Bitcoin in a Multi-Asset Portfolio
Stefan Hubrich
Stefan Hubrich
is the head of systematic
investing in the multi-asset
division and co-portfolio
manager of the T. Rowe
Price Multi-Strategy Total
Return Fund at T. Rowe
Price Associates, Inc., in
Baltimore, MD.
stefan.hubrich@
troweprice.com
KEY FINDINGS
n
Bitcoin’s extreme volatility and short historical track record create unusual challenges
when integrating it into the risk/return framework of a traditional multi-asset portfolio.
n
These challenges can be addressed in a revised portfolio construction framework that
is used to extract the return threshold required for motivating a small (1% or 5%) bitcoin
allocation.
n
The return thresholds so obtained are surprisingly low, in many cases implying negative
geometric means. The ability of a traditional portfolio to diversify away bitcoin’s extreme
volatility is critical to understanding its merits as an investment.
ABSTRACT
Investors looking to integrate digital assets into a traditional, diversified multi-asset portfolio
need to formulate appropriate risk and return assumptions for them. Using the case of bitcoin, we argue that due to the short duration of available returns and the extreme volatility
of the asset, historical returns are an unreliable basis for directly formulating forward return
expectations. We also show that bitcoin’s return characteristics require an emphasis on such
portfolio construction considerations as rebalancing frequency that are often peripheral
in traditional asset allocation studies. We then demonstrate an allocation approach that
addresses these concerns. The main idea is to extract required return thresholds for a small
bitcoin investment (1% or 5%) that need to be underwritten by the investor, rather than
relying on explicit return expectations as the input. We show that these return thresholds
are surprisingly low, illustrating that the broader multi-asset portfolio perspective is critical
when making investment decisions regarding high-volatility assets like bitcoin.
T
his article seeks to answer one question: assuming that one is considering a
portfolio allocation to bitcoin, how much should that be? What must someone
believe to allocate 1% or 5% to this new asset? We assume that a fundamental
investment thesis for bitcoin already exists, that the reader believes that bitcoin has
utility, and that its fair value and return-generating potential exceed zero.1 We will
demonstrate that bitcoin’s extraordinary volatility creates unique portfolio construction
The views expressed in this article are those of the author and do not necessarily reflect the views
of T. Rowe Price Associates, Inc. Further information can be found towards the end of this article under
the heading “Disclosure Statements.”
1
Readers actually considering an investment in digital assets need to consider a host of underwriting considerations, such as the inner workings of blockchain based cryptocurrencies, regulatory
concerns, custody, liquidity, and other trading issues that are outside the scope of this article.
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64 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
complications that merit important modifications when embedding it in a broader
multi-asset portfolio analysis. Since we only rely on historical returns, our analysis
also can serve as a template for integrating other high-volatility assets with short
return histories.
Bitcoin price data in USD are only available starting in 2010,2 and bitcoin did
not exceed $1 billion and $20 billion in market capitalization until March 2013 and
April 2017, respectively. Over this brief period, its realized return and risk have been
extraordinary, with an arithmetic mean of 182% and a volatility of 83% annualized over
the period starting in 2015, which is the focus of this article. It is obvious that any
attempt to answer the portfolio allocation question while relying on these historical
returns needs to contend with their reliability as a basis for informing forward-looking
investment decisions.
The existing literature examining bitcoin or portfolios of “digital assets”3 in the
broader traditional portfolio context has a mixed track record on that front. White
papers by industry practitioners fix a small bitcoin allocation (typically 1%–5%) of
the portfolio and simply show the substantial portfolio improvement that would have
resulted if that decision had in fact been made back in 2014 (Coverdale 2021) or
2015 (Bhutoria 2020, Butterfill Bendiksen 2020, or Duffin 2021). Hougan and Lawant
(2021) follow the same approach within a broader study that emphasizes bitcoin’s
volatility and correlation characteristics.
Earlier peer-reviewed papers obtain their digital asset allocation via mean-variance
optimization on the full sample, while also executing spanning tests to assess the
significance of the value-add from adding these assets (Briere et al. 2015 and Chuen
et al. 2017). More recent papers have contended more head-on with the lack of longterm data for digital assets by implementing more sophisticated portfolio optimization
techniques, many of which are entirely risk-based and do not require expected returns,
and by assessing portfolio performance out-of-sample. Eisl et al. 2015, Trimborn
et al. 2017, Platanakis and Uruqhart 2020, and Pethukina et al. 2021 fall into this
category. With very few exceptions, all these studies present a very favorable role for
digital assets in traditional portfolios.4
For a practitioner pondering a bitcoin allocation, these prior studies are likely of
limited utility. Many of the performance improvements seem “too good to be true,”
and many optimized portfolios feature unrealistically large digital asset allocations
when considering the volatility and existential fragility of this largely unregulated and
technology-dependent new asset class. Additionally, many of the more sophisticated
and/or purely risk-based portfolio construction techniques are highly unlikely to be
adopted by practitioners guided by investment consultants and answering to pension
committees.5
The biggest issue of all, however, is that all the studies mentioned essentially
assume that bitcoin’s return history so far is representative of what investors should
2
Among the articles listed in the references, the earliest sample period used began in July 2010
for Briere et al. (2015) and Eisl et al. (2015).
