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	<title>Fusion Investing and Analysis &#187; Probability</title>
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	<link>http://www.fusioninvesting.com</link>
	<description>Fusing Fundamental and Technical Analysis with lashings of Behavioural Finance. Investing in Australia and North America.</description>
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		<title>Ahead of the Buffett Curve</title>
		<link>http://www.fusioninvesting.com/2009/07/ahead-of-the-buffett-curve/</link>
		<comments>http://www.fusioninvesting.com/2009/07/ahead-of-the-buffett-curve/#comments</comments>
		<pubDate>Thu, 23 Jul 2009 12:00:01 +0000</pubDate>
		<dc:creator>Dean Morel</dc:creator>
				<category><![CDATA[Commentary]]></category>
		<category><![CDATA[Probability]]></category>
		<category><![CDATA[MCO]]></category>
		<category><![CDATA[MQG]]></category>

		<guid isPermaLink="false">http://www.fusioninvesting.com/?p=2686</guid>
		<description><![CDATA[Buffett sells down Berkshire's Moody holding with a hat tip to Fusion Investing which sells down MQG ahead of the trading halt.

<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2009/10/buffett-does-what-i-say/' rel='bookmark' title='Permanent Link: Buffett &#8211; Does What I Say'>Buffett &#8211; Does What I Say</a></li>
<li><a href='http://www.fusioninvesting.com/2010/07/a-simple-method-for-buffett-like-returns/' rel='bookmark' title='Permanent Link: A Simple Method for Buffett Like Returns'>A Simple Method for Buffett Like Returns</a></li>
</ol></strong>]]></description>
			<content:encoded><![CDATA[<div class="fblike" style="height:25px; height:25px; overflow:hidden;"><iframe src="http://www.facebook.com/plugins/like.php?href=http%3A%2F%2Fwww.fusioninvesting.com%2F2009%2F07%2Fahead-of-the-buffett-curve%2F&amp;layout=standard&amp;show_faces=false&amp;width=450&amp;action=like&amp;font=arial&amp;colorscheme=light" scrolling="no" frameborder="0" allow Transparency="true" style="border:none; overflow:hidden; width:450px;"></iframe></div><p><img class="alignright" style="margin: 6px;" title="Warren Buffett" src="http://www.fusioninvesting.com/wp-content/uploads/2008/09/buffett-300x226.jpg" alt="" width="300" height="226" />I like to review my share sales after three months. No particular reason for three months, it&#8217;s a quarter, it&#8217;s both long enough and not too long.  Today I&#8217;m going to make a tiny exception and review my Moody&#8217;s sales four days early. Back in April <a href="http://www.fusioninvesting.com/2009/04/moodys-corp-mco-musings/">I sold MCO</a> at $28+, today it closed at $26.52, 7.5% lower than when I sold. Meanwhile the S&amp;P 500 is up 11.3%. Wait a second while I pat myself on the back&#8230; What I feel even more chuffed about is Warren Buffett now selling down Berkshire&#8217;s stake in MCO. Talk about confirmation bias overload. As John Hempton at Bronte Capital points out Buffet is a <a href="http://brontecapital.blogspot.com/2009/07/not-buy-and-hold.html">fantastic seller of stocks</a>.</p>
<p>I normally focus on absolute performance first and relative performance second, but when it comes to my share sale reviews I reverse that. The relative performance is what counts. I find this often saves me from both unwarranted self flagellation and praise.</p>
<p>Take Netflix as an example. I sold Netflix back in February only to see it move ever higher for the next two months and despite a pull back it was 10% higher at my three month review. Fortunately the S&amp;P and Nasdaq were up respectively 17% and 20% so I was able to leave my cat-o-nine-tails in the draw that day. For people who&#8217;ve been reading for a couple months I&#8217;m still not ready to talk about Leucadia, but I assure you my back has almost healed.</p>
<p>On a less self congratulatory note I was ahead of the Macquarie (<a href="http://www.fusioninvesting.com/tag/mqg/">MQG</a>) curve today. Selling 40% of our holding in MQG an hour or so before the trading halt. The price movement before the halt indicates that some people had more than an suspicion that there&#8217;s good news afoot. After opening at $39.89 MQG had moved up to $41.10 by 2.26pm, a healthy 3% gain for the day. Then five minutes later $41.66, two minutes later $42.04. MQG requested a trading halt at 2.37pm with the price at $41.80.  Still as MQG force fed me the shares for $26.60 six weeks ago I won&#8217;t be complaining.</p>
<p>I&#8217;m currently eyeing up a couple companies which appear to have a better risk/reward profile than MQG and which I can actually understand. There&#8217;s also yet another SPP that I may be taking up. These Australian companies are sucking me dry, still you won&#8217;t hear me complaining about it like so many other commentators seem to be. Suck it up guys, if you portfolio is so large that $15k bites are too small for you bother with then you&#8217;ve really got nothing to complain about have you. Take the lambo out for a spin, have a cry into your Crystal and suck it up.</p>


<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2009/10/buffett-does-what-i-say/' rel='bookmark' title='Permanent Link: Buffett &#8211; Does What I Say'>Buffett &#8211; Does What I Say</a></li>
<li><a href='http://www.fusioninvesting.com/2010/07/a-simple-method-for-buffett-like-returns/' rel='bookmark' title='Permanent Link: A Simple Method for Buffett Like Returns'>A Simple Method for Buffett Like Returns</a></li>
</ol></strong>]]></content:encoded>
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		<slash:comments>4</slash:comments>
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		<item>
		<title>How to Size Individual Positions Using Kellyesk Formula</title>
		<link>http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/</link>
		<comments>http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/#comments</comments>
		<pubDate>Wed, 17 Jun 2009 00:02:42 +0000</pubDate>
		<dc:creator>Dean Morel</dc:creator>
				<category><![CDATA[Better Investor]]></category>
		<category><![CDATA[Commentary]]></category>
		<category><![CDATA[Probability]]></category>
		<category><![CDATA[Kelly]]></category>

		<guid isPermaLink="false">http://www.fusioninvesting.com/?p=2079</guid>
		<description><![CDATA[It's over a year since I read any of the articles I linked in yesterday piece on the Kelly formula and portfolio management. After reading the brilliant article by Michael Mauboussin Size Matters - The Kelly Criterion and the Importance of Money Management I realised that both he and I did not explain how to actually use the Kelly formula in sizing individual positions. So let's try to fix that clearing ommision.

