Quantitative trading - page 34

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High-Level Design of a Homegrown Trading Environment!

High-Level Design of a Homegrown Trading Environment!

Hello, everybody! My name is Denis, and you are watching "Close to AlgoTrading."

In our last video, we discussed the common structure of an algorithmic trading system. Today, I want to create a simple high-level design for a home-grown trading environment.

Let's start by identifying what we expect from our environment. We want the ability to add new strategies and data sources in the future. Scalability is important, as we currently focus on the stock market but may venture into trading cryptocurrencies in the future. Therefore, we need to support new markets. Additionally, we want the flexibility to easily change components within the system.

Now, let's take a look at the trading environment's overall structure.

The two main components are the data and the trading system. We also want to include a user interface to control the environment effectively. As an additional block, let's consider incorporating a simulation system.

The data component drives the entire system. Initially, we plan to use data from our broker and files on the hard drive. However, we may later decide to access different databases and news agencies. Since we want to support multiple data sources, we need to accommodate different drivers.

Drivers provide simple functionality such as reading and writing data, opening or closing connections, and more. However, we don't want our trading system to directly interact with the drivers' API. To address this, we introduce a layer of data handlers.

Data handlers work directly with the drivers and implement logic functionality such as data serialization, deserialization, and conversion to a special format. While we have a set of data handlers, we still need a unified interface for our application layer.

To provide these interfaces, we introduce a Data Manager and an Order Manager. The Data Manager offers a common interface for working with data sources, while the Order Manager provides a common interface for order execution. By utilizing both interfaces, the system can access various data sources and broker services.

Here comes an important aspect. We define components that provide a common interface to access data sources. This means that if our trading system wants to retrieve news, for example, it will use the same interface, regardless of whether the source is our broker or a database. This design ensures that the interface remains unchanged when we switch data providers. Therefore, we can add new data sources or change brokers without disrupting the application's functionality.

Now that we have designed the data handling portion, let's focus on the main user of our data—the trading system. If you've watched my previous video, you're familiar with the five blocks of the trading system: Alpha model, Risk model, Transaction cost model, Portfolio construction model, and Execution model. These blocks utilize the Data Manager and Order Manager to access data and broker services.

We have described the data and trading system components, but now we need to define who will control it all. We need a component responsible for the entire system—a foundation application. This application will handle the startup sequence, driver and component initialization, and implement the main state machine.

After defining the main application, we need to consider the user interface. It could be a PC interface or a web interface, but it's generally a good practice to separate the frontend from the backend. To achieve this, we create a communication layer between the GUI and our core application.

Additionally, we need to include a logger component. This component implements the logger interface and stores logs in a database or log file. We can also utilize a web-based monitoring system to track our environment, which can directly access the database server and read data from there. The frontend application can use the Data Manager to access data and the Order Manager to interact with broker services.

Lastly, let's not forget the possibility of manual order execution. We can integrate that functionality into the system as well.

Although we didn't delve into the simulation environment in this video, it could be implemented similarly to an additional broker service without any changes to the existing system.

Looking at the design of our trading environment, we can confidently say that it enables us to achieve our goals. We've divided our application into separate domains and connected them through interface layers. The Data and Order Managers provide common interfaces, simplifying access to resources. With this design, we can easily change data vendors or brokers and use different user interfaces. We can also make changes to specific parts of the system without having to update the entire application.

That's all for today. If I missed something or if you have any thoughts about this design, please leave your comments and don't forget to subscribe to this channel. See you in the next video!

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Trading environment IB API - QT C++ - Flask - RL Model. Complete working example.

Trading environment IB API - QT C++ - Flask - RL Model. Complete working example.

My name is Denis, and welcome to Close to Algo Trading. Today, I have a special video where I will demonstrate how all the components we have discussed in previous episodes can work together in real trading. You can find all the source code on GitHub and try it out for yourself.

For this demonstration, I will be using the Interactive Brokers API and my old project called IBTrader. I started working on IBTrader many years ago but never had the time to finish it. However, I'm considering renewing the project now. If you're interested in this special project, please let me know in the comments section of this video. So, let's dive into what we are going to do in this video. We'll start with a brief update and overview of our reinforcement learning environment and agent. Then, I'll provide a quick introduction to deploying the model with Flask.

Since we will be using a Qt application to connect to the broker, we need to use a REST API to interact with our model. Towards the end of the video, we'll see how all these parts work together seamlessly.

Before we begin, I recommend watching my earlier videos to better understand the concepts discussed here. If you've already seen those videos, you may recall how to create a RL agent using the TF-Agent library. I have made some changes to both the agent and the environment for this demonstration. The agent is still DQN, just like we used before, but now we will use QNetwork instead of QRnnNetwork. Similarly, I have simplified the environment. Instead of using multiple days of historical data, we now only have the price observation from the current day.

