Bitcoin: An Asset Allocation Perspective

It’s no secret that 2021 has started off well for Bitcoin. Having breached a new all time high of $61,788.45 on March 13th it seems that each passing month brings with it a new milestone, new players, and greater acceptance. Recently, significant news has focused on the pace of institutional adoption. In fact, in a survey of 800 institutional clients, investment giant Fidelity reported that a full 36% of clients across the US and Europe were currently investing in digital assets and 60% reported having interest in adding digital assets to their portfolios. Looking out five years, 91% of respondents expect to have at least 0.5% of their portfolio allocated to digital assets with Bitcoin, presumably, representing a substantial share of this newly allocated capital.

Institutional adoption has always been recognized as a major hurdle for Bitcoin and while progress has certainly been made, obstacles remain. Amongst participants in the Fidelity survey: price volatility, possible market manipulation, lack of accepted fundamentals and no established investing framework were cited as areas of concern.

It is my own observation that Bitcoin still vexes most investors and surveys like Fidelity’s confirm this view. Part of the confusion is technological. Bitcoin (and crypto more generally) represents a paradigm shift in the structure of financial markets that investors will naturally have to learn and become comfortable with. Another reason, which will be the focus of this piece, is educational. While the space has developed and matured considerably from the heady days of 2017, investors have not kept pace. Investment frameworks and valuation criteria remain, by and large, inadequate. No pension fund manager is going to run the risk of defending a crypto allocation to their board without the empirical firepower to back it up.

In this post and several more to come, my objective is to present an asset allocator’s view of Bitcoin. We’ll begin by focusing on the statistical properties of Bitcoin and the state of its relationship with other major asset classes. We’ll then transition into portfolio construction and risk management where we’ll use standard techniques to measure how the inclusion of Bitcoin in a portfolio can impact risk, return, and expected outcomes.

Let get to it!

Modeling Bitcoin

Model Considerations

For the purposes of the analysis to follow I will be analyzing Bitcoin over the period of October 2014 to February 2021. This provides us with 2330 daily and 330 weekly observations of Bitcoin’s price and return series. I’ll primarily be working with the weekly returns series as it is the highest level of granularity we can achieve in a portfolio setting (i.e. Bitcoin trades 24/7/365 while stocks and bonds only trade during weekdays according to set hours).

One of the challenges we encounter upfront with modeling Bitcoin is what approach to take. In the world of institutional risk management there are four general approaches to model “risk”:

1. Fundamental: seeks to measure “fundamental” factors which drive risk and return. For stocks this includes financial variables such as earnings, cash flow, sales, and growth. It also includes variables in the Fama-French paradigm such as Book-to-Price, market capitalization (i.e. size), and momentum.
2. Macro: These models seek to measure a security’s exposure to broad macro trends such as inflation, interest rates, credit spreads, valuations, etc.
3. Statistical: These models are solely based on the statistical properties of an asset by analyzing time series data. Statistical models tend to focus on measuring volatility and the correlation amongst assets to infer the risk of a portfolio.
4. Hybrid: Hybrid models combine elements from some or all of the above approaches. For example, it is common to combine Fundamental and Macro factors into a single risk model and are often informed by theory (ex. Energy stocks are likely subject to both credit spreads and the price of oil).

For this analysis I will adopt a purely statistical framework. Given that Bitcoin has only existed in a post-GFC world, attempting to build a macro model is an ambitious endeavor indeed (I tried here). The statistical framework is attractive as it gives us flexibility to pick the granularity of the analysis (i.e. daily, weekly, monthly, etc.) and we have ample data to work off of.

Bitcoin Returns Distribution

We’ll begin by examining the distribution of weekly returns for Bitcoin in isolation.

