The Sortino ratio is a performance metric that measures the risk-adjusted return of an investment, focusing only on downside risk (negative returns). Unlike the Sharpe ratio, which penalizes both upside and downside volatility, the Sortino ratio distinguishes harmful volatility from beneficial volatility. It is calculated as: Sortino Ratio = (R_p - R_f) / σ_d, where R_p is the average periodic return, R_f is the risk-free rate, and σ_d is the downside deviation (standard deviation of negative returns below a target threshold). The target return is typically the minimum acceptable return, often set to 0% or the risk-free rate. A higher Sortino ratio indicates better risk-adjusted performance relative to downside risk. This metric is particularly valuable for evaluating investments with asymmetrical risk profiles, such as hedge funds, private equity, and options strategies, where investors are more concerned about downside losses than upside volatility.

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. If Y = ln(X) follows a normal distribution with mean μ and standard deviation σ, then X follows a lognormal distribution. This distribution is characterized by positive skewness and is bounded below by zero, making it suitable for modeling quantities that cannot be negative, such as stock prices, income distributions, asset prices, failure times, and environmental concentrations. The probability density function (PDF) is f(x) = (1/(x·σ·√(2π))) · exp(-(ln(x)-μ)²/(2σ²)). The cumulative distribution function (CDF) is derived from the standard normal CDF: F(x) = Φ((ln(x)-μ)/σ). The lognormal distribution has important properties: the mean is exp(μ + σ²/2), the median is exp(μ), and the mode is exp(μ - σ²). This calculator computes CDF, PDF, mean, median, mode, variance, and standard deviation for given parameters.

The median is the middle value in a dataset when the numbers are arranged in ascending or descending order. Unlike the mean (average), the median is not affected by extremely large or small values, making it a robust measure of central tendency. To find the median, first sort all numbers from lowest to highest. If there is an odd count of numbers, the median is the middle number. If there is an even count, the median is the average of the two middle numbers. For example, in the set {1, 3, 5}, the median is 3. In the set {1, 3, 5, 7}, the median is (3+5)/2 = 4. The median is widely used in statistics, economics, and data analysis, especially when dealing with income distributions, house prices, or test scores where outliers could skew the average.

Student's t-distribution is a continuous probability distribution used when estimating the mean of a normally distributed population with a small sample size and unknown standard deviation. It is the foundation of many statistical tests, including the t-test. The theoretical calculation involves the probability density function (PDF) and cumulative distribution function (CDF). The CDF gives the probability that a random variable following the t-distribution is less than or equal to a given t-value. The distribution is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. As degrees of freedom increase, the t-distribution approaches the standard normal distribution. This calculator computes one-tailed and two-tailed p-values from a t-value, and also finds critical t-values for a given significance level (alpha=0.05). It is widely used in hypothesis testing, confidence intervals, and regression analysis.