I have seen a lot of posts today about hedge funds that are currently beating the market through investing in Bitcoin/cryptocurrencies.

Recently, I wrote a research paper on whether it is better to invest in stocks or cryptocurrencies using game theory applications.

It is something I would like to share so I have attached it below.

I know it is a lot to read, so if you don't feel like going through 10 pages of game theory, the graphs and tables are at the end that show the dominant strategy of investing in a Bitcoin heavy portfolio.

However, it was written about two months ago, the conclusion was that the expected value for investing it a bitcoin heavy portfolio is the highest. With the recent market changes I believe that some recalculations would possibly show that investing in bitcoin and alt coins would provide the highest expected value for firms.

Regardless, hedge funds that diversify their investments with cryptocurrencies will outperform their competitors.

Cryptocurrencies Vs. Stocks

Investors want to make as much profit as possible.

The reason people invest money is so that it will grow over time and not be wasted due to inflation. However, finding the most profitable portfolio is a challenging task, there are many factors that go into it. In 2016, the average investor returns for U.S stock funds was 10.33%. This underperformed the S&P 500 by almost 2%, so as people continue to try and beat the market, it may be time to put money into alternative investments (Anderson, 2017). A more profitable solution may be to include cryptocurrencies in portfolios. Cryptocurrencies are digital currencies such as Bitcoin, Ether, Zcash etc. they are used for transactions just like regular currency but can also serve various other purposes. People can buy and sell Bitcoin and other currencies on various online exchanges just like stocks. Many of these currencies have shown tremendous growth, Bitcoin was at one time priced under a dollar and earlier this year hit close to $1250 per coin. If you would have invested 5 years ago into Bitcoin you would have a 5 year return of close to 300% (Shin, 2017). It seems obvious from the numbers that you should dump your money into cryptocurrencies and down the road you’ll be rich, however there are concerns. Extreme volatility, no regulation, and criminal activity are all huge factors that cause people to stay away from currencies like Bitcoin. Nonetheless, as Bitcoin grows and becomes more stable it may be worth investing in for sizeable returns that are much larger than that of the average stock portfolio. Whether or not any investor should weigh their portfolios heavier towards Bitcoin than stocks will be the focus of this paper. Through game theory analysis I will show that the expected value of a Bitcoin heavy portfolio is more likely to create higher returns over time than the average U.S stock portfolio.

The game is sort of an n-person game of the market versus any number of investors. However, you could also view it as two investors competing to make the most money. In order to make the models as realistic as possible, the two portfolios are diversified with different weights of Bitcoin and stocks. This is because no investor will put all their eggs in one basket, most will try and diversify to help hedge against risk. The Bitcoin heavy portfolio consists of a 70% weight to Bitcoin and a 30% weight to the average stock portfolio. The stock heavy portfolio is simply the opposite, 70% weight to stocks and a 30% weight to Bitcoin. These are the two options for any investor with the goal of making the most profit, they will choose the portfolio that has the largest expected value. Through analysis of historical data and future predictions I was able to create two separate expected value equations for both portfolios. The solution is simple, the portfolio with the greatest expected value will be chosen and is what investors should pick in attempts to make the most profit. In order for this to work we must assume investors are risk neutral because there can be large risk when it comes to cryptocurrencies.

The first equation for the Bitcoin heavy portfolio consists of various weights and probabilities for future returns. The first step was trying to figure out what Bitcoin could do in the future, specifically this year to make things simpler. I researched different price estimates from Bitcoin professionals and found a wide range with the general consensus that Bitcoin would increase in value this year. I found anything ranging between $1400 and $3000 estimates for the year end price in 2017 (Bovaird, What Will the Bitcoin Price Be in 2017?, 2017) .I also found reasoning behind why many people think that Bitcoin is going to increase and stabilize in the future. One source from Forbes discussed that Bitcoin’s current decreasing volatility actually correlates with its increase in price (Shin, 2017). Therefore, as Bitcoin continues to grow it will become more secure, and if it becomes more secure it will attract more investors creating even more growth and demand. Another source from Forbes argues that Bitcoin can be used to hedge against global uncertainty in times of increasing volatility in world markets (Eberhart, 2017). An older article that I found explained that some people might put their money in Bitcoin just like many put their money in gold or other commodities as a safety net (Cawrey, 2014). Bitcoin has no ties to the worlds governments so many may look to Bitcoin as a fallback if the economy were to collapse again. So there is some substance to back up the price estimates for 2017.

