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# Normal Distribution

## Definition

Normal distribution, also known as Gaussian distribution, is a type of probability distribution that is symmetric about the mean, showing data near the mean are more frequent in occurrence than data far from the mean. In finance, it’s commonly used to represent the distribution of many kinds of variables, including asset returns and interest rates. The shape of a normal distribution is a bell-shaped curve.

### Phonetic

The phonetics of the keyword “Normal Distribution” is: Normal: /ˈnɔːr.məl/Distribution: /ˌdɪs.trɪˈbjuː.ʃən/

## Key Takeaways

1. Characteristics: Normal Distribution, also known as Gaussian Distribution, is a type of continuous probability distribution for a real-valued random variable. It is characterized by its bell-shaped curve, where the mean, median and mode of the distribution coincide.
2. Symmetry: It’s a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions. This symmetry means that the normal distribution can be described by just two parameters, the mean (μ) and the standard deviation (σ).
3. Commonly Used: Normal distribution has wide applications in natural, social and behavioral sciences. It’s often used in the fields of psychology, business, and even astronomy to make conclusions about the data due to the Central Limit Theorem which states that the distribution of many random variables tends to be a normal distribution.

## Importance

The Normal Distribution, often referred to as the bell curve, is a vitally important concept in business and finance due to its widespread application in statistical analysis and risk management. It is the basis for many business-related metrics, such as forecasting, investment modeling, quality control, and even calculating probabilities and expectations. Financial analysts often assume normal distribution to model market price changes, portfolio returns, and risk exposure. The assumption of normality makes the math and computations easier due to the property characteristics of normal distributions: symmetry, mean-median-mode equality, and the empirical rule. Although the real world data may not be perfectly normally distributed, using normal distribution makes the analysis more manageable and provides a useful approximation for more complex statistical distributions.

## Explanation

The normal distribution, also known as the Gaussian distribution, is a tool used in finance and business to describe variance in a set of data, or in the financial world, risk and return. It’s a probability function that describes the likelihood that a random variable will fall within a certain range, based on the mean (or average) and standard deviation of the data set. Often referred to as the “bell curve,” it’s used to summarize data and predict future outcomes. The shape of the curve with its peak at the mean denotes that variables are more likely to be closer to the mean than far away, which can be useful for identifying outliers in data.In the world of finance and business, normal distribution plays an instrumental role in the pricing of derivatives, portfolio theory, risk management, and trading strategies. Most financial models, like the Modern Portfolio Theory (MPT) and the Black-Scholes Model for options pricing, assume that returns follow a normal distribution due to its simplicity and ease of calculation. Risk management in companies also use normal distribution to assess the probability of unfavourable outcomes such as stock price falls, debt defaults, or cost overruns. However, it’s important to bear in mind that real world data can behave abnormally due to factors like market crashes or unexpected news, therefore the assumptions of normal distribution should be interpreted with caution.

## Examples

1. Stock Market Returns: Stock market returns usually follow a normal distribution over a period of time. Different factors affect the stock prices which go up or down. Over a long period, when you plot these changes, you often get a shape that is very close to the normal distribution. It is advantageous in estimating the probabilities of different ranges of return.2. Credit Scores: Credit scores of individuals within an economy often are in a state of normal distribution. Most people have a score around the average, while fewer people have very low or very high scores. Lenders and creditors use these scores to evaluate the creditworthiness of individuals.3. Company Performance: When evaluating companies in terms of sales, profits, or other key performance indicators, the values across a certain industry or within the entire economy often resemble a normal distribution, with very few high performers and low performers, and most companies falling around the average.

What is a Normal Distribution in finance?

Normal Distribution, also known as Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. In finance, it is often used in modeling asset returns, portfolio theory, and the pricing of options. The graph of the Normal Distribution is a bell-shaped curve where the mean, median, and mode are all equivalent.

Why is Normal Distribution important in finance and business?

Normal Distribution is critical in finance because it simplifies the modeling of security prices and helps in making predictions about future pricing and risk assessment. It allows portfolio managers and other finance professionals to make informed investment decisions and risk management steps.

Are financial markets always normally distributed?

Not necessarily. Though the concept of Normal Distribution is widely used, real-life financial markets often demonstrate skewed or fat-tailed distributions, which means they have higher probability of extreme changes (either positive or negative) than what is predicted under a normal distribution.

What is a standard deviation in Normal Distribution?

Standard deviation in a Normal Distribution is used to quantify volatility or market risk. It measures the dispersion of a set of values from the mean. If the data is closely clustered around the mean, the standard deviation is low; if the data is spread out over a larger range of values, then the standard deviation is high.

What does it mean when we say a stock return is normally distributed?

When we say a stock return is normally distributed, it means the probabilities of the possible outcomes are distributed in a way that forms a symmetrical bell-shaped curve. The most likely outcome is the mean, or peak of the distribution, with less likely outcomes tapering off symmetrically on either side.

Are there any criticisms of Normal Distribution in finance?

Yes, although Normal Distribution is widely used, it has its critics. Some argue that it fails to capture the true representation of financial market dynamics as financial markets frequently exhibit ‘fat tails’ or extreme deviations from the mean, which are more frequent than what the Normal Distribution would predict.

What is the relationship between Normal Distribution and Black-Scholes model?

The Black-Scholes model, used for pricing options, assumes that underlying security prices follow a geometric Brownian motion with constant volatility, resulting in a log-normal distribution. This indirectly aligns with the concept of Normal Distribution due to the property that the logarithm of a log-normally distributed variable is normally distributed.

## Related Finance Terms

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