<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2009/06/kelly-formula-meet-portfolio-management/' rel='bookmark' title='Permanent Link: Kelly Formula Meet Portfolio Management'>Kelly Formula Meet Portfolio Management</a></li>
<li><a href='http://www.fusioninvesting.com/2010/03/position-sizing-size-really-does-matter/' rel='bookmark' title='Permanent Link: Position Sizing &#8211; Size Really Does Matter'>Position Sizing &#8211; Size Really Does Matter</a></li>
<li><a href='http://www.fusioninvesting.com/2010/07/can-individual-investors-outperform/' rel='bookmark' title='Permanent Link: Can Individual Investors Consistently Outperform?'>Can Individual Investors Consistently Outperform?</a></li>
</ol></strong>]]></description>
			<content:encoded><![CDATA[<div class="fblike" style="height:25px; height:25px; overflow:hidden;"><iframe src="http://www.facebook.com/plugins/like.php?href=http%3A%2F%2Fwww.fusioninvesting.com%2F2009%2F06%2Fhow-to-size-individual-positions-using-kellyesk-formula%2F&amp;layout=standard&amp;show_faces=false&amp;width=450&amp;action=like&amp;font=arial&amp;colorscheme=light" scrolling="no" frameborder="0" allow Transparency="true" style="border:none; overflow:hidden; width:450px;"></iframe></div><p><img class="alignright size-full wp-image-1983" style="margin: 6px;" title="How to size individual stock positions" src="http://www.fusioninvesting.com/wp-content/uploads/2009/06/sumo.jpg" alt="How to size individual stock positions" width="300" height="421" />It&#8217;s over a year since I read any of the articles I linked in yesterday piece on the <a href="http://www.fusioninvesting.com/2009/06/kelly-formula-meet-portfolio-management/">Kelly formula and portfolio management</a>. After reading the brilliant article by Michael Mauboussin <a href="http://www.capatcolumbia.com/MM%20LMCM%20reports/Size%20Matters.pdf">Size Matters &#8211; The Kelly Criterion and the Importance of Money Management</a> I realised that both he and I did not explain how to actually use the Kelly formula in sizing individual positions. So let&#8217;s try to fix that glearing omission.</p>
<p>I&#8217;ve now copied a few more columns from my portfolio spreadsheet to yesterday&#8217;s Kelly Spreadsheet. As I stupidly never documented my reasons for using these formulas, the following is my best guess and why I choose them.</p>
<p><strong>Geometric Mean</strong>:  As I am in the market for the long term, have no intention of withdrawing funds and will compound my investment I choose geometric mean instead of the average (also known as mean or arithmetic mean). For a discussion on this read the Mauboussin article.</p>
<p>Percent of Portfolio: This calculates what percent of my portfolio I should invest in one position. Based on dividing the geometric mean by the sum of the geomeans.</p>
<p><strong>Actual Percent of Portfolio</strong>: Let me stress again all the figures in my example are made up. However, I do hold positions in all of the stocks except Microsoft. This column is exactly what it says on the can. The actual percent of my portfolio that I have invested in that stock.</p>
<p><strong>Adjust Actual</strong>: This is the amount I need to change my investment to align my actual with my calculated amount.</p>
<p>Hopefully that makes it clearer how I use a Kellyesk Formula to determine my individual position sizing in my portfolio. If not please feel free to ask questions. Also, please note that I do not solely rely on the expected return, reward/risk, geomean or percent of portfolio to determine my actual investment. I consider them all and then use my gut for a final decision. Perhaps as time goes on I&#8217;ll get more comfortable relying solely on one or a combination of those calculations, but I&#8217;m not there yet. However, I have found these calcuations incredibly helpful for both my buying and selling of indivudal stocks. I hope you also do.</p>


<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2009/06/kelly-formula-meet-portfolio-management/' rel='bookmark' title='Permanent Link: Kelly Formula Meet Portfolio Management'>Kelly Formula Meet Portfolio Management</a></li>
<li><a href='http://www.fusioninvesting.com/2010/03/position-sizing-size-really-does-matter/' rel='bookmark' title='Permanent Link: Position Sizing &#8211; Size Really Does Matter'>Position Sizing &#8211; Size Really Does Matter</a></li>
<li><a href='http://www.fusioninvesting.com/2010/07/can-individual-investors-outperform/' rel='bookmark' title='Permanent Link: Can Individual Investors Consistently Outperform?'>Can Individual Investors Consistently Outperform?</a></li>
</ol></strong>]]></content:encoded>
			<wfw:commentRss>http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/feed/</wfw:commentRss>
		<slash:comments>0</slash:comments>
		</item>
		<item>
		<title>Kelly Formula Meet Portfolio Management</title>
		<link>http://www.fusioninvesting.com/2009/06/kelly-formula-meet-portfolio-management/</link>
		<comments>http://www.fusioninvesting.com/2009/06/kelly-formula-meet-portfolio-management/#comments</comments>
		<pubDate>Tue, 16 Jun 2009 03:56:40 +0000</pubDate>
		<dc:creator>Dean Morel</dc:creator>
				<category><![CDATA[Investing Insights]]></category>
		<category><![CDATA[Probability]]></category>
		<category><![CDATA[Headline]]></category>
		<category><![CDATA[Kelly]]></category>

		<guid isPermaLink="false">http://www.fusioninvesting.com/?p=2011</guid>
		<description><![CDATA[Using the Kelly Formula for Portfolio Management:

I have two main uses for the Kelly formula, neither of which are the intended purpose. My third and less important use is the intended purpose of Kelly, an aid in determining my indiviudal investment size.

I use Kelly as a comparitive tool and as and aid to focusing on probabilities, risk and returns. Before we dive into each of those I should explain how I use a method that is designed to be used with binary outcomes with known odds for investing, where their are multiple possible outcomes with unknown odds.

