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Simple and Effective Portfolio Construction with Crypto Assets
Since the introduction of cryptocurrencies in 2009 nakamoto2008peer, the field of crypto trading has rapidly grown. In this note we consider the problem of constructing a portfolio of assets, including a combination of traditional financial assets such as stocks and bonds with crypto assets such as Bitcoin and Ethereum. Our conclusion is that despite the well known extreme swings of crypto currency values, simple standard portfolio construction methods suffice to realize the benefits of including crypto assets in a portfolio.
Figure (ref) shows the normalized prices of Bitcoin (BTC), Ethereum (ETH), and the S&P 500 index (SP500) over the past six years. To the eye it reasonably seems that crypto returns are quite different from traditional returns. Table (ref) lists some metrics for these asset returns over the same period, which also suggest that crypto asset returns fundamentally differ from traditional asset returns. For example, ETH at one point dropped in value by a factor of almost $20\times$, while the maximum drop in value of SP500 is only one third.
Stylized facts of crypto returns, such as extreme volatility, heavy tails, excess kurtosis, and skewness, have been well documented liu2021risks, hu2019cryptocurrencies, karim2024exploring. Figure (ref) shows quantile-quantile (QQ) plots of the log returns of BTC, ETH, and SP500 over the last six years.
All three return distributions have tails that deviate significantly from the normal distribution, with the crypto asset returns exhibiting more extreme tail behavior than the market index. The 1st and 99th return percentiles are shown in the plots as dashed green lines; here too we see that crypto asset returns have considerably bigger tails than the market index, even when normalized to have the same volatility.
The documented characteristics of crypto asset returns have made conventional investors hesitant to include them in their portfolios, due to perceptions of risk and unpredictability. While asset managers may have additional concerns, such as legislative risk and other fundamental factors like the ongoing debate about the legitimacy of crypto as a real asset yarow2013krugman, smith2013bitcoin, krugman2011golden, ciolli2013bitcoin, we will focus here on the concerns related to the statistical properties of crypto asset returns.
These characteristics of crypto asset returns have also led to a wealth of research on how to extend traditional portfolio construction methods to include crypto assets, including methods similar to those used for traditional assets holovatiuk2020cryptocurrencies, brauneis2019cryptocurrency, burggraf2019risk, hu2019modelling, as well as more complex machine learning driven methods ruiz2022cryptocurrencies, jiang2017cryptocurrency, lucarelli2020deep, jiang2017deep, ramkumar2021cryptocurrency, ruiz2022cryptocurrencies.
Despite the documented differences between crypto and conventional asset returns, some authors have argued that the two asset classes are fundamentally similar palomar2024portfolio, even though the crypto asset returns and volatilities are much higher. We agree with this perspective. In this note we show a simple method for constructing a portfolio of traditional and crypto assets using a risk allocation framework, (hopefully) debunking the idea that novel and complex machine learning approaches are necessary to manage a portfolio that includes crypto assets. Based on a back-test of the risk allocation method, we propose an even simpler portfolio construction method, reminiscent of the traditional 60/40 stocks/bonds split, which consists of a 90/10 split of traditional and crypto assets, followed by dynamic (time-varying) dilution with cash, to achieve a given ex-ante risk. We refer to this simple portfolio as DD90/10.
\paragraph{Outline.} In \S(ref) we review previous and related work. We introduce in \S(ref) an approach for constructing a portfolio that combines traditional and crypto assets within a risk allocation framework. We illustrate the performance of this method on historical data in \S(ref). Finally, in \S(ref) we propose the DD90/10 portfolio allocation strategy, and show that it has performance similar to the risk allocation strategy.
Stylized facts of financial return series refer to a set of empirical observations that are consistently observed across different financial markets and asset classes. We review some of the most prominent stylized facts for financial assets, and refer the reader to palomar2024portfolio for a more comprehensive overview.
