Cryptocurrencies have been acknowledged as an emerging asset class with a relatively low correlation to traditional asset classes and independent drivers of their long-term performance (see for an example excellent papers by Harvey et al (2022) and Adams at al (2024)).
A year ago in Summer of 2023, I published research article in Risk Magazine (SSRN draft) on quantitative methods for optimal allocation to cryptocurrencies within alternative and balanced portfolios. The metrics for consideration include metrics for portfolio diversification, expected risk-return relationships and skewness of the returns distribution. Using roll-forward historical simulations, I showed that all four allocation methods produce a persistent positive allocation to Bitcoin and Ether in alternative and balanced portfolios with a median allocation of about 2.7%.
This time, I would like to present the updated outcomes from my model given that Bitcoin and Ether had a strong performance of 95% and 35%, respectively, since the last update to today (from 30Jun2023 to 16Aug2024).
Spoiler: the performance of all four methods for balanced and alts portfolios have been in line with what has been reported in the article with optimal allocation weights to Bitcoin and Ether largely unchanged. Python code for this analysis is available in OptimalPortfolios packadge github repo.
First I start with the analysis of annual rolling performance. In Subplot (A) of Figure 1, I show Sharpe ratios (through the paper and this post, the Sharpe ratio is computed using monthly log-returns adjusted by 3m UST rate) 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 realized from the investment period from 31Dec2020 to 16Aug2024 is 0.29.
Clearly, the early periods before 2017 are characterized with higher realized Sharpe ratios. What is remarkable that any investment period that starts at the end of each calendar year from 2010 to today generated positive Sharpe ratio. In Subplot (B) of Figure 1, I show the realised skeweness of monthly returns. In early periods, the monthly performance exhibits highly positive skewness. Also more recently the skeweness became positive again.
Figure1. Realized Sharpe ratios 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.
Methodology
The long-term positive performance and positive skeweness of cryptocurrency returns pose well for quantitative allocation methods.
In the paper I consider four quantitative allocation methods for construction of optimal portfolios:
1) Two risk-based methods which include portfolios constructed using equal risk contribution and with maximum diversification methods.
2) Two risk-return based methods which include portfolios constructed using maximum Sharpe ratio and maximum CARA-utility methods.
For the investment universe, I consider the two mandates:
1) Alternatives (Alts) or unconstrained mandate that targets absolute returns by investing into alternative assets. This mandate is typical for high net worth private investors and family offices.
2) Benchmarked (Balanced) mandate which targets excess returns over a benchmark by allocating to a balanced equity/bond portfolio with additional overlay to alternative assets. Such a mandate is typical for institutional investors such as pension funds, insurance companies, and endowments.
As the balanced benchmark, I use the classic 60/40 equity/bond portfolio. I fix the target weight of the balanced portfolio for this mandate to 75% and assign $25%$ allocation to alternative assets. As a result, I consider the modern 70%/30%$approach for allocation portfolio of institutional mandates (see, for an example, McVey et al (2022)) with 30% allocation to bonds, 45% to public equities and 25% to alternative assets.
I refer to the paper for the investment universe of this mandates (In this analysis I change the benchmark for macro funds from NEIXMTI Index to HFRIMDT Index). For each allocation method, I evaluate the following portfolios given in Table 1 below. Portfolios 1, 2, 3 provide insights into the marginal contribution of including cryptocurrencies to investable universe alternative portfolios. Portfolios 4, 5 and 6 provide with insights into including cryptocurrencies to alternatives for blending with the 60/40 equity/bond portfolio. The marginal contribution of including cryptocurrencies is estimated using 4 portfolios with either BTC or ETH using 4 allocation methods, with total of 16 different portfolio schemes allocated to cryptocurrencies. I sue spot returns for performances of cryptocurrencies. This provides a sufficient depth for making insights.
Table 1. Simulated mandate portfolios with cryptocurrencies.
Optimal Portfolios and Their Performances
I use quarterly rebalancing and roll-forward analysis for generation and backtest of optimal portfolios. I describe the methodology in the paper and in github package
Here, I present the result of roll forward simulations from 31Mar2016 t0 16Aug2024. I will present some key figures here, all outputs can be found in pdf report of backtests.
Maximum Diversification
Maximum Diversification is my favorite method because it takes into account only the covariance matrix. Also, unlike Equal Risk Contribution method, Maximum Diversification method may produce zero weights to unattractive instruments. In Table 3, I show the risk-adjusted performance of the simulated portfolios without crypto and with inclusion of BTC and ETH cryptocurrencies. The Sharpe ratio is computed using monthly log-returns adjusted by 3m UST rate, beta and (annualised) alpha are computed by regression of monthly returns against 60%/40% equity/bond (Balanced) portfolio.