At the bottom I link some great articles from Ed Thorp, John Kelly, Michael Mauboussin and Paul Samuelson

<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/' rel='bookmark' title='Permanent Link: How to Size Individual Positions Using Kellyesk Formula'>How to Size Individual Positions Using Kellyesk Formula</a></li>
<li><a href='http://www.fusioninvesting.com/2009/11/kelly-capital-launches-100x-leveraged-etfs/' rel='bookmark' title='Permanent Link: Kelly Capital launches 100x leveraged ETFs'>Kelly Capital launches 100x leveraged ETFs</a></li>
<li><a href='http://www.fusioninvesting.com/2010/05/500-posts-later/' rel='bookmark' title='Permanent Link: 500 Posts Later'>500 Posts Later</a></li>
</ol></strong>]]></description>
			<content:encoded><![CDATA[<div class="fblike" style="height:25px; height:25px; overflow:hidden;"><iframe src="http://www.facebook.com/plugins/like.php?href=http%3A%2F%2Fwww.fusioninvesting.com%2F2009%2F06%2Fkelly-formula-meet-portfolio-management%2F&amp;layout=standard&amp;show_faces=false&amp;width=450&amp;action=like&amp;font=arial&amp;colorscheme=light" scrolling="no" frameborder="0" allow Transparency="true" style="border:none; overflow:hidden; width:450px;"></iframe></div><p><img class="alignright size-full wp-image-2038" style="margin: 6px;" title="NO! Not that Kelly" src="http://www.fusioninvesting.com/wp-content/uploads/2009/06/kelly.jpg" alt="Kelly" width="300" height="409" />The Kelly formula or criterion is best known as a bet optimisation tool. Popularised by Ed Thorpe, the formula which is named after its creator, John Kelly, is used by gamblers to determine the optimal bet based on given odds.   A year ago I mentioned that I use <a href="http://www.fusioninvesting.com/2008/05/using-probabilities/">Kelly criterion in my portfolio management</a> and I promised to expand on that. While I have thought in terms of probabilities for a long time it was only a couple years ago that Mohnish Pabrai, in in his book The Dhando Investor, introduced me to the Kelly Criterion. Let&#8217;s get stuck into how I use a betting tool for money and portfolio management and then I cover some Kelly background. If you&#8217;re a Kelly novice you may want to read the second part first.</p>
<h3>Using the Kelly Formula for Portfolio Management</h3>
<p>I have two main uses for the Kelly formula, neither of which are the intended purpose. My third and less important use is the intended purpose of Kelly, an aid in determining my individual investment size.  <strong>I use Kelly as a comparative tool and as and aid to focusing on probabilities, risk and returns</strong>. Before we dive into each of those I should explain how I use a method that is designed to be used with binary outcomes with known odds for investing, where there are multiple possible outcomes with unknown odds.  I mentioned yesterday how I was struggling to articulate my use of the Kelly Criterion. The reason for this lies in my own confusion. I could not reconcile how my simplistic spreadsheet calculations mapped to the Kelly Formula. I have spent countless minutes trying to reconcile my spreadsheet to this site <a title="Kelly Criterion Bet Sizing" href="http://cisiova.com/betsizing.asp">which computes the Kelly Criterion for multiple outcomes</a>. If only I&#8217;d annotated my spreadsheet with my reasoning I&#8217;d have saved myself those minutes.  I took a different tact this morning and looked at my spreadsheet and thought &#8220;what the hell was I thinking and what are those figures telling me.&#8221; I then recalled I had dismissed the use of Kelly and settled on a much simpler approach of expected average return. I&#8217;ll call it Kelly for dummies. Here are the steps in my process and (hopefully) an embedded spreadsheet to illustrate.</p>
<ol>
<li><strong><span style="font-family: mceinline;">For each current and possible investment determine three target prices</span></strong>. I unimaginatively call the targets, low, medium and high. You may wish to call them forkit, it&#8217;lldo and yeforkingha, I&#8217;ll leave that detail up to you. The idea is you <strong>conservatively</strong> determine the worst, best and most probable outcomes (low, high, medium). In investing the worst outcome is always zero, i.e. a 100% loss. You can decide what to use for worst, but I suggest continuously using 100% would not be very useful in the calculation or you analysis of probable downside. Think of worst as worst probable outcome.</li>
<li><span style="font-family: mceinline;"><strong>Then give each outcome a probability.</strong></span> My default is 30%, 40%, 30% for worst, probable and best respectively. I alter those defaults when I have a strong conviction.</li>
<li>Then based on the current asset price your spreadsheet calculates the average expected return.</li>
</ol>
<h2> Example Spreadsheet.</h2>
<p>If missing try <a href="http://public.sheet.zoho.com/public/moreld/kelly">this link</a>.<br />
<iframe width="580" height="300" frameborder="0" scrolling="no" src="http://sheet.zoho.com/publish/moreld/kelly"> </iframe></p>
<p>I use the average expected return for an investment to compare  all my actual and possible investments. The relatively higher the expected return the more likely I am to invest and the larger my investment will be. Steps one and two are by far the most important. They are the focusing steps. Determining targets and especially the downside worst case focuses my efforts on a key aspect in investing, managing risk and reward. Ascribing percentages to those targets is a double check. In general I find a lot of investors spend too much time focusing on intrinsic value to the detriment of risk and reward analysis. <strong>Investing is an exercise in probability not in mathematics</strong>.  The expected returns is also part of my selling criteria. It quickly highlights any stocks I should be considering selling. That&#8217;s a topic I&#8217;ll cover another day.  Another benefit of this approach is avoiding delusion. You don&#8217;t have to be delusional for you mind to fool you into distorting history. We all like to think we&#8217;re better than we are and our mind is happy to oblige with lashings of <a href="http://en.wikipedia.org/wiki/Hindsight_bias">hindsight bias</a>. Recording your targets and other observations up front and then revisiting them is an excellent way to keep hindsight bias at bay.<br />
