The Sizing of a Bet – or the Art of Not Blowing Up While Getting the Best of It

In many areas of life size is overrated in my opinion. The wife thinks otherwise – which is always a cause for concern as evening approaches… Now, when it comes to investing you cannot afford to close your eyes to uncomfortable facts. And one such fact is that if your strategy to outperform the market relies on high concentration on a relatively small number of high conviction ideas you will blow up at some point in time if your understanding of the mechanics of bet sizing is flawed.

The better the odds, the bigger the sizing – then readjust for risk

The main idea is very intuitive to most. We size each bet according to the expected value of it. EV being the difference between the stock price and what we perceive to be fair value after careful calculation, which is the difficult part. Only then do we readjust for risk by scaling down when the risk is high and conversely scale up when it is not.

(Two notes: 1) EV in this context should not be confused with Enterprise Value, and 2) The time horizon is unimportant. A ”bet” could easily be a 20 year long position. I use the term ”bet” in the title for two reasons. The more important one is the wife thinks ”position” is better used in other contexts. And so being sensitive to risk I opted for ”bet”. The secondary and less important reason is that on a musical level ”bet” is more pleasing to the ear!)

Why the highest EV idea should not always be our biggest position

Assume that we are magically confronted with two bets that both have a 3x expectation and the outcome of both will be decided tomorrow. Bet both our own house and that of our grandma, right? You know the answer: Not so fast – it depends on the risk involved. So let’s quantify:

In example A there is a 30% chance the company gets bought tomorrow at 10x our investment and a 70% chance the deal fails and as a result that the company goes belly up and our investment is worth 0.

In example B the numbers are 4x 75% of the time, and the remaining 25% of the time it goes bust and we get 0.

The Expected Value of each is the same:

EV in example A: (10 x 0,3) + (0 x 0,7) = 3x

EV in example B: (4 x 0,75) + (0 x 0,25) = 3x

Let’s ignore the fact that both of these very hypothetical bets rank in the ”72 virgins in the sky”-sphere, ie. not gonna happen, but really, really nice to think about nonetheless and let’s instead agree that bet A is way riskier than bet B though the average expectation for each is the same. Neither proposition are in the ”bet grandma’s house”-range, but for some very risk tolerant dudes or gals bet B may actually be within the ”bet our own house”-range.

Size really is everything!

The EV is massive for both bets and therefore no-brainers that are criminal not to make. But since we ruled out betting grandmas house on the grounds it was too risky (though plenty of people had no problem taking risks at the expense of others in the years leading up to 2008!) the question then becomes one of sizing.

The Kelly criterion not suitable for everyone

How much then? Some use the so-called Kelly criterion, which is a mathmatical formula that provides the theoretically optimal way to size one’s bets to maximize long-term growth of capital. I would caution against it, however, since it does not take your own personal situation into account, such as cash flows from your job, risk preference, age, mental constitution and last but not least errors in judgement. To add a margin of safety some will use fractional Kelly (mentioned in the link above).

What I do

I don’t use Kelly myself but I do factor in both my personal cash flow situation, the number of opportunities I currently am aware of, my conviction level in each as well as the risk inherent in each – all to best of my ability.

The gold standard: many non-correlated high EV bets

The gold standard is to have a ton of non-correlated high conviction bets. For that to be the case one of two things need to happen.

One is you need to first narrow the field of ideas intelligently and then do a huge amount of work on all of the most promising ones. Only a rare few possess both the intelligence and the inclination for constant hard work such as a Warren Buffett, a Peter Lynch or a Mike Burry (think The Big Short), so many high EV ideas will be hard to come by for most.

Investment ideas from people you know to have the right process in place can be a source but remember you will need to have a thorough understanding of the drivers and the details of the business in order to know when to sell so work is always necessary unless you take to index investing, which is probably preferable for most people – though I will provide another solution that almost certainly will beat the market over time at the end of this post without too much work. It probably won’t be by a huge amount but it will beat indexing.

Absent that you may need to wait for the market to take a big dive and suddenly the waters may be ripe with opportunity.