The marginal gain of including BTC and ETH is of +0.24 (=0.70-0.46) and +0.29 (=0.75-0.46) in Sharpe ratio for Alternative portfolios and of +0.23 and +0.21 for Balanced portfolios, which is significant.
In the last 4 rows I show the weight allocated to cryptocurrencies. The median allocation weight is 2.2%/1.9% and 3.13%/3.04% for BTC or ETH in alternatives and balanced portfolios, respectively.
Table 3. Risk-adjusted performance of Maximum Diversification allocation method.
In Figure 2, I show the time series of cumulative performances and drawdowns of Maximum Diversification portfolios. Adding cryptocurrencies to the portfolio universe did not materially impact realised drawdowns.
Figure2. Cumulative performance of portfolios computed using Maximum Diversification allocation method.
In Figure 3, I show the stack plot of optimal weights for BTC for alternatives and balanced mandates. We observe that the optimal weight of BTC has been persistent through the backtest period, in contract to other asset classes. It is interesting, that the optimal allocation to alternatives within balanced portfolio includes only Bitcoin and SG Trend instruments for the past two years.
Figure 3. Optimal Allocation weights for alternative and balanced mandates with universe including BTC.
Equal Risk Contribution
Equal risk contribution allocates equal buckets for risk (for Balanced mandate, 75% of risk is assigned to the balanced portfolio). We observe that adding cryptocurrencies improves the risk-adjusted performance of alternatives mandate. Interestingly, from the standpoint of the equal risk contribution method, allocations to BTC and ETH are almost same.
Table 3. Risk-adjusted performance of Equal Risk Contribution allocation method.
Maximum Sharpe Ratio
I use the rolling window of 5 years to estimate asset return and covariances for the estimation of the Sharpe ratio. For alternatives portfolio, the contribution to the performance (+0.80 and +0.67 in Sharpe) from adding cryptocurrencies is significant with their median weights of 9% and 4% for BTC and ETH. It is clear that using past returns as inputs to the optimiser may not be robust, however increasing the universe may lead to better results because of higher degree of freedom.
Table 4. Risk-adjusted performance of Maximum Sharpe Ratio allocation method.
Carra Mixture Utility
To estimate the 3-state mixture of returns distribution for the Carra Mixture utility, I also use the rolling window of 5 years. As I explain in the paper, the Carra Mixture Utility allocation method favors instruments with positive skeweness. Similarly to the Maximum Sharpe ratio, adding cryptocurrencies to the alternatives portfolio improves the realised Sharpe ratio considerably by +0.84 and +0.64 with BTC and ETH, respectively. The median allocated weight is 21% and 8% for alternatives mandate and 19% and 8% for the balanced mandate. The higher weights are the result of overweighting instruments with positive skeweness.
Table 5. Risk-adjusted performance of Carra Mixture Utility allocation method.
Summary of Weights
In the summary, I would like to the review the optimal weight to cryptocurrencies. The major goal of my article is to show that cryptocurrencies deserve an allocation for broad portfolios. In my analysis, I did not impose any allocation constraints to make a fair argument.
In Figure 4 I show the time series of optimal allocations to BTC and ETH by each method and for each mandate. In Table 6, I show summary of weights aggregated from time series.
Carra Mixture (CARRA-3) allocation method assigns the highest allocation to cryptocurrencies because it favors assets with high positive skewness.
We observe that the Maximum Sharpe ratio and Carra Mixture, which take into account the rolling performance of assets, have been producing smaller allocation weights in recent years following smaller the risk-adjusted performances of cryptocurrencies.
However, the risk based methods including Equal Risk Contribution (ERC) and Maximum Diversification (MaxDiv) produce largely stable allocation to cryptocurrencies, which stay largely intact in past couple of years.
The median of the time series median allocation is 5.7%, 3.8%, 3.0%, 2.4%, which gives a “median”allocation of 3.4% which slightly increased from 2.7% which I reported originally in the paper.
Figure 4. Optimal weights to BTC and ETH by allocation methods.
Table 6. Summary of weights
Further reading
Enjoy reading the paper and experiment with Python code
Disclosure
This research is a personal opinion and it does not represent an official view of my current and last employers.
This paper and the post is an investment advice in any possible form.
Cryptocurrencies are associated with high risk.
Importantly, the valuation of options on these assets is not feasible using conventional stochastic volatility models applied in practice such as Heston, SABR, Exponential Ornstein-Uhlenbeck stochastic volatility models, because these models fail to be arbitrage-free (forwards and call prices are not martingals). Curiously enough, the topic of no-arbitrage for SV models with positive return-volatility correlation has not received attention in literature, despite a large number of assets with positive return-volatility correlation.
