Data Analysis. Financial markets. Programming.

Welcome to my portfolio website!

Code: https://github.com/felipeOL10/Option_data_Russell2000/blob/main/Option_data_analysis.ipynb

# The Black-Scholes model

The Black-Scholes model is a funtion of time, stock price, implied volatility and the interest rate.

The sensitivity of an option to the variables above is called the "Greeks".

## 1st order greeks:

Delta: The delta of a call option indicates how much its price changes in response to changes in the underlying asset price. It can range from 0 to 1 for a call option, and from -1 to 0 for a put option. The delta tells us how closely the option price tracks the movements of the underlying asset.

Theta: represents the time decay of an option's value. It tell us how much the option price decreases over time due to the passage of time. In theory limits for theta are not explicitly defined because they depend on multiple factors like time to expiration, volatility, and interest rates.

Rho: It indicates how much the option price changes when interest rates fluctuate. Values for rho generally range from 0 to positive infinity, and for put options, it ranges from negative infinity to 0.

Vega: Shows how much the option price changes in response to changes in the volatility of the underlying asset. The limits for vega are not explicitly defined as they depend on the pricing model and other factors. Vega is typically positive for both call and put options.

## 2nd order greeks

Gamma: Indicates how quickly the delta of an option adjusts to movements in the underlying asset. Gamma limits are not specifically stated, but it tends to be higher for options that are close to the current market price and decreases as options move further away from that price.

Volga: Volga, also known as vega convexity, captures the sensitivity of vega to changes in volatility. It shows how vega itself changes as volatility fluctuates. Limits for volga are not explicitly defined and depend on the pricing model and other factors.

Vanna: shows how the delta of the option changes in response to shifts in volatility. The values vanna can take are not explicitly stated and depend on the pricing model and other factors.

## Analysing volatility

In the Black-Scholes model, the volatility is assumed to be constant. Therefore, by ploting the volatility of the option we should get a flat line, as well as a flat volatility surface. However, as we can see on the plots below this is not in accordance with reality.

*   The smile: It describes the situation where implied volatility tends to be higher for both out-of-the-money (OTM) and in-the-money (ITM) options compared to at-the-money (ATM) options with the same expiration date. This results in a U-shaped curve when plotting implied volatility against strike price. The smile indicates that the market assigns higher probabilities to extreme price movements or "fat tails" in the underlying asset.

![](/images/newplot-cd3429bc.png)The Skew: In a volatility skew, the implied volatility tends to be higher for OTM options compared to ATM options. This means that the market expects larger price movements in one direction (e.g., downside) compared to the other direction. The skew is often observed in equity markets where there may be a higher demand for protective put options, leading to increased implied volatility for OTM puts.
![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%2004eedde.png)

## Analysing the greeks

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20042634.png)

*   The delta shows that ITM options have higher deltas then OTM options, in fact, this is expected because the delta can also be represented as the probability of a options expire ITM;

*   The theta is negative for both ITM and OTM options, especilly for OTM options with more negative values. This is because ITM options already have intrinsic value, reducing the impact of time decay on their overall value. OTM options, on the other hand, have higher Theta values and are more affected by time decay due to their higher likelihood of expiring worthless if they remain out of the money.

*   For the vega, ITM option have higher value than OTM, meaning that changes in implied volatility have a more significant impact on the value of ITM options. This is because ITM options have a higher likelihood of being exercised and, therefore, their value is more influenced by changes in volatility.

*   In term of the rho, both ITM and OTM options benefit with rising interest rates.

*   ITM options exhibit higher Gamma values, meaning their Delta and value change more rapidly in response to changes in the underlying asset's price. OTM options, on the other hand, have lower Gamma values and are less affected by small changes in the underlying price.

*   OTM options show higher volgas than ITM, meaning their Vega and value change more rapidly in response to changes in implied volatility.

*   ITM options have higher Vanna values compared to OTM options. This means that as implied volatility changes, ITM options experience larger shifts in their Delta, resulting in a more significant impact on their value.

![](/images/newplot-5b0a66a7.png)

*   Delta is higher for lower strike prices and lower for higher strike prices. This means that in-the-money call options have a higher sensitivity to changes in the underlying asset's price compared to out-of-the-money call options.

*   Theta is higher for options with shorter time to expiration and decreases as the time to expiration increases. Therefore, options with higher strike prices (OTM) tend to have higher Theta values compared to options with lower strike prices (ITM). OTM call options are more susceptible to time decay.

*   Options with lower strike prices tend to have higher Rho values compared to options with higher strike prices.

*   OTM options are more sensitive to changes in implied volatility given by the evga. This could be due to higher implied volatility for OTM options, smaller intrinsic value in OTM options, higher reliance on implied volatility for OTM options' value, or market expectations of future volatility.

*   Strike prices closer to the current price have higher Gamma values compared to options with strike prices further away.

*   The closer to the market price the lower the sensitivity to changes in the vega. Also, OTM options show the highest values, meaning that are the most sensitive to changes in the implied volatility.

*   Options with higher Vanna values near at-the-money will have larger changes in their Delta for small price movements in the underlying asset and changes in implied volatility.



Analysis of Russell 2000 options data using web scrapping and the Black and Scholes model

## Context

##### This project is a part of my thesis, "Bitcoin and Ethereum capabilities against investment philosophis".

## Tools

This part of the thesis was developed using Python.

## Goal

The goal of this part of my thesis was to better understand the implication of adding cryptocurrencies to personalized investment strategies, namely, value investing, growth investing, real estate, bonds and ESG.

## Methodology

The portfolios optimizations as well as the performance ratios were obtained using the pyfolioopt (https://pyportfolioopt.readthedocs.io/en/latest/) model and by using the code in my github repository (https://github.com/felipeOL10/Portfolio_opt_thesis/blob/main/Portfolio_optimization.ipynb)

Note: the following pictures were taken to the teses itself.

Portfolio optimizations

*   Mean variance portfolio optimization

*   Quadratic utility

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20222650.png)

*   Efficient CVaR

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20222744.png)

*   Efficient CDaR

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20222810.png)

Performance ratios:

*   Sharpe ratio
*   Sortino ratio

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223002.png)

## Data used

*   Bitcoin and Ethereum as them were, at the time of the study, controlling close to 60% of the market share.

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223245.png)

*   BlackRock etf as representative of the investment philosophies

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223313.png)

The data used was from Yahoo Finance between [10/11/2017 and 19/07/2022]().

## Data analysis

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20223547.png)

The summary statistics reveal some differences between assets. The annual return of cryptocurrencies is substantially bigger than other assets, where Bitcoin has a return of 25.4% and Ethereum of 33.6%, thus reflected in a much higher annual volatility than other assets.

Furthermore, growth investing has an average of 47.9% annual expected returns compared to value investing with an average of 31.6%. Moreover, large-cap growth was the dominating type of investment during this period with 11.8%, on average. Conversely, the small-cap value was the lowest annual return with 4.2%. In addition, mid-capitalization performed better than small-cap. In terms of annual volatility, small-cap has the highest value with 27.2% for both value and growth and large-cap was the least risky investment in terms of growth and value with 21% and 23.9% for large-cap value and growth, respectively.

In terms of the other investment types, ESG was the highest performer with 9.5%, outperforming every investment philosophy except for large-cap growth firms. However, it achieved such performance with lower annual volatility than most investment philosophies. Furthermore, dividend investing had an annual return of 4.1%, outpacing real estate and bond investing. Bond investing was the only asset class with negative returns, however, it had the lowest recorded volatility with 7.5% on average.

