Last week’s article generated a significant amount of positive interactions-–many QS subscribers visited the forums and shared their own investment portfolio. I want to thank everyone who participated in the exercise. Hopefully the experience was an insightful one.
I for one learned something about myself. In the past my bets have generally been conservative in nature. I’ve disclaimed before my hesitation to go deep in an investment because I am eager to reduce risk through diversification. On paper this sounds like a great theory, but it may surprise you to hear that this strategy may not yield the highest possible payout. Diversification is designed to prevent a portfolio from going to zero, not infinity.
Now, my most recent investments have followed a different style to maximize productivity. This new strategy, which I have been implementing informally without realizing, is known as the Kelly Criterion.
Whoa, the Co-Founder of QS is Famous?
Kelly Reid may be famous, but if he is it has nothing to do with the Kelly Criterion. The original theory was developed by J. L. Kelly Jr. in 1956–-I doubt our Kelly is this old.
From Wikipedia, the Kelly Criterion, also known as the Kelly Bet,
“is a formula used to determine the optimal size of a series of bets…The Kelly strategy will do better than any essentially different strategy in the long run.”
The Wikipedia entry goes on to explain that Kelly’s strategy may yield optimal gambling outcomes but there may be individual investing constraints that interfere with obtaining optimal growth rate.
Despite this, I feel the underlying concept should be explored further for potential reapplication to MTG speculation.
Borrowing again from Wikipedia, the Kelly Criterion states:
The equation may look fairly simple, and in reality it does require some assumptions. In MTG speculation, we don't often run the risk of losing everything, but we do risk losing money. The application to MTG finance may be imperfect, but an example is at least worth a try.
A First Test
I’m going to test my Innistrad booster box investment here. Immediately there are some difficult assumptions we must make--this is no exact science.
For the sake of exercise, let’s say I expect Innistrad boxes to hit $200 in a reasonable time frame of a few years. Since boxes can be purchased on eBay for $149 a box, this represents odds of 1.34:1 meaning b = 1.34 ($200 / $149). I feel the probability of earning the $200 is fairly reasonable if we’re willing to wait long enough, so let’s say the probability is 75%. This would make p = 0.75 and q = 0.25.
Plugging in the numbers we get an estimate for what fraction of our portfolio should comprise of Innistrad booster boxes based on our specific assumptions. In this case that fraction equals (0.75 * (1 + 1.34) – 1) / 1.34, or about 56%.
This means that if our assumptions are correct and there’s a 75% chance we will sell at $200 for every Innistrad booster box we purchase at $149, we should take 56% of our MTG funds and place them into this investment. Wow, that’s steep!
Another Example - Shocklands
The example above seems difficult to execute in real life. I don’t have the courage to go that deep on Innistrad boxes. Remember, my current position represents only about 25% of my portfolio. So either I am not 75% confident in making bank or I am not honestly expecting to sell these at $200 per box. One of these assumptions must be incorrect, or else I’m not behaving rationally to maximize profits.
Let me try another example with shocklands, this time comparing scenarios before and after the recent PTQ schedule changes.
Before the change in schedule was announced, Modern season would have lined up perfectly with Standard rotation to drive shockland prices higher. Also, there was not enough time for any reprints and we are fairly confident shocklands aren’t in Theros.
Therefore, let’s assume we were 90% confident in making bank on shocklands with the old PTQ schedule. As for returns, demand would have increased fairly rapidly in the fall, driving prices upward. For this fall, I was expecting to sell my shocklands with a 1.7:1 payout, meaning I would get $12 for every $7 shockland I purchased ($12 / $7 = 1.7).
Applying the equation discussed earlier, I calculate a portfolio fraction of (0.95 * (1.7 + 1) - 1) / 1.7 = 92%! Wow, this would have been incredibly deep! But if we were truly this confident in such a solid return, logic would dictate we should have been this eager to invest.
Even though it seems crazy to invest in anything this deeply, I do want to point out that many QS subscribers did have very significant investments in shocklands.