3
We use the term “digital assets” to refer to the broader universe of blockchain-based cryptocurrencies and other digital assets that are exchanged in a decentralized fashion based on algorithmic
protocols, such as Ethereum (ETH), Monero (XMR), and the like.
4
Trimborn et al. (2017) find that the value-add from digital assets disappears when performance
is evaluated out-of-sample.
5
Additional issues shedding doubt on the applicability of previous studies are the use of unrealistically short rebalancing horizons (weekly in Briere et al. 2015 and Platanakis and Uruqhart 2019) and
the lack of a clean baseline portfolio excluding digital assets. Most investors, including many institutional investors, hold broadly diversified stock/bond portfolios (such as the famous “60/40” portfolio
consisting of 60% equities and 40% bonds), but generally, the literature fails to explicitly use such a
realistic portfolio as the starting point. This makes it difficult to disentangle the impact of alternative
portfolio construction techniques from the impact of adding digital assets to the universe.
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The Journal of Alternative Investments | 65
Winter 2023
expect going forward. This is true even in cases where the portfolios were evaluated
“out-of-sample”—the “out-of-sample” period here is still drawn from the historical
return distribution observed so far. Any portfolio including any amount of bitcoin over
the available 2010–2021 historical return period will look superior to its traditional
counterpart, no matter how exactly that allocation was derived. As importantly, these
studies generally also use historical traditional asset returns from this short period,
potentially introducing further biases in light of the truly long-term performance history
available for them.6
Of note, the recent paper by Czasonis et al. (2021) is more robust to many
of these issues. The authors explicitly examine how the optimal bitcoin allocation
varies depending on the expected return assumption, given the risk profile. In doing
so, they explicitly model risk preferences that—absent digital assets—produce
a traditional 60% equity/40% bonds portfolio as the starting point, making their
results relevant to traditional investors. Much of their focus, however, is on the
correlation properties of bitcoin and on the use of sophisticated preferences that
feature a preference for lottery-like outcomes. While using a more traditional preference structure, this article continues their line of inquiry by examining a wider range
of rebalancing regimes, putting more emphasis on the assumptions for traditional
asset class returns, and by avoiding the use of historical returns to directly project
forward expected returns.
Our approach takes head-on the challenges arising from bitcoin’s extraordinary
volatility, combined with the short period of its existence. We demonstrate exactly how
this volatility presents material challenges when applying traditional asset allocation
techniques that extrapolate historical risk and return experience into the future. We
then demonstrate a path forward that helps to sidestep these issues. The article’s
outline and main findings are as follows:
1. The “historical” approach of using realized historical bitcoin returns to formulate expected returns leads to entirely unrealistic and—arguably—“too good
to be true” portfolio allocations and improvements.
2. This approach is questionable because the brevity of the available historical
bitcoin return sample, combined with its extraordinary volatility, makes it difficult to assess its true mean return with anything approaching the accuracy
required for sensible portfolio analysis.
3. A better approach is to take only bitcoin’s risk characteristics as given, and
back into the level of return that would be required to rationalize a 1% or 5%
bitcoin allocation given those risk characteristics. Then the reader essentially
can decide independently whether the available investment bitcoin thesis can
support that kind of return expectation.7
4. The return assumptions needed to motivate small bitcoin allocations in this
setting are strikingly low when considering its volatility. In most cases, the
implied geometric average return is negative, suggesting that a small allocation can be justified even if one believes that the value of bitcoin may trend
toward zero over time.
5. Results are critically impacted by the rebalancing philosophy.
6
A recent paper by Hu et al. (2021) is much more robust to these issues, explicitly modeling
complex, non-normal return dynamics and then relying on simulations to reduce sample dependency.
However, its focus is on allocating among digital assets rather than sizing a digital asset allocation in
a broader multi-asset portfolio.
7
This approach also allows us to utilize more realistic, long term return expectations for traditional
assets in the exercise, rather than relying on their realized performance during the shorter bitcoin
sample.
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66 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
THE HISTORICAL APPROACH
EXHIBIT 1
Historical Arithmetic Mean Returns
In this section, we illustrate how a traditional historical asset allocation analysis using bitcoin would
normally proceed. We use bitcoin as the proxy for
the digital asset space, given its dominance in terms
of market capitalization.8 This traditional approach
directly uses historical sample risk and return metrics as inputs and is a natural first pass at the asset
allocation implications of digital assets. To align our
exposition with the practitioner studies surveyed,9 we
14.7%
2.2%
start the analysis in 2015 and utilize simple calendar
Equity
Bonds
Bitcoin
monthly returns with monthly rebalancing. The dataset
here spans the January 2015 to October 2021 period.
