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You should review your portfolio at least every quarter unless you have a financial planner who monitors your investments. Looking at performance can be time-consuming and complex, so it’s important to find the most efficient way of doing so. You can save time by analyzing your portfolio with quantitative investment concepts.
This article will discuss the different concepts of quantitative investments. Included is information on distributions, diversifications, and different risk measurements. In particular, topics such as kurtosis, variance, covariance, and beta will be discussed.
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Distribution of Returns Types
A normal distribution is represented by a bell-shaped curve. It has a single peak in the center consisting of mean, median, and mode.
A mean is the sum of observations divided by the number of observations.
The median is the midpoint of the values after they are ordered from smallest to largest.
The modes are the observation that has the greatest frequency.
Symmetric distribution is where there is a 50% chance that an observation selected at random will fall to the right of the mean, and a 50% chance that it will fall to the left of the mean.
Skewness or Non-Symmetrical Distribution
A non-symmetrical distribution can either be skewed to the right or skewed to the left. When the distribution is skewed to the right, it means that it is positively skewed. The mean is the highest of the three averages, followed by the median, then the mode.
A distribution that is skewed to the left means that it is negatively skewed. In this case, the mode is the largest of the three averages, followed by the median, then the mean.
Kurtosis measures whether the distribution is more peaked or less peaked than normal distributions. It measures the degrees of exceptional values that occur more frequently or less frequently. A leptokurtic means that the distribution is more peaked than a normal distribution.
More observations are clustered closely around the mean, which is good for investors that want to minimize volatility in their portfolios. Platykurtic is when the distribution is less peaked than a normal distribution. There is more observation with large deviations from the mean.
Lognormal Probability Distributions
A lognormal distribution has a greater than 50% chance that an observation that is selected at random will fall to the left of the mean. The distribution is bound by zero, but in theory, the value of the portfolio could rise towards infinity. The distribution is skewed to the right and is often used to analyze the probability distribution of security and other asset prices. This distribution can be used as a representative of the expected future value of a client’s portfolio.
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Diversification is how to structure an investment portfolio to maximize risk and return. An investor should try to realize the amount of return that they are receiving per unit of risk. This can be done by structuring a portfolio that contains a low correlation to each other and having a longer investment time horizon. There is an inverse relationship between the number of stocks and the amount of unsystematic risk within the portfolio. As the number of stock increases, the level of unsystematic risk will decline.
Covariance measures the extent to which two variables move together, either positively or negatively. Covariance is the non-standardized version of the correlation coefficient. It ranges from negative infinity to positive infinity.
The Correlation Coefficient (R or ρ)
A correlation coefficient measures the extent to which the returns on two securities are related. It measures the strength of the straight-line or linear relationship between the two variables. It has a range of +1 to -1.
Negative 1 means that the securities move in the opposite direction at the same rate.
Positive 1 means that the movement between the securities is moving at the same rate and in the same direction
A value of 0 means that the security movements are unrelated to one another.
A correlation coefficient is a standardized version of a covariance. Combining securities that are not perfectly correlated will reduce the overall portfolio risk.
Coefficient of Determination (R-Squared)
It is calculated by squaring the correlation coefficient. This describes the percentage of variability of the dependent variables that are explained by the changes in the independent variable. If R squared is equal to 1 between the portfolio and the market, it means that the portfolio contains no unsystematic risk.
If R2 risk = 1 between a portfolio and the market, the portfolio contains no unsystematic risk.
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Beta is the relative measure of systematic risk, and it is useful for portfolios that are highly correlated with the market or benchmark portfolio. The market has a Beta of 1.
A beta of 1.25 will be considered to be 25% more volatile than the market.
A beta of less than 1 for a portfolio will be considered less risky than the market.
Another type of beta is the portfolio beta, which is a weighted average. Betas for individual securities are not considered stable over time. However, betas for portfolios are usually stable over time.
Statistical Risk Analysis
Statistical risk analysis is the distribution of returns for different assets that are symmetrically distributed around the mean of the returns. The distribution will form to something that is referred to by statisticians as the normal distribution or bell-shaped curve.
Standard deviation is the dispersion of outcomes that is around the mean that is used to measure the risk for normal distribution. The greater the standard deviation, the greater the risk or volatility. The total risk of systematic and unsystematic risk is measured by the standard deviation. The mean is the average return for the sample data. Variance is the standard of deviation squared. Standard deviation is calculated by taking the square root of the sum of the squared differences between the mean, and the individual's observations divided by the number of observations minus one. If the market has a standard deviation of 15%, then the stocks with a standard deviation greater than 15% are considered to be riskier than the market.
Consequently, stocks with a standard deviation of less than 15% are considered less risky than the market. A z-statistic or z-score measures the number of standard deviations a data value is from the mean. The score is calculated by subtracting the mean from the given data value and dividing the result by the standard deviation.
If the data value is higher than the mean, the z-score will be positive. A z-score of 1.3 means that the data value is 1.3 standard deviation above the mean. If the data value is smaller than the mean, then the z-score will be negative. A z-score of -3.3 means that the data value is 3.3 standard deviations below the mean.
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A semivariance considers the downside volatility of an investment and measures the variability of returns that are below the average or the expected return. When it is used in an analysis, the lower the semivariance of a security, the less likely that security can incur a substantial loss in value.
Critics of variance and standard deviation state that investors are not concerned about the volatility above the average return. Instead, investors are concerned about the volatility below the average return. Thus, a portfolio manager with returns that have a large variance can be punished for having superior positive returns. Semivariance attempts to correct this flaw by only considering the returns and volatility below the expected or average return.
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