Because it would be unrealistic to go all in on Bitcoin more than doubling its price in the next few months, I created my own bear bull estimate. I took the expected return of each scenario, then multiplied it by the probability that it would occur, and then summed the numbers for the total expected value. This also included the return on the stock portion of the portfolio. I used the lowest price estimate of $1400 as my base estimate which is about a 17% return on Bitcoins current price. I used around a 150% return for my bull estimate since the highest estimate on Coindesk was around $3000 dollars a coin. Now in order to account for the risk of investing in Bitcoin, my bear estimate was negative 79.25%. The reasoning here was because at the end of 2013 to the end of 2014 there was a major pullback in price from about $1035.703 to $214.8425 (Bitcoin Price Index Chart, n.d.). Now obviously this has not happened every year so the probability that this will occur is low but there is still concern. In 2013, there was a major scandal on an online exchange for trading Bitcoin called Mt. Gox where millions of dollars were stolen. Due to the unregulated nature of Bitcoin, many are concerned that it is not a safe investment and that this type of scandal could happen again. However, I feel that Bitcoin has shown that it can recover from events like this since the price is now over the 2013 high. I also feel that as the currency grows and becomes more popular it will become more secure. Another concern is a possible hard fork which would split Bitcoin into two separate currencies. This could possibly devalue the currency greatly however there’s a slim chance this occurs as 95% of the Bitcoin community would have to vote for this fork to happen (Price, 2017). Even though it is unlikely that a major pullback happens, I think it is important to account for the possibility of a pullback as big as the largest one that’s occurred (79.25%). The expected return used for the average stock portfolio was taken from Morningstar’s 10-year average return of the S&P 500 at about 6.98%. The 10-year average return was used because we cannot say accurately that the market will continue to grow at its current rate of 13% (S&P 500 Index, n.d.).

With the expected return rates figured out for each investment I had to then assign probabilities to the likelihood that the base, bull, or bear estimates would occur. By looking at the past 6 years on Bitcoins price chart I created probabilities for its estimates. There were about 2 stagnant years within the past 6-year cycle that showed little growth compared to some of the more explosive years. For the base estimate I said that there’s a 2 out 6 or 33.33% chance that Bitcoin increases 17%. Further evidence for this assumption comes from a Forbes article that talked about a Bitcoin investment 3 years ago only generating 9% returns (Shin, 2017). This is because there was the major pull back after 2013 then a year of small consolidation and growth. The other year of small growth can be observed before the 2013 spike in price in 2012. For the bull estimate I concluded that 3 out of the past 6 years have shown significant growth by taking away the years of small growth and the year with a major pullback. This gives us a 50% chance that Bitcoin will increase by 150%. Historical data from the price chart does show 3 specific explosive years in Bitcoins price. For the bear estimate I used the 1 year at the end of 2013 where the priced tanked giving us a 1 out of 6 chance of another huge pullback. So there is a 16.67% chance that Bitcoins price decreases by 79.25%. This does seem like a large percentage however I am assuming the worst scenario. The likelihood of a near 80% pullback would require a huge negative event to occur so this really is just calculating in the absolute worst scenario to look at the risk of the investment. I then assumed a 100% chance that the stock market goes up a minimum of 6.98% as it is the 10-year average. This is also a pretty pessimistic outlook for 2017 however, it is always good to veer on the side of caution as we cannot be certain that the market will continue to increase at 13%.