However, relying solely on the current observation is not ideal. To address this, we will transform our observation from one day to include three historical days. We can achieve this by using the history wrapper from the tf-agent environments package and setting the history_length parameter to 3. Now, if we check the environment's observations, we will find that we have three of our normal observations combined. Each observation contains the OHLC data (open, high, low, close) along with the volume.

To ensure a consistent range, I performed some transformations, and now our input data should represent log price percentage changes. Finally, we need to feed our model with three observations: the current day and the two historical days. We also need to remember the position states of the past days. Once the agent is trained, we will have a policy that can predict actions such as buy, sell, skip, or close position. The policy is implemented using a neural network, and we can use a direct API to work with it.

However, what if we don't want to tightly couple our application with the recommendation model? Or what if our model is implemented in Python and we want to use it in different applications? To address this, we can deploy our model as a small web service using Flask. Flask is a micro web framework written in Python. I chose Flask for this video because it is simple and doesn't require much setup time. It also allows us to use JSON to transfer data between our application and the model.

To debug and deploy our Flask application, we can use a Dockerfile. Microsoft has a good tutorial on how to create a Dockerfile, so please check the link in the description for more details. Our Flask application will have a route called "predict," which will handle POST requests containing input data in JSON format. The response will also be in JSON format. We will write code to transform the JSON data and pass it to the policy model for prediction.

Now, let's move on to the application that connects to the broker, receives data, and sends orders to the server. Although we could implement our strategy directly in the application, for this example, we will not do that. Instead, we will be using the QtIBTrade application, which is connected to the broker and communicates with our strategy and action selection model using the REST API. QtIBTrade is a Qt-based application that I created some time ago to explore algorithmic trading using C++. It is cross-platform and well-documented.

However, please note that QtIBTrade is still in development because I haven't had enough time to work on it. If anyone is interested in collaborating on this project, please let me know or check the code on GitHub (link provided in the description).

Now, let's see how all these components work together. First, we need to start our Flask application. I have started it in debug mode, so we need to remember the port number and update it in our Qt application. Next, we need to connect to the IB application or gateway. Before clicking on the "Connect" button, let's check if the API is activated and if we have the correct port configured.

Once everything is set up, we can start our test application, which uses our model. As you can see, every five seconds, we send a request to the server with different price data. Each time we receive price data from the broker, we trigger our model and receive the corresponding action back.

I hope you found this demonstration interesting. If you did, please subscribe to the channel for more videos. Thank you for watching, and see you in the next video!

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How Easily And Simply to Backtest a Stock Portfolio

How Easily And Simply to Backtest a Stock Portfolio

Hello, everyone! Welcome to "Close to AlgoTrading." I'm Denis, and today I want to share with you a very useful and simple framework that I've developed for backtesting different portfolios. It's based on an earlier idea that I discussed in one of my previous videos.

While there are more powerful implementations available, such as the one provided by quantconnect, I believe it's always good to have a local, simple solution that allows us to test ideas without relying on additional services.

The core of this framework is Backtrader, a Python framework for backtesting and trading. I've implemented a generic rebalancing strategy that we will be using, but the framework is designed in a way that allows testing different ideas without having to dive into the specifics of Backtrader.

Let's take a closer look at the structure and components of this framework. The rebalancing strategy is the key component, and it follows a specific structure. First, we need to select the set of assets we will be working with. For this, we have a model that implements a selection process based on our universe, called the selection model. Once we have the selected assets, they are passed to the alpha model. The alpha model is the core of our algorithm and generates the trading signals. These signals are then used to construct our portfolio.

The portfolio construction model takes the signals generated by the alpha model and generates an allocation of assets. This allocation is a dictionary where the key is the asset ticker and the value is the weight. After constructing the initial portfolio, we move to the rebalancing step. Here, we may remove old assets, add new ones, or adjust the amount of assets in the portfolio based on the defined weights.

Next, we may want to check some risk conditions using the risk model. This model receives the output from the rebalancing step, makes any necessary changes, and then passes it to the execution model. The execution model, by default, places orders to buy or sell the requested number of assets.

If you take a look at the code of the rebalancing strategy, you will see the implementation of each of these steps. Before the selection model, I call a function that returns a dataframe with only the close prices. This is because, for this strategy, we only need the close prices.

Now, let's briefly go over the implementation of the models within the framework. The selection model is straightforward. Its main purpose is to drop assets with missing or zero prices to ensure that all the data we use is valid. The alpha model is a simple generic momentum strategy. In this case, we buy assets with the highest momentum. For the allocation model, we implement a model that assigns equal weights to all assets.

Finally, the rebalancing model performs a simple rebalancing based on the assigned weights. Once we have implemented all the models, let's see how to use them.

First, we need some data. For this example, I'm going to use the current list of S&P 500 tickers as our universe. We will download the historical price data for all these stocks starting from January 2007. Next, we fill in our configuration. We define the structure of our data source, specifying that we will use only the open and close prices. The open price is necessary because we want to open and close positions based on the open price of the next day.