A history of three 50%+ positions – and a gut wrenching loser

In the past five years I have had three 50%+ positions. Most people I know would never even consider this a possibility and it certainly isn’t the gold standard you want to aim for most of the time. However, I would argue there are times and situations when huge bets are very rational and not especially risky.

The first of those three times was in early 2012 in a Danish company called United International Enterprises, or UIE. This is a holding company where the discount to net asset value (NAV) had risen irrationally to 46% while the historical average was in the 20-30% range. The assets were stable in nature and on top of that the brothers who held the reins had recently bought shares for 1.5 million USD in the company. To me this seemed like a situation with both significant upside and a lot of safety on the downside attached to it, which is what you want out of a big bet. It worked out well and after 7-8 months the discount had narrowed to more normal levels and so I sold.

The second situation was a Swedish company called East Capital Explorer in late september 2013. The situation is similar in many respects, except the assets were more diversified and it required some work to ascertain the NAV as the holdings consisted of 40+ companies in 10 different countries listed on obscure exchanges. The company would issue a quarterly update of the NAV in 8 days time and due to heavy insider buying (8 million USD) I thought there was a good chance the NAV would increase significantly and that I could front run the update by doing the work myself and know the result ahead of time. I concluded it was at 42% while the historical average was at around 25% so I took a large position and sold it 2 months later when the discount went below 25% and received a 30% gain in exchange for a little effort.

The third case is Wilhelmsen Holding. I’ve written extensively about that so no need to go into detail here except to say that since it is also a conglomerate with a huge discount to NAV as the other two cases the risk is greatly reduced when comparing to a one trick pony type of company that sells products within one sector only. Leverage at the holding level is also very limited in this case.

Huge insider buying adds to safety as well. Also fraud becomes less of a concern in conglomerates and investment companies with many holdings, especially when insider buying has been prevalent.

Greed gave rise to this post 

Back in October 2014 I got greedy and loosened up on my bet sizing requirements (read: I screwed up on them). A Norwegian company in the oil service sector, Polarcus, had essentially been reduced to an option because of declining revenue coming from the big oil companies. The upside was huge but so was the downside risk due to high leverage. Still I thought the EV was attractive enough that it warranted a bet, especially since pension funds were selling out desperately in what seemed to me to be a forced fashion.

My thinking was that these are not the sharpest knives in the industry and that the reason for their selling might be due to institutional market cap constraints following the decline in price or managers simply trying to protect their job by not having a loser in their books. Seeing this I got greedy and violated my own rules and put 20% in it.

The oil price declined further in the weeks that followed after an OPEC meeting in which the Saudis decided they would go for market share rather than cut production. The oil companies cut their capital expenditures more and more as time passed and it started to seem likely that Polarcus would need a cash injection to stay afloat which would probably wipe out the existing shareholders given the size of the loan in comparison to the market cap.

I got out ahead of that, luckily, and took a tough 35% loss on the position. It turned out to be the right decision but still I was taught a painful lesson about sizing.

I don’t regret the bet which I still think was good in terms of risk/reward but hopefully I can control my greed the next time a similar one presents itself and size these option-like situations more appropriately.


The big bets should be reserved for cases where the downside is extremely well protected along with significant upside to protect the opportunity cost. The gold standard is to have many non-correlated high EV bets because then downside risk suddenly becomes much less relevant. This is difficult to obtain at any specific point in time due to both time restraints and due to the market seldom cooperating in providing you with tons of non-correlated high EV bets. And non-correlation is genereally hard to come by in the interconnected world we live in.

So how can someone who isn’t well versed in the markets use the above info and expect to outperform the market? By zooming in on investment companies in particular (and perhaps also conglomerates) and buy when the discount to NAV is above the historical average, preferably by a significant amount, and sell when it goes below. This is an easy way to buy a significant portion of the market while doing it at a discount, which ought to lead to outperformance. Seems almost too easy but it really does work.

Here is a list of Swedish investment companies you might want to look into so you can strike when the discount is out of whack.