Both portfolios have a lump sum of a million dollars for simplicity. This allows us to make the larger portion of each portfolio $700,000 and the smaller portion $300,000. I then took these numbers and plugged them into the expected value formulas (Appendix A,B). The expected value for the Bitcoin heavy portfolio came out to be $493,148.30 and the expected value for a Stock heavy portfolio came out to be $251,235. The state variable here would be around 49% EV for Bitcoin heavy and around 25% for stock heavy. Any investor looking at the game would see this and know to choose a Bitcoin heavy portfolio for greater returns. So the answer to which portfolio would be more profitable is Bitcoin assuming that the investors are risk neutral. Because they are risk neutral they choose the option with the highest expected value. Obviously this is only one model but when adjusting the weights and probabilities, it can be observed that it would take a substantially larger chance of a major pullback occurring for a Bitcoin heavy portfolio to be the least valuable option. Another observation can be made by changing the return for the stock portion of the portfolio. Even if the market continues to grow at its current rate of 13% the heavy stock portfolio still has less expected value. Really the return of the market is not a huge factor even if it increases because of the high potential of Bitcoin. Because Bitcoins current state isn't as volatile as it used to be the reward appears to outweigh the risk in terms of expected value giving investors a green light to invest heavier in the cryptocurrency.

Adding on to this conclusion, it may make investors more comfortable to know that Bitcoin has long term growth in store. There are numerous projections as too were Bitcoin could be valued years from now and the general opinion is that it will continue to grow. One projection actually values Bitcoin at $500,000 per coin in 2030. They argue that increased interest in Bitcoin along with increased political uncertainty and a decrease in cash transactions will push Bitcoin to an astronomical value down the road (Business insider). So far we have calculated that Bitcoin would be a good investment in 2017, but not long term. If Bitcoin were to reach $500,000 per coin within the next 13 years, that would mean a $700,000 investment now would be worth around $280 million. This is quite the estimate for Bitcoin, however, even if it were to get to a fraction of the 500,000-dollar prediction investors would still see exponential growth if bought now. Bitcoin has no intention of slowing down and could revolutionize currency and how transactions are made. If interest continues to grow and Bitcoin stabilizes, these predictions are not far out of reach.

On the other hand, not everyone is risk neutral when it comes to their investment decisions. So for someone that is risk averse they would need a differently weighted portfolio. In order to create one that would make a risk averse person indifferent between portfolios, we need to make the possible loss from Bitcoin less than or equal to the profits from the average stocks. By adjusting the amount of the portfolio invested in Bitcoin a risk averse investor can hedge the risk of Bitcoin. By setting the amount invested in Bitcoin equal to the profits earned from investing in stocks, you can avoid a loss if a pullback in Bitcoin were to happen. For example, if you still invested $700,000 in an average stock portfolio and your expected return was about 6.98%, you would calculate the expected return of that portfolio to be $48,860. You would then simply invest $48,860 in Bitcoin so that if Bitcoin hit zero you wouldn’t lose money due to the gains from stocks. So if you were risk averse, you could invest $700,000 in stocks and what’s equivalent to the stocks expected returns in Bitcoin to be indifferent about the risk of investing in Bitcoin. If Bitcoin tanked, you wouldn't lose overall in your portfolio and if bitcoin went up the 150% you would gain an extra $36,216 dollars on your investment this year. Also, with a $48,860 investment now, in 13 years you could have close to 29 million dollars' worth of Bitcoin.