You’ll notice upfront that the returns are not normally distributed. The empirical distribution appears leptokurtic with excessive “peakedness” around the mean and fat tails. This is confirmed by the summary statistics which indicate a slight negative skew (a Normal distribution has 0 skew) and excess kurtosis (a Normal Distribution has 0 excess kurtosis). However, the mean at 1.33% is distinctly positive; this is unsurprising given Bitcoin’s lifetime performance. The standard deviation (henceforth volatility) of ~10% is quite high for a weekly statistic and is more in line with what one might expect from a blue-chip stock annually.

From a purely statistical perspective, Bitcoin is unquestionably high-risk, high reward. You earn, on average, 1.33% on your investment weekly, but also expose yourself to large tail risks. The return distribution suggests that <-40% returns are substantially more common that a normal distribution would predict. However, a one-dimensional analysis rarely tells the full story. Asset allocators and portfolio managers need to look beyond isolated metrics when constructing a portfolio. At minimum there are three criteria that should be considered when evaluating an asset for inclusion in a portfolio:

1. Volatility
2. Correlation to other assets and potential diversification benefits
3. Weight: which is the primary variable that the PM has control over.

In the sections to follow we’ll dive head long into each and I think you’ll gain some valuable insights.

Volatility and Correlation Analysis

Volatility

Volatility is our most basic measuring stick of dispersion. In the preceding section I remarked that BTC’s weekly volatility is ~10.64%. On an annualized basis, Bitcoin’s volatility stands at ~76% which should give any investor pause. To study how BTC’s volatility has developed over time, the below chart depicts a rolling 120-day calculation presented on an annualized basis.

The range of BTC’s volatility is quite striking with a maximum of ~120% and a low of ~35%. Notice the most recent spike beginning in March 2020 where volatility topped out at over 100%. Curiously, despite the emergence COVID and ensuing meltdown across global financial markets, BTC’s volatility topped out at a lower level than what we observed during the frenzy of 2017–2018. As of this writing, Bitcoin’s volatility is hovering right around the long-run average of ~75%.

For comparison, below is a plot of the rolling 120-day volatility for the S&P 500, Barclays Agg. Bond Index, Long Term Treasury Index (20+) and Gold.

Among the major indices we observe that the S&P 500 is generally the most volatile (but not always), followed by long dated Treasuries, gold and, finally, the Agg. bond index. Again, there is the prominent spike as a consequence of coronavirus where we see that the S&P’s volatility surge to ~50%.

However, even including the dramatic events of COVID, BTC’s volatility is substantially higher than that of the four other major asset classes.

As mentioned previously and cited in the Fidelity survey, one of the primary barriers for institutional adoption of Bitcoin is its volatility. The volatility is considerable and notably more unstable than other major asset classes, but this does not necessarily doom our analysis. Let us continue with an examination of correlations.

Correlations

When we think about building portfolios, volatility is only one consideration. In fact, from the perspective of Markowitz portfolio theory, it is primarily the correlation amongst assets that should be our principle concern. Correlation is a measure of how closely two variables “move” together (or “co-move”). Correlations range from (-1,1) with a correlation of 1 indicating two assets that move in perfect tandem and a correlation of -1 implying two assets that move in complete opposition.

In the extreme, consider the following example where we have two assets (A & B) each with standard deviation of 100%, but are perfectly negatively correlated. What is the vol of an equally weighted portfolio?

From our example, we can see that when combined the negatively correlated assets produce a portfolio with half the risk of either asset individually. This is essentially the situation we face with Bitcoin.

The below chart plots the rolling 120-day correlation of Bitcoin and the S&P 500.

Over its lifetime we can see that Bitcoin has been generally uncorrelated with US stocks. The correlation of returns has varied and, at times, has been slightly positive or slightly negative. The COVID spike presents itself once more, but more recently the correlation has been trending down toward a more normal range.

Let’s expand the correlation analysis to our other assets: long dates Treasuries, Agg bonds, and gold.

