Expected Value, Variance, and Standard Deviation: Risk Management Basics

Welcome to understanding expected value, a crucial concept for making informed investment decisions.Expected value represents the anticipated average outcome of an investment or probability distribution.It's calculated using this formula, where we multiply each possible outcome by its probability and sum the results.Let's look at a practical investment example.Consider an investment with two possible outcomes: a sixty percent chance of gaining one thousand dollars, and a forty percent chance of losing five hundred dollars.We can visualize these probabilities as proportional bars.To calculate the expected value, let's break down each scenario.For the gain scenario, we multiply one thousand dollars by zero point six, giving us six hundred dollars.For the loss scenario, we multiply negative five hundred dollars by zero point four, giving us negative two hundred dollars.Adding these together, our expected value is positive four hundred dollars.Now that we understand how to calculate expected value, let's explore how to measure the variation in these potential outcomes.To understand investment risk, we need to measure how much outcomes can vary from our expected value.With a sixty percent chance of gaining one thousand dollars and a forty percent chance of losing five hundred dollars, our expected value is four hundred dollars.To calculate variance, we first find how far each outcome deviates from the expected value.Next, we square these differences to make all values positive and emphasize larger deviations.Then multiply each squared difference by its probability.Finally, we sum these products to get our variance of five hundred and forty thousand dollars squared.Let's visualize how these outcomes spread around our expected value.Here are our two possible outcomes, and our expected value at four hundred dollars.The variance measures how far these outcomes spread from the expected value. Larger spreads mean higher variance and more risk.This high variance indicates significant volatility in our investment outcomes.Standard deviation helps us understand investment risk by measuring how much returns typically deviate from the expected value.Here's the formula for standard deviation - it's the square root of the average squared differences from the mean.Let's compare two investments with the same expected return but different levels of risk.Investment A, shown in blue, has a smaller standard deviation of zero point five. This means its returns tend to stay closer to the expected value.Investment B, shown in red, has a larger standard deviation of one point five. Its returns are more spread out, indicating higher volatility and risk.Let's compare these investments side by side. While both have the same expected return of ten percent, their risk profiles are quite different.In a normal distribution, we can predict the probability of returns falling within certain ranges.Let's review the key points about standard deviation and risk assessment.Remember: investments with the same expected return can have very different risk levels. A lower standard deviation means more predictable returns. Always choose investments that match your personal risk tolerance.Thanks for learning about standard deviation and risk assessment with Spark.E!