# On Expected Value

### Res ipsa loquitur - The thing speaks for itself

The first wealth is health. — Ralph Waldo Emerson

In 2018 I found myself in the main trading room of a crypto hedge fund in sunny Las Vegas, Nevada. A newly *minted* orange McLaren was in the parking lot next to a new Tesla next to a new… You get the point,it was the Wild West in more ways than one. As I sat inside the air conditioned office in Lululemon ABC pants (best pants one can have) I thought about the best way to maximize LP returns and optimize trading strategies in a volatile environment.

2018 marked the beginning of that cycles “crypto winter”. Asset prices skyrocketed in 2017 and sharply fell through the next two years. It’s hard to raise money when prices are constantly red. So, we, like many others in similar positions, were “*building*”. Building is something entrepreneurs, innovators, etc say when the market is down and the pace of each day has slowed. It’s a time to focus on what comes next.

My firm was consulting on implementation of trading algorithms on various crypto exchanges. *Como, what*? We had a great algo and another fund wanted to pay us to use it. *C’est la vie*. The best time to create, deploy, and refine trading algorithms is during down periods.

Going back, during the day the atmosphere in the office was relaxed as we discussed trading algorithms, systems theory, and yes - expected value (EV). Did I mention La Croix was seemingly on tap. Ah, the good ol days…

Suddenly, the vibe of the room dramatically shifted. What once was a relaxed setting that involved conversations regarding MACD transformed to a frenzied shouting match about expected value, the Kelly criterion, and bet sizes. Notice I didn’t say anything risk management related, HA, this was a hedge fund in Las Vegas, a crypto hedge fund at that. Insanity was a daily occurrence.

What spurred the frenzy? It was rumored Z Cash was being going to be listed on Coinbase. The fund we were consulting thought this was a buy the rumor sell the news event and began moving funds, liquidating positions, and placing large bets on Z Cash’s price appreciation. In what seemed like five minutes, tens of millions of dollars worth of trades and moves had occurred. Why? Because this firm thought buying Z Cash at this time was a high EV bet. I thought otherwise, but hey, that’s why I’m still in the game and writing on Substack and the other guys are talking about dominance hierarchies amongst lobsters somewhere on Fremont Street.

“My life seemed to be a series of events and accidents. Yet when I look back, I see a pattern.”― Benoît B. Mandelbrot

# What is Expected Value (EV)?

Expected value is a concept in probability theory that represents the average value that one can expect to obtain from a random event over a large number of trials. It is calculated by multiplying each possible outcome by its probability of occurring, and then summing up all of these products. In other words, it is a weighted average of all possible outcomes, where the weights are given by their probabilities of occurrence.

**Expected value can be used to make decisions or predictions about uncertain events, and is an important concept in fields such as statistics, economics, and finance.**

Expected value is a fundamental concept in probability theory and statistics, and is used to describe the long-term behavior of random events. Seems like a useful concept to grasp, no? It is denoted by the symbol E(X), where X is a random variable representing the outcomes of the event. The expected value is calculated as the sum of the products of the possible outcomes and their corresponding probabilities:

E(X) = ∑[x * P(X=x)]

where x represents the possible outcomes of the event, and P(X=x) is the probability of each outcome occurring.

Expected value can be used in many different ways. For example, in finance, it can be used to calculate the expected return of an investment, where the possible outcomes are the different returns that the investment might generate, and their probabilities are estimated based on historical data or other factors. In insurance, the expected value can be used to calculate premiums, where the possible outcomes are the different losses that might occur, and their probabilities are estimated based on actuarial models.

One important property of the expected value is that it is a linear operator. This means that if we have two random variables X and Y, and a constant c, then:

E(cX + Y) = cE(X) + E(Y)

This property makes it easy to calculate the expected value of more complex events, by breaking them down into simpler components and adding up their expected values.

**Overall, expected value is a powerful tool for analyzing and predicting random events, and is used in a wide range of applications in science, engineering, and business.**

The brain highlights what it imagines as patterns; it disregards contradictory information. Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it.― Benoît B. Mandelbrot, The (Mis)Behavior of Markets

## Finance

Expected value is commonly used in finance to calculate the expected return of an investment. For example, suppose an investor is considering investing in a stock that has two possible outcomes: it could either go up by 20% or down by 10%, with probabilities of 0.6 and 0.4, respectively.

To calculate the expected return of this stock, we would multiply each possible outcome by its probability and sum up the products:

E(Return) = (0.6 * 0.2) + (0.4 * (-0.1)) = 0.12 - 0.04 = 0.08

So, the expected return of the stock is 8%. This means that, on average, the investor can expect to earn an 8% return on their investment over a large number of trials. Of course, this is just a prediction, and the actual return could be higher or lower depending on the performance of the stock.

Expected value can also be used to compare different investments and decide which one is likely to generate the highest return. For example, if an investor is considering two stocks with different expected returns, they can use the expected value to estimate which one is a better investment choice.

Here's an example of how expected value can be used to compare different investments:

Suppose an investor is considering two different investments: Investment A and Investment B. Investment A has a 60% chance of generating a 10% return, and a 40% chance of generating a 5% return. Investment B, on the other hand, has a 70% chance of generating a 7% return, and a 30% chance of generating a 3% return.

To compare these two investments, we can calculate the expected return of each one using the formula:

Expected Return = (Probability of Outcome 1 * Return of Outcome 1) + (Probability of Outcome 2 * Return of Outcome 2) + ...

Using this formula, we can calculate the expected return of Investment A as:

Expected Return of Investment A = (0.6 * 0.1) + (0.4 * 0.05) = 0.08 or 8%

Similarly, the expected return of Investment B is:

Expected Return of Investment B = (0.7 * 0.07) + (0.3 * 0.03) = 0.058 or 5.8%

Based on these calculations, Investment A has a higher expected return than Investment B. Therefore, if the investor is solely interested in maximizing their expected return, they would choose Investment A over Investment B.

However, it's important to note that expected return is just one factor to consider when making investment decisions. Other factors, such as risk, liquidity, and diversification, also need to be taken into account when evaluating different investment options.

For instance, suppose you offer somebody a choice: They can flip a coin to win $200 for heads and nothing for tails, or they can skip the toss and collect $100 immediately. Most people, researchers have found, will take the sure thing. Now alter the game: They can flip a coin to lose $200 for heads and nothing for tails, or they can skip the toss and pay $100 immediately. Most people will take the gamble. To the imagined rational man, the two games are mirror images; the choice to gamble or not should be the same in both. But to a real, irrational man, who feels differently about loss than gain, the two games are very different. The outcomes are different, and sublimely irrational.― Benoît B. Mandelbrot, The (Mis)Behavior of Markets

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