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</p>
<h3>Putting it all together</h3>
<p>Here is an example from my investment in Biota. I starting investing in Biota in October 2008, with the price around $0.40. I calculated the worst probable outcome as $0.30 with a probability of 40%, the probable outcome of $0.75 and the high of $2.00 both with 30% probabilities. [Note: I did publish those figures, though I actually presented four outcomes.] The average expected return from that was a high 136% or an average target price of $0.95. For me that is a very high average return and combined with other factors led me to make a large investment in what many people would have considered a speculative stock.  So what were some of those other factors?</p>
<ol>
<li><strong>Return to risk.</strong> With the exception of my dividend producing core stocks, I don&#8217;t get out of bed for a return to risk of less than three. That is I&#8217;m looking for a return of three times my risk, 3:1. For Biota I <a title="Biota Analysis and Valuation" href="http://www.fusioninvesting.com/2008/11/biota-analysis-and-valuation/">commented</a> &#8220;<em>Like beauty valuations are also subjective, a combination of art and science. My range of values for Biota range from $53M &#8211; $215M, with per share $0.70 &#8211; $1.20 looking like fair value and $0.30 as downside value.</em>&#8220;  So return was $0.30 to $0.80 with risk $0.10 for a return to risk ratio between 3:1 and 8:1. Not only do I get out of bed for that I leap out and sprint naked to my computer to place a buy order. I apologise for that mental image <img src='http://www.fusioninvesting.com/wp-includes/images/smilies/icon_wink.gif' alt=';-)' class='wp-smiley' /> </li>
<li><strong>Buffett&#8217;s Rule Number Two: Don&#8217;t forget rule number one, never loss money</strong>. I admit to playing fast and loose with those rules. As my teachers often commented &#8220;could try harder&#8221;. Despite not being great at implementing rule number two, I do get excited when I find an opportunity that has very little downside and plenty of upside. The investing world is full of opportunities with considerable upside, but they are predominately accompanied by plenty of risk of loss of capital. Opportunities like Biota are beautifully illustrate what Mohnish Pabrai meant when he coined the phrase &#8220;<strong><span style="font-family: mceinline;">Heads you win a lot, Tails you lose a little</span></strong>&#8220;. Low risk opportunities with possible high returns. In general I consider myself a value growth investor. I buy growth when it is on sale.</li>
<li><strong>Confirmation bias</strong> is considered a cognitive bias and as such is tainted. Heck who wants a cognitive bias! While I don&#8217;t want a bias I am happy to utilise them. The only cognitive bias I am terrified of is <a title="Bias blind spot" href="http://en.wikipedia.org/wiki/Bias_blind_spot">bias blind spot</a> — the tendency not to compensate for one&#8217;s own cognitive biases. While I am a smart guy, I recognise the investing world if full of smart people and that being the case I am unlikely to be the first person to &#8220;discover&#8221; a wonderful opportunity. I like to look at a share register to see if any investors I respect are on the register. When I looked at Biota I discovered not only was management buying shares back at almost twice the current price, but Hunter Hall International (HHL.AX, who I mentioned <a href="http://www.fusioninvesting.com/2009/06/australian-stocks-im-looking-at/">yesterday</a>) had also been buying shares at considerably higher prices and then held 12% of shares. As nothing had fundamentally changed since their recent purchases I took a big hit on the confirmation bias bong.</li>
<li><strong>Circle of confidence</strong>. Tick, I know and like biotech investing.</li>
<li><strong>Possible catalysts</strong>. Tick, tick, tick. While the swine flu outbreak was not predictable a large one quarter uptick in Relenza sales certainly was probable and Biota had a pipeline which could deliver good news.</li>
<li>There&#8217;s more, but I am now so far off topic that I must stop and get back to Kelly.</li>
</ol>
<p>Here is an example spreadsheet which you can copy to Excel. I did this in Zoho sheet so you can play around with a few examples and easily copy it to Excel. Unfortunately Zoho sheet has a few limitations so if you&#8217;re entering new shares you need to copy down the formulas and change the current formula to your ticker (unlike Google Docs, zoho does not let you reference other cells for stocks updates). Also note, I plugged in the first numbers that sprung to mind, these are not intended to be actual target for any of the stocks.</p>
<p>[<strong>UPDATE:</strong> I have now updated the above spreadsheet. Check out how I use <a href="http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/">Kellyesk calculations in my sizing of individual stock positions</a> for an explanation of the new columns.]</p>
<h3>Kelly Background and some Notes I&#8217;ve copied over the years.</h3>