\paragraph{Equities.} Stylized facts for equity returns include the non-normal distribution of returns, characterized by heavy tails and excess kurtosis malmsten2010stylized, bulla2006stylized. Another fact is volatility clustering, where large price movements tend to be followed by additional large movements, regardless of direction. While returns themselves are generally uncorrelated over time, the absolute or squared returns often display strong autocorrelation, highlighting a pattern in the magnitude of fluctuations. Furthermore, the leverage effect is a notable feature, where negative returns increase future volatility to a greater extent than positive returns of the same size. Equity return distributions have also been shown to exhibit asymmetry between positive and negative returns jiang2020stock.
\paragraph{Cryptocurrencies.} Cryptocurrencies exhibit several patterns similar to traditional financial assets, but with more extreme behavior hu2019cryptocurrencies, ghosh2023return. They are highly volatile, with large price fluctuations and heavy-tailed return distributions. Volatility tends to cluster, with periods of high volatility followed by more volatility. While returns show no autocorrelation in the short term, return magnitudes do exhibit autocorrelation. Cryptocurrencies also exhibit asymmetry in returns karim2024exploring.
Portfolio construction involves selecting a combination of assets by balancing the trade-off between expected return and portfolio risk. Here we discuss key historical contributions to the field, along with some recent advancements. For more extensive overviews, refer to texts such as grinold2000active, narang2024inside, and the studies by cornuejols2006optimization, kolm201460, chakrabarty2023mathematical, gunjan2023brief, boyd2024markowitz.
\paragraph{Markowitz portfolio construction.} Before Markowitz's seminal work in 1952, portfolio construction was largely based on heuristics and rules of thumb. Markowitz introduced a quantitative framework for portfolio construction, where the return of a portfolio is modeled as a random variable, and the expected return is maximized for a desired level of risk markowitz1952portfolio. Despite its simplicity, and having over 70 years of history, the Markowitz model remains the foundation of quantitative investing to this day boyd2024markowitz.
Extensions of the Markowitz model include the Black-Litterman model black1990asset, black1992global, black1989universal, fully flexible views meucci2010fully, and conditional value-at-risk (CVaR) optimization rockafellar2000optimization, to name a few. The Black-Litterman model is a Bayesian approach to portfolio construction, where the prior distribution is based on the equilibrium market portfolio, and the posterior distribution is updated with user-specific views on the expected returns of the assets. Fully flexible views is a generalization of Black-Litterman, allowing for nonlinear views on the returns. CVaR optimization replaces the variance in the Markowitz model with the conditional value-at-risk of the portfolio, penalizing the tail risk of the portfolio, and directly addresses the issue of a non-normal return distribution.
\paragraph{Machine learning based portfolio construction.} Typically, portfolio construction is split into two parts: data modelling and portfolio optimization palomar2024portfolio. Data modelling is concerned with predicting the expected returns and covariances of the assets, and portfolio optimization concerns selecting a portfolio of assets that trades off expected return and risk. However, with the growing popularity of machine learning, and documented criticism of the Markowitz model michaud1989markowitz_enigma, in recent years several studies have proposed machine learning based portfolio construction methods as alternatives to this traditional framework gurrib2022machine, bartram2021machine. In particular, it has become popular to combine the two parts of portfolio construction into a single end-to-end machine learning model, where market features are fed into the model, and the model outputs a trade list. The argument for this has been that splitting portfolio construction into two parts is suboptimal, as there are uncertainties in the return forecasts that are not accounted for in the portfolio optimization component of portfolio construction kelly2023financial. Although theoretically appealing, we have yet to see a wide-spread adoption of these methods in practice, and the Markowitz model remains the dominant framework for portfolio construction boyd2024markowitz. We refer the reader to kelly2023financial for a detailed review of end-to-end machine learning models for portfolio construction.
\paragraph{Risk-based portfolio construction.} Risk-based portfolio construction methods rely on estimates of the asset return covariances, but do not require an estimate of the expected returns of the assets, making them attractive for practitioners who do not have access to data sources for estimating expected returns reliably. A trivial example of a risk-based portfolio construction method is the equally weighted portfolio. Another popular portfolio is the minimum variance portfolio, which is also the mean-variance efficient portfolio when expected returns are equal. Other risk-based portfolio construction methods include risk parity Qian119, where the risk contribution of each asset is equal, and maximum diversification portfolios choueifaty2008toward. These portfolios can all be computed via convex optimization johansson2023covariance, which makes them reliable, fast, and practical boydconvex. These portfolio construction methods can be implemented in just a few lines of domain specific languages for convex optimization such as CVXPY cvxpy.
