −5.884 * | −21.629 * |
Treasuries | −2.698 * | −4.643 * | −22.898 * | −28.674 * |
Corporate | −0.749 | −2.436 * | −5.664 * | −14.833 * |
Gold | −1.673 | −6.647 * | −7.596 * | −22.766 * |
Silver | −2.173 * | −3.595 * | −10.423 * | −16.799 * |
Commodities | −2.653 * | −3.631 * | −14.771 * | −19.675 * |
Rare earth metals | −2.165 * | −3.059 * | −11.790 * | −16.523 * |
Luxury | −4.711 * | −5.976 * | −38.189 * | −39.154 * |
Hedge funds | −2.411 * | −4.920 * | −31.721 * | −40.084 * |
Forestry | −3.297 * | −4.926 * | −24.574 * | −31.888 * |
Sustainability | −3.095 * | −4.463 * | −35.441 * | −39.048 * |
ETF | AveVol | Amihud | KO | CVVol |
---|---|---|---|---|
Energy | 53.042 | 2804.286 | 662.593 | 0.788 |
Materials | 13.982 | 649.869 | 530.547 | 0.567 |
Industrials | 10.497 | 762.341 | 513.471 | 0.674 |
Consumer discretionary | 8.822 | 608.777 | 488.460 | 0.540 |
Consumer staples | 9.203 | 523.939 | 575.928 | 0.635 |
Healthcare | 17.570 | 1667.368 | 682.307 | 0.888 |
Financials | 36.115 | 3048.578 | 745.118 | 1.142 |
Information technology (IT) | 47.320 | 3474.009 | 744.759 | 0.671 |
Communication services | 17.662 | 1251.854 | 625.536 | 0.812 |
Utilities | 13.303 | 987.124 | 575.319 | 0.723 |
Real estate | 837.437 | 95,391.560 | 2158.982 | 2.161 |
Treasuries | 150.716 | 20,344.820 | 3112.376 | 0.641 |
Corporate | 1850.648 | 441,898.500 | 4974.119 | 1.485 |
Gold | 1299.584 | 207,527.000 | 3304.449 | 1.696 |
Silver | 351.183 | 23,266.910 | 1395.116 | 0.912 |
Commodities | 15.531 | 564.065 | 543.929 | 0.666 |
Rare earth metals | 10.035 | 89.438 | 354.054 | 0.716 |
Luxury | 0.491 | 0.879 | 171.743 | 0.258 |
Hedge funds | 5.005 | 1281.754 | 913.640 | 0.739 |
Forestry | 2.239 | 97.895 | 286.287 | 0.684 |
Sustainability | 1.088 | 28.508 | 221.752 | 0.464 |
AveVol | 0.5800 | 1.96 × 10−5 | −3.39 × 10−5 | −0.5420 | −0.0054 |
Amihud | 0.6104 | 8.06 × 10−8 | −1.32 × 10−7 | −0.4507 | −0.0045 |
KO | 0.7579 | 1.55 × 10−5 | −2.05 × 10−5 | −1.0364 | −0.0104 |
CVVol | 0.5192 | 0.0375 | −0.0723 | −1.3622 | −0.0136 |
Mean Return | Return SD | Maximum Drawdown | VaR | AveVol | Amihud | KO | CVVol | |
---|---|---|---|---|---|---|---|---|
MV | 0.00037 | 0.0041 | 0.0645 | 0.0060 | 439.68 | 135,810 | 1883.90 | 2.04 |
MV-L low | 0.00042 | 0.0046 | 0.0652 | 0.0071 | 833.82 | 242,160 | 2162.20 | 2.83 |
MV-L medium | 0.00047 | 0.0054 | 0.0768 | 0.0083 | 1624.30 | 406,190 | 2433.40 | 3.28 |
MV-L high | 0.00049 | 0.0071 | 0.1121 | 0.0111 | 1299.70 | 255,040 | 1880.00 | 3.07 |
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Barro, D.; Basso, A.; Funari, S.; Visentin, G.A. The Effects of the Introduction of Volume-Based Liquidity Constraints in Portfolio Optimization with Alternative Investments. Mathematics 2024, 12, 2424. https://doi.org/10.3390/math12152424
Barro D, Basso A, Funari S, Visentin GA. The Effects of the Introduction of Volume-Based Liquidity Constraints in Portfolio Optimization with Alternative Investments. Mathematics. 2024; 12(15):2424. https://doi.org/10.3390/math12152424
Chicago/Turabian StyleBarro, Diana, Antonella Basso, Stefania Funari, and Guglielmo Alessandro Visentin. 2024. "The Effects of the Introduction of Volume-Based Liquidity Constraints in Portfolio Optimization with Alternative Investments" Mathematics 12, no. 15: 2424. https://doi.org/10.3390/math12152424