<blockquote><p>I think Munger&#8217;s recent book recommendation, <strong>Fortune&#8217;s Formula</strong>, and many of the papers referenced in the bibliography are pertinent to this discussion. The book is about <strong>Kelly</strong>&#8216;s criterion, which is a formula for sizing bets to maximize long-term compound return from a series of bets where the winnings are reinvested.  <strong>Kelly</strong>&#8216;s criterion has two components: edge and odds. Edge is the amount of profit that YOU BELIEVE that you will make if you could repeat this bet many times with the same probability. Odds are the market place or tote board odds. Your odds must be different from the market place odds or you don&#8217;t have an edge.The optimal return is obtained by betting a fraction of your portfolio equal to the edge/odds.  Overbetting and underbetting result in sub-optimal results. However, overbetting is more serious because it leads to a large variation in returns and to eventual blowups (LTC and Eifuyu were examples of overbetting). Underbetting reduces returns and variance. The book presented Ed Thorpe&#8217;s (one of the best <strong>Kelly</strong> criterion investors) hedge fund results which were spectacular for the degree of over-performance and the small variance over a 20 year period. Ed Thorpe always underbet – specifically made bets of half the <strong>Kelly</strong> criterion because he was worried about being too confident about his assumptions in investing and unconsciously overbetting.  Long story short, the book and especially the papers show that if you really good at finding investments with large edges, you could get decent returns without a lot risk holding only four or five positions. If you can only find investments with smaller edges, you will get less return and need more positions to reduce risk.  From reading the book and the papers, I found that I was intuitively doing the right thing. However, I am planning to now track the edge and odds on investments to see if I can improve.  <a title="Kelly Criterion and Fallacy of Large Numbers" href="http://boards.fool.com/wsj-how-the-big-money-picks-a-manager-24100780.aspx?sort=whole">via TMF : Liquid Lounge</a></p></blockquote>
<p><a href="http://www.casact.org/pubs/forum/94sforum/94sf049.pdf">Risk and Uncertainty:  A Fallacy of Large Numbers</a></p>
<blockquote><p>Basically the idea is that if a single play is not acceptable, then no sequence should be acceptable, when the goal is to maximize the expected utility. The theory is basically that people, when thinking about the law of large numbers, tend to forget that even if you play 50 million or more you still don&#8217;t have full certainty that you will win, but at the same time <strong>your potential losses</strong> grow accordingly (you could sequentially play 100 million times and still not win, losing a lot of money if the idea was to be risk averse).  Mathematically the expected value of return is the same no matter how many times you would play, it&#8217;s a simple multiplication; but psychological humans behave very different. I don&#8217;t remember the source off the top of my head, but there have been some experiments asking what people were willing to pay for the probability of winning something (i.e. how much they would pay to increase a single percentage point in their chance of winning). The increase from 0% to 1% (lottery) or from 99% to 100% (insurance, the assurance of winning) will always command much higher prices than, for example, paying from 34% to 35%; even though the expected return is the same in all cases.  also <a href="http://boards.fool.com/when-do-you-play-a-lottery-22927950.aspx?sort=whole">via TMF: Liquid Lounge</a></p></blockquote>
<p><strong>But wait, there&#8217;s more, how about a set of steak knives..</strong> Each of the following articles and sites are excellent resources well worth taking a look at<strong>.</strong></p>
<ul>
<li>For those disappointed with my spreadsheet here is the <a href="http://cisiova.com/betsizing.asp">Kelly Bet Sizing for Multiple Outcomes</a> program.</li>
<li>The best site on the web I know of for <a href="http://www.bjmath.com/bjmath/kelly/kelly.htm">papers and discussion of the Kelly Formula</a>. The site includes a link to the original <a title="Original Kelly article A New Interpretation of Information Rate" href="http://www.bjmath.com/bjmath/kelly/kelly.pdf">1956 article, A New Interpretation of Information Rate</a>, which appeared in the Bell Systems Technical Journal.</li>
<li>Ed Thorp <a href="http://www.bjmath.com/bjmath/thorp/paper.htm">The Kelly Criterion in Blackjack, Sports Betting, and the Stock market</a>.</li>
<li>Michael Mauboussin <a href="http://www.capatcolumbia.com/MM%20LMCM%20reports/Size%20Matters.pdf">Size Matters &#8211; The Kelly Criterion and the Importance of Money Management</a></li>
<li>I already linked <a href="http://www.casact.org/pubs/forum/94sforum/94sf049.pdf">Risk and Uncertainty:  A Fallacy of Large Numbers</a> by Paul Samuelson, but in case you missed it, there it is again.</li>
<li>A <a href="http://www.businessweek.com/magazine/content/05_39/b3952138.htm">Business Week review of Fortune&#8217;s Formula</a> with good background into Kelly.</li>
<li>A suggested method for using the <a href="http://www.investopedia.com/articles/trading/04/091504.asp">Kelly Formula in money management</a>.</li>
</ul>
<p>[<strong>KELLY PART TWO</strong>: Check out how I use <a href="http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/">Kellyesk calculations in my sizing of individual stock positions</a>.]<br />
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<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2009/06/how-to-size-individual-positions-using-kellyesk-formula/' rel='bookmark' title='Permanent Link: How to Size Individual Positions Using Kellyesk Formula'>How to Size Individual Positions Using Kellyesk Formula</a></li>
<li><a href='http://www.fusioninvesting.com/2009/11/kelly-capital-launches-100x-leveraged-etfs/' rel='bookmark' title='Permanent Link: Kelly Capital launches 100x leveraged ETFs'>Kelly Capital launches 100x leveraged ETFs</a></li>
<li><a href='http://www.fusioninvesting.com/2010/05/500-posts-later/' rel='bookmark' title='Permanent Link: 500 Posts Later'>500 Posts Later</a></li>
</ol></strong>]]></content:encoded>
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		<title>Shorting now? You&#8217;ve got to be kidding</title>
		<link>http://www.fusioninvesting.com/2008/11/shorting-now-youve-got-to-be-kidding/</link>
		<comments>http://www.fusioninvesting.com/2008/11/shorting-now-youve-got-to-be-kidding/#comments</comments>
		<pubDate>Fri, 14 Nov 2008 03:44:12 +0000</pubDate>
		<dc:creator>Dean Morel</dc:creator>
				<category><![CDATA[Investing Insights]]></category>
		<category><![CDATA[Philosophy]]></category>
		<category><![CDATA[Probability]]></category>
		<category><![CDATA[Trade]]></category>

