A Different Degree of Wealth

Dice and Dollars: Understanding Probability Distribution in Investments

Investing inherently involves uncertainty, making the understanding and quantification of this uncertainty crucial for informed decision-making. Probability distributions serve as mathematical tools that help investors assess the likelihood of various outcomes, thereby enabling more strategic investment choices. This article explores the key types of probability distributions used in investments, their applications, and the critical interplay between risk and return.

Probability Distributions

A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. The shape and characteristics of these distributions depend on factors such as the mean (average value), standard deviation (measure of variability), skewness (asymmetry of the distribution), and kurtosis (tailedness of the distribution).


  • Normal Distribution: The normal distribution, often referred to as the bell curve, is one of the most commonly used probability distributions in finance. It is symmetric around the mean, with most data points clustering around this central value. The normal distribution is particularly useful for predicting stock returns because it allows for the calculation of probabilities within specific ranges. For instance, it can help estimate the likelihood of a stock’s return falling within one standard deviation of the mean.
  • Binomial Distribution: The binomial distribution is used when there are only two possible outcomes – success or failure. This distribution is valuable in scenarios like option pricing, where the probability of a specific outcome (such as a stock price reaching a certain level) must be evaluated before making an investment decision. For example, it can help determine the likelihood of a stock option finishing in the money.
  • Poisson Distribution: The Poisson distribution models the probability of a given number of events happening in a fixed interval of time or space. It is often employed for predicting rare events, such as natural disasters or cybersecurity breaches. In investments, the Poisson distribution can be used to estimate the probability of rare but impactful events like stock market crashes.

Risk Assessment and Management

While probability distributions provide a framework for estimating potential returns, they are equally critical for assessing and managing risk. One widely used risk metric derived from probability distributions is Value-at-Risk (VaR), which estimates the maximum potential loss over a specified period with a given confidence level.

However, it’s essential to recognize the limitations of these models. Real-world outcomes may deviate significantly from theoretical probabilities, especially during extreme market conditions. With that in mind, investors should not solely rely on historical data and probability models but also consider qualitative factors and broader economic indicators that could influence an investment’s performance.

Practical Applications


  • Portfolio Optimization: By understanding the expected returns and associated risks of various assets, investors can construct diversified portfolios that optimize the trade-off between risk and return. Techniques like mean-variance optimization rely heavily on the properties of normal distributions.
  • Risk Mitigation: Investors can use probability distributions to identify potential risks and develop strategies to mitigate them. For instance, understanding the likelihood of rare events through the Poisson distribution can inform the inclusion of hedging instruments in a portfolio.
  • Performance Evaluation: Assessing the performance of investment strategies often involves comparing the expected returns against the actual outcomes. Discrepancies can highlight areas for improvement or adjustment in the strategy.


Mastering Distributions

Mastering probability distributions is essential for any investor aiming to make informed decisions and manage risks effectively. While these mathematical tools provide valuable insights into potential outcomes, they should be used in conjunction with a comprehensive plan and consider other risk factors.

If you have any questions, give us a call, or read “Wealth on Purpose” by Bryan Ballentine.

Have a great weekend!

Sources: Located at the bottom of the article.

Golf Tip of the Week

Is Tiger Woods’ viral no-divot claim actually possible? Here’s what experts say

Tiger Woods loves a good Tiger-ism, and golf fans got another great one this week.

Getting “zeroed.”

Woods used the term this week when TaylorMade Golf asked Tiger about his viral video last year, when he left Scottie Scheffler gob smacked after Woods claimed that when he’s swinging his very best, he doesn’t take divots. 

What does Tiger mean?

This isn’t a term Woods invented. It’s a phrase that has been kicking around for a while. When golfers talk about “zeroing out” their golf swing, they’re usually talking about getting one or multiple of their swing paths, clubface angle, face-to-path, angle of attack, or other metrics to zero on a launch monitor.

“He could be talking about his swing path, a zero angle of attack, or clubface-to-club path,” Tim Briand, the Director Of Business Development of the launch monitor company Foresight Sports. “But a zero-clubface angle, for instance, means it’s perfectly square to the target line.”

For that reason, getting zeroed represents a kind of perfection for some golfers.

In this case, Tiger is talking about his angle of attack. A golfer who hits down or up on the ball has a negative or positive angle of attack, respectively. If a golfer’s angle of attack is zero, then you’re striking the ball exactly level, which is how Tiger says he could hit shots without divots.

… Ok, but is it truly possible?

The thing is that even if you’ve zeroed out your angle of attack, it’s still possible to take a divot.

It could be as simple as a ball position issue. If your ball position is too far forward, for instance, your club could bottom out behind the ball and be moving up by the time it actually gets to the ball. That’s what a drop-kicked drive is.

But assuming that’s not an issue for Tiger, there’s a more scientific explanation.

“It’s possible to take a small divot with zero angle of attack and a low point [of the swing] at the top of the grass blades because of the club-ball interaction,” Golf Digest Best in State Teacher Michael Finney says.

The reason why is because when a clubhead hits a golf ball, the face bends slightly—or “deflects”—in the opposite direction. A bit like how a rifle recoils after it’s fired. So, on higher-lofted clubs especially, it means that as the club is pushing the ball up into the air, the ball is also pushing the club down into the ground.

“For every force, there’s an equal and opposite force,” Briand says. “The loft of the club is enacting on the ball, and that means there’s an equal and opposite force pushing the club down.”

This force, even on a zeroed swing, is enough to create a divot.