\paragraph{Risk and covariance estimation.} Risk-based portfolio construction methods rely on estimates of the portfolio risk. There are in general two ways to estimate portfolio risk. The first is to use a realized measure of the variance of the portfolio. There are many such methods, including the exponentially weighted moving average (EWMA), methods based on mean absolute deviation or the rolling median geary1936moments, geary1947testing, as well as autoregressive conditional heteroskedasticity (ARCH) and generalized ARCH (GARCH) models engle1982autoregressive, bollerslev_1986, engle1986modelling. The second way to estimate portfolio risk is to leverage a covariance matrix of the asset returns. The literature on covariance estimating in finance is vast, but probably the most popular method is to use an iterated covariance matrix DCC, cov_barrat_2022. This method decomposes the covariance $\Sigma$ as $\Sigma = V R V$, where $V$ is a diagonal matrix with the asset standard deviations on the diagonal and $R$ is the correlation matrix. Typically, $V$ and $R$ are estimated separately, using EWMAs with different half-lives. We will refer to this method as the iterated EWMA (IEWMA) johansson2023covariance. It is also possible to dynamically adjust the half-lives of the EWMAs, to account for time-varying market conditions johansson2023covariance.
Several methods to managing portfolios of cryptocurrencies have been proposed, and these tend to be separated from investment strategies proposed for traditional assets.
\paragraph{Diversification.} Many studies have noted that the returns of cryptocurrencies are uncorrelated with traditional assets bakry2021bitcoin, yermack2024bitcoin, holovatiuk2020cryptocurrencies. This means that they can be leveraged to diversify a portfolio of traditional assets, and thus increase the risk-adjusted return of the portfolio holovatiuk2020cryptocurrencies, bakry2021bitcoin.
\paragraph{Portfolio construction.} Much of the literature on cryptocurrency trading is focused on machine learning or deep learning ruiz2022cryptocurrencies. In jiang2017cryptocurrency an end-to-end convolutional neural network, taking in raw price data and outputting a trade list, is proposed. A deep Q-learning portfolio management framework is proposed in lucarelli2020deep. The authors of jiang2017deep introduce a financial-model-free reinforcement learning framework, incorporating convolutional neural networks, recurrent neural networks, and long short-term memory models. In ramkumar2021cryptocurrency ARIMA models, convolutional neural networks, and long short-term memory methods are used for cryptocurrency price forecasting and multiple portfolio construction strategies are evaluated with the forecasted prices as input. For a more detailed review of machine learning in cryptocurrency portfolio management, see {\it e.g.}, ruiz2022cryptocurrencies.
Some crypto studies use traditional portfolio construction methods, such as the Markowitz framework. In holovatiuk2020cryptocurrencies, the authors show that crypto assets can improve the performance of a mean-variance optimized portfolio. The authors of brauneis2019cryptocurrency show how to implement a mean-variance optimized portfolio of cryptocurrencies and that it outperforms an equally weighted portfolio as well as single cryptocurrencies. The paper giudici2020network proposes an extension of the Markowitz model, combining random matrix theory and network measures to manage a portfolio of crypto assets. In platanakis2019portfolio the authors design crypto portfolios using variance-based constraints in the Black-Litterman model, to account for estimation uncertainties.
Other studies focus on risk-based portfolio construction methods, {\it i.e.}, those that do not rely on an estimate of the expected returns of the assets. In burggraf2019risk multiple risk-based portfolio construction methods are evaluated, and the authors find that most of them outperform single cryptocurrencies and the equally weighted portfolio. The authors of hu2019modelling find that minimizing the variance and conditional value-at-risk of a portfolio of cryptocurrencies, yields a portfolio that outperforms the market.
