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What do we mean when we talk about risk?

Risk deals withĀ uncertainty, or when a single choice leads toĀ multiple potential outcomes. In this article I will discuss some of the frameworks and ideas that can inform our investments in cards when the future is unknown.

## Problem A: Comparing Investments Against Each Other

We can think of each investment as a lottery. It hasĀ multiple potential payouts that are distributed in some probabilistic fashion. So how would we compare two different lotteries against each other assuming we know the exact distributions of their payouts?

A natural first approach would be to pick the lottery with the highest average payout. In general, a higher average is a good sign, but focusing solely on the average can lead usĀ into some problems. Consider the St. Petersburg paradox:

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second (...) and so on. What would be a fair price to pay the casino for entering the game?

Let's assume we stack lotteries against each other just based on their average payout. Since the average payout of the St. Petersburg lottery is infinite,Ā we would always prefer it to another lottery with a finite average payout, for example a sure bet of winning a million dollars.

So ask yourself now, assuming the prices of both lotteries were the same, would you prefer the St. Petersburg game or a sure bet of a million dollars? I think the answer is quite clear, which says a lot about our preferences---particularly our aversion to risk.

The solution to the St. Petersburg paradox seems to imply the idea that not every dollar won is equally important. Consider the two following lotteries:

- Lottery A: 50% chance to win $10,000 and 50% chance to win $990,000
- Lottery B: 50% chance to win $0 and 50% chance to win $1,000,000

People will almost always choose lottery A over lottery B, because going from $0 to $10,000 is a more significant jump than going from $990,000 to $1,000,000.

This idea implies the rule that given two lotteries with equal average payouts, we will prefer the lottery with the lowest variance in payouts. This leads us to a popular school of thought in economics---to judge an investment based on both its mean and its variance.

The more interesting question is then: how do we choose to trade off between mean and variance? How do we pick between two investments, one with a higher mean and variance and one with a lower mean and variance? This will vary depending on the individual investor, as different investors will have different risk preferences.

To consistently make the best investments for your own specific situation, it will be a good idea to figure out your own preferences towards risk. You can ask yourself some questions like:

Would you rather a sure bet of $100 or 50% chance of $40 and 50% chance of $180?

Would you rather a sure bet of $100 or a 50% chance of $60 and 50% chance of $150?

Would you rather a sure bet of $100 or a 90% chance of -$200 and 10% chance of Ā $3,000?

And so on. Understanding your preferences towards risk will help you decide whether for example you want to make a bet on an already-proven staple with a low but likely payoff or a risky bet on an unproven card with a potentially big payoff.

## Problem B: Allocating Funds Across Different Investments

We now have some idea of how to compare investments against each other, but how do we allocate between them?Ā Thinking from a mean/variance perspective, diversification can beĀ a useful tool to manage variance, usually at the cost of theĀ mean payoff, but sometimes at the cost of nothing.

Think of the following problem:

- Card A: Price $5; in one year has a 50% chance to be worth $5, and a 50% chance to be worth $10.
- Card B: Price $10; in one year has a 50% chance to be worth $10, and a 50% chance to be worth $20.

Let's say you have a budget of $200, if you invest entirely into card A or entirely into card B, you will have the following payoff:

**50% chance to profit $0, 50% chance to profit $200**

By using diversification, we can reduce the risk (variance) without changing the average payoff. We can do this if we buy 20 copies of card A and 10 copies of card B. The payoff is now:

**25% chance to profit $0, 50% chance to profit $100, 25% chance to profit $200**

In this way, diversification can be a very powerful thing, but I would warn against blindly applying it to all situations.

Diversification is a tool that is used only to manage variance, and often comes at the price of your average return. This comes from the assumption that the quality of your investments will not be uniform as they were in the previous example. If we assume that investments all fall on some spectrum of "good" to "bad" then in order to reduce the risk of your best investments through diversification, you will have to introduce lower quality investments into the picture, thus lowering the average return.

For this reason, in my own investments, I choose to diversify as little as possible. I would rather put all my money into the best bet that I can find. This also happens to be the strategy that is heralded by legendary investors such as Warren Buffet and Charlie Munger.

When Warren Buffet gives investment advice, he often has told people to imagine they can only make 10 investments from now until the end of their lives. Such a restraint would force people to put a lot of thought into a small number of high-quality investments. Putting all of your eggs in one basket can reap high rewards if you are a disciplined investor that goes only after the best bets. I highly recommend it!

## Problem C: When Investments Can't Be Infinitely Scaled

Sometimes it is not possible to put all of your money into one investment because the scale of your entire portfolio is bigger than any single investment that you can make. This relates to the concept of "market depth."

Market depth is the degree to which a market can be influenced by a single investor. In very large (deep) markets, for example the New York Stock Exchange, you can sink millions of dollars into a single asset without having a large effect on its price. This is clearly not the case when you invest in Magic cards!

I'll give a personal example. A few days ago, I was interested in buying as many Abbot of Keral Keep as I could find for under 3.4 tix but was only able to secure about 80 copies. This is because the market is so shallow, I am effectively raising the price of the card by a few cents every time I buy a playset.