In general, every risk measurement shows similar results, where Bitcoin and Ethereum present higher values and bond investing the lower values, suggesting similar results as with the descriptive analysis of returns. The mean absolute deviation shows that bond investing was 0.002 and 0.003 from the mean returns (AGG and LQD, respectively), and Bitcoin and Ethereum were 0.033 and 0.043 from the mean. Looking at the downside risk of an investment, measurements such as semi-deviation, Value at Risk (VaR), Conditional Value at Risk (CVaR) or expected shortfall (ES), Drawdown at Risk (DaR), and Conditional Drawdown at Risk (CDaR) can be very informative. In terms of the semi-deviation, it looks at the dispersion of an asset from an observed mean, which shows that for most values the semi-deviation was less than the standard deviation, meaning that negative returns have less extreme values. The historical VaR, where in the case of Ethereum and Bitcoin 9.6% and 7.5%, suggests that at a 95% confidence level the cryptocurrencies will not have larger losses than 9.6% and 7.5% in a day contrasting to bond investing where the losses do not pass the 0.4% in a day. The CVaR, represents the expected loss if the worst case happens, reaching 15.6% and 12.1% in the case of Bitcoin and Ethereum. The maximum drawdown shows that Ethereum and Bitcoin dropped 273.5% and 177.4% from the highest point in the sample and the average drawdown was 115% and 69.2% respectively. Furthermore, the DaR indicates that Bitcoin and Ethereum take the longest to recoup their losses and the CDaR shows that both assets also have a bigger average of drawdowns.

## Dynamic beta

![](/images/Picture2.png)

As expected, the results show similar behavior of the risk measurements. Both Bitcoin and Ethereum show much larger values than one, indicating much higher riskiness than the overall market. Also, most investment strategies are relatively riskier than the overall market, yet to a lower extent than cryptocurrencies.

## Portfolio analysis

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20225248.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20225432.png)
![](/images/Captura%20de%20ecr%C3%A3%202023-06-04%20225512.png)

From the tables above we can conclude

1.  Firstly,annualized returns are better with only one or both cryptocurrencies, except for the minimum variance portfolios and very risk averse investors relative to the benchmark portfolios (only ETFs). However, in most cases, this was at the expense of an increase in annualized volatility (Baur et al., 2022).

2.  Secondly, when comparing portfolios with Bitcoin and with Ethereum, when investors seek performance, Ethereum-only portfolios are better, although, with higher risk aversions, investors should focus more on Bitcoin.

3.  Thirdly, the investors that benefit the most from the adoption of cryptocurrency are bond investors with the highest improvement in the Sharpe ratio, followed by real estate, dividend investors, value investors, growth investors, and ESG investors. However, for bond and real estate to get such improvement, investors should only invest in cryptocurrency.

4.  Fourthly, bond and real estate should, in most cases, allocate close to 50% in each cryptocurrency, dividend investors with Wbtc=0.389 and Weth=0.418, value investors should allocate Wbtc=0.348 and Weth=0.383 and ESG and growth investors are the ones that allocate the lowest percentage in cryptocurrencies with Wcryptocurrencies=0.091 for ESG and Wbtc=0.220 and Weth=0.246 for growth investing. Therefore, the allocations found for growth investing are relatively close to what Conlon et al. (2020) found for CSI 300 of 16% and 14% for Bitcoin and Ethereum, respectively.

Overall, the portfolio results are in line with those presented by Baur et al. (2022), Borri (2019), Demiralay and Bayracı (2021), and Kajtazi and Moro (2019).


Portfolio optimization using the mean variance framework of Markowitz, efficient CVaR and efficient CDaR optimizations

## Context

##### This project is a part of my thesis, "Bitcoin and Ethereum capabilities against investment philosophis".

## Tools

This part of the thesis was develop using STATA and Excel.

## Goal

This paper analyses the hedge, safe haven, and diversifier capabilities of cryptocurrencies against investment philosophies from 10/11/2017-19/07/2022, COVID-19, and the structural break period.

## Methodology

*   To analyze the hedging, safe haven, and diversification capabilities of Bitcoin and Ethereum for investment philosophies, it will be used the framework proposed by Ciner et al. (2013).

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20092829.png)

Where the DCC is the pairwise correlation between each of the cryptocurrencies and each of the ETFs representative of the investment philosophies, the D(r_etf,q) represents a dummy variable that is equal to one when the returns of the ETFs are lower than the lower 10th, 5th, and 1st percentiles of the ETFs.
Therefore, cryptocurrencies are a diversifier for ETFs when c0 is positive and significant, a weak hedge when is zero, and a strong hedge when is negative and significant. The cryptocurrencies are a weak safe haven if c0 is negative and the c1,c2 and c3 are zero, and a strong safe haven when these coefficients are negative.

*   Hedge Ratio

Following Kroner and Sultan (1993), the hedge ratios (HR) are computed based on the conditional volatility and covariance estimates and are given by the following expression:

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20092906.png)

Where  cov(R_Crypto,R_etf) represents the conditional covariance between Bitcoin or Ethereum and the returns of the ETF, and Var(R_etf) represents the conditional variance of the ETF. Also, positive hedge values indicate reverse positions to hedge against the risk of each asset and negative values indicate that investors must take the same position for the two assets in the portfolio (Chemkha et al., 2021).

*   The hedge effectiveness (HE) index is given the percentage reduction in the variance of the hedge portfolio to the unhedged portfolio (Ku et al., 2007):

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20093006.png)

Where the 〖var〗_hedged is the conditional variance of the hedge portfolio, using equation (4), and 〖var〗_unhedged is the conditional variance of the ETF. Therefore, the hedge portfolio variance is obtained by the following expression:

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20093106.png)
Where R_etf, represents the return of the individual ETF, R_CRYPTO denotes the returns of the cryptocurrency and h_t represents the HR.

*   Optimal Portfolio Weights

Moreover, to determine the optimal holding weight of cryptocurrencies the study will use the Kroner and Ng (1998) framework, which can be expressed as follows:

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20093221.png)

Where, h_t^etf represents the conditional volatility of the ETF returns, h_t^Crypto and h_t^(etf,Crypto)refers to the conditional volatility of the cryptocurrency at time t. Under the assumption of the absence of short selling, the mean-variance portfolio optimization approach imposes the following constraints on the optimal weight of the cryptocurrencies:

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20093142.png)

## Data used

The data used was the same as the portfolio part (https://my-website-57330.stackbit.app/blog/postlayout-gy45g5jll/)

## Empirical results

*   Strutural break

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20100258.png)

To check changes in the mean and/or other parameters of the process that produces the series, structural break tests were performed. The table shows different structural break tests with no suggested date for the model and tests with different dates for the COVID-19 pandemic  and the Ukraine and Russia war (11/02/2022). Therefore, none of the dates suggested are significant for the existence of structural breaks. However, using the supremum Wald test, for both cryptocurrencies indicated that for Bitcoin there is a structural break on 21/10/2021 and for Ethereum on 17/12/2018. Using stationary tests and ARCH-LM tests, both series have a unit root at 1% for both periods and show ARCH effects for all lags, except for the ARCH(1) for Ethereum before and Bitcoin after the structural break.

*   Daily dynamic correlations for BTC, ETH and ETFs using DCC-GARCH for the full sample , respectively:

![](/images/Picture1-7d010afe.png)![](/images/Picture2-a345fc35.png)

Most of the sample presents positive values, especially at the end of the sample for both cryptocurrencies. The period between 2019 and 2020 presented three moments of lower correlations, namely at the beginning of 2019, mid-2019, and the beginning of 2020, especially in the case of Ethereum. In the case of Bitcoin, the mid-2019 period presents the lowest values found, with -0.34 and -0.33 between IWP and IJK, respectively. Therefore, this period for both cryptocurrencies may suggest hedge properties for investment philosophies. Furthermore, at the beginning of the COVID-19 crisis correlations were negative and turned positive rapidly, suggesting a volatile period for correlations between cryptocurrencies and ETFs, losing possible hedge capabilities. This suggests similar findings to Corbet et al. (2020), where Bitcoin acted not as a safe haven or hedge but as an amplifier with sharp elevations of the dynamic correlations. Also, at the end of the sample, there are mostly large positive values, reaching correlations of 0.6 in the case of IWO, IWF, IVW, SUSA, IWP, and IJK for Ethereum and IVW, IWO, and IWP for Bitcoin. In addition, this illustrates a varying relationship between these assets, which is in line with Su et al. (2020).