The most common approach to portfolio construction is to formulate the problem as a trade-off between expected return and risk, as suggested by Markowitz in 1952 markowitz1952portfolio. The main practical challenge with this framework is that it requires estimating expected returns of the assets. These are very difficult to estimate and, for obvious reasons, successful estimation techniques are proprietary; large hedge funds and asset managers have entire teams dedicated to estimating expected returns, using data sources for which they pay large premiums palomar2024portfolio. Here we describe a risk allocation approach, which does not require an estimate of the expected returns of the assets. It is based on the idea of risk parity Qian119, with additional constraints on the portfolio weights and a risk limit. We refer to this method as constrained risk allocation (CRA).
We consider a portfolio of $n$ non-cash assets, plus cash. We denote the asset weights as $w\in {\mbox{\bf R}}^n$, with $w \geq 0$, where $w_i$ is the fraction of the total portfolio value held in asset $i$. We let $c\geq 0$ denote the fraction of the portfolio value held in cash, so we have $\mathbf 1^Tw + c =1$, where $\mathbf 1$ is the vector with all entries one. We refer to $\mathbf 1^Tw$ as our asset exposure.
Let $\Sigma$ denote the $n \times n$ (estimate of the) covariance matrix of the returns. Assuming cash is risk-free, the portfolio risk is $w^T\Sigma w$. (The volatility is the squareroot of this.) Inspired by the identity \[ w^T\Sigma w = \sum_{i=1}^n w_i (\Sigma w)_i, \] we define the risk contribution of asset $i$ as $w_i (\Sigma w)_i$. In risk allocation we specify the fraction of risk to be held in each asset, as the vector $\rho \in {\mbox{\bf R}}^n$, with $\rho > 0$ and $\mathbf 1^T \rho=1$. We interpret $\rho_i$ as the fraction of total portfolio risk contributed by asset $i$. (We assume that all entries of $\rho$ are positive; if any were zero, we simply would not include that asset in the portfolio.) Thus we have
The special case $\rho = (1/n)\mathbf 1$ corresponds to risk parity, where all assets contribute an equal fraction of the risk.
The CRA problem is defined as
where $w \in {\mbox{\bf R}}^n$ and $c\in {\mbox{\bf R}}$ are the variables, $\sigma^2$ is the maximum allowed risk, and $F\in {\mbox{\bf R}}^{m \times n}$ and $g\in {\mbox{\bf R}}^m$ describe constraints on the portfolio. In words: We choose the portfolio to minimize cash holdings (or equivalently, maximize asset exposure), subject to a given risk allocation, a total risk limit, and some additional constraints on the weights.
We will assume that $g>0$ and $F\geq 0$ (elementwise), with each row of $F$ nonzero. We will soon see that this implies the CRA problem (ref) always has a unique solution. The weight constraints can be used to enforce a maximum weight on each asset, or a maximum weight on a subset of assets, {\it e.g.}, crypto assets.
The CRA problem (ref) is not itself a convex optimization problem, but it can be solved efficiently via convex optimization. We first consider the risk allocation constraints (ref) alone, together with $w \geq 0$. It can be shown that $w$ satisfies these constraints if and only if it has the form \[ w = \alpha x^\star, \] where $\alpha \geq 0$ and $x^\star\in{\mbox{\bf R}}^n$ is the unique solution of the convex optimization problem
with variable $x \in {\mbox{\bf R}}^n$ (and implicit constraint $x>0$). See, {\it e.g.}, cvx_book_additional. Thus the set of weights that satisfy the risk allocation constraints is a ray, with a direction that can be found by solving a convex optimization problem.
Now we take this very specific form for $w$ and substitute it back into the original CRA problem (ref), dropping the risk allocation constraints and $w \geq 0$ since they are automatically satisfied. This gives us the problem \[
\] with scalar variables $\alpha$ and $c$. Note that the quantities $\mathbf 1^Tx^\star$, $(x^\star)^T\Sigma x^\star$, and $Fx^\star$ are constants in this problem.
