A Modern Portfolio Theory Case for Bitcoin
By: Galaxy Digital Research
Tags: Blockchain, Bitcoin, Investing

Brief introduction to modern portfolio theory

MPT is a theory on how risk-averse investors can construct portfolios to optimize or maximize expected return based on a given level of market risk. Originally formulated by Nobel-prize winning Harry Markowitz in 1952, MPT proposes a framework for how investors can reduce overall risk while maximizing return by holding a diversified portfolio of non-correlated assets. [1] As opposed to viewing the risk-return characteristics of each asset in isolation, MPT assesses risk and return for the broader portfolio based on the cumulative interactions between the portfolio’s assets.

At is core, MPT assumes that investors are rational and generally risk averse, or if given two portfolios that offer the same expected return, investors will prefer the less risky one. An investor should take on increased risk only if they are being compensated by higher expected returns. Conversely, an investor who wants higher expected returns should be willing to accept more risk. Individual investors will evaluate the risk-return trade-off based on individual risk aversion characteristics. Under the model, a portfolio’s expected return is the proportion-weighted combination of the constituent assets’ returns, and the portfolio volatility is a function of the correlations of the component assets for all asset pairs. Concretely, the expected portfolio return, portfolio return variance, and portfolio return volatility are:

An investor can reduce a portfolio’s risk by simply holding combinations of assets that are not perfectly positively correlated. In essence, investors can reduce their exposure to individual, idiosyncratic asset risk by holding a diversified portfolio of assets, potentially allowing for the same portfolio expected return with reduced risk.

Bitcoin in a traditional portfolio

Bitcoin’s history of positive returns and uncorrelated nature make it potential candidate for traditional portfolios. Despite its high volatility (81.5% annualized since January 2014), we find that adding a small portfolio allocation to Bitcoin (BTC) increases the overall expected return and improves a portfolio’s risk-adjusted returns (or Sharpe ratio).

We assume that a “traditional” portfolio consists of a 60% portfolio weight to equities and 40% portfolio weight to fixed income. For the purposes of this research, we proxy equity returns using the S&P 500 and fixed income using the Bloomberg Barclays US Aggregate Bond Index. We begin the return comparison and data series in December 2013 for two primary reasons. First, we attempt to avoid outsized return assumptions by selecting a time series that does not include Bitcoin’s early aggressive price increases from a few cents to hundreds of dollars. The time series we have chosen includes two full bear markets and one bull market (Bitcoin’s last bubble peaked in December 2013). Second, earlier Bitcoin price data (2010–2013) is less transparent and had thinner liquidity, which later evolved in 2013 to include higher volumes, deeper liquidity, and pricing from more transparent exchanges.

Using the generalized formulas in (1), (2), and (3), the expected portfolio return and variance for a three-asset portfolio can be concretely described as:

For our data, we utilize total return indices (including dividends and interest) for indicative equity (S&P 500 or “SPX”) and fixed income indices (Bloomberg Barclays US Aggregate Bond Index or “AGG”) and monthly Bitcoin returns starting in December 2013. For simplicity, we do not consider value-generating forks for Bitcoin such as Bitcoin Cash. We can clearly see how Bitcoin has been uncorrelated to traditional equity and fixed income assets in Table 1.[2]

Table 1: AGG, SPY, BTC correlation table.

We can simulate the expected return, annualized volatility, and Sharpe ratio[3] of a portfolio with an increasing allocation to Bitcoin; for each incremental 1% allocation to Bitcoin, we deduct 50bps from the equities and fixed income allocation respectively and keep the portfolio fully invested. The summary results can be seen in Table 2.

Table 2: AGG, SPX, BTC correlation table.

The data can be graphically illustrated (Fig. 1) using the Markowitz Efficient Frontier, a common depiction of the collection of portfolios that offer the lowest risk for a given level of return. Each plotted point in Fig. 1 is a potential portfolio along the efficient frontier of portfolios using varying portfolio allocations to Bitcoin (denoted with plotted point labels).