*   DCC-GARCH correlation coeficients

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20094252.png)

Most of the correlations presented are positive, indicating that Bitcoin and Ethereum act not as a hedge or a safe haven but rather as a diversifier.

*   Cinner et al. (2013)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20094556.png)

Note: The first part of the table is related with BTC and the secound to ETH. The values presented represent the full sample

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095438-629d4898.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20094854.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20094754.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095525.png)

The Ciner et al. (2013) framework was used to analyze the cryptocurrency's capabilities, allowing us to conclude that using the full sample, both cryptocurrencies have diversifier capabilities for all investment philosophies. The period before the COVID-19 pandemic shows that Bitcoin had strong hedge capabilities for bonds, dividends, mid-cap value, and growth for S&P500 and Russell stocks and real estate investing. Also, it showed that it had strong safe haven capabilities for the lower 10% quartiles for real estate investing and corporate bonds. Furthermore, diversification capabilities also improved for other investment capabilities, apart from ESG investing, mid-cap growth from the Russell index, and for small-caps from the S&P500 index where it lost diversification abilities. During COVID-19, Bitcoin lost hedge and safe haven capabilities, being just a diversifier. In the case of Ethereum, it became a hedge for bond and real estate investors and a safe haven for the lower 10% quartiles in the case of IYR and AGG and the lower 1% quartiles of corporate bonds. Moreover, it had better diversification capabilities than the full sample model. Besides, it also lost the hedge and safe haven properties during the pandemic and lowered the diversifier properties given by bigger constant coefficients. According to the structural breaks, Bitcoin was a diversifier before the break, and after the diversification, capabilities lowered apart from AGG, where it became a better diversifier. For Ethereum, it acted as a hedge for dividend and real estate investors and a safe haven at 10% lower quartiles for bond investors and 1% lower quartiles for corporate bonds, and it showed better diversification abilities when compared with the full sample. However, after the structural break, it was only a diversifier. The results are in line with Conlon et al. (2020), Dutta et al. (2020), Maitra et al. (2022), Choi and Shin (2022), and Corbet et al. (2020), supporting the non-safe haven capabilities and the support of the diversifier capabilities during the pandemic. They contradict Jeribi et al. (2021) and Mariana et al. (2021) that suggest safe haven capabilities during this period. Overall, these results support the conclusions of  Bouri et al. (2017), Sha and Song (2021), and Marobhe (2022) as a diversifier. Also, the findings highlight the time-varying capabilities of cryptocurrencies which justifies the need for dynamic approaches (Corbet et al. 2020).

*   Hedge Ratio, Optimal Portfolio Weights and Hedge Effectiveness

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095220.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095743.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095853.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095918.png)

![](/images/Captura%20de%20ecr%C3%A3%202023-06-05%20095946.png)

During the full sample, the lower HR with cryptocurrencies in a short position was obtained with bond investing. Also, the higher ratios were for mid-cap growth stocks from the Russell 2000 for Bitcoin and small-cap growth from the S&P 600. Conversely, long positions in cryptocurrencies show much higher ratios, especially for bond investors. Also, the lower ratios for long positions in cryptocurrencies were obtained for small-cap value from the S&P 600 for Bitcoin and real estate for Ethereum (USRT). OPW shows very residual values for all assets. Before the COVID-19 pandemic, most of the ratios show very residual values indicating that to hedge the different investment strategies there is a need for very low amounts of investment in cryptocurrencies.
Long positions in Bitcoin show lower ratios than the ones of Ethereum. Furthermore, some ratios show negative results for both short and long positions, indicating that there is a need for both long positions in both assets. In contrast, during the pandemic, all ratios suggest a need for short positions whereas dividend investing, and bond investing show lower ratios. In long positions concerning this period, there is a significant increase in most ratios when compared with the period before the pandemic, suggesting that long positions in cryptocurrencies during the pandemic needed more investment in the other assets to hedge the position, where bond and dividend assets were the highest ratios. These findings are aligned with Chemkha et al. (2021) and Maitra et al. (2022) where the necessary coverage positions were cheaper before than after the pandemic. Furthermore, OPW is residual in both periods, which goes against what was found in Maitra et al. (2022) for both periods. For cryptocurrencies in long positions, investors constructed the hedge using the same position for S&P500 and Bitcoin before and during the pandemic in Maitra et al. (2022), which goes against most of the ETFs analyzed during the pandemic but it is in line with the results found before the pandemic, especially for most of the long positions in Bitcoin. Moreover, when compared to the full sample, it maintains the previous behavior, where short positions are more expensive with Bitcoin and long positions are more expensive with Ethereum. Before the structural break, dividends and real estate needed to be long in both Ethereum and the assets referred. Also, short positions have very low ratios, and long positions have very high ratios. After the structural break, there is an overall increase in short ratios and long ratios. Overall, the results of HR and OPW align with Guesmi et al. (2019) for short positions. Conversely, long positions in cryptocurrencies present significantly higher values than what is found in Pal and Mitra (2019).
The HE also provided interesting insights. For the entire sample period, most of the values are close to zero, indicating the low effectiveness of the hedge by using cryptocurrencies in short positions. However, large-cap growth stocks from the Russell 1000 provided the highest effectiveness for both cryptocurrencies (almost 100% decline in the variance of the unhedged portfolio). With long positions, most provided non-effective hedges with most of them increasing the variance of the unhedged portfolio, however mid-cap value from the S&P400 provided almost a perfect hedge. During the pandemic, Bitcoin and Ethereum short positions improved compared with the period before, however, none provided a big enough effectiveness (apart from small-cap value from the Russell with more than 50%). For long positions, most values were negative, which provided increases in the unhedged portfolio but mid-cap value from the S&P provided a perfect hedge for Bitcoin and Ethereum (only before the pandemic). Regarding the structural break, large-cap growth from the Russell provided almost a perfect hedge for both short positions in cryptocurrencies, especially Ethereum. Also, there was an improvement before to after the structural break in short positions. Long positions showed similar characteristics to the pandemic period. Overall, the results are according to Charfeddine et al. (2020) where low effectiveness with cryptocurrencies in short positions leads to low effectiveness of the hedge, against Bouri et al. (2020) finding high effectiveness for USA-BTC (1.41%). Also, long positions in cryptocurrencies present, in most cases, negative values which go against Pal and Mitra (2019) and Maitra et al. (2022).


Hedging capabilities of cryptocurrencies against investment philosophies



Code: https://github.com/felipeOL10/Historical_VaR_cpp

The goal of the project was to further develop my C++ skills and to create a program that allows to calculate the historicall VaR for a given csv file.


Historical VaR using C++

See github repository: https://github.com/felipeOL10/VaR-CVaR-and-VaR-backtesting/tree/main



This project analyses the Value at Risk, Condicional Value at Risk and VaR backtesting using the historical, parametric and monte carlo methods for a portfolio consisting of 'TSLA','GOOGL', 'AMZN','PLTR', 'SHOP', 'SNOW', 'NVDA' during 2020-10-01 to 2022-01-01, using data from 'yfinance'.



VaR, CVaR and VaR backtesting

See python code:https://github.com/felipeOL10/Corporate-credit-ratings-analysis

Data:https://www.kaggle.com/datasets/kirtandelwadia/corporate-credit-rating-with-financial-ratios

## Goal of the project

*   Analyse the determinants of investable corporate bonds;

*   Using a logistic regression to predict the investible rating.

## Data

Corporate credit ratings, issued by specialist agencies, provide an assessment about the credit worthiness of a company and acts as a pivotal financial indication to potential investors. It helps provide investors with a concrete idea about the risk associated with the company’s credit investment returns. Every company aims to attain a good credit rating for seeking more investment and lower debt interest rates.