We also define our asset list, benchmark ticker, starting cash, option trade on open, and set a warm-up period for our algorithm to calculate momentum based on the last year. Then, we create our strategy configuration. In this configuration, we set the rebalanceDay option to 22, meaning we will rebalance our portfolio once a month. We also have a reserve cash parameter to avoid execution errors due to insufficient funds.

Next, we define all our models and their parameters. Before starting the backtest, we need to assign two datasets to our configuration: one containing the data for all our assets and another containing the benchmark data. Finally, we can call the backtest function and wait for the report to be generated. The report is generated using the quantstats package and provides a lot of useful information.

You can find the complete code on my GitHub page. The link is provided in the description.

That's all for today. Thank you for watching, and if you have any questions or comments, please leave them below. Don't forget to subscribe to my channel for more exciting content.

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How to beat the Market. Momentum and Portfolio Optimization

How to beat the Market. Momentum and Portfolio Optimization

Hello, everyone! My name is Denis, and you're watching "Close to AlgoTrading."

In this short video, we will explore two basic portfolio investment strategies: Momentum of Return and Momentum as the difference between price and trend. We will also see how portfolio optimization methods can help us improve our investment results.

Momentum is essentially the difference between two prices taken over a fixed interval. It represents the speed or rate of change. In the context of the stock market, momentum can be expressed as daily returns. The idea behind the momentum strategy is simple: If the price has been increasing in the past, it is likely to continue increasing in the future. It is convenient to express all markets in the same notation to facilitate comparisons.

For the stock market, momentum is often defined as the difference between today's price and a corresponding moving average value. When the momentum becomes larger, prices are moving further away from the moving average. Conversely, when the momentum becomes smaller, the speed of price change slows down, and prices are moving closer to or even in the negative direction relative to the moving average.

To implement these two types of momentum, I have already created an alpha model. It allows us to test them with different parameters.

Let's calculate the momentum over a one-year period and select the top 10 stocks with the highest momentum. We will rebalance our portfolio once a month, and all assets will receive equal allocations. Now, let's copy our configuration and set another type of momentum. We can add these configurations to our test and backtest them over the period from 2005 to 2015.

You can see that both strategies exhibit similar volatility and drawdown, and they outperform the investment in the index. However, it's important to note that we accept slightly more risk compared to holding the index. To reduce the risk to the level of the index, we can try using some portfolio optimization methods. Instead of the equal allocation model, we will use a model that implements several optimization methods from the pyportfolioopt package, which I have discussed in a previous video on portfolio optimization.

Here, I'll show you how to change our configuration to use the portfolio optimization allocation model. We will test two well-known methods, CLA and HRP, as well as two more specific methods: Efficient CVaR (Conditional Value at Risk) and Efficient CDaR (Conditional Drawdown at Risk). For more detailed information on these methods, please refer to the official documentation. Keep in mind that these optimization methods may take some time to run, as they are not very fast. Let's wait patiently until the backtesting is completed.

You can observe that all the methods improve our results. Our initial momentum of return strategy, which uses allocation based on the conditional value at risk optimizer, reduces drawdown by almost 12%. The same strategy, with the conditional drawdown at risk optimizer, improves our return by 3%. We have seen that simply changing the allocation model can significantly improve our results. There are many other ways to enhance this initial strategy. What's more important is that the simple idea of using momentum works and outperforms the market, and we can implement it with just a few simple steps.

That's all for today. See you in the next video!

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Beyond the Sharpe Ratio: Unveiling Effective Trading Strategy Assessment

Beyond the Sharpe Ratio: Unveiling Effective Trading Strategy Assessment

Hello, everyone! Welcome to "Close to AlgoTrading." Today, we are delving into the exciting world of trading strategy assessment. We will uncover why the Sharpe ratio, while popular, might not always be the best tool for the job.

In this video, we will cover four main areas. First, we will define the Sharpe ratio and explain its formula. Then, we'll discuss the limitations of the Sharpe ratio. Third, we will introduce the Sortino and Calmar ratios as alternative metrics. Finally, we will compare two hypothetical trading strategies using all these metrics. So, what is the Sharpe ratio? Named after Nobel laureate William F. Sharpe, it measures the performance of an investment compared to a risk-free asset after adjusting for its risk. In other words, the Sharpe ratio seeks to characterize how well the return of an asset compensates the investor for the risk taken.

The formula for the Sharpe ratio is as follows: Sharpe ratio = (Expected return - Risk-free rate) / Standard deviation.

Let's consider a strategy with an expected return of 8%, a risk-free rate of 2%, and a standard deviation of 15%. The Sharpe ratio would be (8 - 2) / 15, which is approximately 0.4. However, the Sharpe ratio has several limitations that we should consider. First, it assumes that returns follow a normal distribution, which is often not the case for many trading strategies that exhibit non-normal distributions with skewed or fat-tailed returns. This can lead to misleading results when using the Sharpe ratio.