But wait! There’s a way around this: with his equipment.

Briand explains that Woods was meticulous in working with his club whisperer, Mike Taylor, on the grind with his wedges and irons. Specifically the leading edge of the club, which is the part of that cuts through the turf. A more rounded leading edge, Briand explains, could help counteract and make all this possible.

“Basically, let’s say Tiger’s got about a zero angle of attack, so it’s very level, and the low point is exactly in the middle of his golf swing arc,” Briand says. “The club is hitting the ball, and then deflecting into the ground. And then the leading-edge grind of the club is bouncing.”

Sum it up for me quickly, and simply

So in a word, yes. It’s possible. But it’s really, really, really difficult. Both to do and to repeat with any consistency.

But we’re also talking about Tiger Woods. He’s the golfer who can do anything, so would anything surprise you?

Tip adapted from golfdigest.comi

Recipe of the Week

Lemon Bars

16 Servings


For The Crust

  • Cooking spray
  • 1 c. butter, softened
  • 3/4 c. granulated sugar
  • 2 1/4 c. all-purpose flour
  • 1/4 c. powdered sugar, plus more for garnish

For The Filling

  • 2 c. granulated sugar
  • 1/4 c. all-purpose flour
  • 6 large eggs
  • 3/4 c. lemon juice
  • Zest of one lemon


  • Preheat oven to 350°. Spray a 9″-x-13″ baking pan with cooking spray. Line with parchment paper and spray parchment paper with cooking spray.
  • Make the crust: In a large bowl using a hand mixer, cream together butter and sugar until light and fluffy, 2 to 3 minutes. Add flour and powdered sugar and beat until combined.
  • Spread dough into prepared baking pan and press to flatten. Bake until lightly golden and just set, 20 to 25 minutes. Remove from oven and let cool for 30 minutes.
  • Make the filling: In a large mixing bowl, whisk together sugar, flour, eggs, lemon juice, and lemon zest and mix until fully combined. Pour filling over crust, return to oven, and bake until firm, about 25 minutes.
  • Remove from oven and refrigerate 3 to 4 hours. Slice and garnish with powdered sugar.




Recipe adapted from Delish.comii

Travel Tip of the Week

This Psychologist’s Hack for Falling Asleep Quickly on Planes Is so Simple


Getting a comfortable night’s sleep on a plane is, well, a dream for many. However, those overnight rides include more tossing and turning than anything else (and if you are one of the lucky few who can snooze in the middle seat, good for you).

But there may be one way to help you catch a few ZZZs, and it’s as easy as crossing the “alpha bridge.” 

“Do you want to know a skill that will help you fall asleep on an airplane, or pretty much anywhere else? It’s called ‘going over the alpha bridge.’ It’s very simple,” Erica Terblanche, a psychologist and mental well-being educator, shared in a recent Instagram post. 

According to Terblanche, it takes just four “simple” steps to fall asleep. Step one begins by lying or sitting comfortably (we know this is tough on a plane, but give it a try). Then, close your eyes, and count to 30. 

Step two, “Open your eyes on a little sliver, like a little half-moon, and count to five,” Terblanche said.

Step three is to “close your eyes again, and count to 30.” And lastly, step four, is to “open your eyes again to a little half-moon sliver, count to five, and then close your eyes again and just watch your breathing go in through your nose and out through your nose.” 

Most people, Terblanche noted, “fall asleep in the first cycle.” 

But, why, exactly, is it so hard for some of us to fall asleep on planes? According to a 2020 qualitative analysis, Determining the Predictors for Ease of Sleep While on Aircraft, published in the Journal of Air Transport Management, it’s all the things you’d predict, and more. 

“Unfortunately, it is likely not as easy for individuals to sleep on an aircraft as it is to sleep in daily life, due to in-cabin noise levels, or comfort of the seats,” the researchers noted, citing several other studies on the matter. “Some research has suggested that passengers should combine drug-based interventions with certain ambient or artificial lighting conditions to combat disturbed sleep on intercontinental flights.”

Additionally, for many, the fear of flying can cause a sleepless flight, “as well as anxiety regarding potential sleeping arrangements on an aircraft or mitigating the effects of jet lag.” Additionally, the team said the number of times a person has flown in the past can also affect their in-air sleep pattern. “As an individual becomes more familiar with flying, it is possible that they would find an increased ability to sleep on a flight.” 

But if you’re someone who flies frequently and still can’t sleep, go ahead and give Terblanche’s tip a try. At the very least, it can’t hurt. 

Tip adapted from travelandleisure.comiii 

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Investor’s Guide to Probability Distributions: Key Insights

Probability Distribution Explained: Types and Uses in Investing

Expected Return: How to Estimate the Future Return of Your Investments Using Probability Distributions


Ballentine Capital Advisors is a registered investment adviser. The advisory services of Ballentine Capital Advisors are not made available in any jurisdiction in which Ballentine Capital Advisors is not registered or is otherwise exempt from registration.

Please review Ballentine Capital Advisors Disclosure Brochure for a complete explanation of fees. Investing involves risks. Investments are not guaranteed and may lose value.

This material is prepared by Ballentine Capital Advisors for informational purposes only. It is not intended to serve as a substitute for personalized investment advice or as a recommendation or solicitation or any particular security, strategy, or investment product.

No representation is being made that any account will or is likely to achieve future profits or losses similar to those shown. You should not assume that investment decisions we make in the future will be profitable or equal the investment performance of the past. Past performance does not indicate future results.

Advisory services through Ballentine Capital Advisors, Inc.


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