Not everyone has a million dollars to invest, so it may come off as a pricey investment to the average investor. However, you can buy fractions of bitcoins unlike stocks, you don't have to pay the full $1200 for one Bitcoin. For example, you can purchase 10 dollars’ worth of Bitcoin which can then be held on to as an investment or used for transactions. So if you wanted to split your portfolio this way, you would be able to and would see proportional returns to the previous models. Therefore, all investors should at least consider this option for a much more profitable portfolio within the next 10 years.
Turning away from an n-person game where any number of investors can look to the two expected value formulas for the better investment choice, we can view this as 2 player game. By building a payoff matrix between two competing investment firms, we can figure out the Nash equilibrium. Both investors have two choices, they can invest Bitcoin heavy or stock heavy. The payoff for either investor investing in stock heavy portfolios will always be 2. Investing in a Bitcoin heavy portfolio will give either player a payoff of 5, however if one player invests in stock heavy and the other chooses Bitcoin heavy, the Bitcoin heavy investor will get a payoff of 6. This is because investment firms want to make the most money so that they can attract more people to invest in their firms. If they outperform the other firm by choosing Bitcoin, they will then receive more customers. If we analyze the matrix we can see that both investors always have incentive to switch from a stock heavy portfolio for a more profitable payoff. Therefore, there is a dominant strategy for the investors to both invest in Bitcoin. The assumption here is that both investors want to make the most profit so they will always choose the Bitcoin portfolio. In the long run more people investing in Bitcoin will drive the price up even further (Appendix Table 1).

What hasn't been discussed yet is whether or not to invest in Bitcoins competing cryptocurrencies. Labeled as alt-coins, they make up many alternative choices to Bitcoin that show promising return. For example, the second biggest cryptocurrency in the market is Ether. Developed by Ethereum Ether is one of the fastest growing alt-coins. Within the last couple months Ether prices have seen an increase from around $17 per unit to $50 per unit. In mid-March when a group of Bitcoin investors applied for an ETF, they were denied due to volatility and other risk concerns. Right after this happened Ether saw the beginning of a massive price spike (Bovaird, Ether Price Tear Continues With New All-Time High, 2017). As Bitcoin continues to grow, it is no surprise that people want to find alternatives that have a lower price. This is because there is room for a lot more profit in a shorter time if these currencies ever reach the value of Bitcoin. Also as Bitcoin grows the industry for cryptocurrencies develops and more currencies and usages for blockchain technology are developed alongside it. As profitable as these alt-coins can be, the question remains whether or not they are really worth investing in. With a lot less information on these cryptocurrencies predicting their future can be a lot harder than predicting Bitcoins. There is also a lot more volatility and room for random spikes and crashes of the prices of alt-coins. For example, Zcash, started off trading in the thousands of dollars per unit and is currently trading in the 70-dollar price range. Knowing which cryptocurrency will remain stable and grow can be near impossible as they begin to emerge. Another concern is that many of these alt-coins serve no purpose other than for transactions on the deep web for criminal activity and some are just pump and dump schemes. However, the top alt-coins like Ethereum do show great potential with their specified uses. What sets them apart from Bitcoin is that Bitcoin is just a currency and alt-coins serve a variety of purposes from smart contracts to overseas instant bank transactions. So even with the high risk, assigning a portion of your portfolio to them might not be a bad idea.

By rewriting our expected value formula's and rearranging the weights for our portfolios we can include alt-coins in the mix. I think that investing any more than 5% of your portfolio to alt-coins could be extremely risky just due to their volatile nature. In doing so you hedge your portfolio so that what you could lose should be able to be recovered by what your return is from Bitcoin or stocks. So with a 5% weight alt-coins would take up 50,000 dollars of the million-dollar portfolio. Another way to hedge against risk would be diversifying the altcoin investment. The top 6 alternative cryptocurrencies according to Investopedia are Litecoin (LTC), Ethereum (ETH), Zcash (ZEC), Dash, Ripple (XRP), and Monero (XMR) (Prableen Bajpai, 2017). If you were to split up your investment between the six of these, you would have substantial diversification in case one of them were to tank out and end up being a bust. Now another issue is knowing what each of these are going to do over time. With little information and many of these companies in the early stages, I think we can assume a 50% chance that they will succeed. This may be pessimistic but as history shows there can be scandals and issues within these companies and investing in them right now really can be a gamble. A 50/50 shot of major success isn't too far from the truth. Determining how much success is a lot tougher as none of these coins have showed the same success as Bitcoin. However, searching online, you can find a wide range of predictions as to where prices may fall. I think if these coins have a good year they could double in price and since they are relatively inexpensive that wouldn't be too crazy of a prediction. Also Ether has more than doubled in price since March and so has Zcash, so it's not that bold of a statement (source). However, it is highly unlikely they all will double in price within the year so going off historical data in the past couple months, about 3 out of the main 6 have doubled, Ether, Dash, and Zcash. We could say that there's a 50% chance this investment goes up 50%. The next step is to recalculate the portfolios expected values.

