How Position Sizing Based on Probability of Returns Can Save Your Portfolio


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Jae Jun

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As I was writing this, I got Vitaliy Katsenelson’s latest article on J.C. Penney.

If Katsenelson is a new name to you, he is a prominent investor and fund manager in the value investing community. He is also the author of Active Value Investing where he introduces the Absolute PE model which I’ve modified and incorporated into the OSV Stock Analyzer.

The article Katsenelson wrote was based on the errors of investing in J.C. Penney, but the main point he was conveying was the use of probabilities and position sizing.

Something that isn’t discussed enough.

What I can say is that by including it into your analysis, your decisions become more systematic, emotionless and low risk.

Before I continue, please click on the image below to download a mobile-friendly, PDF version of this artile that you can take on the go.  

Download a PDF of this Article

Download a PDF of this Article

Investing is a Probabilistic Adventure

I don’t think buying the department store chain’s shares was a mistake. Investing is a probabilistic adventure: You assess upside and downside probabilities of a potential investment, and if at the end the balance is significantly favorable, you pull the trigger. – Source

One of the many reasons why Warren Buffett is so successful at what he does is because he is a probability machine. He is able to internally calculate the probabilities of every investment opportunity and act accordingly.

His love for Bridge isn’t surprising because it’s a game where the  most successful players are able to judge mathematical probabilities to beat their opponents.

Same concept with poker.

Not only are you forced to bet based on the probabilities, a strong mental state is required to win at it.

David Einhorn is an avid poker player, even going to Las Vegas to participate in poker tournaments. One thing that he wrote was that every investor should play poker, just to learn about themselves.

Poker tends to bring out your inner enemies to the surface.

  • If you believe that you are good at analyzing, poker will show you that you suck
  • If you believe that you are able to keep calm in the face of losses, poker shows you that you suck
  • and the list goes on

My tip is for you to download a poker game on your phone or play it from facebook and play like it’s real money. Then you’ll see what you need to work on when it comes to investing too.

Back to probabilities.

The Forgotten Analysis of Looking at Probabilities

Let’s say you found a hidden gem, read the reports, identified the risks, listened to management, and calculated the intrinsic value.

Everything points to a buy.

So what happens next?

An order is placed based on how confident you feel about the company.

But there is an additional step that gets forgotten just before the buying phase.

What Buffett says is that you should weigh the probability that certain events will or will not occur.

Simple Probability Questions You Should Ask Yourself Before Buying Anything

Let’s look at Apple (AAPL).

As a shareholder, I argued that Apple is not worth $460. The way I did it was by using reverse stock valuation with the stock analyzer to see what the expectations of the stock were.

By running reverse valuations, you are forced to think and answer simple probability type questions.

  • What is the chance that the stock price is due to a deterioration of the business?
  • What is the probability that Apple will not be able to meet the market expectations?
  • How likely is it that growth will be negative?
  • What is the likelihood that FCF will grow xx% over a 5 year period?

Simple questions that a professional analysts wouldn’t bother asking.

But these are fundamental questions to make rational decisions.

Thinking in terms of probabilities has its advantages. You end up focusing on the longer term instead of quarterly or annual numbers.

I could have easily answered the question of “will Apple exceed market growth expectations this year?” with a no, ultimately not buying the shares.

But when you articulate the question, it ends up coming out to be something like “will Apple exceed current market growth expectations in the next 3 years?”, my answer turns into a yes, and I’m more than happy to hold and ignore the short term noise.

A More Concrete Probability Calculation with InfuSystem

A practical example of applying probabilities comes from analyzing risk arbitrage special situations.

Let’s pick apart my failed arbitrage in Infusystem (INFU) to see what I mean.

INFU Outcome Table

INFU Outcome Table

This table shows how I viewed the end result. I placed a bet that there was a 70% chance of the deal going through and doubled down.

Boy was I wrong about that.

For what it’s worth, once the buyout was canceled, the stock price did fall back to $1.50 on the first day, but with disappointed investors, the stock price fell to as low as $1.30.

INFU after the Buyout is Canceled

INFU after the Buyout is Canceled

If something like this happens to you, your gut tells you to take the loss and walk away.

It’s what most people think as you can see from the comment I was getting.

Use peoples reaction as an indicator

Use people’s reaction as a contrarian signal

My bet that the deal would go through was incorrect, but my bet that InfuSystem was worth more was not.

The investor trying to buy out InfuSystem offered a lowball offer of $1.85 to $2.00. Management came back and said the business was worth more.

So in this case, which probability is higher?

  • Probability A: Stock is worth less than $1.50
  • Probability B: Stock is worth at least $1.85

I placed more odds that option B is correct and bought a little more during the fall.

As mentioned before, thinking in terms of probabilities helps you stay focused and see beyond short term cycles and criticism.

But Your Probabilities are Wrong

In case I’m drawing up a dreamy picture of limiting downside by slapping on probabilities to everything, I want to bring up some realistic view points.

Probabilities are going to be incorrect most of the time.

In Katsenelson’s article, he mentions that he thought J.C. Penney had a 70% of tripling versus a 30% chance of dropping 40%.

