Suppose That The Daily Simple Returns Of A Stock In One Week

A Suppose That The Daily Simple Returns Of A Stock In One Week Were

A Suppose That The Daily Simple Returns Of A Stock In One Week Were

A. Suppose that the daily simple returns of a stock in one week were -0.4%, 0.8%, 1.3%, -1.5%, and 0.9%. What are the corresponding daily log returns? What is the weekly simple return of the stock? B.

Please list out three types of Moments of a Random Variable and briefly explain why they are important for financial data. C. The data file assignment_1.txt contains stock returns for “ge†(general electric), “vw†(value-weighted market returns), “ew†(equal-weighted market returns), and “sp†(Standard Poor composite index). The time span of the data ranges from Jan 1940 to Sept 2011. Please do following work within R/RStudio environment. (a) Compute the sample mean, standard deviation, skewness, excess kurtosis, minimum, and maximum of each simple return series. (b) Transform the simple returns to log returns and redo part (a). (c) Test the null hypothesis that the mean of the log returns of “ge†stock is zero.

Use 5% significance level to draw your conclusion. date ge vw ew sp ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................071762

A Suppose That The Daily Simple Returns Of A Stock In One Week Were

The given assignment involves analyzing stock return data, converting simple returns to log returns, computing summary statistics, and testing hypotheses using data from a text file that spans from January 1940 to September 2011. The tasks include calculating statistical measures, transforming return types, and conducting statistical hypothesis testing within the R/RStudio environment.

Paper For Above instruction

Understanding financial returns and their statistical properties is fundamental for financial analysis, risk assessment, and asset management. This paper addresses the calculation of returns, their moments, and hypothesis testing based on historical stock data, illustrating key concepts in financial econometrics.

Introduction

Financial markets generate vast amounts of data, providing essential insights into asset performance and risk. Accurate measurement of returns, understanding their distributional properties, and rigorous statistical testing form the backbone of quantitative finance. This paper focuses on analyzing stock returns using different methods, transforming return data, and testing hypotheses within the context of historical data from 1940 to 2011.

Part A: From Simple Returns to Log Returns and Weekly Return Calculation

Suppose the daily simple returns of a stock over a week are -0.4%, 0.8%, 1.3%, -1.5%, and 0.9%. To analyze these returns, it is necessary to convert simple returns to logarithmic returns, which are additive over time and better suited for modeling continuous compounding. The formula for daily log returns is:

log return = ln(1 + simple return)

Applying this, the daily log returns are:

  • Day 1: ln(1 - 0.004) = ln(0.996) ≈ -0.004008
  • Day 2: ln(1 + 0.008) = ln(1.008) ≈ 0.007974
  • Day 3: ln(1 + 0.013) = ln(1.013) ≈ 0.012933
  • Day 4: ln(1 - 0.015) = ln(0.985) ≈ -0.015113
  • Day 5: ln(1 + 0.009) = ln(1.009) ≈ 0.008961

These values represent the daily log returns, which when summed over the week approximate the total growth rate.

The weekly simple return is calculated by multiplying the daily simple returns:

Weekly simple return = (1 - 0.004) × (1 + 0.008) × (1 + 0.013) × (1 - 0.015) × (1 + 0.009) - 1

Calculating stepwise:

  • Start: 1
  • After Day 1: 1 × 0.996 = 0.996
  • After Day 2: 0.996 × 1.008 ≈ 1.004
  • After Day 3: 1.004 × 1.013 ≈ 1.017
  • After Day 4: 1.017 × 0.985 ≈ 1.001
  • After Day 5: 1.001 × 1.009 ≈ 1.010

Subtracting 1 yields a weekly simple return of approximately 1.0%. This illustrates the cumulative effect of daily returns on weekly performance.

Part B: Moments of a Random Variable and their Importance in Financial Data

Moments of a random variable are statistical measures that describe various aspects of its distribution. The three primary moments are:

  1. First Moment - Mean: Represents the central tendency or average of the variable. In finance, the mean return is crucial for estimating expected asset performance and guiding investment decisions.
  2. Second Moment - Variance/Standard Deviation: Measures the dispersion or volatility around the mean. High variance indicates higher risk, which investors consider when constructing portfolios and managing risk.
  3. Third Moment - Skewness: Describes the asymmetry of the distribution. Positive skewness indicates a tail on the right, implying the possibility of extreme positive returns; negative skewness suggests potential for extreme losses. Understanding skewness helps in assessing tail risks and designing strategies that manage downside risk.

Higher-order moments, such as kurtosis (the fourth moment), provide additional insights into the tail behavior and the likelihood of extreme events, which are particularly important in financial risk management for identifying "black swan" events and tail risks.

Part C: Empirical Analysis of Stock Return Data

(a) Descriptive Statistics of Simple Returns

The dataset "assignment_1.txt" encompasses returns for General Electric (GE), value-weighted market (VW), equal-weighted market (EW), and S&P index (SP). Computing the sample mean, standard deviation, skewness, excess kurtosis, minimum, and maximum offers a comprehensive understanding of the return characteristics.

Using R, these statistics can be derived via functions such as mean(), sd(), skewness(), kurtosis(), min(), and max() for each series. Such statistics reveal the central tendency, volatility, asymmetry, tail heaviness, and spread of each return series.

(b) Transformation to Log Returns and Recalculation

The conversion from simple to log returns involves:

log return = ln(1 + simple return)

Applying this transformation, the same descriptive statistics are recalculated for log returns. Generally, log returns tend to be more normalized and better suited for modeling, especially over long periods where compounding effects are significant.

(c) Hypothesis Testing on the Mean of Log Returns of GE

The null hypothesis is that the mean log return of GE is zero, indicating no excess return over the period. The alternative hypothesis suggests a non-zero mean. Using an appropriate t-test with a 5% significance level involves computing the test statistic:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(n))

where the hypothesized mean = 0. The p-value derived from this test informs whether to reject the null hypothesis. If p

Conclusion

This analysis underscores the importance of converting simple returns to log returns for better statistical properties, and emphasizes the relevance of understanding distributional moments in financial data. Conducting statistical tests such as hypothesis testing provides critical insights into market efficiency and asset performance over extensive historical periods.

References

  • Campbell, J. Y., Lo, A. W., & MacKinlay, A. C. (1997). An Introduction to Time Series and Forecasting. Princeton University Press.
  • Fama, E. F. (1970). "Efficient Capital Markets: A Review of Theory and Empirical Work". Journal of Finance, 25(2), 383-417.
  • Cont, R. (2001). "Empirical properties of asset returns: stylized facts and statistical issues". Quantitative Finance, 1(2), 223-236.
  • Sharpe, W. F. (1964). "Capital asset prices: A theory of market equilibrium under conditions of risk". Journal of Finance, 19(3), 425-442.
  • Mandelbrot, B. (1963). "The variation of certain speculative prices". Journal of Business, 36(4), 394-419.
  • Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions. Wiley.
  • Tsay, R. S. (2010). Analysis of Financial Time Series. Wiley.
  • Engle, R. F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation". Econometrica, 50(4), 987-1007.
  • Harvey, A. C. (1990). The Econometric Analysis of Time Series. MIT Press.
  • Zivot, E., & Wang, J. (2006). Modeling Financial Time Series with S-Plus. Springer.

Related Assignments