		<guid isPermaLink="false">http://www.fusioninvesting.com/blog/?p=262</guid>
		<description><![CDATA[Someone is always left holding the bag. This is the same concept as the oft quoted patsy at the poker table.
If you invest late into trends then you have a higher probability of being that patsy.
People investing late in trends should examine if they suffer from herd mentality, confirmation bias etc. While numerous studies show momentum strategies are successful, all those strategies attempt to identify major trends early on. One of the most basis the death cross of the 50dMA crossing below the 200day shows when getting on board this ...

<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2008/11/shorting-suntech/' rel='bookmark' title='Permanent Link: Shorting Suntech'>Shorting Suntech</a></li>
</ol></strong>]]></description>
			<content:encoded><![CDATA[<div class="fblike" style="height:25px; height:25px; overflow:hidden;"><iframe src="http://www.facebook.com/plugins/like.php?href=http%3A%2F%2Fwww.fusioninvesting.com%2F2008%2F11%2Fshorting-now-youve-got-to-be-kidding%2F&amp;layout=standard&amp;show_faces=false&amp;width=450&amp;action=like&amp;font=arial&amp;colorscheme=light" scrolling="no" frameborder="0" allow Transparency="true" style="border:none; overflow:hidden; width:450px;"></iframe></div><p>Someone is always left holding the bag. This is the same concept as the oft quoted patsy at the poker table.<br />
If you invest late into trends then you have a higher probability of being that patsy.</p>
<p>People investing late in trends should examine if they suffer from herd mentality, confirmation bias etc. While numerous studies show momentum strategies are successful, all those strategies attempt to identify major trends early on. One of the most basis the death cross of the 50dMA crossing below the 200day shows when getting on board this trend was appropriate, <a href="http://finance.yahoo.com/q/ta?s=^GSPC&amp;t=2y&amp;l=on&amp;z=m&amp;q=c&amp;p=m50,m200&amp;a=&amp;c=" target="_blank">back in January</a>.</p>
<p><a title="TMF post on BMW board" href="http://boards.fool.com/shorting-27187006.aspx?sort=whole" target="_blank">yttire&#8217;s initial post</a> in this thread on TMF spelt out a good case for not going short right now. This was my response to the idea of using an ultra short now.</p>
<p>Stepping into a trade as yttire outlines is a great trading strategy. Most traders I have read recommend a similar strategy and even long term investors express similar sentiment in phrases like water the flowers and pull the weeds.</p>
<p>The only tools that I know which are any use at giving you an edge over the short term are sentiment and chart based. If you use those tools then you may also want to layer on some historical fundamental analysis. From that it seems likely we&#8217;re at fair value now. While markets often overshoot and hence lower prices are a possibility, the probability of lower markets are reducing.</p>
<p><strong>There is no need to swing at every trade. Is this a time you&#8217;re likely to have a high probability of success with an ultra short?</strong></p>


