One year ago, we presented a case for including bitcoin in a traditional investment portfolio. Analyzing historical data, we found that a small allocation to bitcoin substantially increased the historical returns of a diversified portfolio while, surprisingly, adding very little risk. This beneficial impact on the portfolio's risk adjusted returns was a result of bitcoin's strong price performance combined with a sustained lack of correlation with traditional assets.
Since the time of our article twelve months ago, bitcoin has experienced a significant decline in price and increase in volatility. We won't go into all the possible reasons for this, but its reasonable to say that bitcoin proved to be a risk asset and was accordingly vulnerable to the Fed's rapid tightening of monetary policy.
Given the decline in bitcoin's price over the past twelve months and claims of it now being an unsuitable investment, we thought it would be an interesting time to revisit our analysis.
Marking to Market Bitcoin’s historical returns
2022 was the second-worst year for bitcoin's price performance, recording a 65% decrease during the year, beaten only by the 73% decline recorded in 2018. The first quarter of 2023 saw a rebound in all risk assets including bitcoin, but its price remains down 38% year over year. So how do bitcoin's historical returns compare now to traditional assets such as equities and gold?
As Figure 1 shows, despite the well publicized "crypto winter", bitcoin’s annualized returns remain significantly ahead of traditional assets across a range of historical time periods.
Figure 1: Compounded Annual Returns of Bitcoin, Gold, and Equities 
Bitcoin's Contribution to Traditional Portfolio Returns
As part of our analysis last year, we introduced a hypothetical portfolio comprised of 60% equity and 40% fixed income securities ("Traditional Portfolio"). The Traditional Portfolio is meant to represent a diversified investment portfolio against which we can calculate hypothetical portfolio attribution.
We calculated the Traditional Portfolio's returns starting with no allocation to bitcoin and added 10 basis point increments of bitcoin, maintaining a 60/40 equity-to-fixed income split for the balance. With each incremental allocation to bitcoin, we recalculated 3-, 5-, and 10-year annualized returns to understand the impact that a range of allocations would have across various time horizons. So how would allocations to bitcoin have impacted a traditional investment portfolio?
As Figure 2 illustrates, even small allocations to bitcoin would have dramatically increased 3-, 5-, and 10-year returns. In fact, just a 3% investment in bitcoin would have increased the Traditional Portfolio's 10-year annualized returns from 8% to 12%.
Figure 2: Annualized Portfolio Returns with 0-10% Allocations to Bitcoin
What about volatility?
Bitcoin's impact on portfolio returns is not surprising to us given its absolute performance relative to traditional securities. However, bitcoin has been a volatile asset, especially over the past year. So, the question is whether bitcoin’s positive contribution to returns more than offsets the increased volatility.
To answer this, we calculate risk adjusted returns using the Sharpe ratio, a measure of performance per unit of volatility. Portfolios with higher returns or lower volatility produce higher Sharpe ratios, and vice-versa. Sharpe ratios above 1.0 are considered good, as the assets are generating excess returns relative to their combined volatility.
On a trailing 3-year basis, bitcoin has produced a Sharpe ratio of 1.3. This tells us that, despite its high volatility over the past three years, bitcoin has generated attractive risk adjusted returns (i.e., bitcoin's returns have more than compensated for its volatility). Looking at risk adjusted returns rolling over time paints an even rosier picture. As Figure 3 shows, bitcoin has produced a trailing 3-year Sharpe ratio consistently greater than 1.0 over time (77% of observed periods) with a median of 1.6.
Figure 3: Bitcoin Rolling 3-Year Sharpe Ratio 
Bitcoin's Contribution to Risk
While measuring bitcoin's Sharpe ratio over time provides an understanding of its risk adjusted returns on a standalone basis, our ultimate question is how would it impact the risk adjusted returns of a traditional portfolio. The answer to this question depends not only on the returns and volatility contributed by bitcoin but also its correlation with other portfolio assets.
Adding imperfectly correlated assets to a portfolio reduces its overall volatility and, all else equal, increases risk adjusted returns. The practical challenge for traditional portfolios is that very few assets offer a persistent lack of correlation to equity and fixed income securities over long periods of time. Bitcoin, thus far, has been an exception.
The correlation between bitcoin’s daily returns and those of our Traditional Portfolio was 0.36, 0.30, and 0.15 over the past 3-, 5-, and 10-year periods, respectively. This indicates very low correlation between the returns for bitcoin and a portfolio comprised of stocks and bonds. The next question is whether this imperfect correlation is enough to offset the volatility that bitcoin contributes. In other words, how much risk does bitcoin add to a traditional portfolio?
