🎓Normal Distribution and Its Financial Application
Since we have covered basic statistical concepts like mean and standard deviation, we are well-equipped to learn the famous normal distribution.
📝Normal distribution, also known as Gaussian distribution, is one of the most important probability distributions in statistics.
It is bell-shaped and has remarkable properties.
📌It is symmetric about the mean μ.
📌 There is a well-known 68-95-99.7 rule of normal distribution.
🖇️about 68% of the data lies within one standard deviation of the mean
🖇️about 95% of the data lies within two standard deviations of the mean
🖇️about 99.7% of the data lies within three standard deviations of the mean
Please check the last picture to see the visualization of this rule.
📌Additionally, the tails of the normal distribution curves approach the x-axis asymptotically as x approaches positive or negative infinity.
📝 A special case of the normal distribution is the standard normal distribution, where μ = 1 and σ = 0 . We usually denote the random variable in the standard normal distribution as Z.
📝Applications of normal distribution in finance
📌 Modeling of Asset Returns
In finance, the returns of many financial assets such as stocks and bonds are often assumed to follow a normal distribution. The Capital Asset Pricing Model (CAPM) and the portfolio theory of Markowitz both assume normal distribution of returns so that systematic risk of an asset can be analyzed and optimal portfolio can be constructed.
📌 Risk Assessment and Management
Normal distribution is widely used in financial risk management to measure market risk. Value at Risk (VaR) is a common risk measurement method. Under the assumption of normal distribution, VaR can be easily calculated. For an investment portfolio, assuming that its return follows a normal distribution, at a given confidence level (such as 95% or 99%), VaR can measure the maximum potential loss within a specific holding period.
📌Pricing of Financial Derivatives
Many financial derivative pricing models assume that the price changes of the underlying assets (such as stocks) follow a normal distribution. For instance, in the Black-Scholes option pricing model, it assumes that the logarithm of the underlying asset price follows a normal distribution. The model calculates the reasonable price of an option by considering factors such as the volatility of the underlying asset price (measured by standard deviation), the strike price of the option, and the expiration time, providing a pricing reference for option trading in the financial market.
Normal distribution, also known as Gaussian distribution, plays a significant role in statistics and finance. Understanding its properties is essential for financial analysts and investors alike. The mean (μ) and standard deviation (σ) are critical to describing this distribution. In finance, normal distribution is commonly used to model asset returns, which directly influences investment strategies and risk analysis. The Capital Asset Pricing Model (CAPM) relies on this distribution to calculate expected returns based on systematic risk. Furthermore, the normal distribution is integral in risk management through the Value at Risk (VaR) metric, essential for determining potential losses in investment portfolios. VaR calculations assume normally distributed returns, making it a vital tool for financial decision-making. Additionally, the Black-Scholes model, widely used for pricing options, assumes that the logarithm of stock prices follows a normal distribution. This highlights how understanding normal distribution can enhance financial analysis and decision-making processes.