<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2008/11/shorting-suntech/' rel='bookmark' title='Permanent Link: Shorting Suntech'>Shorting Suntech</a></li>
</ol></strong>]]></content:encoded>
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		<title>Edge &#8211; The Way of the Turtle</title>
		<link>http://www.fusioninvesting.com/2008/08/edge-the-way-of-the-turtle/</link>
		<comments>http://www.fusioninvesting.com/2008/08/edge-the-way-of-the-turtle/#comments</comments>
		<pubDate>Fri, 08 Aug 2008 04:22:58 +0000</pubDate>
		<dc:creator>Dean Morel</dc:creator>
				<category><![CDATA[Behavioural Finance]]></category>
		<category><![CDATA[Probability]]></category>
		<category><![CDATA[Review]]></category>
		<category><![CDATA[edge]]></category>
		<category><![CDATA[Faith]]></category>
		<category><![CDATA[Trade]]></category>
		<category><![CDATA[Turtle]]></category>

		<guid isPermaLink="false">http://www.fusioninvesting.com/blog/?p=110</guid>
		<description><![CDATA[Most Friday&#8217;s I drop my daughter at dance class, head to one of Melbourne&#8217;s best cafés and have a latte while I wait for Borders to open at 10. I then have around 20 minutes to choose and scan an investment book. While I am normally a slow reader I can skim a book in 20 minutes and pick out a few lessons which resonate with me.
Today my book of choose was Way of the Turtle, by one of the original turtles, Curtis Faith.
Almost every page I scanned contained a ...

<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2008/05/behavioural-finance-part-one/' rel='bookmark' title='Permanent Link: Behavioural Finance Part One'>Behavioural Finance Part One</a></li>
</ol></strong>]]></description>
			<content:encoded><![CDATA[<div class="fblike" style="height:25px; height:25px; overflow:hidden;"><iframe src="http://www.facebook.com/plugins/like.php?href=http%3A%2F%2Fwww.fusioninvesting.com%2F2008%2F08%2Fedge-the-way-of-the-turtle%2F&amp;layout=standard&amp;show_faces=false&amp;width=450&amp;action=like&amp;font=arial&amp;colorscheme=light" scrolling="no" frameborder="0" allow Transparency="true" style="border:none; overflow:hidden; width:450px;"></iframe></div><p>Most Friday&#8217;s I drop my daughter at dance class, head to one of Melbourne&#8217;s best cafés and have a latte while I wait for Borders to open at 10. I then have around 20 minutes to choose and scan an investment book. While I am normally a slow reader I can skim a book in 20 minutes and pick out a few lessons which resonate with me.</p>
<p><a href="http://www.fusioninvesting.com/wp-content/uploads/2008/08/turtle.jpg"><img class="alignleft" style="margin: 6px; float: left;" title="turtle" src="http://www.fusioninvesting.com/wp-content/uploads/2008/08/turtle.jpg" alt="Way of the Turtle" width="140" height="205" /></a>Today my book of choose was <a href="http://www.amazon.com/Way-Turtle-Methods-Ordinary-Legendary/dp/007148664X">Way of the Turtle</a>, by one of the original turtles, Curtis Faith.</p>
<p>Almost every page I scanned contained a gem. From the introduction where Faith said he has learnt something from almost everyone he has every interacted with to his thoughts on edge and his simple clear rules.</p>
<p>Before I buy this book I thought I should re-read the Turtle Rules, a copy of which is linked at the bottom of this blog post on <a href="http://www.fusioninvesting.com/2008/05/investing-tips-secrets-1/">investment tips</a>.</p>
<p>Faith&#8217;s <strong>Dos and Don&#8217;ts for Thinking Like a Turtle</strong>, are a great example why even strict fundamental investors can learn a lot from trading books like this or <a href="http://www.amazon.com/Trade-Your-Way-Financial-Freedom/dp/0070647623">Tharp&#8217;s Trade your way to Financial Freedom</a>.</p>
<blockquote>
<ol>
<li><strong>&#8220;Trade in the present.</strong> Do not dwell in the past or try to predict the future. The former is counterproductive and      the latter is impossible.</li>
<li><strong>Think in terms of probabilities not predictions. </strong>Instead of trying to be right by predicting the market, focus on methods in which the      probabilities are in your favor for a successful outcome over the long run.<strong></strong></li>
<li><strong>Take responsibility for your trades. </strong>Don&#8217;t blame your mistakes and failures on others, the markets, your broker, and so forth. Take responsibility for your mistakes and learn from then.&#8221;<strong></strong></li>
</ol>
</blockquote>
<p>Jim (BMW) has provided the framework to easily implement number two. The BMW method provides a framework rooted in probabilities for a successful outcome over the long run.</p>
<p>But Dean, what does this all have to do with EDGE? I hope to wrap up the loose ends at and tie it back to many conversations held on the BMW board. First indulge me with one final quote from Faith, this quote spoke to me so strongly that other shoppers looked up as I exclaimed &#8220;Fork yeah!&#8221;</p>
<blockquote><p>&#8220;<strong>the best edges come from the market behaviors caused by <a href="http://www.fusioninvesting.com/2008/05/behavioural-finance-part-one/">cognitive biases</a>.</strong>&#8220;</p></blockquote>
<p>I realise the market is made up of different opinions and communities such as this are a reflection of those opinions. However, I all too often see cognitive biases in TMF posters and no doubt some of mine ooze on to these digital pages.</p>
<p>People talk about price way to much. The worst examples of this can be found in TMF subscriptions publications. Many issues contain phrase like &#8220;XYZ is now a bargain at 30% off its high&#8221; or &#8220;we&#8217;re getting to buy XYZ at a 30% discount form recent prices&#8221;. While XYZ may be a bargain, I find this constant reference to price insidious in publications which aim to educate investors.</p>
<p>Many people fail to recognise that companies are on sale for a reason and those reasons are obvious to all market participants. Once a reason is priced is, you need to invert your thinking and see if a reversal of that reason is an opportunity for a catalyst. Making predictions based on known priced in reasons or worse predicting based on your own views is unlikely to outperform a focus on the probabilities. This mental momentum, make us believe that as everything is going wrong for XYZ it will keep going wrong. While that can occur the BMW method highlights that it probably won&#8217;t keep going wrong. <strong>The most probable outcome for companies with proven track records is that they will bounce back.</strong></p>
<p><strong>Consider your biases and formulate strategies to overcome then.</strong> If you are not buying more now than you were over the last couple years then perhaps you are predicting and letting your cognitive biases rule the day.</p>
<p>FWIW &#8211; in personal and/or model portfolios</p>
<ul type="disc">
<li>I continue to hold SSD and sell calls on it.</li>
<li>I recently sold my trading position in Amgen. While the news flow has changed from negative to positive recent investors are likely to be nervous and have little      conviction. Pocketing 50% in short order was a no brainer and more than      compensated for my only ever loss on Amgen (2008 Leaps vertical spread which due to inexperience with spreads I let slip from profitable to a loss)</li>
<li>I&#8217;ve bought PFE, FDX and GE.</li>
<li>Taken large losses on a number of companies.</li>
<li>Started buying Australian banks (not recommended for US investors due to Fx rate).</li>
<li>Passed on BARE.</li>
<li>Safely <a href="http://www.fusioninvesting.com/2008/05/satety-in-numbers/">profiting with SAFT</a>.</li>
<li>Continue to hold way to many companies.</li>
<li>Trying to make time to look at AKAM and STP, which from a cursory look both appear to offer good value and excellent prospects.</li>
<li>Been very distracted trying to get a couple businesses off the ground when I should probably simply be focusing on My Family Inc. Despite having overcome      monetary desires I still fail to contain my egotistically desires. While I should be the most contented I have ever been, my ego pushes me to achieve more, denying me the enjoyment of here and now. I must find a solution for that! Oh the pain, the pain of being an A type <img src='http://www.fusioninvesting.com/wp-includes/images/smilies/icon_wink.gif' alt=';-)' class='wp-smiley' /> </li>
</ul>