Against fixed income (Agg bonds and Treasuries) we observe that Bitcoin is relatively uncorrelated; usually hovering somewhere around 0 with some periods of variability. Against gold, the correlation pattern is somewhat bifurcated. Between 2015 and 2018 we observe a correlation around 0, but recently correlation has been more pronounced (which presents fertile ground for further research). On balance, Bitcoin appears weakly correlated with gold.

Rather than looking at rolling correlations we can instead opt to examine simple averages. Below is the correlation matrix over the full time period.

Over the full time period, the intuition that we gained from the rolling plots is reinforced: Bitcoin is uncorrelated with Treasuries and weakly correlated with the stocks, Agg bonds and gold. The correlation coefficient for Agg bonds is of particular interest as it is a bit higher than the plot suggests. I would argue that this finding is an artifact of the COVID crisis and not necessarily indicative of a long-term expectation, but must be considered, nonetheless.

We can take our analysis one step further by running formal tests. The claim of interest is: “is Bitcoin statistically uncorrelated with the other major asset classes?”. Tests enable us to formalize the intuition we have gleaned from the plots and provide a range of plausible correlations that we can use when we transition to stress testing portfolios.

There are three basic tests of correlation: Pearson’s product-moment, Kendall’s rank, and Spearman’s rank. Pearson’s test tests the usual correlation coefficient that we calculated in the above correlation matrix:

The test statistic follows a Student t distribution with n-2 degrees of freedom and is calculated as:

A crucial assumption of Pearson’s test is that the data is normally distributed. This poses a potential problem since we have observed that Bitcoin’s returns are not normal. There is a way to adjust Pearson’s test to account for non-normal random variables, but we will not address that here. For our purposes, we’ll still use Pearson’s test, but keep in mind that it may be low power in this setting.

To supplement Pearson’s parametric test, we’ll also calculate the Kendall and Spearman rank statistics. Rank statistics are attractive as they are non-parametric and only consider how well the data “line up”; on a given day that Bitcoin’s return is “high”/”low” was the return for the S&P also “high”/”low”. Running these tests in tandem will provide a well-rounded picture for how correlated the major asset classes are with Bitcoin. The below table contains the p-values for the three tests for each asset pair. The null hypothesis is that correlation equals 0 with a low p-value (<5% being a traditional cut off) indicating the null is rejected (or, equivalently, the pair is statistically correlated).

For Bitcoin and the S&P, Pearson’s test is highly statistically significant which implies the two are, in fact, correlated. However, the Kendall and Spearman tests are more borderline with p-values close to but falling shy of the 5% threshold. Taken together, the results suggest that Bitcoin and the S&P are correlated, but only weakly. In laymen’s terms, they tend to comove on average, but Bitcoin having a “good”/”bad” day when the S&P has a “bad”/”good” day is not uncommon.

The results for Bitcoin and Agg bonds are probably the most interesting. Based on Pearson’s test, Bitcoin and Agg bonds appear statistically correlated. However, the null hypothesis of zero correlation fails to be rejected under the Kendall and Spearman tests. To me this suggests that the argument I made earlier that the apparent correlation of Bitcoin and Agg bonds was really an artifact of the COVID Crisis is true and not representative of a long run expectation.

The tests for Bitcoin and LT Treasuries are in resounding alignment with all three indicating the two are uncorrelated. Similarly, for gold and Bitcoin the tests all indicate a statistically significant degree of correlation which has deep implications for the thesis of Bitcoin as “digital gold”.

At this point, we have gone pretty deep on correlation and, in summary, make the following observations:

• Bitcoin and the S&P 500 are weakly correlated
• Bitcoin and Agg bonds are uncorrelated
• Bitcoin and Treasuries are uncorrelated
• Bitcoin and Gold are weakly correlated

In short, Bitcoin exhibits relatively low correlation across asset classes which will be crucially important for the analysis to follow. The diversification effect from even a small allocation to an uncorrelated asset can have a substantial impact in a portfolio. While Bitcoin’s volatility is likely to increase the absolute level of risk, we’ll find that the added risk can be richly rewarded.