Most of the credit rating agencies have a unique discrete ordinal rating scale. The rating scale of the S\&P is: {AAA, AA+, AA, AA−, A+, A, A−, BBB+, BBB, BBB−, BB+, BB, BB−, B+, B, B−, CCC+, CCC, CCC−, CC, C, D} – a total of 22 grades that are ordered from AAA, the most promising one to D, the most risky one. S\&P broadly classifies the companies with rating higher than BB+ as investment grade companies and others as junk grade companies. Credit ratings are mostly determined by financial ratios coming from balance sheets, income statements, and cash-flow statements.

Therefor, the dataset has 7805 rows with 25 columns, ranging from 2010 to 2016. The data incorporates several financial ratios and the corporate ratings.

`list(corp_credit_rating.columns)`

\['Rating Agency',
'Corporation',
'Rating',
'Rating Date',
'CIK',
'Binary Rating',
'SIC Code',
'Sector',
'Ticker',
'Current Ratio',
'Long-term Debt / Capital',
'Debt/Equity Ratio',
'Gross Margin',
'Operating Margin',
'EBIT Margin',
'EBITDA Margin',
'Pre-Tax Profit Margin',
'Net Profit Margin',
'Asset Turnover',
'ROE - Return On Equity',
'Return On Tangible Equity',
'ROA - Return On Assets',
'ROI - Return On Investment',
'Operating Cash Flow Per Share',
'Free Cash Flow Per Share']

## EDA

\# Companies in dataset

`corp_credit_rating.Corporation.unique()`

array(\['American States Water Co.', 'Automatic Data Processing Inc.',
'Avnet Inc.', ..., 'Xerox Corp.', 'YPF Sociedad Anonima',
'iHeartCommunications Inc.'], dtype=object)

\# Sector in dataset

`corp_credit_rating.Sector.unique()`

Output
array(\['Utils', 'BusEq', 'Shops', 'Manuf', 'NoDur', 'Other', 'Chems',
'Telcm', 'Hlth', 'Money', 'Durbl', 'Enrgy'], dtype=object)

\# Types of bonds in dataset

`corp_credit_rating.Rating.unique()`

array(\['A-', 'AAA', 'BBB-', 'AA-', 'A', 'BBB+', 'BBB', 'BB', 'B', 'BB+',
'B+', 'BB-', 'B-', 'A+', 'CCC', 'AA', 'CCC+', 'CC', 'C', 'CCC-',
'AA+', 'D', 'CC+'], dtype=object)

\# Check for missing values

`corp_credit_rating[pd.isnull(corp_credit_rating)].count()`

Rating Agency                    0
Corporation                      0
Rating                           0
Rating Date                      0
CIK                              0
Binary Rating                    0
SIC Code                         0
Sector                           0
Ticker                           0
Current Ratio                    0
Long-term Debt / Capital         0
Debt/Equity Ratio                0
Gross Margin                     0
Operating Margin                 0
EBIT Margin                      0
EBITDA Margin                    0
Pre-Tax Profit Margin            0
Net Profit Margin                0
Asset Turnover                   0
ROE - Return On Equity           0
Return On Tangible Equity        0
ROA - Return On Assets           0
ROI - Return On Investment       0
Operating Cash Flow Per Share    0
Free Cash Flow Per Share         0
Date                             0
dtype: int64

![](/images/ED.png)

\- The average long term debt/capital is 0.45

\-Debt/Equity ratio is 0.18 on average, suggesting that there is low risk when looking at all sectors.

\-The operating margins is, on average, 11.5%.

\-The ROA and ROI are lower than 10%, however the return on tangeble equti is more than double of ROI.

\-The companies in the dataset have CF to pay debt and/or dividents, suggested by positive FCF per share.

![](/images/1.png)

Good corporate credit ratings were the leading rating in all years analysed.

Till 2015 there was a significant growth of good ratings. In fact, this reflects the better economic outlook after the 2008 crises.

![](/images/2.png)

Egan-Jones were the leading agency to give good ratings with more than 2000 ratings.

![](/images/1.1.1.png)

The utilities sector was the leading sector for good corporate credit ratings with more than 700.

![](/images/1.1.png)

The debt-to-capital ratio is a measurement of a company's financial leverage, in this case, relative to long term-debt. Therefore, the "Durbl" category is the one with the highest ratio suggesting higher risk and possible lower credit rating. In fact, the last bar plot suggest that, in fact, this category is one of the lowest in terms of quality ratings. Moreover, the same case can be seen in the "money" and "telecom" category.

Generally, a good debt-to-equity ratio is considered lower than 1.0, a ratio of 2.0 or higher is usually considered risky. If a debt-to-equity ratio is negative, it means that the company has more liabilities than assets, thus extremely risky. Therefore, we can see that these sectors present all negative ratios, adding to the riskiness of the investment.

FCF per share signals the ability to pay debt. Therefore, the "Durbl" and money sectors presents good ability to pay debt. However, the energy, telecom and others suggest negative ratios, meaning they, in average, are not able to generate sufficient cash to support the business, thus suggesting higher risk.

![](/images/1.2.png)

This bar plot suggest that good credit ratings are associeted with debt/capital ratios below 1, suggesting that this may be a good predictor. However, there is bad credit ratings that also are associeted with lower ratios, such as the D rating.

In terms of debt/equity, good credit ratings are associeted with positive and lower than 1 values, although there are some exceptions. In the other hand, bad credit ratings have generally negative ratios.

The FCF per share also suggest the same ideas as the other ratios, where good ratings, in general, have positive values.

![](/images/corr-e23ac983.png)

There is a high correlation between the binary rating, margin rations and ROA, suggesting that good margins and good financial performance move in the same direction.

There is a high correlation between ROI and long term debt/capita, ROE and gross margin and  ROA and margin ratios. Therefore, having debt can be beneficial for better performance in terms of ROI and margins and performance move in the same direction, which is expected.

## Logistic Regression

A logistic regression is a Machine Learning classification algorithm that is used to predict the probability of a categorical dependent variable. In logistic regression, the dependent variable is a binary variable that contains data coded as 1 (yes) or 0 (no). The classification goal is to predict whether the rating agency will give a investable rating to corporation.

### SMOTE

One approach to addressing imbalanced datasets is to oversample the minority class. The simplest approach involves duplicating examples in the minority class, although these examples don’t add any new information to the model. Instead, new examples can be synthesized from the existing examples. This is a type of data augmentation for the minority class and is referred to as the **Synthetic Minority Oversampling Technique**, or **SMOTE.**

\# Check if data is balanced

balance = data\["Binary Rating"].value_counts()print(balance)

1    543
0    432
Name: Binary Rating, dtype: int64

There is an unbalance of the class zero.

`os_data_X,os_data_y=os.fit_resample(X_train, y_train)`

`os_data_X = pd.DataFrame(data=os_data_X,columns=columns )`

`os_data_y= pd.DataFrame(data=os_data_y,columns=['Binary Rating'])`

`# we can Check the numbers of our data`

`print("length of oversampled data is ",len(os_data_X))`

`print("Number of non investable bonds in oversampled data",len(os_data_y[os_data_y['Binary Rating']==0]))`

`print("Number of investable bonds",len(os_data_y[os_data_y['Binary Rating']==1]))`

`print("Proportion of non investable bonds data in oversampled data is ",len(os_data_y[os_data_y['Binary Rating']==0])/len(os_data_X))`

`print("Proportion of investable bonds data in oversampled data is ",len(os_data_y[os_data_y['Binary Rating']==1])/len(os_data_X))`

Output
length of oversampled data is  778
Number of non investable bonds in oversampled data 389
Number of investable bonds 389
Proportion of non investable bonds data in oversampled data is  0.5
Proportion of investable bonds data in oversampled data is  0.5

\# Check if data is balanced

`balance = os_data_y.value_counts()`

print(balance)

Binary Rating
0                389
1                389
dtype: int64

### &#xA;RFE

RFE is based on the idea to repeatedly construct a model and choose either the best or worst performing feature, setting the feature aside and then repeating the process with the rest of the features. This process is applied until all features in the dataset are exhausted.

