# Markov-Switching Stochastic Processes in an Active Trading Algorithm in the Main Latin-American Stock Markets

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## Abstract

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## 1. Introduction

## 2. Literature Review of the Use of MS-GARCH Models

- A two-asset trading algorithm that determines in which asset to invest, given the probability of being in each regime at $t$ or $t+1$.
- A portfolio management system, in which some tilts in the investment weights are made, given the same probabilities.

- To invest in the simulated stock market index if the investor expects to be in the low volatility ($s=1$) regime at $t+1$ or
- To invest in the U.S. risk-free asset if the investor expects to be in the high volatility ($s=2$) regime.

## 3. Backtest Materials and Methods

#### 3.1. The MS-GARCH Model and Its Use in the Active Trading Algorithm

- To invest in the simulated stock market index if the investor has a regime expectation of ${s}_{t+1}=1$.
- To invest in the U.S. risk-free asset if the investor has a regime expectation of ${s}_{t+1}=0$.

#### 3.2. Data Description and Markov-Switching Tests

- The MSGARCH R package [57] can estimate only a GARCH(1,1) variance process. The maximum lag in the ARCH and GARCH terms are 1. Given this, the use of the Schwarz [62] or the Hannan-Quinn [63] one, are not necessary because we only want to determine the most accurate MS or MS-GARCH model (an issue in which the AIC is better than the other two criterions). This computational restriction sets aside the trade-off between accuracy and parsimony in the model.
- We prefer to use an information criterion such as the AIC because we want to test the goodness of fit of the estimated model. For this reason, it is preferable to use the AIC criterion that uses the LLF as input. The value of this function, along with the number of parameters and the length of the time series (information set), could give us more accurate fitting values than the sample RMSE. In addition, since we used the Viterbi [56] algorithm, one of the outputs of the estimation process is the LLF (the value to be optimized in the estimation process). Given this, it is natural for us to use the AIC as a fitting parameter than the RMSE.

- The 7 January 2000 base 100 value of the simulated country’s stock index. A value that we will assume as the theoretical market price of a zero-tracking error Exchange Traded Fund or ETF (risky asset) of such a simulated MSCI index. This will be our risky asset for normal or good performing time periods.
- The 7 January 2000 base 100 value of a theoretical index calculated with the weekly rate of the 3-month U.S. Treasury-Bill. This will be also considered as the theoretical price of a zero-tracking error theoretical ETF that pays the Treasury-Bill rate. This will be our risk-free asset for distressful time periods.

Algorithm 1. The MS-GARCH based trading algorithm’s pseudocode |

For date 1 to 996:- To determine the current balance in the portfolio (cash balance + market value of holdings).
- To execute the Markov-Switching model analysis in Equation (6) with either GARCH, ARCH, or constant variance (either with Gaussian or t-Student probability density function).
- To calculate, by using Equation (12), the forecasted smoothed probability ${\xi}_{s=2,t+1}$ related to being in a distressful or bad-performing regime at $t+1$.
- If ${s}_{t+1}=0$ (${\xi}_{s=2,t+1}>0.5$), then:a. To invest in the risk-free asset (Treasury-Bill ETF)
b. To invest in the risky asset (The simulated index ETF) - 5.
- To price the value of the portfolio with a market-to-market (with closing market prices at $t$) procedure.
End |

## 4. Simulation Results Discussion

## 5. Conclusions

- To invest in the stock market index if the investor expects to be in the low volatility ($\mathrm{s}=1$) regime at $t+1$ or
- To invest in the U.S. risk-free asset if the investor expects a high volatility ($\mathrm{s}=2$) one.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The historical performance of the six backtested portfolios in the Brazilian stock market.

Refinitiv™ RIC^{®} | Source | Index Name | Ticker in the Paper | Country |
---|---|---|---|---|

.dMIBR00000PUS | Refinitiv™ Eikon™-Xenith™ | MSCI Brazil price index (USD) | MSCIBRLUSD | Brazil |

.dMICL00000PUS | Refinitiv™ Eikon™-Xenith™ | MSCI Chile price index (USD) | MSCICHLUSD | Chile |

.dMIMX00000PUS | Refinitiv™ Eikon™-Xenith™ | MSCI Mexico price index (USD) | MSCIMEXUSD | Mexico |

UST3MT = RR | Refinitiv™ Eikon™-Xenith™ | U.S. 3-month treasury bill rate | USTBILL | U.S. |

**Table 2.**Statistical summary of the weekly returns of the three stock indexes studied here (values as percentages).

Index or Asset | Min | 5% Quantile | Mean | Standard Deviation | 95% Quantile | Max |

MSCIBRLUSD | −28.1500 | −8.1800 | 0.2300 | 5.1900 | 7.7800 | 29.2000 |

MSCICHLUSD | −29.2900 | −4.7800 | 0.1500 | 3.2900 | 4.9000 | 21.0900 |

MSCIMEXUSD | −26.4200 | −6.2000 | 0.2100 | 4.0100 | 5.8900 | 25.3000 |

Index or Asset | K-S Gauss | K-S t Student | A-D Test | Skewness | Kurtosis | J-B Test |

