**4. Portfolio Sampling**

In this section, we perform an extensive sampling procedure to explore how diversification benefits depend on the number of cryptocurrencies held in a portfolio and on the number of decile sectors from which to choose those cryptocurrencies. Motivated by Section 3, we choose the normalized leading eigenvalue *λ*˜ <sup>1</sup>(*t*) to quantify the potential diversification benefit. We investigate the diversification benefits of portfolios that consist of *mn* cryptocurrencies such that *n* cryptocurrencies are drawn from *m* separate deciles. Both the individual cryptocurrencies and the sector deciles are drawn randomly and independently with uniform probability. We draw *D* = 500 portfolios for each ordered pair (*m*, *n*).

To quantify the potential diversification benefit for a portfolio consisting of *mn* cryptocurrencies, we determine the *mn* × *mn* correlation matrix **Ψ** for each draw and calculate the associated normalized first eigenvalue *λ*˜ *<sup>m</sup>*,*n*(*t*). Again, we use a rolling time window of length *τ* = 90 days when determining the cross-correlation matrix. In particular, we record the 50th percentile (median) of the *D* values, which we denote *λ*˜ 0.50 *<sup>m</sup>*,*<sup>n</sup>* (*t*).

We analyze this quantity in two experiments. First, we compute the temporal mean of the median of the normalized first eigenvalue

$$
\mu\_{m,n} = \frac{1}{T - \pi + 1} \sum\_{t=\pi}^{T} \lambda\_{m,n}^{0.50}(t) \tag{5}
$$

as a measure of the diversification benefit of a portfolio with *n* cryptocurrencies in each of *m* decile sectors. Table 2 records these means *μm*,*<sup>n</sup>* for cryptocurrency portfolios across values of (*m*, *n*) for 1 ≤ *m* ≤ 10 and 1 ≤ *n* ≤ 4.

Table 2 shows the mean *μm*,*<sup>n</sup>* of the median normalized eigenvalue *λ*˜ 0.50 *<sup>m</sup>*,*<sup>n</sup>* (*t*) for various combinations of cryptocurrency sectors, and randomly sampled cryptocurrencies within each sector. We also mark in red a "greedy path" to decrease the overall average *μm*,*<sup>n</sup>* (that is, increase the overall diversification benefit of a portfolio) by greedily increasing either *m* or *n* at each stage. There are several key findings from this analysis. First, the overall structural finding with respect to optimal portfolio construction strongly resembles that of the equity market in [86]. We see incrementally greater benefit in diversifying across sectors rather than within them, and we see a significant reduction in marginal diversification benefit once a portfolio reaches a critical mass of securities (sampled from different sectors). This leads to the existence of a "best value" cryptocurrency portfolio, such as that seen in the equity market. This finding is slightly surprising and may support our hypothesis that retail cryptocurrency investors diversify across cryptocurrency market capitalization levels. Indeed, this may occur in the absence of clearly defined sector themes, which may exhibit different performances during different parts of the business cycle. As investors come to better understand cryptocurrencies, and cryptocurrencies related to separate aspects of the digital economy begin to perform differently during various market conditions, this diversification benefit may slightly alter and amplify. That is, rather than cryptocurrency market capitalization being a primary discriminator in diversified performance we may see a closer resemblance to the equity market with cryptocurrency sector themes more closely resembling equity dynamics.

**Table 2.** Average *μm*,*<sup>n</sup>* of the median normalized eigenvalue *λ*˜ 0.50 *<sup>m</sup>*,*<sup>n</sup>* (*t*) for different pairs of *m* sectors and *n* cryptocurrencies per sector. In red we display a greedy path that aims to increase the total diversification benefit (by decreasing *μm*,*n*) at each step. We identify a best value cryptocurrency portfolio consisting of 4 sectors and 4 cryptocurrencies per sector. This (4,4) portfolio has nearly the same diversification benefit as the largest possible (10,4) portfolio, as we will also show in the next experiment.


In the second experiment, we investigate which portfolio combinations (*m*, *n*) share the most similar evolution in their collective dynamics. For this purpose, we perform hierarchical clustering on the distance metric

$$d((m,n),(m',n')) = \frac{1}{T-\tau+1} \sum\_{t=\tau}^{T} |\tilde{\lambda}^{0.50}\_{m,n}(t) - \tilde{\lambda}^{0.50}\_{m',n'}(t)|\,\tag{6}$$

which computes the average absolute difference between the median eigenvalues of two portfolios (*m*, *n*) and (*m* , *n* ). This results in a 40 × 40 distance matrix for 1 ≤ *m* ≤ 10, 1 ≤ *n* ≤ 4. Hierarchical clustering is a convenient and easily visualizable tool to reveal the proximity between different elements of a collection. Here, we perform agglomerative hierarchical clustering using the average-linkage criterion [90]. The algorithm works in a bottom-up manner, where each ordered pair (*m*, *n*) starts in its own cluster, and pairs of clusters are successively merged as one traverses up the hierarchy.

The results of hierarchical clustering are displayed in Figure 3. The resulting structure is interesting. The dendrogram consists of four predominant groups of clusters. There is an easily identified outlier cluster, consisting of the smallest portfolios that provide the least diversification benefit. This cluster, located at the bottom of the dendrogram, includes portfolio combinations such as (1,1), (1,2) and (2,1). The second least diversified cluster is located at the top of the dendrogram and includes portfolio combinations such as (1,3), (1,4) and (4,1). Below this, is a significantly larger cluster consisting of portfolio combinations such as (8,1), (3,3) and (4,2). Finally, the largest, most well-diversified fourth cluster consists of portfolio combinations ranging from (4,3) through to (10,4). The size and range of portfolio combinations within this cluster have interesting implications for risk management in cryptocurrency portfolio diversification. The fact that combination (4,3) is in the same cluster as portfolio (10,4) suggests that comparable risk mitigation can be realized with a portfolio of size 12 when compared to a portfolio of size 40. This finding is not dissimilar to that proposed in [86], where a "best value" portfolio (9,4) is shown to provide comparable diversification benefit to a (9,9) portfolio. Furthermore, the sheer size of this cluster indicates that one may require a lower cardinality portfolio in cryptocurrency portfolio management than in equities when trying to attain a "best value" portfolio.

**Figure 3.** Results of hierarchical clustering applied to (6) between ordered pairs (*m*, *n*). A large majority cluster confirms the finding of Table 2 that the (4,4) portfolio is closely similar to the full (10,4) portfolio in its diversification benefit over time.