\# Chossing variables

`logreg = LogisticRegression()`

`rfe = RFE(logreg)`

`rfe = rfe.fit(X, y.values.ravel())`

`print(rfe.support_)`

`print(rfe.ranking_)`

Output

\[ True  True False False  True  True False False  True  True False False
False  True False  True]
\[1 1 6 4 1 1 2 5 1 1 8 9 3 1 7 1]

Therefore, the variables suited are: 'Current Ratio', 'Long-term Debt / Capital','Operating Margin', 'EBIT Margin','Net Profit Margin', 'Asset Turnover','ROI - Return On Investment','Free Cash Flow Per Share'

### Regression

`X = os_data_X[['Current Ratio', 'Long-term Debt / Capital','Operating Margin', 'EBIT Margin','Net Profit Margin', 'Asset Turnover','ROI - Return On Investment','Free Cash Flow Per Share']]`

`y = os_data_y`

`X = sm.add_constant(X)`

`X_train, X_test, y_train, y_test = train_test_split(X, y)`

`logit_model = sm.Logit(y,X)`

`result = logit_model.fit()`

`print(result.summary2())`

![](/images/6.png)

`The model suggests that:`

`All variables selected are significant at 1%, and therefore can explain the binary rating.`

`Ebit margin, net profit margin, ROI and Free CF per share have all positive coeficients. Therefore, higher levels of all these ratios lead to higher probability of an investable rating, especially in the case of the ebit margin.`

` Current ratio, Long term Debt/Capital, Operating margin and asset turnover have all negative coeficients. Thus,  `higher levels of all these ratios lead to lower probability of an investable rating, especially in the case of operating margin.

`y_pred = logreg.predict(X_test)`

`print('Accuracy of logistic regression classifier on test set: {:.2f}'.format(logreg.score(X_test, y_test)))`

Accuracy of logistic regression classifier on test set: 0.79

### Confusion matrix

`from sklearn.metrics import confusion_matrix`

`confusion_matrix = confusion_matrix(y_test, y_pred)`

`print(confusion_matrix)`

\[\[ 85  32]
\[ 17 100]]

This result sugests that 85+100 are correct predictions and 17+32 are incorrect predictions.

print(classification_report(y_test, y_pred))

![](/images/7.png)

To quote from [Scikit Learn](http://scikit-learn.org/stable/index.html):

The precision is the ratio tp / (tp + fp) where tp is the number of true positives and fp the number of false positives. The precision is intuitively the ability of the classifier to not label a sample as positive if it is negative. Therefore, 80% of prediction were correct.

The recall is the ratio tp / (tp + fn) where tp is the number of true positives and fn the number of false negatives. The recall is intuitively the ability of the classifier to find all the positive samples. Thus, 79% of the investable rating were correctly identified.

The F-beta score can be interpreted as a weighted harmonic mean of the precision and recall, where an F-beta score reaches its best value at 1 and worst score at 0. Therefore, 79% of the positive prediction were correct.

The F-beta score weights the recall more than the precision by a factor of beta. beta = 1.0 means recall and precision are equally important.

The support is the number of occurrences of each class in y_test.

## ROC

![](/images/8.png)

The ROC curve is another common tool used for this type of models. The dotted line represents the ROC curve of a purely random classifier; a good classifier stays as far away from that line as possible (toward the top-left corner).

# Conclusion

Ebit margin, net profit margin, ROI, Free CF per share, current ratio, Long term Debt/Capital, Operating margin and asset turnover are all significant and determinants for the probability of having an investable rating.

Ebit margin, net profit margin, ROI and Free CF per share increses the probability of having an investable rating.

Current ratio, Long term Debt/Capital, Operating margin and asset turnover lower the probability of having an investable rating.

79% of the investable ratings were correct by the model.


Logistic regression analysis of corporate credit rating using Python

See python code: https://github.com/felipeOL10/VaR/blob/main/Value%20at%20Risk.ipynb


VaR using the variance-covariance method in python

See python code:https://github.com/felipeOL10/Stock-program/blob/main/Stock_app.py

# Goal

The goal of the project is to create a program that allows investors to create customizable portfolios based on the maximum sharpe ratio and getting the financial data.

Therefore, the program is divided in two sections:

1.  The first section allows users to input stock tickers in order to construct an optimized portfolio based on the maximum sharp ratio and the respective weights of each stock.

2.  The secound part allows for users to download in csv data the balance sheet statement, cash flow statement, earnings, recommendations, dividends, sustainability and next events for each of the selected stocks.



Stock program

This analysis is a short part of a future publication that I´m working on, namelly the 'Eco-Innovations and greener transtitions: what can be done by public policy?", which was presented in the International Conference of Applied Business and Management (ICABM2022) (certificate in photo of the project).

## Data

The empirical analysis relies on the Community Innovation Survey (CIS) 2018 from Portugal, encompassing the period of 2016-2018. The database includes 13701 firms operating in Portugal with heterogeneous structural characteristics and innovative profiles. This survey is the most comprehensive concerning innovation-related issues providing enlarged evidence for the relevant variables in use.

## Exploratory analysis

This secction of the papers allows for the understanding of firm´s profiles as well as the descriptive analysis of the data, using the crosstab and frequency analysis in IBM SPSS.

Approximately 85,2% of the environmental regulations had no effect on innovative activities. In fact, this effect was felt the most by low technological firms (57,8%) and smaller firms (70,1%) contrasting to higher technological firms and larger firms. Moreover, this effect was the leading effect on all size and technological regime firms. As a result, the implementation of environmental regulations as been largely ineffective in innovative activities across all firms. Secondly, the second most important effect was as a promoter in the case of 8,2% of the environmental regulations, affecting once again mostly low technological firms (8,4%) and smaller firms (7,1%). In effect, this reflects the frequency of this types of firm characteristics on the survey, namely 57,9% in the case of low technological firms and 69% of smaller firms. However, in the case of scale intensive firms the second most important effect was as a barrier to innovative activities, corresponding to 8,1% to all scale intensive firms.

More than half of the financial support was given to smaller, and supplier dominated firms. Moreover, larger, and science-based firms were the least supported by local, national, or European institutions, representing only around 7% of total financial support given. Interestingly, specialized firms were the second most supported by subsidies, surpassing the scale intensive firms. Additionally, 78,5% of all firms that received financial support reported that environmental regulations had no effect on innovative activities. Moreover, 73,9% of firms that reported environmental regulations as a promoter did not receive any financial support and 75,1% of firms that reported as a barrier to innovative activities did not receive any subsidies. Furthermore, 78,5% of all firms that received subsidies reported no effect of environmental regulations on innovative activities.

As expected, most of the firms that did not innovate reported no effect of environmental regulations on innovative activities. However, the most reported effect of innovative firms was “no effect” on innovative activities, namely 69,4% for technological innovators and 75,6% for general innovative firms. Secondly, the most reported effect was as a promoter in the case of innovative firms, were 14,8% of all general innovative firms reported this type of effect and 18,9% of technologically innovative firms. Interestingly, the second most reported effect for firms that did not perform technological innovations reported a “promoter” role of this type of regulations, contrasting to firms that did not innovate were the “barrier” effect was the second most reported. In the case of technological innovations, 71,3% of the firms that receive subsidies did not performed technological innovations. However, 58,1% of those that received subsidies do perform general innovations. Moreover, most of the firms that did not receive financial support are not innovators, namely 89,4% for technological innovations and 70,3% for general innovations.

Regarding marketing innovations, 17,6% of the firms that performed marketing innovation reported a promoter effect of the environmental regulations on innovative activities, indicating the possible existence of the “green washing effect”.

When it comes to management efficiency, namely the analysis of both high share of qualified employees and the organizational capabilities of firms, it is possible to see that most firms with lower organizational capabilities are also the firms with lower share of high qualified employees. On the other hand, firms with higher organizational capabilities are the ones with a higher share of qualified employees.