MSCIBRLUSD | 0.7766 | 0.0000 | 0.0000 | −0.0118 | 3.8109 | 0.0000 |

MSCICHLUSD | 1.9346 | 0.0000 | 0.0000 | −0.6560 | 8.2142 | 0.0000 |

MSCIMEXUSD | 0.0445 | 0.0000 | 0.0000 | 0.0578 | 6.0194 | 0.0000 |

**Table 3.**Summary of fitting the different MS models used as candidates for the stochastic process in each index.

Stochastic Process | MSCIBRLUSD | MSCICHLUSD | MSCIMEXUSD |
---|---|---|---|

Single-Gaussian | −3301.55 | −4285.52 | −3858.55 |

Single-t Student | −3433.74 | −4439.53 | −4038.77 |

MS-Gaussian | −3467.75 | −4492.07 | −4093.52 |

MS-t Student | −3468.11 | −4488.84 | −4098.02 |

MSARCH-Gaussian | −3459.81 | −4494.7929[Best] | −4092.63 |

MSARCH-t Student | −3461.04 | −4487.31 | −4093.82 |

MSGARCH-Gaussian | −3473.4399[Best] | −4493.92 | −4108.1490 [Best] |

MSGARCH-t Student | −3470.60 | −4482.84 | −4102.71 |

**Table 4.**Summary of the weekly recursive fitting analysis of the different MS models tested in each index.

Stochastic Process | MSCIBRLUSD | MSCICHLUSD | MSCIMEXUSD |
---|---|---|---|

Single-Gaussian | −1706.5086 | −2266.615 | −1993.9081 |

Single-t Student | −1767.9515 | −2346.8044 | −2083.8841 |

MS-Gaussian | −1783.3242 [Best] | −2372.2979[Best] | −2109.2868[Best] |

MS-t Student | −1777.5791 | −2366.1851 | −2106.7913 |

MSARCH-Gaussian | −1774.6806 | −2368.5946 | −2104.4816 |

MSARCH-t Student | −1768.8776 | −2359.9256 | −2100.1547 |

MSGARCH-Gaussian | −1771.4864 | −2365.4424 | −2106.9333 |

MSGARCH-t Student | −1765.128 | −2355.3302 | −2099.0413 |

**Table 5.**Summary of the frequency of using each simulated model in the 996 weeks of the recursive simulation.

Stochastic Process | MSCIBRLUSD | MSCICHLUSD | MSCIMEXUSD |
---|---|---|---|

Single-Gaussian | 0 | 0 | 0 |

Single-t Student | 256 | 68 | 184 |

MS-Gaussian | 487 | 394 | 275 |

MS-t Student | 0 | 0 | 0 |

MSARCH-Gaussian | 0 | 511 | 0 |

MSARCH-t Student | 0 | 0 | 0 |

MSGARCH-Gaussian | 253 | 23 | 537 |

MSGARCH-t Student | 0 | 0 | 0 |

Total of dates | 996 | 996 | 996 |

Index or Asset | Accumulated Return | Mean Return | Return Standard Deviation | Max Drawdown |

MSCIBRLUSD | 172.1344 [9.4753] | 0.1006 | 5.0575 | −33.0558 |

MSCICHLUSD | 145.205 [7.9929] | 0.0901 | 3.2291 | −34.662 |

MSCIMEXUSD | 162.0699 [8.9213] | 0.0968 | 3.8617 | −30.6773 |

USTBILL | 38.3854 [2.113] | 0.0374 | 0.0393 | — |

Index or Asset | CVaR (95%) | CVaR (98%) | Sharpe Ratio | |

MSCIBRLUSD | −12.4852 | −16.7693 | 0.0042 | |

MSCICHLUSD | −7.5621 | −10.6382 | 0.0076 | |

MSCIMEXUSD | −9.132 | −12.4010 | 0.0128 |

**Table 7.**Performance summary of the Markov-Switching investment strategy applied in the Brazilian stock market (from a U.S. dollar-based investor perspective).

Markov-Switching Model Used | Accumulated Return | Mean Return | Return Standard Deviation | Max Drawdown |

MS-Gaussian | 299.8539 [16.5057] | 0.1393 | 3.9772 | −15.4177 |

MS-tStudent | 161.3091 [8.8794] | 0.0965 | 4.028 | −19.8727 |

MSARCH-Gaussian | 216.5051 [11.9177] | 0.1158 | 3.9228 | −15.4154 |

MSARCH-tStudent | 289.6302 [15.9429] | 0.1367 | 4.0232 | −19.8827 |

MSGARCH-Gaussian | 441.0209 [24.2764] | 0.1697 | 3.9149 | −15.4184 |

MSGARCH-tStudent | 59.6986 [3.2862] | 0.0470 | 3.1778 | −19.8978 |

Markov-Switching Model Used | CVaR (95%) | CVaR (98%) | Sharpe Ratio | Mean Risky Exposure |

MS-Gaussian | −9.3428 | −11.3154 | 0.0113 | 97.36% |

MS-tStudent | −9.7378 | −12.1689 | −0.0006 | 96.53% |

MSARCH-Gaussian | −9.2274 | −11.2619 | 0.0095 | 95.76% |

MSARCH-tStudent | −9.5271 | −11.7792 | 0.0198 | 96.14% |

MSGARCH-Gaussian | −9.1400 | −11.2266 | 0.0244 | 93.75% |

MSGARCH-tStudent | −8.4226 | −11.3701 | −0.0005 | 89.09% |

**Table 8.**Performance summary of the Markov-Switching investment strategy applied in the Chilean stock market (from a U.S. dollar-based investor perspective).

