An old wisdom says that one should diversify and not put all eggs in one basket. Even religious text like the Talmud highlight diversification as an important rule for investing. According to the Talmud one should invest one third in land, one third in business and one third in gold. But only in 1952 Harry Markowitz presented a mathematical calculus how the optimal diversification can be determined. Markowitz “modernized” portfolio theory which at the time meant to give it a mathematical foundation. For this breakthrough, he was awarded the Nobel Prize in Economics in 1990. Now his portfolio theory is no longer modern but it still important enough to apply it to new assets like cryptocurrencies. Cryptocurrencies have become popular after the great financial crisis. On October 31, 2009, someone (or a group of people) under the pseudonym Satoshi Nakamoto posted a white paper entitled "Bitcoin: A Peer-to-Peer Electronic Cash System" to a cryptography mailing list. Briefly thereafter, Nakamoto published the first release of Bitcoin (BTC) as open source software. Soon copycats of bitcoin appeared, most importantly Ether, Litecoin and Ripple and Bitcoin itself forked into bitcoin cash. Now more than 4500 cryptocurrencies are traded totaling to a market capitalization of around 400 billion USD. Would modern portfolio theory suggest to include crypto currencies in an investor's portfolio?

Modern portfolio theory is based on the expected returns and the variance and co-variance between the returns of assets. Investors try to form a portfolio that achieves the highest expected return but that does not exceed a given level of risk measured by the portfolio variance. While the expected return of a portfolio of assets is the average of the expected returns of its components, the variance of a portfolio of assets can be smaller than the average of the variance of its components because assets might move in opposite direction. If one asset gains while the other asset loses the overall volatility of the portfolio is smaller since the movements of the assets compensate each other.

To illustrate this idea, consider two assets enumerated k=1 and k=2. If we denote the percentage of wealth invested in asset k = 1,2 by and the expected return of asset k by then the return of the portfolio, , is given by . However, the variance, , of the portfolio returns is given by . Thus, if the two assets have a tendency to move in the opposite direction then and the variance of the portfolio is reduced compared to the variance of the individual assets. The benefit of portfolio diversification can be illustrated in a mean-standard deviation diagram. In this figure we have standardized the covariance and displayed the correlation, , instead. Recall that . The figure shows that if both assets are perfectly correlated, i.e. they always move in the same direction then there is no diversification and a reduction of investment in asset E reduces the standard deviation of the portfolio at the same rate as it reduces the expected return. However, for , the risk is reduced more than the return so that in general one benefits from diversification.

Figure 1: Benefit of portfolio diversification for different levels of correlation.

If one does portfolio diversification not only with two assets but with all assets available then one can compute the so called efficient frontier, i.e. the curve of maximal expected portfolio return given a level of risk. Figure 2 shows the expected return and standard-deviations of various asset classes as well as the efficient frontier computed with those.

Figure 2: Efficient frontier computed from various broad asset classes.

The idea of modern portfolio theory is to find uncorrelated assets with positive expected return so as to enhance the efficient frontier. i.e. to obtain higher expected return for any given level of risk. Whether crypto currencies are a suitable asset class in this sense depends whether one assumes a positive expected return going forward and whether they are not to much correlated with the standard asset classes on which the portfolio is usually built. Figure 3 shows that crypto currencies are correlated with each other. Therefore it makes sense to aggregate them into one asset class. This aggregation is usually done in proportion to the relative market capitalization.

Figure 3: Correlation among crypto currencies.

On the other hand, as Ankenband and Bieri1 have recently shown, cryptocurrencies are not correlated to traditional asset classes, as shown in Figure 4.

Figure 4: Correlation of traditional asset classes and cryptocurrencies. Source: Ankenbrand and Bieri1.

Thus, including them in the portfolio analysis enhances the efficient frontier, as figure 5 shows. Depending on the risk profile of an investor he should then allocated also to cryptocurrencies. However, the amount of the allocation also depends on the expected returns. While the expected returns of traditional asset classes can be formed based on economic theory and a long history of data, this is unclear for crypto currencies (see blogpost “Why is bitcoin so volatile?”). Thus with these assets one should be more cautious.

Figure 5: Enhancement of the efficient frontier due to inclusion of crypto currencies. Source: Ankenbrand and Bieri1.

Further reading

- Edwin J. Elton, Martin J. Gruber, Stephen J. Brown and William N. Goetzmann (2017). Modern Portfolio Theory and Investment Analysis. Wiley Custom.

1Thomas Ankenbrand and Denis Bieri (2018). Assessment of cryptocurrencies as an asset class by their characteristics. Investment Management and Financial Innovations, 15(3), 169-181.Harry Markowitz (1952). Portfolio Selection. Journal of Finance, 7(1), 77-91. ↵

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