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# Bell Curve – Definition

### Bell Curve Definition

A bell curve is referred to as a normal distribution because it is the most common type of distribution for a variable. The term ‘bell curve’ is derived from the graph depicting a normal distribution because it consists of a bell-shaped line. The highest point in the curve represents the most probable events in a series of data. All the other possible occurrences are distributed equally around the most probable event and end up forming a downward sloping line on either side of the peak.

### A Little More on What is the Bell Curve

A bell curve refers to the graphical representation of normal probability distribution. The underlying standard deviations of this distribution from the median or the highest point of the curve give it the shape of a curved bell. The standard deviation is a measurement that is utilized in the quantification of the variability of data dispersion in a given set of values, while the mean gives the average of the total data points in the data set.

The calculation of standard deviation is done after that of the mean, and it represents a percentage of the total data. For example, if a series of 100 test scores are collected and used in a normal probability distribution, 68% of the entire test scores are expected to fall below the mean or within one standard deviation.

When two standard deviations are moved away from the mean, 95% of the test scores are included. 99.7% of the test scores are represented when three standard deviations are moved away from the mean. Text scores which are extreme outliers are considered as long tail data points they lie outside the range of the three standard deviations. Such outliers include 100 or 0.

### Bell Curves in Finance

When analyzing the returns of the overall market or a security sensitivity, financial analysts and investors usually use a normal probability distribution. In finance, volatility is used to refer to standard deviations that show the returns of security. For example, blue chip stocks are the ones usually depicting a bell curve, and they have predictable low volatility.

By using the normal probability distribution of the past returns of stock, investors can make assumptions with regards to the expected future returns. In some cases though, stocks and other securities display distributions that are non-normal. Fatter tails than normal ones characterize these non-normal distributions. When the fatter tail is skewed negative, it signals to the investors that there is a higher probability of negative returns. The converse is also true.

### Academic Research on Bell Curve Portfolio

• A multifractal walk down Wall Street, Mandelbrot, B. B. (1999). Scientific American, 280(2), 70-73. This paper explains that individual investors and professional stock traders are aware that the prices usually quoted in any financial market can change very quickly.
• A focus on the exceptions that prove the rule, Mandelbrot, B., & Taleb, N. (2006). Financial Times, 23. This article presents how the current studies of uncertainty, such as economics and statistics, have been staying close to the bell curve which represents a probability distribution.
• Six ways companies mismanage risk, Stulz, R. M. (2009). Harvard Business Review, 87(3), 86-94. This paper presents six ways through which companies mismanage risk and also suggests solutions to these problems.
• Power laws and the new science of complexity management, Buchanan, M. (2004). Strategy+ Business, 34(Spring), 70-79. This is research that is aimed at gathering attention in the world of business since executives and scholars alike are starting to understand that the conventional theories of management forged in the industrialization era cannot cope anymore with the highly complex organizations that have emerged in the last two decades of increasing globalization and decentralization.
• Implementation issues in project web sites: a practioner’s viewpoint, O’Brien, W. J. (2000). Journal of management in engineering, 16(3), 34-39. This study uses the observations of a practitioner involved in the development and use of the first generation of project web sites to summarize the critical issues in implementing the sites on projects.
• On modified Black–Scholes equation, Ahmed, E., & Abdusalam, H. A. (2004). Chaos, Solitons & Fractals, 22(3), 583-587. This paper explains that since it is argued that from several points of view the telegraph equation is more suitable. The Black-Scholes equation is modified and then proposed.
• How Fractals Can Explain What’s Wrong with Wall Street, Mandelbrot, B. B. (2008). Sci. Am, 15. This paper presents the modern portfolio theory as a cornerstone of finance that tries to maximize returns given a specific level of risk.
• The Bell curve is wrong: so what?, Embrechts, P. (2000). Extremes and Integrated Risk Management, xxv-xxviii. This paper explains that the bell curve is considered as erroneous since various studies in different fields have shown that apparently, rare events are more common than predicted by the curve
• Rethinking modern portfolio theory, Warner, J. (2010). Bank Investment Consulting. This study shows how different investors are trying to decide whether to repair or abandon the theoretical foundations on which their portfolios are developed.
• A rational approach to pricing of catastrophe insurance, Dong, W., Shah, H., & Wong, F. (1996). This paper describes a methodology for rational pricing which contains solvency and a stability-based framework as well as a formula to quantify the loss variability driving solvency and stability.
• Portfolio crash testing: making sense of extreme event exposures, Novosyolov, A., & Satchkov, D. (2010). The Journal of Risk Model Validation, 4(3), 53. This article addresses various misunderstandings about stress testing which are common and then provides for its inclusion into the risk process as a supplement to risk measures like value-at-risk and tracking error.