NOTES: Based on calendar monthly data covering January 2015
See the appendix for details on the data sources and
to October 2021, annualized. All returns are expressed in
indexes used. Exhibits 1 thorough 3 below show the
excess of cash. Equity and bonds are represented by S&P 500
historical arithmetic and geometric mean returns as
and Bloomberg US Aggregate index, respectively. Cash returns
are derived from the yield on three-month US Treasury bills.
well as return volatility for US Large Cap Equities
All data sourced from Bloomberg. See the appendix for details
(“Equities”), US Investment Grade Bonds (“Bonds”),
on data sources and calculations. Past performance is not a
and bitcoin over this period.
reliable indicator of future returns.
Clearly, bitcoin’s volatility levels and realized
returns are in a league of their own when compared
EXHIBIT 2
to traditional asset classes. Traditional portfolio construction
relies on the notion that an asset’s role in
Historical Geometric Mean Returns
a broader multi-asset portfolio is chiefly a function of
120%
112.0%
its return level in relation to its risk characteristics
(Markowitz 1952 and Sharpe 1964). The same level of
100%
return is more attractive if it comes along with less vol80%
atility and/or lower correlation to the traditional asset
60%
classes already present in the portfolio. Bitcoin’s very
high risk and return levels are intriguing in that regard.
40%
Bitcoin’s correlation to equities has been low over this
20%
period at around 0.25, lending further support to the
13.5%
2.1%
notion that it might be worth including in a multi-asset
0%
Equity
Bonds
Bitcoin
portfolio for diversification purposes.
Exhibits 4 and 5 below show the portfolio level
NOTES: Based on calendar monthly data covering January 2015
average
return and Sharpe ratio obtained when addto October 2021, annualized. All returns are expressed in
ing
a
small
amount of bitcoin to a traditional 60%
excess of cash. Equity and bonds are represented by S&P 500
and Bloomberg US Aggregate index, respectively. Cash returns
Equity/40% Bonds portfolio.10 A 1% bitcoin allocation
are derived from the yield on three-month US Treasury bills.
would have raised the historical mean return of this
All data sourced from Bloomberg. See the appendix for details
portfolio by 1.1% and raised the Sharpe ratio from
on data sources and calculations. Past performance is not a
1.08 to 1.18. Allocating 5% would have taken tradireliable indicator of future returns.
tional portfolio performance to a different level, adding
5.3% to return (roughly the magnitude of the equity risk
premium itself) and lifting the Sharpe ratio to 1.46—a
risk-adjusted performance out of reach for all but the best hedge funds.
Finally, Exhibit 6 below shows the results from an actual mean-variance portfolio
optimization. The “optimal” bitcoin allocation is one-quarter of the portfolio (24.6%),
200%
180%
160%
140%
120%
100%
80%
60%
40%
20%
0%
181.6%
8
Per www.coinmarketcap.com, accessed on 12/28/21, bitcoin’s $904 billion market capitalization
represents 40% of the entire $2,250 billion market capitalization tracked across 8,624 digital assets.
9
Coverdale (2021), Bhutoria et al. (2021), Butterfill and Bendiksen (2020), and Duffin (2021).
10
The bitcoin allocation is funded by the equity allocation, which is reduced accordingly.
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The Journal of Alternative Investments | 67
Winter 2023
EXHIBIT 3
EXHIBIT 5
Historical Volatility
Portfolio Sharpe Ratios with Fixed Bitcoin Allocations
90%
82.8%
80%
1.2
60%
1.18
1.08
1.0
50%
0.8
40%
0.6
30%
0.4
14.6%
10%
0%
1.46
1.4
70%
20%
1.6
0.2
3.1%
Equity
Bonds
Bitcoin
0.0
0% Bitcoin
1% Bitcoin
5% Bitcoin
NOTES: Based on calendar monthly data covering January 2015
NOTES: Based on calendar monthly data covering January 2015
to October 2021, annualized. All returns are expressed in
excess of cash. Equity and bonds are represented by S&P 500
and Bloomberg US Aggregate index, respectively. Cash returns
are derived from the yield on three-month US Treasury bills. All
data sourced from Bloomberg. See the appendix for details on
data sources and calculations. Past performance is not a reliable indicator of future returns.
to October 2021, annualized. All returns are expressed in
excess of cash. Equity and bonds are represented by S&P 500
and Bloomberg US Aggregate index, respectively. Cash returns
are derived from the yield on three-month US Treasury bills. All
data sourced from Bloomberg. See the appendix for details on
data sources and calculations. Past performance is not a reliable indicator of future returns.
EXHIBIT 4
EXHIBIT 6
Portfolio Returns with Fixed Bitcoin Allocations
Mean-Variance Optimal Portfolio Including Bitcoin
16%
14.8%
14%
12%
10%
9.5%
24.6%
40.0%
10.6%
8%
50.8%
6%
60.0%
4%
24.6%
2%
0%
0% Bitcoin
1% Bitcoin
5% Bitcoin
NOTES: Based on calendar monthly data covering January 2015
to October 2021, annualized. All returns are expressed in
excess of cash. Equity and bonds are represented by S&P 500
and Bloomberg US Aggregate index, respectively. Cash returns
are derived from the yield on three-month US Treasury bills.
All data sourced from Bloomberg. See the appendix for details
on data sources and calculations. Past performance is not a
reliable indicator of future returns.