Second, the Sharpe ratio favors strategies that generate small, frequent profits and assumes that these profits can be scaled up proportionally. This assumption may not hold true for all types of strategies, especially high-frequency trading strategies. As a result, high-frequency trading strategies might appear more successful when assessed using the Sharpe ratio. Third, the Sharpe ratio does not explicitly account for tail risk, which refers to the likelihood of extreme events or significant losses. Strategies with higher tail risk may not be adequately reflected in the Sharpe ratio, potentially underestimating the risk associated with such strategies.

Given these limitations, traders often turn to alternative metrics such as the Sortino ratio and Calmar ratio. These metrics provide additional insights and can help overcome some of the limitations of the Sharpe ratio. The Sortino ratio measures the risk-adjusted return of an investment by considering only downside volatility. It focuses on the deviation of returns below a specified threshold, typically the risk-free rate. This ratio provides a more specific assessment of risk and aligns with the common concern of investors towards downside risks.

On the other hand, the Calmar ratio evaluates the risk-adjusted return by comparing the average annual rate of return to the maximum drawdown. This ratio is particularly useful for strategies where the maximum drawdown is a critical factor. It highlights the return generated in relation to the risk of experiencing significant losses. By considering these alternative metrics, traders gain a more comprehensive perspective on the risk and return characteristics of their trading strategies. The Sortino ratio focuses on downside volatility, while the Calmar ratio measures return relative to the maximum drawdown.

Now, let's compare two hypothetical strategies using these metrics. We'll call them Strategy A and Strategy B. Strategy A has an annual return of 15%, a standard deviation of 10%, downside deviation of 7%, and a maximum drawdown of -20%. This gives us a Sharpe ratio of 1.3, a Sortino ratio of 1.86, and a Calmar ratio of 0.8. On the other hand, Strategy B has an annual return of 12%, a standard deviation of 8%, downside deviation of 5%, and a maximum drawdown of -15%. The Sharpe ratio for Strategy B is 1.25, the Sortino ratio is 2.0, and the Calmar ratio is 0.8.

As we can see, although Strategy A has a higher return, Strategy B outperforms it when we take into account the downside deviation and maximum drawdown, showing lower risk for the return. Hence, Strategy B gives you more return for each unit of risk, which is an important consideration in any investment decision. The choice between the Sharpe ratio and alternative metrics depends on the specific characteristics and goals of your trading strategy. Here are some recommendations to help you decide when to focus on the Sharpe ratio and when to consider other metrics:

When to pay more attention to the Sharpe ratio: The Sharpe ratio can be a valuable tool when assessing strategies that fall within certain categories. For example, if you have a well-diversified long-term investment portfolio focused on traditional assets such as stocks and bonds, the Sharpe ratio provides a suitable measure of risk-adjusted performance. It is particularly relevant for strategies with relatively stable returns, moderate risk, and a focus on overall risk-adjusted returns.

When to consider alternative metrics: On the other hand, alternative metrics such as the Sortino ratio and Calmar ratio come into play for strategies that may not conform to the assumptions of the Sharpe ratio. For instance, if you are engaged in higher-risk trading strategies such as options trading, leverage-based strategies, or strategies with concentrated positions, alternative metrics become more valuable. These strategies often exhibit non-normal return distributions, higher tail risk, and may require a focus on downside risk management. The Sortino ratio and Calmar ratio offer more specific insights into risk-adjusted performance, tail risk, and drawdowns, providing a better assessment of the strategy's viability in those contexts.

Remember, no single metric can fully capture the complexity and nuances of a trading strategy. It's essential to consider a combination of metrics to gain a comprehensive understanding of risk and return. By using multiple metrics such as the Sharpe ratio, Sortino ratio, and Calmar ratio, you can assess the strengths and weaknesses of your strategy from different perspectives, allowing for a more robust evaluation and informed decision-making.

That's all for today's tutorial. Thank you for joining us, and we hope you found this video helpful.

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Fast Monte Carlo Simulation For Estimation Maximum Expected Drawdown (run python code x2000 faster)

Fast Monte Carlo Simulation For Estimation Maximum Expected Drawdown (run python code x2000 faster)

Welcome, everyone! I am Denis, and you are watching CloseToalgotrading.

In this video, we will delve into the process of significantly improving the execution performance of a Python code, potentially by more than a thousand times. Our objective is not only to make it interesting but also highly useful. To achieve this, we will employ the example of Monte Carlo simulation to calculate an expected maximum drawdown.

To begin, let's address the fundamental question of what the maximum drawdown signifies and how it can be calculated. The maximum drawdown serves as an indicator of the downside risk over a specified period. Typically expressed in percentage terms, it is determined using the following formula.

Although the formula appears promising, we can achieve even better accuracy by applying it to real data. Let's take the example of SPY and set the time period to 100 days. The graph displays the closing prices of SPY over these 100 days. Two drawdowns are evident, with the second one appearing to be the maximum. To calculate the drawdown, we can perform a simple calculation. The highest price reached around 212.1, and the lowest was 204.4. Utilizing the previously mentioned formula, we can estimate the maximum drawdown to be approximately 3.6%.

