Rolling returns

References

Is there a quantitative finance metric that aggregates the performance of an asset considering different starting points in a rolling window fashion?

Rolling Window Performance Metrics in Quantitative Finance Yes, there are several quantitative finance metrics that aggregate the performance of an asset over different starting points in a rolling window fashion. These metrics are particularly useful for understanding how an asset’s performance varies over time and in response to different market conditions.

Common Rolling Window Metrics: Rolling Returns:

Calculation: Calculate returns over a fixed window size, shifting the window by a specific interval. Purpose: Provides a view of how returns have fluctuated over time. Rolling Sharpe Ratio:

Calculation: Calculates the risk-adjusted return (Sharpe ratio) over a rolling window. Purpose: Measures the excess return per unit of risk over a specific period. Rolling Sortino Ratio:

Calculation: Similar to the Sharpe ratio but focuses on downside risk (standard deviation of negative returns). Purpose: Provides a more accurate assessment of risk-adjusted returns, especially for assets with skewed return distributions. Rolling Maximum Drawdown:

Calculation: Measures the largest peak-to-trough decline in value over a rolling window. Purpose: Identifies periods of significant losses. Rolling Calmar Ratio:

Calculation: The ratio of annualized return to maximum drawdown. Purpose: Measures the risk-adjusted return, focusing on downside risk. Key Considerations: Window Size: The choice of window size depends on the desired frequency of analysis. A shorter window can capture recent trends, while a longer window provides a more comprehensive view. Overlap: Rolling windows can overlap or be non-overlapping. Overlapping windows provide more data points but can introduce autocorrelation. Data Frequency: The frequency of data points (daily, weekly, monthly, etc.) affects the granularity of the analysis. Example: To assess the performance of a stock over the past year, you could calculate rolling returns, Sharpe ratios, and maximum drawdowns using a monthly window. This would provide insights into how the stock’s performance has varied over different months within the year.

By using these rolling window metrics, analysts can identify patterns, trends, and potential risks associated with an asset’s performance.

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Is there a way to aggregate different window sizes?

Aggregating Rolling Window Metrics Across Different Sizes Yes, there are several ways to aggregate rolling window metrics across different sizes. This can provide a more comprehensive view of an asset’s performance by considering both short-term and long-term trends.

Common Aggregation Methods: Weighted Average:

Calculation: Assign weights to different window sizes based on their importance or relevance. Purpose: Combines the results from multiple windows into a single metric, giving more weight to windows that are deemed more significant. Median or Mode:

Calculation: Find the median or mode of the metric across different window sizes. Purpose: Provides a central tendency measure that can be less sensitive to outliers. Range or Variance:

Calculation: Calculate the range or variance of the metric across different window sizes. Purpose: Measures the dispersion or variability of performance across different time horizons. Trend Analysis:

Calculation: Examine how the metric changes as the window size increases. Purpose: Identifies patterns or trends in performance over time. Example: Suppose you want to assess the risk-adjusted performance of a stock using the Sharpe ratio. You could calculate rolling Sharpe ratios for different window sizes (e.g., 3 months, 6 months, 1 year). Then, you could:

Weighted average: Assign higher weights to longer-term windows (e.g., 6 months and 1 year) to give more emphasis to sustained performance. Median: Find the median Sharpe ratio across all window sizes to get a central tendency measure. Range: Calculate the range of Sharpe ratios to see how much the risk-adjusted performance varies across different time horizons. By aggregating rolling window metrics across different sizes, you can gain a more nuanced understanding of an asset’s performance and identify potential patterns or anomalies that might not be apparent when considering a single window size.

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what about a time average?

Time-Averaged Rolling Window Metrics Time-averaged rolling window metrics provide a way to smooth out short-term fluctuations and focus on longer-term trends. This can be particularly useful for assets with volatile price movements.

Calculation: Calculate rolling window metrics for a range of window sizes. Weight each metric by the length of its corresponding window. Sum the weighted metrics to obtain the time-averaged value. Example: Suppose you want to calculate a time-averaged Sharpe ratio for a stock. You could calculate rolling Sharpe ratios for window sizes of 3 months, 6 months, and 1 year. Then, you would weight each Sharpe ratio by the number of months in its window (3, 6, and 12, respectively), sum the weighted values, and divide by the total number of months (21).

Benefits of Time-Averaged Metrics: Reduced noise: Smoothing out short-term fluctuations can make it easier to identify underlying trends. Focus on longer-term performance: Time-averaged metrics can provide a better assessment of an asset’s long-term risk-adjusted returns. Improved signal-to-noise ratio: By reducing noise, time-averaged metrics can improve the signal-to-noise ratio, making it easier to identify meaningful patterns. Time-averaged rolling window metrics can be a valuable tool for investors and analysts who want to focus on longer-term trends and reduce the impact of short-term volatility.