Baseline
Equity
Optimal
Bonds
Bitcoin
NOTES: Based on calendar monthly data covering January
2015 to October 2021, annualized. Monthly rebalancing.
Equity and bonds are represented by S&P 500 and Bloomberg
US Aggregate index, respectively. All data sourced from
Bloomberg. See the appendix for details on data sources and
calculations.
funded entirely out of equities in order to manage the impact on portfolio volatility.11
For the confessed digital asset enthusiast, it is tempting to end the article here (next
step: Pension Committee!). But there is ample reason to hesitate at these results.
The rest of this article is devoted to revealing the unique challenges of inserting an
extremely high-volatility asset into a traditional portfolio optimization framework and
to illustrating an alternative approach to addressing these challenges.
11
This optimization is done in two stages. We first extract the risk preferences necessary to make
the 60/40 portfolio optimal in our sample when bitcoin is not available. We then re-optimize, permitting
bitcoin, under these same preferences. See the appendix for details.
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68 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
PROBLEMS WITH THE TRADITIONAL APPROACH
Not Enough History
Traditional asset allocation analysis often relies on long-term historical time series
to extract average returns for asset classes in order to then feed those into portfolio optimization models and solve asset allocation problems. Using the historical
averages in this fashion implicitly makes two subtly different assumptions. First, it
assumes that the mean observed in the sample accurately represents the true mean
for the underlying process that generated those returns. Second, it assumes that this
true mean will be the same going forward as it has been historically.
Naturally, this approach is more appropriate the longer the available historical
sample is. The case for the equity risk premium, notwithstanding its solid theoretical foundation, has been bolstered considerably by its long-term consistency over
decades and decades in countries with long capital market histories like the US
(see Siegel 2002). Long-term studies of the equity risk premium are so compelling
because, given the volatility of the asset class, the sample mean has arrived too
consistently, over too long a period of time, to sustain the alternative notion that the
true historical mean has been, say, zero, or that the future mean is likely to be zero.
Unfortunately, the available bitcoin return history is quite short compared to other
asset classes, and bitcoin’s volatility is magnitudes above that of equities. As a result,
the return data available to us is insufficient to make useful estimates of the “true”
bitcoin return generating process that may have produced what little historical data
we have (let alone to provide confident estimates for what the future mean might be).
We demonstrate this problem by making two very innocent modifications to the
analysis in Exhibit 6 and showing that they have a drastic impact on the “optimal”
bitcoin allocation (see the appendix for details on both). First, we let the month begin
on a different day. Why not start our “month” on the 3rd or the 17th of the month?
When we vary the definition of “calendar month” in this manner, the optimal bitcoin
allocations range from below 18% to 30%. A second analysis ends the dataset on
any of the 11 month-ends preceding October 2021 in order to show sensitivity to the
end point of the sample period. Here, results range between a 25% to 33% allocation
to bitcoin.12 Both exercises demonstrate how brittle historical mean return estimates
for bitcoin are.
What about Equity and Bonds?
Despite the long-term data available to support traditional risk premiums, any
simplistic analysis that uses just the period when bitcoin existed also samples a
random draw of equity and bond returns over that shorter period. To illustrate this,
Exhibit 7 below shows their risk and return characteristics during our sample period,
over a longer historical US sample starting in 1983, and alternatively according to
T. Rowe Price’s long-term capital market assumptions.
It is quite clear that the period since 2015 has been an unusual time for traditional
assets, with equity Sharpe ratios two to three times what long-term expectations
would have suggested. All else equal, this means that bitcoin could be even more
attractive than shown earlier if more reasonable and muted return expectations for
traditional assets were applied.
12
An earlier draft of this article was based on a dataset ending in February 2021. The same analysis
here produced a maximum 50% allocation to bitcoin.
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The Journal of Alternative Investments | 69
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EXHIBIT 7
Traditional Asset Risk and Returns over Different Sample Periods
Sample Period
Metric
Equity
Bonds
01/2015–10/2021
Mean Excess Return
Standard Deviation
Sharpe Ratio
14.9%
14.6%
1.02
2.2%
3.1%
0.71
06/1983–10/2021
Mean Excess Return
Standard Deviation
Sharpe Ratio
9.1%
14.9%
0.61
3.3%
4.0%
0.82
T. Rowe Price Long-Term
Capital Markets Assumptions
Expected Mean Return
Expected Standard Deviation
Expected Sharpe Ratio
5.5%
16.5%
0.33
1.5%
6.2%
0.24
NOTES: Based on calendar monthly data. Mean and standard deviation are annualized. All return statistics are based on excess
returns above cash. Sharpe Ratio is defined as Mean Excess Return divided by Standard Deviation. Equity and bonds are represented
by S&P 500 and Bloomberg US Aggregate index, respectively. Cash returns are derived from the yield on three-month US Treasury
bills. All data sourced from Bloomberg. See the appendix for details on data sources and calculations. Past performance is not a reliable indicator of future returns. Capital Market Assumptions data is based on forecasts that are for illustrative purposes only and are
not indicative of future results. Forecasts are based on subjective estimates about market environments that may never occur.