A variety of methods exist to measure risk. For investment portfolios we tend to use Value at Risk ("VaR"), which quantifies the extent of possible losses over a specified period of time. We calculated the daily VaR of our Traditional Portfolio using 3-, 5-, and 10-year data. Then, in order to understand the impact that a range of bitcoin positions would have on the portfolio, we added 10 basis point increments of bitcoin and recalculated daily VaR.
The results were surprising to us. As Figure 4 illustrates, small allocations to bitcoin over the past 3-, 5-, and 10 years resulted in very little increase in the overall risk of the portfolio.
Figure 4: Daily Portfolio VaR with 0-10% Allocations to Bitcoin
Bitcoin's Contribution to Risk Adjusted Returns
Given the substantial increase in returns with very little added risk, it follows that a small position in bitcoin should positively impact the portfolio's Sharpe ratio. Figure 5 confirms this logic. The Traditional Portfolio produced Sharpe ratios well below 1.0 during the past 3-, 5-, and 10-year periods. Adding just a small position in bitcoin would have substantially increased the portfolio's Sharpe ratio. In fact, just a 3% investment in bitcoin would have increased the 10-year historical Sharpe ratio from 0.77 to 1.00.
Figure 5: Historical Sharpe Ratio with 0-10% Allocations to Bitcoin
Given bitcoin's decline in price and increased volatility since our article twelve months ago, we thought it would be an interesting time to revisit our analysis. The results not only confirmed our original conclusions but were surprisingly at odds with much of the "crypto winter" narrative.
Despite a 38% decline in price year over year, bitcoin's historical returns remain significantly ahead of traditional assets. And despite the perceived increase in bitcoin's volatility and correlation with equities, there remains very little increase in risk by adding bitcoin to a traditional portfolio. The fact remains that allocating a small position to bitcoin substantially improved the risk adjusted returns of a traditional portfolio across a range of historical time periods.
About Two Ocean Trust
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 We began with a hypothetical portfolio comprised of 60% equities (S&P 500 Index), 40% fixed income (Barclays Aggregate Bond Index) and 0% bitcoin. We then analyzed hypothetical portfolios blending incremental 0.1% positions in bitcoin while maintaining a 60/40 ratio of equity to fixed income for the balances. Return, risk and Sharpe ratio of each portfolio were measured across 3-, 5-, and 10-year historical periods. A variety of measures exist to ascertain risk. We use Value at Risk (VaR), a statistic that quantifies the extent of possible financial losses within a portfolio over a specified period. To calculate risk adjusted returns we use the Sharpe Ratio which measures the profit of a portfolio in excess of the risk-free rate, per unit of standard deviation.
 As with any asset, the value of bitcoin can go up or down and there can be a substantial risk that you lose money buying, selling, holding, or investing in bitcoin. You should carefully consider whether trading or holding bitcoin is suitable for you in light of your financial condition. Past performance is not indicative of future returns. Prospective investors are not to construe the contents of this document as a recommendation to purchase bitcoin. This information should not be relied upon as research, investment advice, or a recommendation regarding any products, strategies, or any security in particular. This material is strictly for illustrative, educational, or informational purposes and is subject to change.
 As of March 31, 2023. Source: Yahoo Finance
 As of March 31, 2023. Source: casebitcoin.com
 The Sharpe ratio is calculated by dividing the excess return of a portfolio by the standard deviation of its excess returns. Weekly data from April 2017 through March 2022. Source: casebitcoin.com
 Correlations fall within the range of -1 to 1. A correlation of 1 indicates perfect positive correlation. A correlation of -1 indicates perfect negative correlation. A correlation of or near 0 indicates that there is no relationship between asset returns. Daily price data from April 1, 2013 through March 31, 2023. Source: YCharts
 VaR is measured by assessing the amount of potential loss, the probability of occurrence for the loss, and the timeframe. For example, if a portfolio has a Daily VaR 1% (5-year lookback) of 2%, there is a 99% confidence that the portfolio will not have a daily loss larger than 2%, calculated using 5 years of historical data.
 As of March 31, 2023, a 3% allocation to bitcoin increased 3-, 5-, and 10-year annualized returns by 290, 208 and 396 basis points, respectively, while increasing Daily VaR (1%) by only 11, 15, and 25 basis points, respectively.