Lastly, through the analysis of the absorved capacity of a firm (high qualified employees and organizational capabilities) and having performed technological innovations, it is possible to conclude that of the firms that performed technological innovations, only 6.5% did it without any type of high qualified employees and organizational capabilities. On the other hand, 23,9% of those firms did not perform any technological innovations. Consequently, of the firms that performed technological innovations, 93,5% had at least some form of absorve capacity.

![](/images/firm%20dimension.png)

![](/images/knowladge%20intensity.png)

![](/images/env_reg.png)

![](/images/subs.png)

![](/images/enpud.png)

![](/images/abs%20cap.png)


Eco-Innovations and greener transtitions: what can be done by public policy?

S&P500 analysis using Python, MySQL and Tableau

See Tableau visualization:

<https://public.tableau.com/app/profile/felipe.lima6878/viz/Creditcardcustomeranalysis_16520659299920/Painel1>

See MySQL code:https://github.com/felipeOL10/Credit-card-customer-analysis/blob/main/Bank%20customer%20analysis.sql



Credit card customer analysis using MySQL and Tableau

## What´s portfolio optimization?

Optimization can be roughly defined as a quantitative approach for decision making, where we seek to determine a “best” decision from a “set” of possible decisions.

## Goal

With this project my main objective was to find the optimal portfolio with my favorite companies in the stock market, namely Tesla, Amazon, Apple, Google, Nvidia, Qualcomm and Microsoft.

## Tools

###### `import pandas as pd`

###### `import numpy as np`

###### `import matplotlib.pyplot as plt`

###### `import statsmodels.api as sm`

###### `import yfinance as yf`

## Data

###### `stocks = ["AAPL", "AMZ", "MSFT", "TSLA", "GOOGL", "QCOM", "NVDA"]`

###### `data = yf.download(stocks, start="2018-01-01")`

###### `data = data["Close"]`

###### `x = data.pct_change()`

Then I precided to create empty lists for weights, returns, risk and for the sharp ratio.

## Generate random portfolios

> ###### `count = 10000`
>
> ###### `for k in range(0, count):   `
>
> ###### `wts = np.random.uniform(size = len(x.columns))`
>
> ###### `wts = wts/np.sum(wts)   `
>
> ###### `p_weights.append(wts)`

Then I did the same process for the other parameters and preceded to find the max values for the sharp ratio and the weights to each stock in the portfolio, using  np.argmax.

## Optimal portfolio

1.4270321023413632
\[0.3264971  0.01088931 0.01399479 0.22366134 0.03132052 0.020364
0.37327295]

![](/images/transferir%20\(2\).png)

Using the scatter() function with the volatility and return lists as parameters:

> plt.scatter(p_risk, p_returns, c=p_sharpe, cmap="plasma")
> plt.colorbar(label="Sharpe Ratio")
> plt.scatter(p_risk\[max_ind], p_returns\[max_ind], color="r", marker="\*", s=100)
> plt.show()

![](/images/transferir%20\(1\)-6a885277.png)


Sunset fishing is a beautiful time of day to relax and get in touch with nature

Portfolio optimization using Python

This research was based on the assumptions of an equity research of BPI bank.

## Top-down analisis

![](/images/Picture1.png)

We can see that there is similar movements between both the setor and gdp growth. Howerver, the average setor growth was 3% in contrast to the 0,5% growth of the economy. Since the model uses 2017 has the last year of the model, we will assume the growth rate of the economy at that year (2%).

## Financial indicators

![](/images/Pi.png)

![](/images/Pictur-6248b6d0.png)

The company's total sales have been increasing since 2011 and this growth is expected to continue until at least 2017. EBITDA allows us to assess a company's overall financial performance and profitability. We can easily understand that the 2011 EBITDA increase, but in 2014 a decrease is expected (EBITDA variation of the year 2014/2013 of -3.60%), however it is expected a total recovery in 2015. Operating cash flow has been steadily increasing since 2011 with the exception of what was expected for 2014 (659 million euros), demonstrates the company's strong capacity to generate cash. The dividends per share, despite having decreased in 2012, is expected to remain stable at 50%. In fact, the ratio is considered reasonable and a factor for attracting dividend investors.

With regard to turnover, analysts estimate a sustained growth of 2%. These figures are expected, taking into account the history of sustained growth in the company's sales in recent years. As for EBITDA, it is expected a value of 777 million euros in 2015, that is, about 3.74% above the value of the previous year.

Jerónimo Martins has been creating a permanent improvement in efficiency, achieved through the reduction of operating costs. It is estimated that this efficiency will be reflected in EBITDA, which is expected to reach an approximate growth of 8.19% in 2017 compared to the previous year, reaching an amount of 872 million euros.

## Multiples analisis

![](/images/Captura%20de%20ecr%C3%A3%202022-04-25%20030522.png)

Regarding the PER, Jerónimo Martins presents the highest value due to a greater growth with BPI's estimates in relation to other companies. In fact, Jerónimo Martins is expected to grow at 8%, while DIA only at 4% and Sonae at 3%. In this sense, with the PER being the number of years needed to recover the investment, given current results, we found that the most attractive investment in this regard would be Sonae, given that it has a faster return on investment. In the case of the other two stocks, they present values ​​above the average PER of 12.13, being considered more expensive investments. Regarding the PBV, it allows us to understand the value of equity relative to the book value. In this sense, if we consider a benchmark below 1 as an attractive investment, we find that none of these companies meet this criteria. However, considering the average of the three companies (3.39) we found that Sonae is the most attractive investment. Finally, taking into account the P/S, it is observed that Jerónimo Martins is the most attractive investment since investors are paying the lowest value for each euro of sales.

## Sensitivity analises

![](/images/Captura%20de%20ecr%C3%A3%202022-04-25%20031532.png)

![](/images/Captura%20de%20ecr%C3%A3%202022-04-25%20031549.png)

## Free Cash Flow

![](/images/Picture10.png)

In this case, taking in to account that the price of the stock was 8,38 in 17 of outuber 2014, the decision is to buy the stock since it is expected a return of 17%.


Equity analysis- Jerónimo Martins

## Resumo

Este estudo examina a volatilidade de Bitcoin, Ethereum e Litecoin em três períodos, nomeadamente antes da crise de COVID-19 (5/9/2018-31/1/2020), durante a crise (1/2/2020- 28/6/2021) e todo o período. Neste sentido, recorreu-se a modelos multivariados de GARCH como DCC, CCC, VEC e DVECH-GARCH, escolhendo os modelos com maiores log likelihood e significativos de forma global. Neste sentido, conclui-se que modelos com a introdução dos retornos de outras moedas são os mais adequados, verificando-se um impacto positivo e significativo dos retornos entre as moedas, especialmente em períodos com maior correlação. Verificou-se, igualmente, persistência positiva ao nível da volatilidade do dia anterior bem como persistência positiva ao nível dos resíduos em todos os períodos analisados.

## Análise descritiva

A tabela 1 indica as estatísticas descritivas dos retornos (1ª diferença do logaritmo dos preços) das três criptomoedas. Podemos verificar que o valor médio dos retornos das moedas é positivo em todos os momentos analisados, sendo a Ethereum a exceção, no período antes da pandemia. Não obstante, Ethereum é a moeda com maiores retornos em todo o período, bem como durante a crise. Ao nível da volatilidade, Litecoin é a moeda que apresenta maior risco, sendo esta volatilidade maior durante a crise (medida pelo desvio padrão). Na verdade, é de notar que existem maiores volatilidades durante o período relativo à crise comparativamente aos outros períodos analisados, podendo existir indícios de contágio da crise neste mercado, especialmente entre as moedas analisadas. Foram também realizados testes à normalidade da distribuição dos resíduos, concluindo-se que, em todos os períodos, verifica-se a existência de uma distribuição não normal (figura1), dada pelos p-values do coeficiente de assimetria e de curtose, algo que também é reforçado pelo p-value do teste de Jarque-Bera. Neste sentido, relativamente à assimetria da distribuição dos resíduos, observa-se que é leptocúrtica, estando assim associados há existência de maior risco para investidores. Ao nível dos coeficientes de assimetria, verifica-se em todos os períodos a prevalência de valores negativos com a exceção de Litecoin no pré-covid, ou seja, a curva da distribuição de probabilidade associada a cada moeda tem uma cauda maior à esquerda em termos dos retornos, pelo que se constata que elevados retornos negativos são mais comuns que elevados retornos positivos. No âmbito dos coeficientes de curtose, estes indicam que os investidores irão ter retornos mais extremos, tanto positivo como negativos, estando assim associado a maior risco. Desse modo, constata-se que Ethereum apresenta retornos mais extremos em todo o período e durante a crise, no entanto, Bitcoin apresenta os maiores valores antes da crise.