EXHIBIT 8
Bitcoin’s Lottery Aspects: Higher Moments
Skewness of Monthly Asset Returns
Very high volatility assets can be appealing
because they imply an asymmetric return distribution.
1.12
While the maximum loss is capped at a -100% return,
1.0
when volatility is sufficiently high, upside returns that
are multiples of 100% become conceivable. In other
0.5
words, these assets attain “lottery-like” characteris0.0
tics. Traditional mean-variance optimization, however,
–0.17
–0.5
ignores both the abnormal possibility of extreme outcomes (captured in the fourth moment, called kurto–1.0
–1.36
sis) and the asymmetry in those outcomes (measured
–1.5
in the third moment, called skewness). Most invesEquity
Bonds
Bitcoin
tors have a dislike of extreme outcomes and a “like”
NOTES: Based on calendar monthly data covering January 2015
of positively skewed distributions, similar to the way
to October 2021, annualized. All returns are expressed in
we like the first moment (return) and dislike the secexcess of cash. Equity and bonds are represented by S&P 500
ond moment (volatility). As can be seen in Exhibit 8,
and Bloomberg US Aggregate index, respectively. Cash returns
bitcoin’s extreme volatility necessarily creates an
are derived from the yield on three-month US Treasury bills. All
attractive, positive degree of skewness in the return
data sourced from Bloomberg. See the appendix for details on
data sources and calculations.
distribution compared to other asset classes. Interestingly, bitcoin’s overall tail fatness, as measured
by excess kurtosis, is relatively benign (Exhibit 9).
This means that the extreme return events bitcoin regularly experiences are largely
explained by its excessive volatility rather than the shape of the distribution and the
fatness of the tails. Still, an asset allocation study involving bitcoin would do well to
take these higher moments into account.
1.5
The Role of Rebalancing Horizon
Investors typically rebalance their portfolios at regular intervals—for example,
monthly or quarterly. This becomes more relevant when the mix includes highly volatile assets, whose allocations can drift materially between rebalancing dates. When
we rebalance the portfolio at a certain frequency, that is also the frequency at which
we are impacted by the asset returns. For weekly rebalancing, it is the distribution of
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70 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
weekly returns that matters; for monthly rebalancing,
we need to study monthly returns, and so forth. This
Excess Kurtosis of Monthly Asset Returns
should inform the formulation of risk and return expec10
tations: what matters for portfolio construction is the
8.83
9
expected risk and return at the frequency at which we
8
will rebalance the portfolio.
7
In the case of bitcoin, the choice of rebalancing
6
frequency has a material impact on the sampled return
5
4
distribution that informs our portfolio construction.
3.32
3
Exhibit 10 below shows the annualized arithmetic
2.36
2
mean, volatility, and Sharpe ratio obtained from histor1
ical bitcoin returns at daily, weekly, monthly, quarterly,
0
and semi-annual rebalancing frequency. Estimates of
Equity
Bonds
Bitcoin
the mean are only marginally impacted by this dimenNOTES: Based on calendar monthly data covering January 2015
sion. The annualized arithmetic mean is 121% when
to October 2021, annualized. All returns are expressed in
using daily returns and 155% when using semi-annual
excess of cash. Equity and bonds are represented by S&P 500
returns. For volatility however, there is a stark pattern,
and Bloomberg US Aggregate index, respectively. Cash returns
increasing in time horizon. Annualized volatility rises
are derived from the yield on three-month US Treasury bills. All
data sourced from Bloomberg. See the Appendix for details on
from 64% for daily returns to 145% for semi-annual
data sources and calculations.
returns, leading to a Sharpe ratio drop from 1.89 (daily)
to 1.07 (semi-annually). In other words, the attractiveness of bitcoin in terms of risk-adjusted performance
would have historically varied by a factor of 2x depending on the rebalancing period
used. The reason for this difference in volatility is the pronounced momentum pattern
in daily bitcoin returns. It implies that the daily returns that constitute longer period
returns essentially “amplify” each other (good returns likely to be followed by other
good returns), which widens the range of possible longer term return outcomes and
thus leads to greater volatility when measured for longer holding periods.
EXHIBIT 9
A Better Approach: What Does One Have to Believe to Put 1% in Bitcoin?
We now demonstrate an alternative approach that addresses the issues raised
above. This approach makes five specific improvements:
1. We use daily rolled, overlapping windows instead of calendar periods (think of
this as averaging results through all possible ways of defining, say, a monthly
calendar).
2. We show results for one-month, three-month, and six-month rebalancing
regimes.
3. We alternatively use long-term risk and returns assumptions for equity and
bonds.
4. Instead of mean-variance optimization, we use a utility-based portfolio optimization that recognizes the presence of higher moments.13
5. Rather than solving for the optimal allocation given expected returns as an
input, we back into the expected bitcoin return needed to motivate a 1% or
5% allocation, given historical risk characteristics.14
13
Other studies have previously addressed the issue of higher moments, including Czasonis et al.
(2021), who also use a utility function, or Pethukina et al. (2021), Eisl et al. (2015), and Chuen et al.