Markov-Switching Model Used | Accumulated Return | Mean Return | Return Standard Deviation | Max Drawdown |

MS-Gaussian | 304.3449 [16.7529] | 0.1404 | 2.35 | −6.8837 |

MS-tStud | 142.8021 [7.8607] | 0.0892 | 2.3695 | −10.0516 |

MSARCH-Gaussian | 351.4477 [19.3457] | 0.1515 | 2.2433 | −6.8826 |

MSARCH-tStud | 131.6221 [7.2453] | 0.0844 | 2.344 | −10.0567 |

MSGARCH-Gaussian | 240.7838 [13.2542] | 0.1232 | 2.2545 | −14.8974 |

MSGARCH-tStud | 296.9058 [16.3434] | 0.1385 | 2.2653 | −10.0541 |

Markov-Switching Model Used | CVaR (95%) | CVaR (98%) | Sharpe Ratio | Mean Risky Exposure |

MS-Gaussian | −4.9655 | −5.8791 | 0.013 | 96.97% |

MS-tStud | −5.3301 | −6.611 | 0.0085 | 96.96% |

MSARCH-Gaussian | −4.8441 | −5.8324 | 0.0314 | 95.96% |

MSARCH-tStud | −5.3446 | −6.6101 | 0.0099 | 95.74% |

MSGARCH-Gaussian | −5.2099 | −6.6896 | 0.019 | 89.12% |

MSGARCH-tStud | −5.0197 | −6.2081 | 0.0218 | 92.55% |

**Table 9.**Performance summary of the Markov-Switching investment strategy applied in the Mexican stock market (from a U.S. dollar-based investor perspective).

Markov-Switching Model Used | Accumulated Return | Mean Return | Return Standard Deviation | Max Drawdown |

MS-Gaussian | 259.3627 [14.2768] | 0.1286 | 2.9764 | −17.4144 |

MS-t Stud | 255.8699 [14.0846] | 0.1276 | 2.8257 | −10.1805 |

MSARCH-Gaussian | 75.7788 [4.1713] | 0.0567 | 2.7333 | −10.1763 |

MSARCH-t Stud | 95.3431 [5.2482] | 0.0673 | 2.8529 | −17.4112 |

MSGARCH-Gaussian | 154.7175 [8.5166] | 0.094 | 2.8376 | −10.1853 |

MSGARCH-t Stud | 37.3705 [2.0571] | 0.0319 | 2.8956 | −14.0064 |

Markov-Switching Model Used | CVaR (95%) | CVaR (98%) | Sharpe Ratio | Mean Risky Exposure |

MS-Gaussian | −6.9409 | −8.3678 | 0.0147 | 96.76% |

MS-t Stud | −6.6954 | −7.9479 | 0.0223 | 96.36% |

MSARCH-Gaussian | −6.6761 | −7.8676 | −0.0048 | 94.93% |

MSARCH-t Stud | −7.1122 | −8.8926 | 0.0034 | 97.13% |

MSGARCH-Gaussian | −6.6997 | −7.8395 | 0.011 | 91.72% |

MSGARCH-t Stud | −7.182 | −8.8326 | −0.0054 | 93.51% |

Country | Best Model in Full TS Analysis | Best Model in Recursive Analysis | Most Used Model in the 996 Weeks | Best Model for Active Investment |

Brazil | MS-GARCH Gaussian | MS-Gaussian | MS-Gaussian | MS-GARCH Gaussian |

Chile | MS-ARCH Gaussian | MS-Gaussian | MS-Gaussian | MS-ARCH t-Student |

Mexico | MS-GARCH Gaussian | MS-Gaussian | MS-GARCH Gaussian | MS-Gaussian |

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De la Torre-Torres, O.V.; Galeana-Figueroa, E.; Álvarez-García, J. Markov-Switching Stochastic Processes in an Active Trading Algorithm in the Main Latin-American Stock Markets. *Mathematics* **2020**, *8*, 942.
https://doi.org/10.3390/math8060942

**AMA Style**

De la Torre-Torres OV, Galeana-Figueroa E, Álvarez-García J. Markov-Switching Stochastic Processes in an Active Trading Algorithm in the Main Latin-American Stock Markets. *Mathematics*. 2020; 8(6):942.
https://doi.org/10.3390/math8060942

**Chicago/Turabian Style**

De la Torre-Torres, Oscar V., Evaristo Galeana-Figueroa, and José Álvarez-García. 2020. "Markov-Switching Stochastic Processes in an Active Trading Algorithm in the Main Latin-American Stock Markets" *Mathematics* 8, no. 6: 942.
https://doi.org/10.3390/math8060942