Cryptocurrencies have been acknowledged as an emerging asset class with a relatively low correlation to traditional asset classes. One of the most important questions for allocators is how much to allocate to Bitcoin and to a portfolios cryptocurrency assets within a broad portfolio which includes equities, bonds, and other alternatives. I wrote a research paper addressing this questions. I will provide a short summary here and refer to my paper on SSRN for details.

I apply four quantitative methods for optimal allocation to Bitcoin cryptocurrency within alternative and balanced portfolios based on metrics of portfolio diversification, expected risk-returns, and skewness of returns distribution. Using roll-forward historical simulations, I show that all four allocation methods produce a persistent positive allocation to Bitcoin in alternative and balanced portfolios. I find that the median of optimisers’ average weights is 2.3% and 4.8% for 100% alternatives and for 75%/25% balanced/alternatives portfolios, respectively. I conclude that Bitcoin may provide positive marginal contribution to risk-adjusted performances of optimal portfolios. I emphasize the diversification benefits of cryptocurrencies as an asset class within broad risk-managed portfolios with systematic re-balancing.

I start by considering a few drivers that support the allocation to Bitcoin using on statistical properties of its returns (see Harvey et al (2022) for an excellent review of supporting fundamental factors).

**Rolling Performance of Bitcoin returns**

Stellar performances of core cryptocurrencies, including Bitcoin and Ethereum, have been a major supporting factor for investing into cryptocurrencies. However, these performances are realized with high volatilities, so that risk-adjusted performance, for example measured by Sharpe ratio of average log-returns, is not very significant and have been declining over the past years.

In Subplot (A) of Figure (1) I show Sharpe ratios for trailing holding periods with the period start given in the first column and the period end given in the first row. For an example, Sharpe ratio corresponding to the period from 31 December 2017 to 1 September 2022 is 0.10. It is obvious that most of large gains are attributed to periods prior to the end of 2017, when Bitcoin was little known to investment community. As a result, any historical analysis covering the early years of Bitcoin performance should be taken with caution.

*Figure (1). Realized performance from the period start (given in the first column) to the period end (given in the first row). Subplot (A) shows Sharpe ratio using average monthly log-returns; Subplot (B) shows skewness of monthly returns*

**Correlations**

A low correlation with traditional asset classes has been a supporting factor for allocating to cryptocurrencies within broad portfolios. In Figure (2) I show correlation matrices of monthly returns for three different periods: prior to 2018, from 2018 to August 2022, and from 2020 to August 2022. We see that returns of Bitcoin were little correlated to 60/40 portfolio in the early period, however, the correlation between Bitcoin and equities and bonds increased over the past three years. Remarkably, Bitcoin’s correlation with returns on alternative assets has not changed significantly. Thus, the allocation to Bitcoin is still viable within a diversified portfolio of alternatives.

*Figure (2). Correlation matrix of monthly log-returns between assets in the investable universe for three periods. HFs is HFRX Global Hedge Fund Index, SG Macro is SG Macro Trading Index, SG CTA is SG CTA Index, Gold is SPDR Gold ETF (NYSE ticker GLD).*

**Positive skewness of distribution of Bitcoin returns**

Positive skewness of returns of cryptocurrencies is a supporting factor for allocation to this asset class. Indeed, in a very interesting paper, Ang et al (2022) argue that for skewness-seeking investors the allocation to Bitcoin could be optimal even if cross-sectional mean return may be negative. However, we observe that the realized skewness of returns of Bitcoin has been declining, following the decline of its Sharpe ratio, as I show in Subplot (B) of Figure (1). While in early years Bitcoin returns are characterized by high positive skewness, the skewness became negative in recent years. Still, the realized skewness of Bitcoin returns is higher than that of traditional assets. Importantly, Ang et al (2022) apply a two-state Normal mixture model to describe the profile of returns on Bitcoin. Further they apply maximization of CARA utility for skewness-seeking investors using this mixture model. I extend the model of Ang et al to multi-asset universe with N assets including Bitcoin.

I apply Gaussian Mixture model with M clusters to describe the distribution of asset returns conditional on a few clusters. Within each cluster, the distribution of N-dimensional vector of asset returns is normal with vector of estimated means and covariance matrix. I employ Python module *sklearn.mixture* for the estimation of Gaussian Mixture model and, through cross-validation, I have concluded that using 3 clusters is most robust to model the distribution of monthly returns of assets in our universe. In Figure below, I show the scatterplot of Bitcoin returns vs returns of 60/40 benchmark portfolio and one-std ellipsoids of Gaussian distribution in estimated clusters for two periods.

*Figure (3). Scatter plot and model clusters using estimated Gaussian mixture model. Subplots (A) and (B) show returns data from 19 July 2010 and from 18 December 2017, respectively, to 31 August 2022. Subplots (C) and (D) show corresponding cluster parameters for Bitcoin.*

**Portfolio Allocation Methods**

I consider four quantitative asset allocation methods for construction of optimal portfolios.

**Risk-only** based methods which include portfolios with equal risk contribution (denoted by ERC) and with maximum diversification (MaxDiv).

**Risk-return** based methods which include portfolios with maximum Sharpe ratio (MaxSharpe and with maximum CARA-utility.