We can solve this simple problem analytically. Minimizing $c$ is the same as maximizing $\alpha$. Along with $\alpha \geq 0$, all constraints on $\alpha$ are (positive) upper limits: \[ \alpha \leq \frac{1}{\mathbf 1^T x^\star}, \quad \alpha \leq \frac{\sigma}{\left((x^\star)^T\Sigma x^\star\right)^{1/2}}, \quad \alpha \leq \frac{g_i}{(Fx^\star)_i}, \quad i= 1, \ldots, m. \] (Each of the denominators is positive due to our assumptions and $x^\star>0$.) It follows that the solution is
Roughly speaking: Scale the unconstrained risk allocation weights as large as possible with all constraints holding.
\paragraph{Summary.} The two step solution procedure is summarized as follows:
We note that Feng and Palomar have suggested a more sophisticated formulation of the CRA problem, in which the risk allocations need only hold approximately feng2015scrip, wu2020general, palomar2024portfolio. This formulation can be approximately solved by solving a sequence of convex problems. Our simple formulation is, however, good enough for us to make our main point.
\paragraph{Variation.} We can modify the way $\alpha^\star$ is computed in (ref). Instead of estimating the standard deviation of the unconstrained risk allocation portfolio $x^\star$ as $\left((x^\star)^T \Sigma x^\star\right)^{1/2}$, we compute the realized return trajectory of the portfolio $x^\star$, {\it i.e.}, $(x^\star)^T r_t$, $t=1,2, \ldots$, where $r_t$ is the vector of asset returns at time $t$. We then compute an estimate of the standard deviation of the portfolio return trajectory, using, {\it e.g.}, a EWMA. Thus to compute the scaling (which sets the cash dilution) we directly estimate the standard deviation of the return trajectory with the unconstrained risk allocation weights, rather than find it from our estimated covariance matrix (which we use to compute the risk allocation weights $x^\star$ in step 1.) We have found that leads to a modest but significant improvement in portfolio performance.
\paragraph{Data.} We consider daily close prices of two crypto assets, and ETH, with data from LSEG Data and Analytics. We also consider four daily traded industry portfolios: consumer goods and services, manufacturing and utilities, technology and communications, as well as healthcare, medical equipment, and drugs; these were obtained from Kenneth French's data library french_data_lib. The data spans from September 8th, 2017, to September 22nd, 2024, for a total of 2565 days, or 1729 trading days. (Although crypto assets are traded every day, we rebalance our portfolios only on market trading days; we do, however, realize gains and losses on crypto assets on weekends and holidays.) Figure (ref) shows the normalized price evolution of the six assets, and table (ref) lists some metrics for them. The data and code to reproduce the results are available at
\paragraph{Risk model.} We estimate the covariance matrix of the assets using an iterated EWMA, described in detail in johansson2023covariance. We use a 63-day half-life for the volatility estimate and a 125-day half-life for the correlation estimate. The estimated volatilities of the six assets are shown in figure (ref).
Two examples (from different time-periods) of the estimated correlation matrix are shown in figure (ref).
\paragraph{Simulation and parameters.} We simulate three portfolios.
We rebalance the portfolios every trading day, using risk parity. We impose a 10% annualized risk limit on the portfolio, {\it i.e.}, $\sigma = 0.1 \sqrt D$, where $D=250$ is the number of trading days in a year. To estimate the risk of the unconstrained risk parity portfolio $x^\star$, we use a 10-day half-life EWMA of the portfolio return. We also impose a 10% maximum weight constraint on crypto assets, {\it i.e.}, for BTC and ETH combined. These limits were chosen as reasonable values that one might use in practice; the results are not sensitive to these choices.
We describe the metrics used to evaluate the performance of the portfolios over the time interval $t=1, \ldots, T$.
\paragraph{Return.} The (realized) return of the portfolio at time $t$ is given by \[ w_t^T r_t, \] where $r_t$ and $w_t$ are the vector of (realized) asset returns and the portfolio weights at time $t$, respectively. The annualized (realized) return is given by \[ \bar r = \frac{D}{T} \sum_{t=1}^T w_t^T r_t. \]
\paragraph{Volatility.} The (realized) annualized volatility of the portfolio is given by \[ \left(\frac{D}{T} \sum_{t=1}^{T}\left(r_t-\overline r\right)^2\right)^{1/2}. \]
\paragraph{Sharpe ratio.} The Sharpe ratio is the ratio of the annualized return to the annualized volatility.