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what about convolutions or fourier transforms?

Using Convolutions and Fourier Transforms in Rolling Window Metrics Convolutions and Fourier transforms can be powerful tools for analyzing rolling window metrics, especially when dealing with time-series data.

Convolutions: Smoothing: Convolution can be used to smooth out noise and irregularities in the data, making it easier to identify underlying trends. Feature extraction: By convolving the data with different kernels, you can extract features such as trends, cycles, or spikes. Signal processing: Convolutions are commonly used in signal processing applications to filter, detect, and analyze signals. Fourier Transforms: Frequency analysis: Fourier transforms can decompose a time-series into its constituent frequencies, making it easier to identify periodic patterns or cyclical components. Feature extraction: By analyzing the frequency spectrum, you can extract features such as periodicity, amplitude, and phase. Signal processing: Fourier transforms are widely used in signal processing for applications such as filtering, compression, and modulation. Applications in Rolling Window Metrics: Noise reduction: Convolving rolling window metrics with a smoothing kernel can help reduce noise and improve signal quality. Trend detection: By using appropriate kernels, convolutions can be used to detect trends or patterns in the data. Cycle analysis: Fourier transforms can be used to identify cyclical components in rolling window metrics, such as seasonal patterns or business cycles. Feature engineering: Both convolutions and Fourier transforms can be used to extract features from rolling window metrics that can be used for further analysis or modeling. By combining rolling window metrics with techniques like convolutions and Fourier transforms, you can gain deeper insights into the underlying dynamics of your data and develop more sophisticated models.

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Can you give some examples of python code?

Python Code Examples for Rolling Window Metrics Here are some Python code examples using the Pandas library to calculate rolling window metrics:

Calculating Rolling Returns: Python import pandas as pd

Assuming ‘prices’ is a Pandas Series containing stock prices

rolling_returns = prices.pct_change()

Calculate rolling 30-day returns

rolling_30d_returns = rolling_returns.rolling(window=30).sum() Use code with caution.

Calculating Rolling Sharpe Ratio: Python import numpy as np

Assuming ‘returns’ is a Pandas Series containing daily returns

and ‘risk_free_rate’ is the daily risk-free rate

excess_returns = returns - risk_free_rate rolling_sharpe = excess_returns.rolling(window=252).mean() / excess_returns.rolling(window=252).std() Use code with caution.

Calculating Rolling Maximum Drawdown: Python def max_drawdown(series): cumulative_returns = (1 + series).cumprod() peak = cumulative_returns.cummax() drawdown = (cumulative_returns - peak) / peak return drawdown.min()

Calculate rolling 252-day maximum drawdown

rolling_max_drawdown = returns.rolling(window=252).apply(max_drawdown) Use code with caution.

Using the statsmodels Library for Rolling Statistics: Python import statsmodels.api as sm

Calculate rolling mean and standard deviation

rolling_mean = sm.stats.stattools.rolling_mean(returns, window=252) rolling_std = sm.stats.stattools.rolling_std(returns, window=252) Use code with caution.

Note: These examples provide a basic overview of how to calculate rolling window metrics in Python. For more complex calculations or custom metrics, you may need to write additional functions or use specialized libraries.

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what about the time weighted average?

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Time-Weighted Average Returns Time-weighted average returns (TWAR) are a method of measuring investment performance that accounts for the timing of cash inflows and outflows. This is particularly important for portfolios that experience significant changes in asset allocation or cash flows over time.

Calculation: Divide the investment period into subperiods: These subperiods should be defined based on significant changes in the portfolio, such as cash inflows, outflows, or changes in asset allocation. Calculate the return for each subperiod: Determine the return for each subperiod using the beginning and ending values of the portfolio. Geometrically link the subperiod returns: Multiply the returns for each subperiod and then take the nth root, where n is the number of subperiods. Formula: TWAR = (Product of (1 + Return for subperiod i))^(1/n) - 1 Example: Suppose you have a portfolio that starts with $100. After one month, it grows to $120. Then, you invest an additional $50. After another month, the portfolio value is $180.

Subperiod 1: Return = ($120 - $100) / $100 = 0.20 (20%) Subperiod 2: Return = ($180 - $150) / $150 = 0.20 (20%) TWAR = (1.20 * 1.20)^(1/2) - 1 ≈ 0.196 (19.6%)

Advantages of TWAR: Accounts for timing: TWAR considers the timing of cash flows, making it more accurate for portfolios with significant changes in asset allocation or cash flows. Fair comparison: TWAR allows for a fair comparison of investment performance across different time periods, even if the portfolio’s size changes significantly. Used in industry: TWAR is widely used in the investment industry, particularly for measuring the performance of mutual funds and hedge funds. By using TWAR, you can get a more accurate picture of an investment’s performance, especially when dealing with portfolios that experience significant changes in cash flows or asset allocation.

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