However, this result lacks precision due to the assumptions made about the price level. To obtain a more accurate calculation, we will utilize a Python function specifically designed for this purpose. One possible implementation is as follows:

[Python code for calculating maximum drawdown]

This function calculates the maximum drawdown, resulting in a value of 3.5%. The execution time of this function is approximately 16.3 microseconds, which is respectable. But can we further enhance this timing without resorting to complex and intricate techniques? The answer is a resounding yes, and the simplest measure to improve execution is by employing Numba.

Numba is an open-source JIT (Just-In-Time) compiler that translates a subset of Python and NumPy code into highly efficient machine code. By converting our Python code to Numba, we can significantly enhance its performance. However, the initial implementation of our function will not work with Numba because it does not recognize the "accumulate maximum" function. Nevertheless, we can devise an alternative method to calculate the maximum drawdown without using this function. The modified drawdown calculation can be implemented as follows:

[Python code for modified drawdown calculation using Numba]

As you can observe, we have added a Numba decorator above the function definition. With this implementation, Numba raises no complaints, and the execution time is reduced to a mere 4.8 microseconds. This represents an improvement of over three times in terms of speed. It's quite simple, isn't it? Personally, I prefer this implementation because it describes the algorithm in a straightforward manner.

Thus, we can accurately calculate the maximum drawdown, which amounts to 3.5%. Let's refer to it as the "historical maximum drawdown." Based on this risk metric, we can assume that if the market conditions remain unchanged, our maximum risk will be only 3.5%. However, it raises the question of whether we can place our trust solely in a value derived from a single specific observation. Wouldn't it be better to have a few more observations from similar situations? Indeed, it would be beneficial, and this is where the Monte Carlo method comes into play.

Let's take a moment to clarify what the Monte Carlo method entails. It involves describing a process through a mathematical model using random variable generators. The model is then repeatedly computed, and probabilistic characteristics of the process are derived based on the acquired data.

In our case, the process entails a series of stock returns, which we simulate using the Geometric Brown Motion (GBM) model. The GBM model assumes that stock returns follow a continuous-time stochastic process, and it is widely used in financial modeling. By simulating stock returns using GBM, we can generate multiple scenarios and calculate the maximum drawdown for each scenario. This will give us a distribution of maximum drawdowns, providing a more robust estimate of the downside risk.

To implement the Monte Carlo simulation, we need to define the parameters for the GBM model. These parameters include the drift rate (μ), volatility (σ), and time horizon (T). We will also specify the number of simulation runs (N) and the number of time steps (M) within each run. With these parameters set, we can proceed with the simulation.

Here's an example implementation of the Monte Carlo simulation for calculating the expected maximum drawdown using Python:

import numpy as np

def monte_carlo_max_drawdown(S0, mu, sigma, T, N, M):
    dt = T / M

    S = np.zeros((N, M+1))
    S[:, 0] = S0

    for i in range(1, M+1):
        epsilon = np.random.normal(0, 1, N)
        S[:, i] = S[:, i-1] * np.exp((mu - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * epsilon)

    drawdowns = np.zeros(N)

    for i in range(N):
        peak = np.maximum.accumulate(S[i])
        drawdowns[i] = np.max((peak - S[i]) / peak)

    expected_max_drawdown = np.mean(drawdowns)

    return expected_max_drawdown

# Example usage
S0 = 100.0  # Initial stock price
mu = 0.05  # Drift rate
sigma = 0.2  # Volatility
T = 1.0  # Time horizon in years
N = 10000  # Number of simulation runs
M = 252  # Number of time steps (assuming 252 trading days in a year)

expected_max_drawdown = monte_carlo_max_drawdown(S0, mu, sigma, T, N, M)
print("Expected Maximum Drawdown:", expected_max_drawdown)

In this example, we use NumPy to efficiently handle array operations. We initialize an array S to store the simulated stock prices for each run and time step. We iterate over the time steps and simulate the stock price changes using the GBM formula. After simulating all the runs, we calculate the drawdown for each run by finding the peak value and subtracting it from the current price. Finally, we calculate the expected maximum drawdown by taking the mean of all the drawdowns.

By running this Monte Carlo simulation, we can estimate the expected maximum drawdown based on multiple simulated scenarios. This provides a more comprehensive understanding of the downside risk compared to a single observation from historical data.

In conclusion, we have explored the concept of maximum drawdown, its calculation using historical data, and the application of the Monte Carlo method to estimate the expected maximum drawdown. We have also discussed how to enhance the performance of the drawdown calculation using Numba. Incorporating these techniques can greatly improve the efficiency and accuracy of financial calculations, empowering traders and investors with valuable risk metrics for decision-making.

Thank you for watching CloseToalgotrading! I hope you found this video informative and helpful. If you have any questions or comments, please feel free to share them below. Stay tuned for more exciting topics in algorithmic trading and quantitative finance.