For each allocation method, I evaluate the following portfolios:

- 100% Alts w/o BTC is the portfolio including alternative assets excluding Bitcoin;
- 100% Alts with BTC is the portfolio including alternative assets and Bitcoin;
- 75%/25% Bal/Alts w/o BTC is the portfolio with fixed allocation to 75% of balanced 60/40 equity/bond portfolio and 25% allocation to alternative assets excluding Bitcoin;
- 75%/25% Bal/Alts With BTC is the portfolio with fixed allocation to 75% of balanced 60/40 equity/bond portfolio and 25% allocation to alternative asset classes including Bitcoin.

Portfolios 1 and 2 enable us to analyze the marginal contribution of including Bitcoin to the investable universe of alternative portfolios. Portfolios 2 and 3 provide with insights into including Bitcoin to alternatives universe for constructing overlays for 60/40 equity/bond portfolio.

**Optimal weights**

In table below, I show the statistics of time series of optimal weights to Bitcoin produced by the four implemented portfolio optimisers. First, it is notable that all four optimizers produced non-zero weights at all quarterly re-balancing (because the time series minimum is higher than zero) for both portfolios, except for the last quarterly rebalancing of the most diversified 75%/25% portfolio. The optimization of CARA utility produced the highest allocation to Bitcoin for both portfolios, because Bitcoin adds most to the skewness of portfolio returns that is favorable for CARA method. However, the CARA portfolios have the lowest historical allocation to Bitcoin because of declining skewness of its returns. The median of optimisers’ average weights is 2.3% and 4.8% for 100% alts and 75%/25% alts/balanced portfolios, respectively. As a result, including of Bitcoin to the investable universe is beneficial for diversification benefits of broad portfolios.

*Figure (4). Minimum, average, maximum, and last weight (as of last quarterly re-balancing on 30 June 2022) to Bitcoin by allocation methods computed using roll-forward simulations from 30 June 2015 to 31 August 2022. Subplot (A) shows the weight in the 100% alternatives portfolio, Subplot (B) shows the weight in the 75%/25% balanced and alts portfolio. ERC is portfolio with equal risk contribution, MaxDiv is portfolio with maximum diversification, MaxSharpe is portfolio with maximum Sharpe ratio, CARA-3 is portfolio with maximum CARA utility under Gaussian mixture model with 3 clusters.*

**Trailing performance**

In below table I show trailing realized Sharpe ratios of simulated optimal portfolios. I add equally weighted portfolio as a benchmark. For 100% alts portfolio w/o and with Bitcoin, the weight of Bitcoin is fixed to 0% and 2%, respectively, while the rest is equally allocated to alternative assets. For 75%/25% balanced/alts portfolio w/o and with Bitcoin, the weight of Bitcoin is fixed to 0% and 0.5%, respectively, the weight of 60/40 portfolio is 75% and rest is equally allocated to alternatives.

First, comparing 100\% alts portfolio w/o and with Bitcoin, we see that adding Bitcoin to the investable universe increased Sharpe ratios over the past periods of 2, 3, 5, 7 years except for the portfolio with maximum Sharpe ratio. The performance over the last year is better for portfolios without Bitcoin. However, I emphasize a robust positive performance of risk-based portfolios with and without Bitcoin in comparison to a poor performance of the benchmark balanced portfolio.

Contrasting 75%/25% balanced/alts portfolio w/o and with Bitcoin, we see that including Bitcoin benefits most of portfolios over all trailing periods. The exceptions include, first, the portfolio with the maximum Sharpe ratio and, second, for the ERC portfolio which slightly under-performs when Bitcoin is added.

A poor relative performance of portfolios with maximum Sharpe ratio highlights the hazard of relying on past data for forecast of future returns. In contrast, out-performers include risk-based methods that rely on the dynamic update of covariance matrices using most recent data.

*Figure (5) Sharpe ratios for trailing periods of 1, 2, 3, 5, 7 years starting from 31 August 2021, 2020, 2019, 2017, 2016, respectively, up to 31 August 2022. 60/40 is the benchmark equity/bond balanced portfolio, and EqualWeight w/o and with BTC are equally weighted portfolios with fixed 0% and 2% weights to Bitcoin, respectively.*

**Conclusion**

I present empirical evidence that it has been optimal to include Bitcoin to an investable universe for alternative and blended portfolios, using portfolio diversification metrics. Using roll-forward analysis with dynamic updates of portfolio inputs, I also find that adding Bitcoin have improved performances of optimal portfolios.

I conclude that adding Bitcoin, and more generally, a diversified basket of cryptocurrencies, to the investable universe of broad portfolios may be beneficial for both alternative portfolios and blended balanced/alternative portfolios. I emphasize the need for a robust portfolio allocation method with regular updates of portfolio inputs and re-balancing of portfolio weights.

My favorite allocation method is the optimiser of portfolio diversification metric along with the optimiser of the CARA utility under Gaussian mixture distribution for skewness-seeking investors.

Further details are provided in my paper on SSRN http://ssrn.com/abstract=4217841

**Disclaimer**

The views and opinions presented in this article and post are mine alone. This research is not an investment advice.

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