Another casual observation is that in periods of stress, Bitcoin’s correlation tends to increase. Over short periods of time the adage “all correlations go to 1” seems relevant. But a skilled manager knows that it is the long run expectations and not short-term dynamics that win the day. Indeed, the drivers of Bitcoin’s returns are quite different from those of stocks and certainly fixed income and commodities. Thus, if history is any guide then we can expect Bitcoin’s price to continue to diverge which we can use to our advantage.

With these findings in mind we can now finally turn to portfolio construction and asset allocation.

Asset Allocation Analysis

What happens to a portfolio when a new asset is added? How does the overall risk profile change? How do the sources of risk change? Does the expected return justify the change in risk? These are the questions that a manager asks when considering the inclusion or exclusion of a particular asset in a portfolio.

The key question in our context becomes, “what role (if any) does Bitcoin have to play in a portfolio?” To help us answer this question we’ll look at four possible allocations that you can easily imagine employed in a real-world asset management context. The portfolios and asset allocations are broken out in the table below:

The Base portfolio does not allocate to Bitcoin and will (unsurprisingly) be used as the basis for comparison. The Pro-Rata portfolio allocates 5% to Bitcoin taken pro-rata from the weights of stocks, bonds, and gold. The Risk portfolio allocates 5% to Bitcoin by allocating away from stocks (the second most risky asset). Finally, the No Gold portfolio swaps the 5% allocation to gold in the Base portfolio for Bitcoin; the idea being that Bitcoin is “digital gold” and we want to see how the two differ when you remove physical gold entirely.

The four portfolio variations I’ve defined are very simple. Of course, you can imagine extending the analysis to include allocations to Developed market equity, Emerging markets, small caps, a broader cross section of commodities, real estate, hedge funds, private equity, and currency. However, before adding complexity we first need to determine if Bitcoin is useful in a portfolio context at all and for that to happen the Bitcoin portfolio needs to beat the simple 60:40.

Let’s begin by looking at the estimated risk for each portfolio:

The Base portfolio has an annualized volatility of 10.33%. Amongst the portfolios that allocate to Bitcoin (Pro-Rata, Risk, and No Gold) the Risk portfolio has the lowest risk. This result makes intuitive sense as the Risk portfolio has the same Bitcoin allocation, but the lowest stock allocation of the three alternatives. With a volatility of 11.11% the Risk portfolio is approximately 7.5% more volatile relative to the Base portfolio.

The No Gold portfolio has the highest standard deviation at 11.65%; approximately 12.7% higher than the Base. This also make sense as we have removed an entire asset class and hence reduced the overall diversification benefit.

Let’s dig a little deeper now and examine the contribution of each asset to total portfolio risk. Risk contribution is another way to view portfolio diversification that enables us to decompose how an individual asset drives the total risk.

As the below chart and accompanying table demonstrate even in the supposedly “diversified” Base portfolio, a full 90% of the total portfolio risk is attributable to stock. The implication of this is that in a situation where the market tumbles, but bonds hold up well your portfolio will still be dragged down as stock is still the dominant source of risk. Contrast this with the ‘Risk’ portfolio where the contribution from stock has been reduced to 72% and redistributed to Bitcoin. In effect this creates a more balanced portfolio with risks more evenly distributed.

By now we have a good feel for how Bitcoin impacts the risk profile of a portfolio. In general, a 5% allocation to Bitcoin results in a fairly modest increase in volatility compared to the Base. The portfolio is also less concentrated and less reliant on equity. Finally, Bitcoin does better in a diversified setting. Having Gold and Bitcoin in a portfolio (like the Risk portfolio) dampens the overall volatility as we are broadening the menu of uncorrelated assets for Bitcoin to “bounce off” of.