The new portfolio will contain a 5% alt-coin investment. In adjusting the Bitcoin and stock weights to the portfolio, I figured it is best to take the 5% out of the Bitcoin investment in order to secure the steady returns from stocks. So the new portfolio will be weighted 65% Bitcoin, 30% stocks, and 5% alternative cryptocurrencies. Adding this option gives every player in the n-person game and the 2 player game three choices between a Bitcoin heavy, stock heavy, and alt-coin mixed portfolios. In the n-person game, the new expected value equation for the alt-coin portfolio contains the new adjusted weights for the investment. As discussed there is a 50% chance of 50% return on the alt-coin investment so the 50,000 dollars invested will be multiplied by .5 and then by .5 again and this is added to the EV equation. The expected value of an alt-coin mixed portfolio was $471,919 (Appendix C). This is actually lower than a Bitcoin heavy portfolio by close to $20,000. So investors that are risk neutral would probably stay away from investing in alt-coins. The reason behind all of this is that alt-coins are generally too unpredictable. With little known about the companies behind the currencies and their insane volatility, alt-coins are currently a poor investment. In the long run they could increase in value greatly like Bitcoin, but until that point we need to see more price stabilization to feel comfortable buying into them due to the high risk of collapse. Looking at the adjusted 2 player game we now have a third choice for players to choose the alt portfolio, their payoff would be a 4.7. Since there is still no incentive to switch from a Bitcoin heavy portfolio for either player, we still see the same dominant strategy towards Bitcoin. This leaves the Nash equilibrium in place at (Bitcoin Heavy, Bitcoin Heavy). Firms investing in alt-coins could be seen as too risky for many customers so people would possibly veer away from them and stick with firms investing solely in Bitcoin (Appendix Table 2).

Going back to the n-person game, you can see the dominate strategy to choose a Bitcoin heavy portfolio more clearly on a graph. By using the amount invested on the X-axis and the returns/expected value on the Y-axis I created two graphs of the expected value formulas. The first graph contains just the Bitcoin and Stock heavy portfolios. You can see that no matter how much you invest in the stock portfolio; the graph will never intersect with the Bitcoin portfolio. This shows that the Bitcoin heavy portfolio will always be a dominant strategy in this game. By adding in the Alt-Mix portfolio you can see the same results as before. The Bitcoin heavy portfolio will never be intersected by the other two options so it remains a dominant strategy (Appendix Graphs).
In conclusion, investors that are trying to maximize their profits should always choose a Bitcoin heavy portfolio. As we have seen the expected value for that portfolio is higher than the other two options, stock heavy and alt-coin mixed. In terms of what’s expected any number of investors that invest any amount of money and are risk neutral should always weight their portfolios more towards Bitcoin. However, in a real world setting we know that people are not so risk neutral and want to assure themselves or other investors not risk but profits. Those looking to minimize risk can look towards their returns from the stock portion of the portfolio and then invest an equivalent amount into Bitcoin. In doing so they essentially will lose no money if Bitcoin were to tank to zero and any smart investor would sell before Bitcoin hits that low of a price anyways. Looking towards future investments alt-coins could definitely be something of consideration. Ethereum is one to watch as it has recently spiked in price and received great interest this year from several large companies such as Microsoft and JP Morgan (Hackett, 2017). Anyone investing in these currencies must be well in tune with the news surrounding them as well since the only thing that drives their value is supply and demand. In the end, game theory analysis shows us that investing in Bitcoin is definitely something firms and investors should at least take into consideration to help boost their returns in years to come. My final argument for Bitcoin is that when I began writing this paper Bitcoin had been priced around $1200 a coin and is currently trading around $1600 dollars a coin beating the lowest two estimates for this year that I found on Coindesk.