We believe position sizing should be driven not by reward but by risk. J.C. Penney had a terrific upside, but it still had a 40 percent downside, with a meaningful 30 percent probability. However, when we sold the stock at a loss, the impact on the total portfolio was less than 1 percent.

The trouble is that quantifying 70% or 30% is impossible.

Random events cannot be quantified.

Then what’s the point?

The alternative method that I apply is by determining what the max upside and downside is.

I did it in the InfuSystem case.

Here it is again.

Position Sizing INFU

INFU Upside vs Downside

If ever I need to quantify a probability, I make sure that I know what the upside is versus the downside.

In this case, it was a 30% upside compared to a 13% downside.

Breaking it down like this then makes the decision of how much to buy easier.

Probability of Returns and Position Sizing

This all ties in together with position sizing.

Positions are bought in two ways.

  1. Systematically following a rule or
  2. Gut feelings and confidence

The most common way of building a position is option 2.

When I came across Dacha, I dreamed of the profits I could be making with this stock.

Long story short, it was one of my worst picks and also one of my worst position sizing.

Instead of performing the probability check and then applying a position size accordingly, my eyes flipped over in greed, and I bit off way more than I could chew.

I sold at a mega loss.

In fact, that post is still listed as #8 on the most popular posts on the blog page. I’m ashamed that it’s still there, but it’s a sober reminder to screw my head on the right way.

my wall of shame

Thankfully, my position sizing has improved drastically since.

Going back to Katsenelson’s thesis on J.C. Penney, he saw that the upside was 300% vs downside of 40%.

A possible drawdown of 40% for any stock is a high risk stock, but with a tempting upside of 300%, it would have been tempting to load up and build a massive position. So I applaud Katsenelson’s ability to not be led astray by the upside and to limit the impact to his portfolio to less than 1%.

Here’s some more insight into how he sizes positions now.

Each company in our portfolio gets a rating for the quality of its business: the size of its moat, the strength of its balance sheet, how it fits in its industry. We assess its management in two dimensions: how good it is at running the business (building moats around it) and at allocating capital. Last, there is an X factor, where we judge business cyclicality, complexity and transparency (banks, for instance, would never get a high score there). Then we balance the totality of these factors against the cheapness of the stock: Should we take a starter position or a full position?

The Kelly Criterion for Portfolio Allocation

If you want a more mathematical approach, check out the Kelly Criterion.

It was first widely used by the gambling society but has since spread to the investing community as well.

It’s a formula that will tell you how much you should put into a single bet or position based on your stock winning probability and win/loss record.

Here’s a Kelly Criterion calculator you can use for position sizing.

For extra reading, here is one from Investopedia that dives into the details of the Kelly Formula.

F Wall Street also discusses a practical way of applying the Kelly Formula based on Mohnish Pabrai’s The Dhando Investor which I also list as a top 15 book to read.

Ultimately the Price You Pay is the Most Important

Quantifying events is something that you need to think about, but close to impossible to get right.

Where does that leave you?

The only thing that you do have control over with a realistic chance of applying probabilities is the intrinsic value.

Too many investors purchase stocks without knowing what they are buying and what it’s worth. Whatever the next person says is the intrinsic value.

That’s where the OSV stock analyzer comes in as well as the other free spreadsheets and calculators I have up for you to use. By calculating the stock value and knowing what it’s worth before buying, it makes it easier to size your position.

I don’t bother reading what others are saying Apple is worth. All I know is that it wasn’t worth $460 when I bought it.

It’s worth more.

InfuSystem wasn’t worth $1.30 when it hit bottom this year.

It’s worth more.

On the flip side, I’m afraid to say the same thing for LinkedIn ( LNKD) or Netflix (NFLX). The downside is just as real as the upside with equal probabilities on the downside.

Tesla could do no wrong for investors. The consensus was that hitting $200 was a 100% probability. Downside was 0%.

Learn from my mistakes and start applying probabilistic thinking into your analysis.

It ties directly into position sizing, and you will definitely see less drawdowns and save your portfolio from the possibility of a blowup due to one mistake.

Tell me Your Position Sizing Method

Leave a comment because I’m interested to hear your thoughts. Do you analyze in terms of probabilities?

How have you been sizing positions?

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16 responses to “How Position Sizing Based on Probability of Returns Can Save Your Portfolio”

  1. Beau Kramer says:

    I tend to size positions based on how long I expect the position to play out. I tend to make larger bets on longer horizons because I feel like rationality will win in the long term and the stock will price.

    I haven’t used probabilities before, but after reading Vitaly’s article last night and yours this morning, I am definitely incorporating it into my process. Both remind me of Nate Silver’s book, the signal and the noise, which I think is actually a great read for investors to understand a probabilistic mindset. Anyone else read it?

  2. Thanks for the book rec. Another one to look up and add to my already long list 🙂

    I pretty much did what you do as well. If it’s cheap and it has good potential over a long time, I buy bigger positions.

    The problem I found is that, even my best ideas are wrong. By keeping it systematic and limiting it to rules, it makes the whole sleeping at night so much easier.