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Portfolio optimization for maximizing Sharpe Ratio using R Studio (CRAN)

Portfolio optimization for maximizing Sharpe Ratio using R Studio (CRAN)

Hello friends, welcome to Macro Sapiens. In today's tutorial, we will learn how to optimize a portfolio to maximize the Sharpe ratio. The Sharpe ratio measures the portfolio's returns per unit of risk taken. To accomplish this, we will be using the quantmod, PortfolioAnalytics, PerformanceAnalytics, and NSE2R libraries. The quantmod library will help us fetch data from Yahoo Finance, PortfolioAnalytics will provide the optimization function, PerformanceAnalytics will assist with visualization, and NSE2R is a dedicated library for the National Stock Exchange of India.

First, we define a variable called "stock_names" to store the names of the stocks. We use the quantmod library to fetch the data from Yahoo Finance for the 50 stock symbols of Nifty 50 index in India. We extract the symbol column and store it in a new variable called "symbols".

Next, we need to specify the start and end dates for the data we want to retrieve. We use the "since" variable to define the start date as January 1, 2018, and the end date as the current date. We focus on the adjusted closing prices as they provide a better representation of the stock's performance, and store them in a variable called "prices_data".

To calculate the portfolio return, we use the daily returns formula: (Closing price of day 2 - Closing price of day 1) / Closing price of day 1. We modify the function to handle any missing values (NAs) in the data.

Next, we define the column names of the portfolio return dataset as "funds". This will be used later in the optimization process.

Now, we can start the optimization process using the PortfolioAnalytics library. We create an initial portfolio and add constraints. One constraint ensures that the total weightage of the portfolio is equal to 1, and the other constraint specifies that we want a long-only portfolio, meaning no negative weightages.

We add the objective function, which is the mean return of the portfolio using the prices_data. We set the maximum number of iterations for the optimization process to 2000.

To calculate the Sharpe ratio, we need to define the risk-free rate of return. In this case, we consider it as 0%. We use the PerformanceAnalytics library's function to analyze the returns of the portfolio and calculate the Sharpe ratio.

We create a table to display the results, including the maximum Sharpe ratio return achieved through random optimization and the returns from the ROI optimizer. We plot the efficient frontier to visualize the optimized portfolios.

Additionally, we compare the portfolio returns with the returns from investing in the Nifty 50 index. We calculate the cumulative returns for each approach and plot them to analyze the performance.

Based on the results, we observe that the random portfolio optimization method provides a higher Sharpe ratio and cumulative returns compared to the ROI optimizer and the Nifty 50 index.

These optimized weightages can be used to create a basket of stocks for investment. By tracking the performance of the basket over a period of time, investors can make informed decisions about investing in the selected stocks.

We hope you found this tutorial helpful. If you have any questions, please feel free to ask. Don't forget to like, share, and subscribe to our channel. Thank you!

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Spot Rate vs. Forward Rates (Calculations for CFA® and FRM® Exams)

Spot Rate vs. Forward Rates (Calculations for CFA® and FRM® Exams)

Good day, everyone! I would like to introduce today's concept for our discussion. We will be exploring the calculation of forward rates from spot rates, but instead of using the formula, we will employ the timeline method. This approach will eliminate the need to memorize complex formulas and make the process more intuitive.

Before we delve into the details, let's briefly recap the definitions of forward and spot rates. The spot rate refers to any interest rate available in the market at present. It represents the rate at which one can invest for a specific period, such as two, three, four, or five years starting from today. The spot rate is an investable rate, allowing individuals to earn returns by investing in the market.

On the other hand, the forward rate is a theoretical rate often referred to as the implied forward rate. It represents the projected interest rate between two future time periods. For example, we might want to determine the rate between year three and year four if we are currently at time period zero. The forward rate is calculated based on current spot rates and serves as a forecast of the interest rate at a future point.

It's important to note that the forward rate is not an investable rate unless it is locked in using a derivative instrument such as a forward contract or future contract. Until then, it remains an implied rate, meaning it may or may not exist in reality when the specified time period arrives in the future.

To make the implied forward rate investable, one must enter into a forward contract. This ensures that the rate is fixed and can be utilized in financial transactions at the designated future time.

Now, let's explore the timeline method for calculating forward rates. We will first examine the formula, but remember that the goal is to move away from relying on formulas and embrace the timeline method. By comparing both approaches, you will realize that the timeline method yields the same results without the need for formula memorization.

The formula for calculating forward rates is as follows:

(1 + za)^a * (1 + ifr) = (1 + zb)^b

In this formula, "a" represents the shorter maturity, "b" denotes the longer maturity, "za" and "zb" refer to the respective spot rates for the shorter and longer maturities, and "ifr" represents the implied forward rate between time period "a" and "b."

Now, let's illustrate an example to solidify our understanding. We are given the following spot rates: the one-year spot rate is 5%, and the two-year spot rate is 6%. Our objective is to determine the one-year forward rate one year from today.