(2017), who use tail risk sensitive risk metrics like Conditional Value-At-Risk (CVar) in portfolio construction.
14
See Hubrich (2021) for a broader exploration of this approach when allocating to liquid alternatives. Our approach further advances the use of hurdle rates illustrated in Chambers et al. (2020) by
targeting a specific allocation size and by incorporating higher moments via explicit utility optimization
(as contrasted with mean-variance optimization).
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The Journal of Alternative Investments | 71
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EXHIBIT 10
EXHIBIT 11
Impact of Return/Holding Period on Annualized
Statistics
Bitcoin’s Correlation Characteristics
Panel A: Arithmetic Mean
Bitcoin Correlation
Properties
180%
160%
140%
120%
100%
80%
60%
40%
20%
0%
120.8%
122.9%
138.0%
143.0%
154.5%
Daily
Weekly
Monthly
Quarterly
120%
101.1%
100%
64.0%
66.4%
Daily
Weekly
78.9%
40%
20%
Monthly
Quarterly
SemiAnnually
Panel C: Sharpe Ratio
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.249
0.008
0.313
0.011
0.338
–0.026
SemiAnnually
144.8%
140%
0%
6M Holding
Period
10:2021. Equity and bonds are represented by S&P 500 and
Bloomberg US Aggregate index, respectively. Cash returns are
derived from the yield on three-month US Treasury bills. All data
sourced from Bloomberg. See the Appendix for details on data
sources and calculations.
160%
60%
3M Holding
Period
NOTES: Based on daily rolling multi-period returns, 01:2015 –
Panel B: Volatility
80%
Equity
Bonds
1M Holding
Period
1.89
1.85
1.75
1.41
1.07
Daily
Weekly
Monthly
Quarterly
SemiAnnually
NOTES: Based on daily rolled 1, 5, 21, 63, and 126 trading day
returns 2015-2021. All returns are expressed in excess of cash.
Equity and bonds are represented by S&P 500 and Bloomberg
US Aggregate index, respectively. Cash returns are derived from
the yield on three-month US Treasury bills. All data sourced
from Bloomberg. See the Appendix for details on data sources
and calculations. Past performance is not a reliable indicator of
future returns.
Improvements #1 and #2 directly address some
of the sample and timing sensitivities illustrated earlier. In terms of improvement #3, we show results
for two different sets of risk and return assumptions,
labeled A and B. Scenario A uses unmodified historical
1:2015-10:2021 returns for all three assets, while
scenario B replaces the return properties of equities
and bonds with the T. Rowe Price long-term capital
markets assumptions shown in Exhibit 7. The correlation properties in both scenarios are based directly on
the short sample period and are shown in Exhibit 11
above.15
Our last two improvements are related to how the
portfolio itself is constructed. In order to incorporate
higher moments into the analysis, we use a direct
utility maximization approach to obtain optimal portfolios (see the appendix for details). Finally, improvement #5 recognizes that the short period of bitcoin
returns available cannot be used to reliably assess
its return generating potential going forward. If that is
the case, the traditional approach of using historically
based return assumptions as inputs and solving for
the optimal allocation suffers from a “garbage in-garbage out” problem. By contrast, it is more defensible
to assume that past bitcoin returns do provide a guidepost for its risk characteristics going forward.16 Thus,
we invert the traditional portfolio optimization process.
Recognizing that a realistic bitcoin allocation is likely
to be quite small, we simply ask: given what we have
observed about bitcoin’s risk characteristics, in what
level of return would one have to believe such that a
1% or 5% allocation would be optimal? Readers then
15
We also examined scenarios analogous to A and B, but using a shorter dataset starting in
January 2017, in order to address the possible impact of the more recent “institutionalization” of bitcoin
adoption on correlation properties. Correlations and overall results however were substantially similar
to the ones shown here, so they were omitted from the article.
16
See Hubrich (2021) for a demonstration of the stylized fact that in general, the risk characteristics of financial assets are easier to ascertain with limited amounts of return data than their return
characteristics.
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72 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
EXHIBIT 12
Bitcoin Return Levels Required to Justify a 1% Bitcoin
Allocation
Panel A: Arithmetic Mean
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
45.6%
28.2%
20.8%
12.4%
9.7%
7.5%
1M
3M
6M
Panel B: Sharpe Ratio
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.42
0.33
0.28
0.13
0.12
0.10
1M
3M
6M
Panel C: Geometric Mean
10%
6.0%
5%
0%
–5%
–2.6%
–5.8%
–10%
–15%
–20%
–16.2%
1M
–16.6%
3M
Scenario A
–18.2%
6M
Scenario B
NOTES: Based on daily rolling multi-period returns, January
2015–October 2021, augmented by T. Rowe Price capital
market assumptions in Scenario B (see Exhibit 7). All returns
expressed in excess of cash. Equity and bonds represented
by S&P 500 and Bloomberg US Aggregate index, respectively.
