1.1 Mean

• Means are “Point Estimates” meaning all the data is collapsed into one point
• Point Estimates Can Be Deceiving since you may lose important information

2.1 Range

• Range = Max - Min

2.2 Mean Absolute Deviation (MAD)

\(MAD = \frac{\sum_{i=1}^n |X_i - \mu|}{n}\)

• MAD is average absolute difference of each datapoint from mean

2.3 Variance

\(\sigma^2 = \frac{\sum_{i=1}^n (X_i - \mu)^2}{n}\)

• Variance(\(\sigma^2\)) is average squared difference of each datapoint from mean
• prefer Variance over MAD because squared function is smooth and differentiable but absolute-value function is not

2.4 Std dev

• Std dev(\(\sigma\)) is sqrt of variance
• Easier to interpret since it is same units as datapoints

2.5 Semivariance and semideviation

• Stocks we worry more about deviation downwards,
• Semivariance(\(\sigma^2_{<\mu}\)) and semideviation(\(\sigma_{<\mu}\)) is like variance and std dev but only for datapoints less than or equal to the mean(\(\mu\))

(1,2,3,4,5) has mean of 3. Semivariance only looks at data points (1,2,3) and ignore (4,5). \(\sigma^2_{<\mu}=\frac{(1-3)^2 + (2-3)^2 + (3-3)^2}{3}\)

• Another related concept is target semivariance or target semideviation, which is deviation below our chosen target value.

3.1 Kurtosis

• kurtosis = 3, All normal distribution regardless of mean and variance have kurtosis = 3
• kurtosis > 3, called leptokurtic distribution, many spikes away from mean
• kurtosis < 3, called platykurtic distribution, fewer spikes away from mean

SP500 returns is leptokurtic, meaning returns have many spikes away from mean

3.2 Normality Testing Using Jarque-Bera

Example taking the mean of a bi-modal distribution is useless
Typically we assume the distribution of asset returns are normally distributed but they arent so many statistical tools are actually flawed

• The Jarque-Bera test is a common statistical test that compares whether sample data has skewness and kurtosis similar to a normal distribution.
• The Jarque Bera test’s null hypothesis is that the data came from a normal distribution. Because of this it can err on the side of not catching a non-normal process if you have a low p-value.

4.1 Covariance vs Correlation

• Correlation = Normalized(Covariance)

4.2 Rolling window correlation

• Correlation may vary over time period so it may be useful to do a rolling window correlation w/ assets

5.1 Sharpe Ratio

\[R = \frac{E[r_a - r_b]}{\sqrt{Var(r_a - r_b)}}\]

• One statistic often used to describe the performance of assets and portfolios is the Sharpe ratio, which measures the additional return per unit additional risk achieved by a portfolio, relative to a risk-free source of return such as Treasury bills

5.2 Moving averages

• You can take moving average of sharpe ratio, means, standard deviation
• Moving average of mean, +1 std dev, -1 std dev are called Bollinger bands

6.1 Binomial distribution

\[p(x) = P(X = x) = \binom{n}{x}p^x(1-p)^{n-x} = \frac{n!}{(n-x)! \ x!} p^x(1-p)^{n-x}\]

• Either it happens or it doesnt with probability p and 1-p
• \(\binom{6}{2} = 6 \times 5\)

6.2 Stock movement as Binormial Distribution

• Example: Assume we know a stock moves UP or DOWN with probability of 50%. In 5 days, we can predict the chance there are 1,2,3,4, or 5 UP days using a binomial distribution. Ofc the downside is order doesnt matter, example, 5 choose 3 tells us there are 3 UP days but they can be in any order.

6.3 Linearity of Normal Distributions

\[N(\mu_1, \sigma_1^2) + N(\mu_2, \sigma_2^2) = N(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)\]

• The mean and variance of a normal distribution is linear meaning you can add two normal distributions (aka two normal random variables) and the mean and variance of the resultant normal distribution will be the sum

• In modern portfolio theory, stock returns are generally assumed to follow a normal distribution. One major characteristic of a normal random variable is that a linear combination of two or more normal random variables is another normal random variable. This is useful for considering mean returns and variance of a portfolio of multiple stocks. Up until this point, we have only considered single variable, or univariate, probability distributions. When we want to describe random variables at once, as in the case of observing multiple stocks, we can instead look at a multivariate distribution. A multivariate normal distribution is described entirely by the means of each variable, their variances, and the distinct correlations between each and every pair of variables. This is important when determining characteristics of portfolios because the variance of the overall portfolio depends on both the variances of its securities and the correlations between them

7.1 Deriving Linear Regression using OLS

\[OLS=\sum_{i=1}^n (Y_i - a - bX_i)^2\]

• Linear Regression model is an optimization function using the Ordinary Least Squares (OLS) as the objective function aka likelihood function
• We use \(a\) and \(b\) to represent the potential candidates for \(\alpha\) and \(\beta\). What this objective function means is that for each point on the line of best fit we compare it with the real point and take the square of the difference. This function will decrease as we get better parameter estimates.
• Regression is a simple case of numerical optimization that has a closed form solution and does not need any optimizer. We just find the results that minimize the objective function.

We will denote the eventual model that results from minimizing our objective function as:

\[\hat{Y} = \hat{\alpha} + \hat{\beta}X\]

7.2 Standard Error

• We can also find the standard error of estimate, which measures the standard deviation of the error term \(\epsilon\) by getting the scale parameter of the model returned by the regression and taking its square root. The formula for standard error of estimate is

\[s = \left( \frac{\sum_{i=1}^n \epsilon_i^2}{n-2} \right)^{1/2}\]

9.1 Regime changes

• In this case our regression model will not be predictive of future data points since the underlying system is no longer the same as in the sample. In fact, the regression analysis assumes that the errors are uncorrelated and have constant variance, which is often not be the case if there is a regime change.

9.2 Multicollinearity