Além disso, o teste de ARCH-LM é conduzido de modo a investigar a existência de efeitos ARCH nos retornos dos preços. Observa-se que estes apenas existem antes da crise, tendo em conta os desfasamentos analisados com uma significância de 5%, com a exceção de Litecoin com cinco desfasamentos. Apesar disso, verifica-se que só desfasamentos de quatro a cinco dias são significativos para todo o período e apenas a cinco dias durante a crise, com um nível de significância de 10%.

![](/images/descr.png)

![](/images/desc.png)

Através da análise da tabela 2, pode-se inferir a existência de correlações fortes e significativas a pelo menos 5% entre todas as moedas, destacando-se o par de moedas Ethereum-Litecoin como aquele ao qual está associada a maior correlação verificada em todos os períodos. Consequentemente, existe uma relação de complementaridade entres todas as moedas, isto é, quando os retornos de uma moeda aumentam, os retornos das restantes tendem também a subir, movimentando-se no mesmo sentido. Depreende-se, de igual modo, a persistência destas relações em diferentes contextos económicos e financeiros, tendo esta dependência aumentado com a crise, com a exceção da correlação entre os retornos de Bitcoin e Ethereum. Na verdade, a forte correlação é expectável visto que estão todas expostas às mesmas inovações de mercado e choques macroeconómicos.

![](/images/corr.png)

Seguidamente, procedeu-se á elaboração dos testes de estacionaridade, designadamente o teste de Dickey-Fuller Aumentado (ADF) e de Phillips-Perron (PP). Sendo que as estatísticas de teste assumem valores negativos, sendo menores que o valor crítico, e o correspondente pvalue é significativo (inferior a qualquer nível usual de significância), tais factos denotam a rejeição da hipótese nula, segundo a qual as séries de retornos apresentam uma raiz unitária. As estatísticas de teste apresentam valores negativos altos, possuindo significância a todos os níveis, o que indica a rejeição da hipótese nula. Desta forma, os retornos das cotações de fecho são estacionários, isto é, as séries são convergentes e movimentam-se em torno da sua média amostral, pelo que não apresentam qualquer tendência ou sazonalidade.

### ![](/images/esta.png)&#xA;&#xA;

Denota-se um comportamento bastante semelhante entre as três moedas ao nível dos preços e retornos respetivos. Antes da crise pandémica verificamos que existe uma elevada flutuação dos preços, sendo esta mais extrema em Litecoin, verificada pela volatilidade na tabela 1. No entanto, durante a crise verifica-se um movimento muito semelhante entre as moedas com o um aumento do preço e com a elevada volatilidade no fim de 2020 a 2021, nomeadamente com uma queda drástica em relação aos valores máximos verificados nesse período. Neste sentido, podemos concluir a possível existência de correlação entre estas, sendo confirmado pelas correlações efetuadas na tabela 2. As quedas abruptas verificadas durante a crise vão ao encontro dos maiores valores verificados nos coeficientes de curtose, ou seja, a existência de retornos mais extremos relativamente aos outros períodos, sendo estes, pela observação gráfica,
mais negativos.

![](/images/pre%C3%A7os.png)


## Modelos de volatilidade

A estimação dos modelos multivariados de GARCH foi elaborada ao nível de todas as especificações, designadamente, DCC, CCC, VCC e VECH-GARCH. Neste sentido, verificou-se que em maior parte dos casos os parâmetros, bem como os logs likelihood e pvalues eram todos iguais, optando-se pela descrição nas tabela de MGARCH. Tendo em conta a tabela 4, observa-se a insignificância dos retornos anteriores nos retornos atuais de Bitcoin, ocorrendo o mesmo com a variância do dia anterior na variância do dia seguinte, com a exceção do período durante a crise. Já ao nível dos resíduos passados, nomeadamente do período anterior, verifica-se a significância destes na variância condicional, indicando a persistência do que não é explicado pelo modelo.

![](/images/modelo%201.png)

![](/images/modelo%202.png)

Podemos verificar com as tabelas 4 e 5 que o melhor modelo para explicar os retornos de Bitcoin, considerando os valores de Log likelihood e a significância global dos modelos, é o modelo com os retornos das restantes moedas, o que era de esperar pela elevada complementaridade verificada em todos os períodos na tabela 2. Neste sentido, os retornos de Ethereum são significativos em todos os períodos, destacando-se no pré-crise, onde uma variação de 1% nestes induz a uma variação no mesmo sentido de 0,42% nos retornos atuais de Bitcoin. Já os retornos de Litecoin destacam-se no período durante a crise com um impacto de 0,277%. Estes resultados vão de encontro á maior correlação analisada entre os pares de moedas (Tabela 2). Relativamente aos retornos anteriores, verifica-se uma insignificância em todos dos períodos, ou seja, o retorno anterior não tem impacto nos retornos atuais de Bitcoin. Tendo em conta aos parâmetros de ARCH (1), observa-se a existência de impactos ao nível da variância histórica na presente, com destaque para o período pré-pandemia. Neste seguimento, uma variação de 1% da variância do dia anterior induz uma variação de 0,64% na variância atual dos retornos de Bitcoin, indicando persistência em relação à variância anterior na atual. Já em relação aos resíduos do modelo, verifica-se que são significativos em todos os períodos com destaque para durante a crise onde a variação de 1% dos resíduos do dia anterior induzem uma variação no mesmo sentido de 0,78% na variância condicional, sendo persistente o que não se encontra no modelo apresentado.
Com a tabela 6 depreende-se que os retornos passados de Ethereum são apenas significativos durante a crise e em todo o período, sendo a um dia de desfasamento significativo antes da crise e em todo o período. Por sua vez, a variância anterior é apenas significativa para explicar os retornos de Ethereum durante a crise e em todo o período. Já os resíduos são significativos em todos os períodos.

![](/images/modelo%203.png)

![](/images/modelo%204.png)

Comparativamente ao modelo apresentado na tabela 7, verifica-se um melhor ajustamento neste último com um log likelihood superior, bem como a significância global a 1% em todos os períodos. Neste sentido, verifica-se que os retornos de Bitcoin e Litecoin são significativos em todos os períodos a 1% com impactos positivos, tendo Bitcoin um maior impacto antes da crise e no período todo. Não obstante, Litecoin apresenta um impacto superior durante a crise, indo ao encontro da maior correlação verificada em todos os períodos (Tabela 2). Relativamente à variância do dia anterior, verifica-se que esta é significativa, com exceção no período durante a crise onde não foi persistente nos retornos atuais de Ethereum. Ao nível dos resíduos do dia anterior, estes verificam-se significativos com destaque para o período durante a crise, onde induzem uma variação de 0,87% nos resíduos atuais, aquando uma variação de 1% nos resíduos passados.

