Portfolio Expected Value and Risk

Expected Returns

• The discussion revolves around calculating expected returns and risks associated with investments in stocks X and Y, both individually and as a portfolio.

• Initial data includes potential losses and gains for stock X ($-25,000$, $50,000$, $100,000$) and stock Y ($-200,000$, $60,000$, $350,000$).

• Probabilities are assigned to each of these return values, forming a joint probability distribution.

Portfolio Definition

• A portfolio (P) is defined as a linear combination of stocks X and Y,:
  $P = wX + (1-w)Y$
  where $w$ represents the weight or proportion invested in stock X, and $(1-w)$ is the proportion invested in stock Y.

• This is analogous to constants A and B from a prior lecture.

Risk Assessment

• Risk is quantified using standard deviation.
• The goal is to calculate standard deviation for X, Y, and the portfolio.

Formulas

• The presenter refers to an uploaded example with formulas for expected value, variance, covariance, and correlation.

• The expected value of X is calculated using the formula:
  $E[X] = \sum x<em>i * P(x</em>i)$, where $P(x<em>i)$ is the marginal probability of $x</em>i$.

• The variance of X is calculated using: $Var(X) = E[X^2] - E[X]^2$

Simplified Table

• The table is simplified with marginal probabilities for X and Y.
• Joint probabilities are considered, noting that some probabilities are zero, simplifying calculations.

Calculation Examples

• Expected value of X is calculated as:
  $E[X] = (-25,000 * 0.2) + (50,000 * 0.3) + (100,000 * 0.5) = 50,000$

• Expected value of Y is calculated similarly.

• Variance is calculated using the formula, then standard deviation (risk) is derived.

Risk vs. Return

• Stock Y has a higher expected return ($95,000$) but also higher risk ($148.32$).
• Stock X has a lower return ($50,000$) and lower risk ($43,012$).
• Investing in a portfolio might offer a balance between risk and return.

Portfolio Expected Value

• The expected value of the portfolio is a weighted average of the expected returns of X and Y:
  $E[P] = w * E[X] + (1-w) * E[Y]$.

Portfolio Variance

• The formula for the variance of the portfolio is:
  $Var(P) = w^2 * Var(X) + (1-w)^2 * Var(Y) + 2 * w * (1-w) * Cov(X,Y)$.

Covariance Calculation

• Covariance between X and Y is calculated using the formula:
  $Cov(X,Y) = \sum \sum (x<em>i - E[X]) * (y</em>i - E[Y]) * P(x<em>i, y</em>i)$.

• Due to zeros in the joint probability table, the calculation simplifies.

Portfolio Example

• If 40% is invested in X and 60% in Y, the expected return of the portfolio is $77,000$.
• The risk is also somewhere in the middle, potentially making the portfolio an attractive option.

General Concepts

• Expected values and standard deviations are used to measure predictions and volatility.
• Confidence intervals can be estimated using expected value and standard deviation, helping assess risk.

Visualization

• Investment returns can be visualized over time, with a predicted value and confidence intervals representing risk.

Chapter 6 Introduction

• Chapter 6 will cover continuous variables.
• Key distributions: Normal, Exponential and Uniform.

Normal Distribution

• The normal distribution (bell-shaped curve) is common in nature, e.g., weight distribution.
• It is foundational for many statistical techniques, including Ordinary Least Squares (OLS) regression.

Other Distributions

• Logistic Regression: Used for binary classification problems.
• Poisson Regression: Used to model count data (e.g., number of phone calls).
• Exponential Distribution: Used to model lifetimes or time duration (e.g., length of phone calls, battery life).
• Uniform Distribution: Primarily used for simulation purposes, can be transformed to generate data for other distributions.

Simulation

• Simulation involves generating data based on a theory to make predictions.
• It can be used for weather forecasting, where different scenarios are simulated to estimate future conditions.