\paragraph{Drawdown.} Let $V_t$ denote the portfolio value in time period $t$, starting from $V_1=1$, with returns compounded or re-invested. These are found from the recursion $V_{t+1} = (1+w_t^Tr_t) V_t$, $t=1,\ldots, T-1$. The (maximum) drawdown of the portfolio is defined as \[ \max_{1 \leq t_1 < t_2\leq T} \left(1- \frac{V_{t_2}}{V_{t_1}} \right), \] {\it i.e.}, the maximum fractional drop in value form a previous high.
\paragraph{Portfolio weights.} The portfolio weights of the three portfolios are shown in figure (ref).
As expected, the crypto portfolio holds mostly cash, due to the 10% crypto limit. The industry and combined portfolios have more diversified weights, varying over time. As expected these portolios hold a lot of cash during the turbulent 2020 period. On average, the industry portfolio holds 25% cash, the crypto portfolio 90% cash, and the combined portfolio 33% cash.
\paragraph{Performance.} Figure (ref) shows the value of the three portfolios over time, normalized to one at the start of the simulation. (SR denotes Sharpe ratio.)
The aggregate performance of the three portfolios is shown in table (ref).
Including crypto assets in the portfolio clearly gives a noticeable boost, despite allocating less than 10% of the portfolio to them. The combined portfolio has a higher return and Sharpe ratio than the industry portfolio, while having similar volatility and drawdown. To get a better understanding of the performance differences, figure (ref) shows the return, volatility, and Sharpe ratio of the three portfolios over each year of the simulation.
We would like to attribute the performance of the portfolio to the different asset classes. Shapley values account for each assets's contribution to the portfolio, ensuring a fair allocation. They are uniquely characterized by satisfying a collection of desirable properties, including fairness, monotonicity, and full attribution shapley1953value, huettner2012axiomatic, zhang2023did, fryer2021shapley, owen2017shapley, moehle2021portfolio. We will now look at the Shapley attribtutions to each industry and to crypto as a whole, for the combined portfolio. The Shapley attributions of the different asset classes for the combined portfolio are shown in table (ref).
Crypto assets have the highest attribution to return and Sharpe. All assets other than crypto have around a 2% contribution to volatility; crypto has a noticeably lower volatility contribution. Crypto assets have the highest contribution to drawdown.
Figure (ref) shows the relative non-cash weights, {\it i.e.}, $w/\mathbf 1^Tw$ for the combined portfolio over time.
We see that, apart from 2020, the relative weights are relatively stable and evenly distributed with about 10% in crypto assets (equally split between BTC and ETH), and 90% roughly equally split between the four industry portfolios. This motivates an even simpler portfolio construction method, akin to the popular 60/40 stocks/bonds allocation.
\paragraph{Volatility estimators.} We evaluate the performance of the DD90/10 portfolio using two different volatility estimators: a 10-day half-life EWMA, and a GARCH(1,1) model refitted every day on the last 250 days of data, using the arch package in Python sheppard2024arch. (We tried several other volatility estimators; all gave similar results.) The volatility estimates of the 90/10 portoflio are shown in figure (ref).
\paragraph{Weights.} The weights of the DD90/10 portfolios are shown in figure (ref).
The weights are quite similar, with the GARCH estimator being noticeably more reactive during some periods. \paragraph{Performance.} The performance of the DD90/10, compared to the (combined) CRA portfolio, is shown in table (ref).
Figure (ref) shows the value of the three portfolios over time.
We have illustrated that, despite the documented extreme behaviors of crypto assets, simple traditional portfolio contrstruction techniques can be used to include them in a diversified portfolio. We show this using two standard portfolio construction methods, one based on risk parity, and the other a fixed set of relative weights, with each one dynamically diluted with cash to achieve a target ex-ante risk. The addition of even a modest crypto weight of 10% increases the return and Sharpe ratio of the portfolio significantly, without significantly increasing volatility or drawdown.
\paragraph{Acknowledgments.} This work was supported by the IOG Research Hub.