A análise da tabela 8 indica que retornos do período anterior de Litecoin não têm significância em todos os períodos analisados, no entanto, ao nível de um dia, observa-se que estes têm significância antes da crise e em todo o período a 10%. Ao nível da volatilidade do dia anterior, verifica-se que apenas têm significância durante a crise e em todo o período a 5%, tendo um impacto positivo. Em relação aos resíduos passados, apresentam significância em todos os períodos, com enfâse antes da crise. Tal como nos outros modelos, verifica-se que com a introdução dos retornos das outras moedas, os modelos tornam-se significativos a 1% em todos os períodos, verificando-se aumentos do log likelihood. Tendo em conta os parâmetros do modelo podemos observar que os retornos de Bitcoin e Ethereum são significativos e têm um impacto positivo nos retornos atuais de Litecoin. Neste sentido, os retornos de Bitcoin destacam-se durante a crise onde se verificou um aumento de complementaridade entre os retornos (Tabela 2) com um impacto de 0,49% quando Bitcoin varia 1%. No caso de Ethereum, observa-se que o maior impacto antes da crise. Verifica-se igualmente que a variância do dia anterior tem um impacto positivo e significativo na variância atual, onde uma variação de 1% na variância do dia anterior induz
uma variação no mesmo sentido de 0,25% no caso da crise. Já os resíduos apresentam significância a todos os períodos com impactos positivos, com destaque para antes da crise.

![](/images/modelo%205.png)

![](/images/modelo%206.png)

O coeficiente de persistência da volatilidade é dado pela soma dos parâmetros ARCH e GARCH e define de que forma os riscos persistem na série de retornos. Este coeficiente procede á mensuração da duração dos choques de volatilidade, sendo que, quanto maior (mais próximo de 1) for o seu valor, mais lento será o processo de reversão á media para a variância. Valores mais próximos de 0 indicam que os choques na volatilidade se dissipam mais rapidamente. Observa-se que o somatório dos coeficientes relativos aos modelos GARCH dos retornos das três criptomoedas situam-se sempre mais próximos de 1 do que 0, o que constitui um indicador de que os choques na volatilidade revelam persistência. O maior valor da soma a diz respeito aos retornos da Bitcoin, pelo que se conclui que os choques sobre a volatilidade dos retornos dessa moeda são os que mais lentamente se dissipam. O menor coeficiente de persistência que se obteve foi o da Ethereum, indicando que um choque inicial sobre a volatilidade provoca efeitos mais rápidos, isto é, o período de tempo necessário para o choque ser eliminado é menor. A soma dos coeficientes de Bitcoin é superior a 1, ao passo que a soma dos coeficientes de Ethereum e Litecoin é inferior a 1. Estes resultados evidenciam que as informações presentes sobre a volatilidade serão importantes para a previsão da variância condicional dos retornos das moedas por um horizonte indefinido de tempo, indicando um elevado grau de risco associado ás séries de retornos.

## Correlações condicionais

Verifica-se que os resíduos condicionais das moedas apresentam movimentos muito similares, com 5 períodos de volatilidade definidos, nomeadamente, em meados de 2018, fim e início de 2019, fim de 2019, meados de 2020 e início de 2021. Na verdade, devido ás fortes correlações positivas e significativas entre os retornos das três moedas (Tabela 2), era de esperar estes movimentos semelhantes. Esta volatilidade dos resíduos está também ao encontro da distribuição dos retornos, uma vez que seguem uma distribuição leptocúrtica, associado ao maior risco e volatilidade.
As correlações entre os três diferentes pares de criptomoedas foram estimadas a partir do método DCC. Os gráficos com as correlações condicionais entre os três diferentes pares de criptomoedas encontram-se expostos nas figuras . A representação gráfica confirma a existência de correlações condicionais dinâmicas entre os pares, com as correlações a tomarem tanto valores positivos como negativos, apesar de não se observarem correlações negativas no par Ethreeum-Litecoin.

![](/images/fig%203.png)

![](/images/fig%204.png)

Em relação á correlação entre Bitcoin e Litecoin, verifica-se que assume tanto valores negativos como positivos, existindo duas picos de queda notáveis. No ínicio de 2019, os valores caíram quase até -0,4, tendo subido até 0,4 durante esse mesmo ano. No final de 2019 existe outra queda pronunciada, regressando a valores negativos. Com o íncio de 2020 e o despoltar da pandemia, observa-se um aumento exponencial das correlações, sendo depois corrigido. No que diz respeito á correlação entre Bitcoin e Ethereum, observa-se que as correlações são predominantemente negativas e situadas no intervalo entre -0.4 e 0. Existe uma grande queda nos valores entre o meio de 2018 e o ínicio de 2019. Tal como verificado no par BTC-LTC, os valores vão subindo com o decorrer do ano de 2019, subida essa que termina assim que atinge valores positivos. Constata-se um grande pico de subida coincidente com os meses de ínicio da pandemia, sendo prontamente corrigido. Mais recentemente, no final de 2020 e observa-se outro pico repentino.

Relativamente á correlação no par LTC-ETH, predominam os valores positivos, denotando-se que estes se encontram bastante concentrados no intervalo entre 0.4 e 0.6. Perto do fim de 2018 existe um grande pico de descida, com a correlação a atingir valores negativos, revertendo rapidamente para os valores anteriormente verificados. De notar, também, a tendência de descida dos valores sensivelmente entre os primeiros meses de 2020 e o meio do ano. Observa-se um crescimento dos valores no final do ano, quase atingindo valores máximos. As correlações entre Litecoin e Ethereum apresentam evidência de estacionaridade, ao contrário do que acontece com os pares anteriormente analisados. Verifica-se que a correlação condicional entre os retornos de Litecoin e os retornos de Ethereum é aquela que assume valores mais elevados, ao passo que a correlação condicional entre o par Bitcoin-Ethereum é a que assume valores inferiores.

## Conclusão

Em termos de análise descritiva, observou-se que Ethereum apresenta maiores retornos médios, bem como retornos mais extremos durante a crise e Bitcoin antes da crise. Em termos de volatilidade, Litecoin destaca-se, especialmente durante a crise. Foi também observado a existência de uma distribuição leptocúrtica, ou seja, de maior risco. Conclui-se, também, a existência de maior volatilidade durante o período de crise indo ao encontro de Salisu & Ogbonna (2021). Verifica-se que os modelos mais adequados para explicar os retornos são modelos que têm em conta os retornos das outras moedas, existindo aumentos do log likelihood, bem como uma significância global a 1% ao nível de todos os períodos analisados. Os retornos das moedas têm impactos positivos nos retornos das restantes, em todos os modelos analisados, sendo significativos e apresentando valores maiores em períodos onde a correlação é também maior (Tabela 2). Constata-se, de igual modo, que os resíduos do período anterior são significativos em todos os modelos realizados, indicando a falta de variáveis explicativas, uma vez que existe persistência ao nível do que não está presente nos modelos apresentados.
Consequentemente, verifica-se a existência de interdependência entre as moedas, indo ao encontro de Katsiampa (2019). A variância da própria moeda no período anterior tem, em todos os modelos entre moedas, um impacto positivo, em que, no caso de Bitcoin, é maior antes da crise, tal como Ethereum e Litecoin durante a crise. Verifica-se assim a existência de persistência ao nível do impacto da variância do dia anterior na variância atual, especialmente nos períodos referidos para cada moeda. Esta persistência é também verificada ao nível dos resíduos. De modo a entender qual a direção dos choques em todos os períodos, procedeu-se á estimação da causalidade de granger. Verificou-se que nada se pode concluir neste âmbito devido à insignificância dos paramentos.


Welcome to my portfolio website!

Recent projects

Analysis of Russell 2000 options data using web scrapping and the Black and Scholes model

Portfolio optimization using the mean variance framework of Markowitz, efficient CVaR and efficient CDaR optimizations

Hedging capabilities of cryptocurrencies against investment philosophies

Historical VaR using C++

VaR, CVaR and VaR backtesting

Logistic regression analysis of corporate credit rating using Python

VaR using the variance-covariance method in python

Stock program

Eco-Innovations and greener transtitions: what can be done by public policy?

S&P500 analysis using Python, MySQL and Tableau

Credit card customer analysis using MySQL and Tableau

Portfolio optimization using Python

Equity analysis- Jerónimo Martins

Análise da volatilidade das criptomoedas: evidência do Covid-19

Home

Welcome to my portfolio website!

Recent projects