Using the formula, we can substitute the given spot rates into the equation:

(1 + 0.05)^1 * (1 + ifr) = (1 + 0.06)^2

Simplifying further, we get:

1.05 * (1 + ifr) = 1.1236

Now, let's explore the timeline method for the same calculation. Draw a timeline with time period zero, one, and two. Plot the spot rates accordingly. For the two-year spot rate, mark 6% from zero to two. For the one-year spot rate, mark 5% from zero to one. Our goal is to calculate the one-year forward rate one year from today, denoted as "f."

To determine the implied forward rate using the timeline method, we leverage the principle of no arbitrage. This principle asserts that regardless of the chosen route on the timeline, we should end up with the same future value. In this case, we can invest $1 for two years at 6% interest or invest $1 for one year at 5% interest, then reinvest that amount for another year at the forward rate "f."

To calculate the one-year forward rate, we start by investing $1 for one year at the one-year spot rate of 5%. This investment grows to $1.05 after one year.

Now, we take the $1.05 and reinvest it for another year at the forward rate "f." The future value of this investment should be the same as investing $1 for two years at the two-year spot rate of 6%.

Let's assume the forward rate "f" is x%. We can set up the equation as follows:

(1 + 0.05) * (1 + x%) = (1 + 0.06)^2

Simplifying further, we have:

1.05 * (1 + x%) = 1.1236

Dividing both sides by 1.05:

1 + x% = 1.1236 / 1.05

1 + x% = 1.07

x% = 1.07 - 1

x% = 0.07

So, the one-year forward rate one year from today, denoted as "f," is 7%.

By using the timeline method, we were able to calculate the forward rate without relying on the formula. This approach provides a visual representation of the timeline and allows for a more intuitive understanding of the implied forward rate.

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Sharpe Ratio, Treynor Ratio and Jensen's Alpha (Calculations for CFA® and FRM® Exams)

Sharpe Ratio, Treynor Ratio and Jensen's Alpha (Calculations for CFA® and FRM® Exams)

Ladies and gentlemen, I extend my warm greetings to all of you. Today, we will delve into an important topic, namely the various measures of portfolio performance. Specifically, we will explore the Sharpe ratio, the Treynor ratio, and Jensen's alpha. While there exist several other portfolio performance measures, these three are widely recognized as pivotal. Understanding the interrelationship among these measures and their practical significance is crucial, not only for your upcoming CFA or FRM examinations but also for their real-life application. These concepts permeate all three levels of the CFA curriculum, emphasizing their significance throughout your exam journey.

Let us commence with the Sharpe ratio, which, to this day, remains the most esteemed ratio in the field. Its appeal lies in its simplicity, despite some inherent limitations. Nevertheless, it continues to be the go-to ratio when comparing funds, as it is commonly reported by hedge funds and mutual funds. Similarly, both the Treynor ratio and Jensen's alpha are extensively used in the industry. Hence, it is essential to grasp these concepts, not solely for your exams but also for their practical relevance.

The Sharpe ratio, formulated as follows, warrants our attention:

Sharpe ratio = (Portfolio return - Risk-free rate) / Standard deviation of the portfolio

In the numerator, "Portfolio return (rp)" represents the excess return over the risk-free rate (rf). When investing in a portfolio, one expects returns higher than the risk-free rate, as taking on risk implies seeking greater rewards. Therefore, we focus on the excess return, which signifies the return obtained beyond the risk-free rate. In the denominator, we have the standard deviation, which serves as a measure of risk. Here, it is essential to note that the standard deviation accounts for both diversifiable and non-diversifiable risks. Diversifiable risks can be eliminated through diversification, while non-diversifiable risks persist. Consequently, the Sharpe ratio evaluates the excess return per unit of total risk, combining both systematic and non-systematic risks.

It is crucial to highlight that the Sharpe ratio's value is most relevant when compared to the Sharpe ratios of other portfolios. It finds significance when evaluating portfolios relative to one another. In fact, industry professionals often refer to Sharpe ratios as "sharps." For instance, portfolio managers might say, "I'm returning two sharps" or "three sharps," indicating their respective Sharpe ratios.

A higher Sharpe ratio is considered favorable. A higher ratio indicates a portfolio's ability to generate more return for the same level of risk, making it a preferable choice. Thus, when selecting a portfolio based on the Sharpe ratio, opt for the one with the highest ratio.

Now, let's turn our attention to the Treynor ratio, which bears a striking resemblance to the Sharpe ratio in its numerator but diverges in its denominator. The Treynor ratio measures excess return per unit of systematic risk, denoted by beta. Beta represents the non-diversifiable, systematic risk inherent in an investment. This narrower measure focuses solely on systematic risk, unlike the broader scope of the Sharpe ratio. Similarly, the Treynor ratio is more valuable when compared to the ratios of other comparable funds. Selecting a portfolio based on the Treynor ratio entails choosing the one with the highest ratio, as a higher value indicates a greater excess return per unit of systematic risk.