    If you are correct, you may miss out on huge concentrated gains, but again, the idea comes back to limiting downside and not losing money.

    Dacha for me was the real eye opener.

  3. Great post.

    “The alternative method that I apply is by determining what the max upside and downside is.”

    This is pretty much the same thing I do. People are horrible at judging probabilities; but, you can also look at historic probabilities if you use a mechanical investing strategy and just assume that things are they same this time around, as well.

    In the end the whole exercise boils down to buying at as deep of a discount as possible.

  4. yes margin of safety is king.

    But also have to consider that there are black swan events that can occur with any company. No way of predicting something like that. It’s a small chance that it could occur, but it does occur and that’s where blowups can occur.

    With your net nets, how big of a position do you buy?

    are you like Graham in that you try to buy as many as possible?

  5. Hi Jae,

    No, actually I embraced the diversification strategy Greenblatt wrote about in his book, “You Can be a Stock market Genius”. I buy the best possible net net candidates based on how low the company is selling for versus the firm’s net current asset value or net cash value, how much debt the company has, how stable the net current asset value is, and other qualitative factors that tend to boost returns.

    Ideally, I try to have no more than ten positions and have started putting more money into my best ideas instead of distributing my funds equally among all positions.

  6. Jacob says:

    Jae, since you are investing companies mainly for their steep discount to intrinsic value you should be very diversified. Your investment style more closely follows Walter Schloss not Warren Buffett and I think using more of Schloss’s investment philosophy instead of Buffett’s will help you maximize your returns

  7. well that is partly true. I have holdings where they are more Buffettesque than Graham or Schloss 🙂

  8. cse says:

    How you are able to end up with the exact probability figures like the 300% upside and 40% downside , And how based on these figure do you find the size of investment to be mande

  9. Those were Vitaliy’s numbers so I cant say for sure, but what I do know is that you need to value the stock and that way you can determine the upside and downside.

  10. JohnnyB says:

    I’m now 3 years into investigating trading systems in addition to trading value stocks. Position sizing is the most important aspect of success investing/trading but conveniently gets ignored because few people can handle the complexity of something that seems as easy and straightforward as figuring out how much of your portfolio to risk.

    The best author on position sizing is Van Tharp’s Definitive Guide to Position Sizing. This book is a beast to handle at first but is worth multiple reads to fully understand the importance of position sizing. Another great book of his is Super Trader. I would suggest starting with Super Trader.

    Although I’m sure most readers here see themselves as investors and not traders, there is great knowledge to gain when viewing your investing as really speculation, and thus trading. Unless you plan on holding a stock for at least 10 years, then you should view yourself as a speculator that trades stocks.

  11. Mr. Ed says:

    I loved this article and the discussions in the comments section. I find that most position sizing articles I find on the Web deal with technical analysis and aren’t very helpful to someone who is value oriented.
    I’m not sure if this falls under position sizing or not but my dilemma seems to revolve around taking the dollar amount I’ve allocated for the stock based on the probability it will work out and actually buying it. Even with a margin of safety I’m certain the price I buy isn’t going to be the low before a move higher. How do I manage this process? Do I buy it all at once? Should I use some of the money I allocated for this stock to make an initial buy and use the rest to try and average down? If I average down what criteria do I use? Should I instead pull cash from my overall portfolio cash bucket to buy more stock to get a better average price?
    A future article or book recommendation that deals with this topic would be great.

  12. Mike says:

    I am mostly a swing trader, not an investor, but I love reading about the way you think about things and your approach. Additionally, position sizing and probability assessment is something that regardless of approach matters quite a bit.

    Please take note: the Kelly criterion is based upon the assumption that you have an infinite time horizon. The very aggressive position size weathers the vicious downswings because it has an infinite amount of time until it strings together enough compounding wins to compensate. However, I have created my own simulations of position sizing, and found that based upon a fixed time frame of 300 trades, the distribution (histogram) of monte carlo simulation of 1000 traders begin to show a very strong skew right as risk increases with an unacceptably high probability of catastrophic results. Increase it to 1000 trades and it only reduces the skew slightly. Trading at a full Kelly showed drawdowns of 80% or more. Even trading at 1/5th the Kelly criterion showed vicious drawdowns of over 30% at times.
    Not to mention the relationship between risk and results is bell shaped. In other words 1/2 the full Kelly risk will produce 3/4ths the return with half the volatility.

  13. thanks for sharing that Johhny

  14. Buying into positions all come from personal experience and what you are comfortable with. I only wish there was a book that had a step by step method. But since investing is more art than science, it all has to be personally experienced.

    I used to buy all at once, but I’ve changed to slowly building my position. That way, I can buy more if it goes down, and since the stocks I buy are supposed to be cheap, even if it goes up a little, I don’t mind because I prefer not to fuss over cents in the long run.

  15. ahh very interesting. I dont heed exactly to the Kelly criterion, but it does help. Interesting to hear about your simulation though.

  16. Daniel Victor says:

    There is another reason to limit size in a particular stock – liquidity.Of course,this doesn’t apply to mega-caps.But with small caps,buying too many shares can force the price up,and make it harder to get out at the right price at the right time.

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