To me, this is the true role of diversification in MagicĀ investment. Since the markets are often quite shallow, you will have diminishing returns on the profit associated with each subsequent purchase of the same card. The diminishing effect is twofold because the next copy you speculate on will both cost more and sell for less.

Taking this into account, my personal strategy is to choose both a maximum price I'm willing to pay and a maximum number of copies I'm willing to buy for each individual investment. The maximum number of copies will vary depending on the market depth of the associated card. A four-of staple in Standard, for example, will have much more market depth than a one-of in Modern.

Keep this in mind when you make your investments. I think it is prudent not to think of the total amount invested into each card, but the amount of each individual card you are buying. I would recommend starting at a reasonably low number and working your way up as you gain more experience investing.

## Conclusion

In order to make the best investments for your own personal situation, you should ask yourself how you feel about risk, or how much variance you are willing to put up with for a higher average return. Once you have your risk preferences figured out, I recommend only diversifying if you are very risk-averse or if you believeĀ you're dealing with multiple investments with very similar potential outcomes.

An exception to this rule is when issues of market depth come into play. In shallow markets, it makes sense to diversify because there are steep diminishing returns to profits for each additional card that you buy.

Great article! I appreciated the discussion on how the mtgo market has some very real diminishing returns when it comes to investing. When you read in the forums/articles or research about good positions, it can be hard to judge how deep do you invest in it. Even when I know I’m onto a great opportunity, I’ll limit myself, because once you get to a certain point diminishing returns on buying and selling becomes very real. I’ve had a few friends that casually play magic, that have wanted to bankroll me on mtgo after hearing of my early successes, that I’ve had to turn down for the time being (until I get better at scaling up, full sets is the next frontier). It’s tricky to explain this concept to them why injecting 1000 tickets into my bankroll, won’t result in the same growth rate I’ve been achieving. I feel this article can help explain those pitfalls.

Thank you! I had a very similar experience to the one you mentioned. When my bankroll was low in the beginning, it was easy to get great returns that are hard to keep as your bankroll gets bigger and bigger. I think one of the solutions to a big bankroll is to invest in items that are very expensive (like full sets for example). Like I mentioned in the article, I think diminishing returns are a function of the number of items you buy, not the amount of money spent on the item. So naturally, with more expensive items there is less diminishing returns per dollar spent.

I highly recommend checking out Sylvain’s new series with a huge bankroll as well! I will be learning a lot from it I can already tell.

In paper Magic you can at some point scale up to buying larger and larger collections, I don’t think there’s an equivalent to that on MTGO?

You have managed to articulate many concepts that I have instinctively been using. It is extremely helpful to have a more conscious understanding so that I can apply more reasoning to my finance approach. I really enjoy these types of articles and I hope to see more in the future.

Keep up the great work!

Thanks so much for the kind words! It means a lot!

Very interesting article. Regarding the scaling, in some cases you may avoid pushing up the price to an extend by pacing yourself when buying: pick up a few, wait for more to go on the market, pick up a few more, etc.

In my opinion pacing only helps you if the bot algorithms increment the price too quickly (many bots ramp up exponentially when they sense speculation). Like any market, the only thing that will push the price back down after you buy is other people that are trying to sell.

In this sense, pacing is a good strategy, and I implement it often, but I don’t think it’s a cure-all. If you buy 100 copies of a card and wait a few days, the price will always be higher than if you never bought the 100 copies in the first place. In the end, there is no way to really cheat the system, since every copy of the card you buy increases the price by about the same amount, no matter how spread out the purchases are (IMO).

Great article, please keep them coming. One dimension yet to be mentioned though is correlation (perhaps a future article?). Reduction of a portfolios variance is achieved through investing in assets with low correlation, and we could identify specific examples both via common sense (e.g. modern vs standard, opposing archtypes, etc.), or by crunching price numbers.

An other thought that has been bugging me on the approach of risk management to magic is that the mean return of (many) magic cards in the long run can’t be assumed positive, unlike stocks. Paper legacy (i.e. collectors) and modern might be the case, but redeemable MTGO standard isn’t – the majority of standard cards mean daily return eventually flatten to zero (or neg) as rotation and redemption approaches. When using any numerical approach we have to be mindful that the numbers only make sense for a short time context.

Correlation definitely plays a huge role in diversification. I omitted the discussion to keep things simpler, but like you mentioned, it can be used to manage risk if we buy cards of different archetypes or formats.

I think seeing a magic card as a stock has a lot of merit. A stock pays out dividends while a magic card pays you dividends in terms of your ability to play it in tournaments over time. Yes, this means the dividends from a standard staple will be short-lived, but no company lasts forever either. Things in Magic are just happening on a shorter time frame (but not fundamentally different).

I don’t think you can ever assume the mean return of either stocks or magic cards will be positive (or negative). Return is a function of price which will vary depending on investor sentiment. In other words, if the market is overoptimistic, you will have negative returns on average and vice versa.

You could have the most profitable company in the world, and it could still bring you a negative return if the stock is overpriced. The same goes with a really insane magic card that is overhyped. I talk about a lot of these ideas in my 2nd article.

Hi Luca,

Meliora?

-Jesse