Before we explore Jensen's alpha, let's review the Capital Asset Pricing Model (CAPM). CAPM assists us in understanding Jensen's alpha, as it helps determine the expected return or required return for a portfolio. CAPM calculates the expected return by starting with the risk-free rate and adding beta times the market risk premium (the difference between the market return and the risk.

Jensen's alpha, also known as the Jensen performance index or simply alpha, is a measure of the excess return of a portfolio compared to its expected return based on the Capital Asset Pricing Model (CAPM). The CAPM relates the expected return of an asset or portfolio to its beta, which represents the systematic risk or sensitivity to market movements.

Jensen's alpha is calculated as follows:

Jensen's alpha = Portfolio return - [Risk-free rate + Beta × (Market return - Risk-free rate)]

In this formula, the portfolio return represents the actual return earned by the portfolio, the risk-free rate is the return on a risk-free investment such as a government bond, beta measures the portfolio's sensitivity to market movements, and the market return is the average return of the overall market.

Jensen's alpha indicates whether the portfolio outperformed or underperformed its expected return based on the CAPM. A positive alpha suggests that the portfolio has generated excess returns beyond what would be expected given its systematic risk, while a negative alpha indicates underperformance. Therefore, investors and portfolio managers typically seek positive alphas when evaluating investment performance.

It is important to note that Jensen's alpha considers only systematic risk and does not account for the total risk or the specific risks associated with the portfolio. As a result, it is recommended to use Jensen's alpha in conjunction with other performance measures, such as the Sharpe ratio and Treynor ratio, to gain a more comprehensive understanding of a portfolio's performance.

In summary, the Sharpe ratio, Treynor ratio, and Jensen's alpha are all valuable measures for evaluating portfolio performance. The Sharpe ratio assesses the excess return per unit of total risk, while the Treynor ratio focuses on excess return per unit of systematic risk. Jensen's alpha compares a portfolio's actual return to its expected return based on the CAPM, considering only systematic risk. These measures provide different perspectives on portfolio performance and can be used together to make informed investment decisions.

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Covariance and Correlation (Calculations for CFA® and FRM® Exams)

Covariance and Correlation (Calculations for CFA® and FRM® Exams)

Hello everyone, let's begin by discussing the concept of covariance and correlation. Today's topic can be confusing for many people because correlation is a term that is commonly heard, while covariance is often unfamiliar when it comes to calculations. Additionally, both covariance and correlation are intended to measure the same thing, which can be confusing. We will explore why we have two different measures for the same purpose and determine when to use covariance and when to use correlation. Furthermore, we will examine how to calculate both covariance and correlation.

Before diving into covariance, let's quickly review how to calculate variance because it forms the basis of our discussion. Once we understand how to calculate variance, we can proceed to covariance and explore the relationship between the two measures. This will help us gain insight into the origin of these measures and their relationship with correlation.

Now, let's consider an example to understand the calculation of variance. We have a data series representing portfolio returns for five years. The returns are given as percentages for each year. To calculate the variance, we first need to determine the mean or average of the data series. We sum up all the returns and divide the sum by the number of observations, which in this case is five years. This provides us with the mean of the data series.

Next, we calculate the deviation from the mean for each observation. We subtract the mean from each return value. This gives us the deviation from the mean for each observation. The squared deviations are then calculated by squaring each deviation. We sum up all the squared deviations and divide the result by the number of observations to obtain the variance. Finally, we take the square root of the variance to find the standard deviation, which is a related measure.

It's important to note that while we are calculating variance manually here, in real-world scenarios or exams like CFA or FRM, these calculations are typically done using built-in functions in calculators like the BA2 Plus or BA2 Plus Professional.

Moving on to covariance, it is a measure of the co-movement or relationship between two different data series. Unlike variance, which deals with a single data series, covariance allows us to examine how two data series move together. For example, we can use covariance to analyze the co-movement between an ETF and a benchmark index. Positive covariance indicates that the two variables move in the same direction, while negative covariance suggests opposite movements. A covariance of zero indicates no relationship between the variables.

To calculate covariance, we subtract the mean of the first data series from each observation of that series and multiply it by the deviation from the mean of the second data series. We repeat this process for all observations, multiply the deviations, sum up the results, and divide by the number of observations to obtain the covariance.

It's worth noting that covariance shares similarities with variance, but it involves two different data series instead of just one. In fact, variance can be considered a special case of covariance where the two variables are identical.

However, there is a limitation to using covariance alone. While covariance provides insight into the relationship between two variables, it does not provide a sense of the magnitude of the relationship. This poses a challenge when comparing relationships between different data series. This is where correlation comes into play.

Correlation is a standardized version of covariance. It is calculated by dividing the covariance by the product of the standard deviations of the two data series. This normalization process enables us to compare relationships on a standardized scale, ranging from -1 to +1. A correlation of +1 indicates a perfect positive relationship, -1 represents a perfect negative relationship, and 0 denotes no relationship.

Covariance and correlation are measures that help us understand the relationship between different data series. Covariance provides an indication of co-movement between variables, while correlation standardizes this measure and allows for easy.