Fig 1: Efficient Frontier with BTC Allocation

We can also plot (Fig. 2) a portfolio’s allocation to Bitcoin and its expected Sharpe ratio to visualize potential optimal allocations to Bitcoin based on risk-adjusted returns.

Fig 2: Sharpe Optimization with BTC Allocation

Unsurprisingly, adding a small portfolio allocation to Bitcoin improves a portfolio’s expected return, and an increasing allocation at small levels improves a portfolio’s expected return at a faster rate than increases to the portfolio’s volatility; Table 3 shows the point derivative change in expected return with respect to portfolio volatility relative to the prior portfolio. We see that a portfolio’s Sharpe is optimized somewhere near a 6% allocation to Bitcoin, suggesting that a traditional portfolio could benefit from a small allocation to Bitcoin based on the investors’ risk tolerance.

Table 3: First order expected return change with respect to volatility.

Bitcoin in a macro portfolio

The prior section assumes a traditional, simplistic portfolio, but modern portfolios are rarely simple and have an infinite number of permutations across complex instruments. Many modern portfolios include broad-based diversification across geographies and asset classes, as well as both long and short exposure to assets. For macro and institutional investors, there is no indicative standard portfolio, though we can still see the benefits that an allocation to bitcoin may have on complex and diverse portfolios.

Tables 4 and 5 illustrate the correlative relationships between Bitcoin and equity, fixed income, sovereign currencies, and commodities globally; Bitcoin maintains its low correlation to long exposure of global assets with all correlations ranging between -0.1 to 0.2. Since the correlation coefficients are low in absolute terms, Bitcoin remains uncorrelated and a portfolio diversifier in conjunction with portfolios that are short global assets; correlation coefficients of long Bitcoin exposure and short exposure to global assets range from -0.2 to 0.1 (Table 5).

Table 4: BTC correlation with global asset classes, long exposure to assets relative to USD.[4]

Table 5: BTC correlation with global assets, long and short exposure.

Final remarks

There are known limitations to modern portfolio theory. Research findings in other fields of study, particularly within behavioral economics, argue that the assumption that investors are rational actors is wrong. Studies in behavioral finance have challenged the notion that investors rationally make risk-return tradeoffs and suggest that investors are biased. In addition, the systematic relationships between assets are not constant, so the assumption that the implied covariance of future returns and correlation between assets remains fixed and predictable become less useful during times of uncertainty when correlations tend to trend to 1. Furthermore, investors may not be as concerned about upside volatility but rather may focus more on the potential for downside volatility, and so measurements of standard deviation may be a poor metric for portfolio risk. Since some of the variables used to measure correlation coefficients may be flawed, the actual risk level of an asset may be mispriced.

Future implied correlations and volatility notwithstanding, future expected returns are also obvious limitations of modern portfolio theory. Namely, past performance cannot simply be projected into the future. While this naturally applies to traditional assets, it is even more relevant for Bitcoin, an asset that has historically had near-exponential outsized returns since its inception. We simply cannot guarantee that Bitcoin, or any risk asset for that matter, will maintain its historical performance, its volatility profile, and its relationships with other risky assets.

Despite the imprecise models of modern portfolio theory and its considerations, Bitcoin’s fixed supply and programmatic limit to stock-to-flow ratio changes offer potential diversification to the systemic risk of rapid monetary supply increases present in modern portfolios. Government-issued money is susceptible to rapid supply increases compared to its existing stock through central bank activity, and therefore has the potential to lead to a rapid loss of salability, diminishment of purchasing power, and wealth depreciation of its current holders as we have seen throughout the late 1900s and 2000s with the suspension of the gold standard. This systemic risk exists in all portfolios that own traditional assets given the inherent links between global economics and financial markets, and therefore an allocation to Bitcoin that is expressly resistant to inflation and untethered from the global sways of central bank activity, global economies, and financial markets can potentially diversify some of the systemic risk that exists in modern portfolios. Moreover, an application of modern portfolio theory and review of the data suggests that a small allocation to Bitcoin can improve a portfolio’s risk-adjusted returns.