Cash returns are derived from yield on three-month US Treasury
bills. All data are sourced from Bloomberg. See the Appendix
for details on data sources and calculations. Past performance
is not a reliable indicator for future returns.
can consult their investment thesis around bitcoin,
and gauge whether their own conviction lies above or
below that threshold.17
Exhibit 12 summarizes the results of this exercise for motivating a 1% bitcoin allocation. The panels
show three metrics: arithmetic mean return, Sharpe
ratio, and geometric mean return. These are the
thresholds in which one would have to believe to
make a 1% investment optimal in either Scenario A
or B, and for different rebalancing assumptions. In
general, the return thresholds are much higher in
scenario A (using unmodified, recent return data for
all assets) than in scenario B (which uses more moderate, long-term assumptions for traditional assets).
The extraordinarily attractive equity returns observed
over the past seven years (shown in Exhibit 7) have a
material impact on this Scenario A, truly “raising the
bar” that bitcoin must clear in order to be included in
the portfolio. Arithmetic mean thresholds in Scenario
A range from 21%–46%, necessary to produce Sharpe
ratios in the 0.28–0.42 range, in order to “keep up”
with Equity (whose recent historical Sharpe ratio was
1.02 per Exhibit 7) after accounting for diversification
benefits. By contrast, arithmetic mean thresholds
under the more realistic assumptions underlying Scenario B are much lower, ranging from 7.5%–12.4%,
and implying Sharpe ratios at or slightly above 0.1.
Since Equity here is a less attractive asset compared
to Scenario A, bitcoin returns only need to be moderately positive to merit portfolio inclusion at the 1%
level given its strong diversification benefits (equity
correlations of 0.25–0.34; see Exhibit 11).
These numbers are positive and large, yet meaningfully smaller than the realized average return shown
in Exhibit 1. We saw earlier that taking bitcoin’s large
historical returns at face value, optimal allocations
should be much larger than 1%. Part of the reason
for these low thresholds is that in a long-only portfolio high volatility actually is an advantage, because
a new asset has to displace other return generating
other assets. When asset volatility is high, a smaller
investment is required to make a risk contribution,
leaving less of a need to displace incumbent assets.
Bitcoin’s high volatility makes it an efficient tool from
a portfolio construction perspective, thus lowering the
Sharpe ratio required to justify an allocation.
The last row in Exhibit 12 shows the geometric
mean required to motivate an allocation. In the case
of bitcoin, it would capture the expected annualized
17
The expected improvement from adding just 1% or 5% bitcoin will very much depend on the
degree to which the investment thesis supports return expectations beyond the thresholds derived
in this study.
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Winter 2023
The Journal of Alternative Investments | 73
growth rate in the price of bitcoin itself. Unlike the arithmetic mean, the geometric
mean is also a function of risk. For the same average return (arithmetic mean), the
rate of capital accumulation (geometric mean) will be lower the higher the volatility of
the investment is. This is called the “volatility drag.” It creates the possibility that, for
sufficiently high volatility, an asset with a positive average return can have a negative
geometric mean. And indeed, we see in Exhibit 12 that this is the case here. With
the exception of the six-month rebalancing period case in Scenario A, the geometric
returns implied by the thresholds are negative.
If returns are so low that an asset’s value likely trends to zero over time, why
would anyone want to have it in the portfolio?18 This is where the broader portfolio
perspective becomes relevant. Recall that it is the volatility drag that pushes bitcoin’s
geometric return thresholds into negative territory. But unlike return, the volatility of
the portfolio is not just the sum of the volatilities of the underlying assets. When
we add a new asset, its return is added one-for-one, but its risk is “added” to a
much lesser degree if its correlation to the existing portfolio is low (the principle of
diversification). As a result, the portfolio holding 1% or 5% bitcoin does not suffer
the volatility drag to which bitcoin itself is subjected. This is related to the so-called
“diversification return”: when investing in risky but imperfectly correlated assets,
regular rebalancing lifts the growth rate of the portfolio above that of the underlying
individual investments.
This benefit increases with asset volatility, rebalancing frequency, and lack of
correlation among the constituent assets. Its high volatility and low correlation thus
make bitcoin a highly effective asset in a broader portfolio context, assuming regular
rebalancing. This in turn lowers the portfolio inclusion threshold for a multi-asset
investor whose decisions are guided by arithmetic rather than geometric mean considerations.19 Note that arithmetic mean thresholds in Exhibit 12 increase with the
rebalancing period, as less frequent rebalancing reduces the diversification return
benefit introduced by bitcoin. Hougan and Lawant (2021) similarly find that bitcoin’s
high volatility and the rebalancing frequency surrounding it greatly affect its impact
on a diversified portfolio.
Exhibit 13 below shows how thresholds change when targeting a larger, 5% bitcoin
allocation. Unsurprisingly, all thresholds here are meaningfully larger as well. Arithmetic means and Sharpe ratio thresholds roughly double in Scenario A and increase by
about 50% in Scenario B. Implied geometric means turn distinctly positive in Scenario
A, but remain negative in Scenario B.