While 92% seems excessive, some application of utility theory may have yielded a more practical number that our emotions could handle. Just look at Zendikar fetchlands, which would have had similar assumptions while they were Standard-playable, and you will see how the Kelly Criterion can yield superior results!
Now let’s test different numbers because Modern season is a year away. Demand won’t be as high and there’s some reprint risk now. I’m actually thinking of selling these this fall anyway once Standard rotates, but I am far less excited about the potential profit. Perhaps I expect $10 now for each $7 shockland, a payout of 1.4:1 ($10 / $7 = 1.4). Also, I am less confident in such a payout so soon after a reprint–-I’ll use a likelihood of 60%.
Now my portfolio fraction is (0.60 * (1.4 + 1) - 1) / 1.4 = 31%. This is a severe drop! With these assumptions, I should invest 31% of my portfolio in shocklands instead of 92%. This is a full 2/3 less the investment!
Can This Really Be Useful?
I don’t have the fortitude to withstand the risk associated with investing 92% of my MTG portfolio in one position. Because of the human aversion to losing everything, I’m guessing I’m not alone in this regard. That’s where Utility Theory and Fractional Kelly Betting come into play–-but that’s an article for another week.
Perhaps the Kelly Criterion can still be useful in a comparative sense, however. The shockland example above illustrates what I mean. With two different sets of criterion, we can use an unemotional, proven strategy to identify where we should make larger bets and where we may need to scale back. Likewise if our assumptions need to change due to a WOTC announcement, a reprint, or a shift in metagame we can use this equation to help us estimate what adjustments are needed.
This tool can also help us sort through the hundred’s of new speculation targets we face each month. If you come across an opportunity that seems risky, consider the likelihood of turning a profit and how much profit, and see how the resulting Kelly Bet would compare versus a more tried-and-true option.
Although I often advocate diversification, sometimes it’s better to just buy another Birthing Pod rather than possibly overpaying on Keen Sense. Of course, if you find Keen Sense or any other cards below mtg.gg buy prices–-well, this equates to 99% likelihood of profit. You should know what to do here, and if it were possible to put 100% of my collection into guaranteed profit I would.
The equation can also help us identify situations where selling is best because it can yield negative numbers! A negative portfolio fraction can be obtained should the payout be too low and unlikely. In this case, we would actually want to sell our entire position (and theoretically, sell short even further if it were possible). Should you end up with this outcome when doing a test, perhaps it’s time to bail and move on.
The Kelly Criterion isn’t the answer--at least not for me. But it is a proven, numerical strategy that is set up to maximize outcome. In the game of MTG there is no guarantee, and we are left with many assumptions on price trajectories. But by putting our best guesses down on paper, the Kelly Criterion can be a valuable tool to help us evaluate if we’re thinking along the right path or if we should move onto the next target.
I cannot bring myself to embrace this tool completely, but I have certainly implemented it lately with extensive investment in Innistrad boxes and shocklands. Now I can finally use numbers to back up my rationale.
- I mentioned Elesh Norn, Grand Cenobite recently because the card was sold out at SCG for $9.99 and I expected a price bump. That price bump has happened and now Elesh Norn is $14.99. But she is still sold out! I expect to see $19.99 next.
- Horizon Canopy continues its run--I wish I hadn’t sold mine already! The card is up to $30 on TCG Player and SCG is sold out at $29.99. What interests me is that the foils on SCG are only twice the nonfoils, and there are a couple in stock. If Modern is going to become a mainstay, that multiplier needs to go higher. Especially for a card like Horizon Canopy which is also played in Legacy. Perhaps I will buy these last couple SCG copies…wait let me check the Kelly Criterion…nah, the potential gain is not worth the difficulty moving it given fees.
- Chord of Calling is even stranger. They are selling on SCG’s site for $29.99 but foils are only $49.99. This is a multiplier less than 2x for a major Modern card. I wonder why the multiplier is so low–-if the price bump in Chord is legit, then the foils should at least hit $60 retail to obtain that 2x multiplier.