<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2008/05/behavioural-finance-part-one/' rel='bookmark' title='Permanent Link: Behavioural Finance Part One'>Behavioural Finance Part One</a></li>
</ol></strong>]]></content:encoded>
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		<title>You Probably Should Be Using Probabilities</title>
		<link>http://www.fusioninvesting.com/2008/05/using-probabilities/</link>
		<comments>http://www.fusioninvesting.com/2008/05/using-probabilities/#comments</comments>
		<pubDate>Tue, 13 May 2008 12:42:47 +0000</pubDate>
		<dc:creator>Dean Morel</dc:creator>
				<category><![CDATA[Probability]]></category>
		<category><![CDATA[Philosophy]]></category>

		<guid isPermaLink="false">http://www.fusioninvesting.com/blog/?p=27</guid>
		<description><![CDATA[Investing is all about probabilities, the probability of reward versus risk. It doesn’t mater whether you’re a value, growth, fundamental or technical trader or investor you should be using probabilities to help you be a better investor. I use the Kelly Formula for comparative analysis of my portfolio and new opportunities and will discuss that in depth another day. Today I’d like to focus on short term probabilities. If you sit firmly and unflinchingly in the long term buy to hold camp then move along there is nothing for you ...

<strong>Related posts:<ol><li><a href='http://www.fusioninvesting.com/2008/11/is-netflix-on-your-top-ten-list/' rel='bookmark' title='Permanent Link: Is Netflix on your top ten list?'>Is Netflix on your top ten list?</a></li>
</ol></strong>]]></description>
			<content:encoded><![CDATA[<div class="fblike" style="height:25px; height:25px; overflow:hidden;"><iframe src="http://www.facebook.com/plugins/like.php?href=http%3A%2F%2Fwww.fusioninvesting.com%2F2008%2F05%2Fusing-probabilities%2F&amp;layout=standard&amp;show_faces=false&amp;width=450&amp;action=like&amp;font=arial&amp;colorscheme=light" scrolling="no" frameborder="0" allow Transparency="true" style="border:none; overflow:hidden; width:450px;"></iframe></div><p><strong>Investing is all about probabilities</strong>, the probability of reward versus risk. It doesn’t mater whether you’re a value, growth, fundamental or technical trader or investor you should be using probabilities to help you be a better investor. I use the <a href="http://en.wikipedia.org/wiki/Kelly_criterion">Kelly Formula</a> for comparative analysis of my portfolio and new opportunities and will discuss that in depth another day. Today I’d like to focus on short term probabilities. If you sit firmly and unflinchingly in the long term buy to hold camp then move along there is nothing for you to see here. For everyone else I hope to illustrate how using probabilities helps me answer the question of whether to buy or sell prior to an earnings release.</p>
<p>There are two main reasons why we should even consider the question of buying or selling prior to earnings. Behaviourally investors experience heightened emotional turmoil prior to earnings. This is because earnings act like steroids for the competing emotions of Fear and Greed. This is especially true for volatile companies, for smaller companies or any company that you are considering selling or buying.</p>
<p>Before we go on I’d like to instill some faith in you that probability is one of keys to investing. Charlie Munger, Warren Buffett, Mohnish Pabrai and countless other superstar investors have talked and written extensively on the probability. It is one of the essential tools in their belts and they even pursue probability in their leisure activities. Buffett is drawn to Bridge as it is a game of probabilities.</p>
<p>I’d like to use this quarters reporting season as an example of my use of probability. I’m currently trying to trim down my bloated personal portfolio and get back to my concentrated ways. Concentrated for me means 10-20 companies, while for various reasons I found myself holding 49 companies late in 2007. My diet is going OK, I’m now down to 44 companies with 21 of those being meaningful positions. Anyway, back to the point.</p>
<p>Prior to earnings my thinking went like this. I want to sell at least a few companies. I want to do it prior to earnings as a significant portion of price movements occur around earnings time. In general the prices of stocks are static for 70% of the time with real movements occurring 30% of the time. I want to sell companies with uncertain long term futures.</p>
<p>My initial list was eResearchTechnology (ERES), Netflix (NFLX), CryptoLogic (CRYP), American Reprographics (ARP), Take-Two Interactive (TTWO), Harmonic (HLIT), Alvarion (ALVR), American Eagle Outfitters (AEO), II-VI Inc (II-VI) and Nuance Communications (NUAN).</p>
<p>I posted on a number of those companies and <a href="http://boards.fool.com/q1-2008-preview-26594025.aspx">this post on ARP</a> illustrates my thinking. ARP had good earnings and as it was so beaten down and investors had low expectations the price gained significantly for just good earnings.</p>
<p>I didn’t post the following as by writing my question I answered it. (If you don&#8217;t already write down yhour investing thoughts, then start an investing journal today.)</p>
<p>So tell me, what would you do with the following? No this is not a trick question it is a real life question and I am interested in your opinion, naturally everyone should feel free to reply.<br />
Netflix hit my final sell point of <a href="http://stockcharts.com/charts/gallery.html?nflx">$40 last Thursday</a> [April 17]  so I stayed up Friday night to sell it. I was selling a part positions in a couple other companies as well, II-VI and NUAN, so when I turned to NFLX to enter my limit order I was happy to see it trading at $40.85 bid and so entered my sell limit at $40.85. The bid then started falling away quickly, very quickly. I decided, like me, there would be a lot of people who had $40 as their price target and that the selling would quickly abate and my order still had a good chance of being filled later that day. I went to bed content with the dual thought that I always use limits orders and due to intra-day volatility they are hit 90% of the time, plus I always liked the saying that amateurs open the market and professionals close it.<br />
I awoke the next morning to see NFLX down at $38.56, my order had not filled. Both of my other orders had been filled and they were entered at prices higher than the stock was trading at.</p>
<p>NFLX earnings are out today, 21st April. I expect good earnings as Netflix has beaten all competition into a pulp. While NFLX isn’t quite priced for perfection it is certainly priced with good things in mind, so excellent earnings and guidance will be required to push NFLX higher.<br />
Long term I like the Netflix strategy, but it has been a volatile company and I am confident it will continue to be volatile as competitors will continue to assault the NFLX castle. Hence I sell NFLX when it approaches my valuation targets, this has worked well before and I’d wager it will again. I sold half of my large position at around $33, my cost basis is around $20 from June last year. It is not in a tax deferred account, so capital gains will bite a little.”</p>
<p>By the time I had finished typing that and ran some quick numbers I decided to sell NFLX and placed an order for $39.95. <strong>Like most investors I am plagued by fear and greed. I use numbers to stop the pathetic, debilitating, irrational arguments.</strong> The probabilities supplemented with an understanding of investor behaviour drummed in me to David Dreman led me to sell.</p>
<p>This is already getting to long and I’m gagging for a cuppa so here is how my decisions this earnings season turned out.<br />
<strong> NFLX</strong> – Sold. Excellent, I feel good about selling, but as I like the long term potential of NFLX I am now faced with a decision on when to get back in. I consider the current price good, but not a slam dunk.</p>
<p><strong>ARP</strong> – Held. Excellent, but I am still concerned about their lack of organic growth and the probability of a commercial building slowdown. I need to look closer, but my gut says sell.</p>
<p><strong>ERES</strong> – Sold. This didn’t work out so well as ERES is now around 30% higher than where I sold. What went wrong, what can I learn. I invested in ERES as a turnaround opportunity. They sell cardio monitoring for clinical trials and the FDA had mandated heart monitoring. Their previous management were clowns and investors had no faith. I set firm targets for the new management and tracked what they said and delivered very closely. The new management proved to be excellent, my thesis was validated I profited. I sold based on a high valuation. If they had of delivered OK or worse earnings the price was going to tank. I should fell OK about my decision as based on the probabilities of investing it was the smart thing to do and ERES could easily trip in a coming quarter and present another opportunity for me. However, I don’t feel good, as I believe I didn’t rank the chances of this earnings being good as high as I should. I had grown to admire the new management, but was blinded by my valuations. I believe my valuations affected my probabilities. This is a mistake I make over and over again and one day hope to counter it. Here are <a href="http://boards.fool.com/2007-review-and-q1-08-preview-26563580.aspx">my pre-earnings thoughts</a>.</p>
<p><strong>CRYP</strong> – Sold. Lucky. I had been looking to sell CRYP as it was never a company I was happy holding. It is the only company I’ve ever held that I would never mention to my partner or friends. I think less of myself for holding it and am glad to rid myself of it. As I dislike it so much I ignored it. A couple days prior to spreadsheet highlighted the upcoming earnings and coincidentally they announced a deal. The price shot up I sold. I didn’t run numbers, but my gut told me to grab the opportunity.</p>
<p>This is getting way to long, my mouth is so fury that I’m coughing up fur balls.<br />
<strong> TTWO</strong> – Held, they’re worth more. Though my probabilities tell me to sell if I find a better opportunity.<br />
<strong> HLIT</strong> – Added to, good earnings, market stupidly focused on guidance and price is now a bit lower.<br />
<strong> ALVR</strong> – Added to, good earnings and guidance, market loved it.<br />
<strong> AEO</strong> &#8211; Held<br />
<strong> II-VI</strong> – Sold half<br />
<strong> NUAN</strong> – Sold half</p>
<p>I am not advocating short term trading here. I’ve never calculated my average holding period, but it is probably around two or three years. Quickly checking my current portfolio I see the average purchase date is August 2006. I don’t run the probabilities for each my company every quarter, though I do try to update my Kelly targets after each earnings release. I find both my pre and post earnings probability calculations help me:<br />
•	contain fear and greed<br />
•	to have a rational basis to my decisions which I can check and hopefully improve on<br />
•	and force me to think about the company and its prospects at least once a quarter.</p>
<p>For anyone wanting to read more here are a couple articles to get you started</p>
<p>http://www.wallstraits.com/main/viewarticle.php?id=1397</p>
<p><a href="http://en.wikipedia.org/wiki/Probability">http://en.wikipedia.org/wiki/Probability</a></p>
<p>Cheers<br />
Dean</p>


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