Appendix
A. EV for Bitcoin Heavy​
= 0.167(700,000-0.7925)+(700,0000.17)0.33+(700,0001.5)0.5 +(0.0698*300,000)= $493,148.30 ​

B. EV for Stock Heavy​
= (.0698700,000)+.167(300,000-.7925)+(300,000.17).33+(300,0001.5)*.5= ​$251,235 ​

C. EV For Alt-Mix Portfolio=
0.167(650,000-0.7925)+(650,0000.17)0.33+(650,0001.5)0.5+0.0698(300,000)+0.5(50,000*0.5) ​= $ 471,919.20​

Works Cited
Anderson, T. (2017, January 5). Most investors didn't come close to beating the S&P 500. Retrieved from CNBC: http://www.cnbc.com/2017/01/04/most-investors-didnt-come-close-to-beating-the-sp-500.html
Bitcoin Price Index Chart. (n.d.). Retrieved from Coindesk: http://www.coindesk.com/price/
Bovaird, C. (2017, March 16). Ether Price Tear Continues With New All-Time High. Retrieved from CoinDesk: http://www.coindesk.com/ethereum-price-tear-continues-setting-new-time-high/
Bovaird, C. (2017, January 1). What Will the Bitcoin Price Be in 2017? Retrieved from CoinDesk: http://www.coindesk.com/what-will-the-bitcoin-price-be-in-2017/
Cawrey, D. (2014, August 24). Stability and Taxes: Could Bitcoin Be a Replacement for Gold? Retrieved from CoinDesk: http://www.coindesk.com/stability-taxes-bitcoin-replacement-gold/
Eberhart, A. (2017, April 6). SEC Rejects Bitcoin ETFs: Should You Reject Bitcoin Investments? Retrieved from Forbes: https://www.forbes.com/sites/allaneberhart/2017/04/06/sec-rejects-bitcoin-etfs-should-you-reject-bitcoin-investments/#14fb4f9c79bd
Garber, J. (2017, May 4). Bitcoin just soared to a new $1,600 high — but the first investor in Snapchat thinks it could hit $500,000 by 2030. Retrieved from Business Insider: http://www.businessinsider.com/bitcoin-price-could-be-500000-by-2030-first-snapchat-investor-says-2017-3
Hackett, R. (2017, February 27). Big Business Giants From Microsoft to J.P. Morgan Are Getting Behind Ethereum. Retrieved from Fortune: http://fortune.com/2017/02/28/ethereum-jpmorgan-microsoft-alliance/
Prableen Bajpai, C. (2017, April 21). The 6 Most Important Cryptocurrencies Other Than Bitcoin. Retrieved from Investopedia: http://www.investopedia.com/tech/6-most-important-cryptocurrencies-other-bitcoin/
Price, O. W. (2017, March 26). A Bitcoin civil war is threatening to tear the digital currency in 2 — here's what you need to know. Retrieved from Business Insider: http://www.businessinsider.com/bitcoins-hard-fork-bitcoin-unlimited-segregated-witness-explained-2017-3
S&P 500 Index. (n.d.). Retrieved from Morningstar: http://quicktake.morningstar.com/index/IndexCharts.aspx?Symbol=SPX
Shin, L. (2017, January 9). Bitcoin's Price Was Volatile Last Week, But Not Last Year. Retrieved from Forbes: https://www.forbes.com/sites/laurashin/2017/01/09/bitcoins-price-was-volatile-last-week-but-not-last-year/#49adc7ff126f

I am not a professional advisor and this is not something you should put all your money into. I am not responsible for any gains or losses this is just an opinionated article about what I think could happen.