Let’s now transition to consider returns. If the inclusion of Bitcoin in a portfolio requires us to take slightly more risk, then we need to be compensated for doing so. Below is a plot of the annual return for each of the four alternative portfolios:

The figures show just how dramatically a 5% allocation to Bitcoin can impact a portfolio’s returns. Historically the additional risk has been richly rewarded with the Pro-Rata, Risk and No Gold portfolios outperforming substantially in 2016, 2017 and 2020. However, 2018 saw underperformance of the alternative portfolios as Bitcoin shed over 80% of 2017 high.

Viewed a slightly different way, we can examine the cumulative return for each portfolio:

The three alternatives outperform the Base portfolio across the board; each by over 30%. Interestingly, there is not much of a difference in cumulative return between the Pro-Rata, Risk and No Gold portfolios. The No Gold portfolio does the best, but only slightly with Pro-Rata and Risk (respectively) following closely behind.

Given the similarity of return profiles for each of the three alternatives it’s natural to ask, “which allocation is superior?”. To that end we can turn to risk adjusted measures of return, namely, the Sharpe and Sortino ratios. Both Sharpe and Sortino attempt to quantify how well you are compensated for taking a unit of risk. The Sharpe Ratio is defined as follows:

Where:

• E(rp) is the expected return of the portfolio
• σp is the portfolio volatility
• r(rf) is the risk-free rate of return

A criticism of the Sharpe ratio is that is punishes volatility symmetrically. In other word, “upside” volatility is penalized the same as “downside” volatility which doesn’t really make sense as investors are presumably quite content for an asset to be volatile if the price is rising.

To address this short coming we can employ the Sortino ratio, which is defined similarity to the Sharpe ratio, but instead has the “downside deviation” in the denominator. Downside deviation represents the volatility of returns, but only below a threshold (usually 0). In this way the Sortino Ratio accounts for asymmetric upside.

During the timeframe under examination, the risk-free rate was approximately 0 so I elect to drop it for the purpose of calculation.

As can be seen from the above tables, the Sharpe Ratios for the alternatives are considerably higher than the Base portfolio which implies that the additional risk that Bitcoin adds to a portfolio is very well compensated. The Risk portfolio has the highest Sharpe ratio which indicates the best risk reward trade-off.

When viewed through the lens of the Sortino ratio and downside deviation a very interesting picture emerges. Once again, the Risk portfolio emerges as having the highest Sortino ratio. With respect to downside deviation, the difference between the Base portfolio and alternatives is remarkably small. The downside deviation of the Risk portfolio is only ~ 46 bps higher than the Base (~5.8% in a relative sense). If we consider what implications this has for portfolio tail risk, even a 3 standard deviation move would imply <1.5% underperformance of the Risk v. Base portfolio, which certainly seems tolerable for the opportunity to notch higher returns.

On a risk-adjusted basis it is clear that an allocation to Bitcoin is justifiable if not prudent. Bitcoin’s returns are high and generally uncorrelated. Consequently, investors are able to reap substantial rewards while only taking on marginally higher risk. Amongst the alternative allocations, the Risk portfolio appears to be the superior choice. By incorporating four asset classes and allocating away from stock to Bitcoin the portfolio offers maximum diversification benefits and a better risk-return tradeoff.

Concluding Remarks

The key question that I sought to address in this piece was: “what role does Bitcoin have to play in an institutional caliber portfolio?”. In short, Bitcoin offers a highly attractive opportunity for managers who are long term focused and willing to expand their investment universe. One of the key takeaways is that Bitcoin’s reputed volatility is very real and will necessarily require that portfolio managers and boards allocate in a way consistent with their risk tolerance. But managing that risk appropriately is more than achievable.

Allocating to crypto assets is going to become increasingly mainstream in coming years. Allocating to Bitcoin provides managers with an opportunity to gain exposure and get comfortable with this emerging technology, hedge portfolios against disruption and get results for their clients.

Hopefully you’ve found this piece interesting and informative. For me, it was a lot of fun to write!

Until next time, thanks for reading!

-Aric Lux.

Originally published at https://lightfinance.blog on April 1, 2021.