Next post

Stay tuned for my next post around New Year. It will be about the mistakes I committed in 2016 as well as the composition of my portfolio going into 2017. Having made almost no changes all year I made some significant ones following the insane Trump victory.

3 thoughts on “The Sizing of a Bet – or the Art of Not Blowing Up While Getting the Best of It

  1. Hi there,

    We very much liked your posts of WWIB & TRE. I commented under another nickname, i.e. Belgian Guy. We have a large position in WWIB as well and agree on the downside protection focus when taking large positions.

    –Kelly formula–
    An interesting result is that error of judgment (without bias) introduces variability around the “optimal” size and that the overbetting side is much worse than underbetting, indeed, in the below source you can find that overbetting harms you twice on a risk-adjusted basis: it doesn’t only lead to higher variability but also lower log returns.

    I tried messing around with an online equation solver (setting the derivative equal to zero) to model multiple outcomes for Treasure ASA (similar equations as noted in the second link below). I like the formula most for relative bets (long-short) although I am not short on Glovis. However this teached me that generally the Kelly formula is very sensitive on how you model the downside (do you include a ruin scenario? if yes how much probability?).

    In short, I think using the Kelly formula in real-life is very problematic to determine absolute % of portfolio. I think the closest applications Merger arb situations and closed-end fund.

    I’m interested to hear if you agree with my “real-life risk-taker” interpretation on how to use the Kelly formula: throw away absolute estimations for % portfolio & “fixed fractional Kelly”. Only look at Kelly estimations on a relative basis *between positions*.

    For example you have:
    1) 6 portfolio positions you want to keep A,B,C,D,E,F
    2) Kelly size estimations for each (A = 20%, B = 40%, C = 35%, D = 50%, E = 40%, F = 60%)
    3) Several reasons to keep X% in portfolio cash (e.g. 10%, see my next point)
    4) Rescale all kelly sizes to get to 90% equity positions

    What this means in fat tony language is that you should only use your individual upside per downside estimations on a *relative* basis with other estimations (throw away the absolute numbers).

    –Cash as a % of portfolio–
    How do you reconcile your “other income” with your absolute net worth “NW”?
    How many months or years of income do you add to your NW to get to an “equivalent NW” for someone who doesn’t have any “other income” (e.g. filthy rich person)?

    If one’s savings from salary is e.g. 20% of one’s net worth and still has many many years to go, how much less cash & more concentrated positions can one hold for the same “unit of risk”*?

    I think this “other income” is very important to estimate true risk and not many people have written on this in finance (I might be wrong? maybe wealth planning literature?)

    *”unit of risk” I hate this term as I sounds like pseudorisk talk lol

    –Closed-end funds/Holding companies–
    I think your investment philosophy very much aligns with mine. I like deep value and I love relative bets in simple situations such as holding company NAV discounts. I’ve done many holding company investments over the years now and I daresay I get most of my dependable returns from this class.

    How can I get in touch with you to share some personal & sell-side research on pan-European holding companies with you? I didn’t understand your Avanza link very well. Can the discounts to NAVs be found somewhere on that page? Do you have a nice research to follow up these discounts? (e.g. Google Sheets)

    Many thanks,


    1. Great points, Belgian guy. I think Kelly is more suitable when the odds are knowable such as in certain gambling situations. Fair value calculations of a stock is ripe with judgment error and can only be approximations so I agree Kelly can be dangerous and underestimate your risk of ruin which is why I don’t use it. But perhaps it could work better in a long/short situation like you outlined.

      I agree the ”other income”/NW ratio ought to be a major factor when determining bet sizing, especially in high volatility situations such as Bet B in my post or stocks that behave like options (such as the company I mentioned, Polarcus). When ”other income” is low or 0 sizing should go down for those significantly, unless you are in the filthy rich category.

      The links to the investment companies was simply to show the names so that people could look into each case for themselves. I don’t know of a site that tracks the discounts. I’m not invested in any of them currently myself.

      Let’s connect by mail to discuss ideas 🙂


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