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### Abstract

As one of the finance classics, Modern Portfolio Theory is one of the most important and influential economic theories dealing with finance and investment. The theory provides a mathematical framework to build a portfolio of assets that maximizes expected return for a given level of risk taken. This article applies the theory to cryptocurrency trading and evaluates its effectiveness in risk diversification and investment return improvement.

### Full Report

We built mathematical models to backtest the risk diversification effectiveness of Modern Portfolio Theory by using historic data of the top 10 cryptocurrencies (BTC, ETH, XRP, BCH, EOS, LTC, ADA, XLM, TRX, IOTA). The result demonstrated that a well-proportioned portfolio diversification within Cryptocurrency investing can indeed reduce idiosyncratic risk effectively.

#### Data Preparation

We used the historical pricing data of the top 10 cryptocurrencies (by market cap) captured from Coinmarketcap from Nov 30th 2017 to May 21st 2018.

#### Mathematical Modeling

We built the portfolio based on Modern Portfolio Theory and assigned each top 10 Cryptocurrency with different weight, with a total weight adding up to 1. By doing so, we should be able to achieve higher returns for a given risk level. Here, we used standard deviation of daily return to evaluate risk and used Sharpe ratio to evaluate effectiveness of portfolio diversification. According to Modern Portfolio Theory, adding assets to a diversified portfolio that have correlations of less than 1 with each other can decrease portfolio risk without sacrificing return. Such diversification will serve to increase the Sharpe ratio of a portfolio.

Sharpe ratio = (Expected Portfolio Return − Risk-free Rate)/Standard Deviation of Portfolio Return

Under two conditions (all weights are greater than 0, and all weights add up to 1), we built a stochastic model by calculating the standard deviation of portfolio return, expected portfolio return and sharpe ratio respectively based on 500,000 random values. As a result, we were able to find a set of optimal investment portfolios.

#### Results

Below is the scatter plot created from 500,000 stochastic simulations, with x-axis representing expected volatility (standard deviation of portfolio return) and y-axis representing expected portfolio return. The size of sharpe ratio is represented by different colors of the dots, with deeper color representing a larger sharpe ratio. From the plot, we can see that dots close to the bottom-right corner of the graph have larger sharpe ratios that represent better portfolio diversifications. Dots in the upper left corner are bounded by “efficient frontier” — a set of optimal portfolios that offers the highest expected return for a defined level or risk or the lowest risk for a given level of expected return.

We can then draw the efficient boundary and find the best portfolio with largest sharpe ratio, marked by the red star in the figure below.

We can then connect the red star to the the origin (since we assume Risk free return to be 0), and draw the capital market line — the best portfolio composed of risk and risk-free assets along the efficient boundary line.

We then plotted efficient boundary, expected returns of optimal portfolio and single currency, standard deviations on the same graph in order to compare risk diversification.

#### Conclusion

Based on the evaluation above, we can see it is indeed possible to use Modern Portfolio Theory to construct an efficient frontier of optimal portfolios offering the maximum possible expected return for a given level of risk. However, we do realize that there are many other finance theories and models worth studying, and we will continue to explore these theories and models in our future reports.

References

1. Zvi Bodie, Investments 10th edition;
2. Jonathan Berk, Peter DeMarzo, Corporate Finance 3th edition;
3. Harry Markowitz, Portfolio Selection;
4. Wes McKinney, Python for Data Analysis

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