In summary and focusing on Scenario B, the overall analysis suggests that if
readers can reconcile themselves to a Sharpe ratio expectation in the 0.10–0.12
range for bitcoin, a 1% allocation would seem justifiable, and a 5% allocation could
come into view when the outlook implies a Sharpe ratio in the 0.15–0.20 range. And
perhaps counter-intuitively, as long as bitcoin retains its enormous levels of volatility,
that minimum threshold could be entirely consistent with a downward trend in the
price of bitcoin.
It bears re-emphasizing that the thresholds shown above assume regular rebalancing. Given bitcoin’s volatility, this can imply sizeable rebalancing trades relative to
18
As pointed out in Chambers and Zdanowicz (2014), the mathematically expected return implied by
the ending value of the asset (in this case, the expected long term price of Bitcoin) is purely a function of
the arithmetic mean return (assuming independence) and would remain positive in our scenarios. But this
is driven by the highly skewed shape of long term cumulative return distributions, where a small subset
of unlikely but extremely large ending values give great lift to the average of the ending distribution. The
geometric mean is a better indicator for likely evolution of the asset (e.g., the median ending value).
19
See Booth and Fama (1992), Willenbrock (2011), and Lueneberger (2014, chapter 18) for more
detailed discussions of diversification returns, the role of rebalancing, and implications for portfolio
growth.
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74 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
EXHIBIT 13
EXHIBIT 14
Bitcoin Return Levels Required to Justify a 5% Bitcoin
Allocation
Rebalancing Trade Size Statistics at Three-Month
Holding Period (1% bitcoin allocation)
Panel A: Arithmetic Mean
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
89.8%
Bonds
Bitcoin
1.03%
1.22%
6.06%
1.27%
1.44%
6.05%
0.24%
0.41%
2.16%
NOTES: Based on calendar month returns, January 2015 to
12.7%
1M
20.5%
15.7%
3M
6M
0.80
0.72
0.70
0.50
Equity
Median
Mean
Maximum
50.0%
37.6%
Panel B: Sharpe Ratio
0.60
Statistic
0.50
October 2021, aggregated to quarterly frequency on rolling
monthly basis. The exhibit shows statistics of the absolute
value of the trades required to rebalance a portfolio with a 59%
equity, 40% bond, and 1% bitcoin target allocation at the end
of each rolling three-month period. Equity and bonds are represented by S&P 500 and Bloomberg US Aggregate index, respectively. All data are sourced from Bloomberg. See the appendix
for details on data sources and calculations.
0.56
its small allocation. Exhibit 14 is based on quarterly
rebalancing of a portfolio that includes 59% equity,
0.30
0.21
0.19
40% bonds, and 1% bitcoin. While the key statistics
0.17
0.20
shown for the size of the quarterly rebalancing trades
0.10
seem small for bitcoin in absolute terms, they are
0.00
1M
3M
6M
very large when viewed in light of the small underlying
Panel C: Geometric Mean
allocation. The mean quarterly bitcoin trade for rebal50%
ancing amounts to 0.41% of the portfolio, almost half
38.2%
40%
as big as the target 1% allocation itself.20
Let us finally turn one more time to the raw historical
30%
data.
How likely is it that the true mean generating those
20%
14.0%
past
returns
satisfied the various thresholds shown
7.2%
10%
above? We used a simulation analysis (described in
0%
the appendix) to estimate the p-values of the arithmetic
–10%
mean thresholds in Exhibits 12 and 13. These p-values
–12.1%
–12.2%
–12.3%
–20%
represent the likelihood that the true, data-generating
1M
3M
6M
historical mean (as distinguished form the sample
Scenario B
Scenario A
mean) was below the threshold level required for
investment. For the 1% case, all p-values were close
NOTES: Based on daily rolling multi-period returns, January
to or well below the customary 0.05 level (5% signif2015 to October 2021, augmented by T. Rowe Price capital
icance), indicating that it is highly likely that at least
market assumptions in Scenario B (see Exhibit 7). All returns
under the return generating process that produced its
expressed in excess of cash. Equity and bonds are represented
return history up to this point, a 1% investment can be
by S&P 500 and Bloomberg US Aggregate index, respectively.
Cash returns are derived from yield on three-month US Treasury
motivated with ease. For a 5% investment, there are
bills. All data are sourced from Bloomberg. See the Appendix
meaningful differences by holding period. For monthly
for details on data sources and calculations. Past performance
rebalancing the p-values were 0.02 or lower, while for
is not a reliable indicator for future returns.
semi-annual rebalancing, the p-value for Scenario A
rose to 0.17, no longer giving us overwhelming confidence that even bitcoin’s historical return generating
process would justify this larger investment.
This finding indicates that the dimensions emphasized in this article play an
important role in making an informed investment decision. Further, this line of
0.40
20
The comparatively large rebalancing trades associated with the bitcoin allocation could also have
meaningful tax implications in taxable accounts.
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The Journal of Alternative Investments | 75
Winter 2023
reasoning assumes that the return generating process for bitcoin remains the same
in the future as it has been in the past. Despite the high likelihood that the historical
bitcoin return process has satisfied the mean return requirement for a moderate
investment, readers still need to consult their bitcoin investment thesis to assess
whether it remains likely to do so in the future. This article takes no view on whether
that is the case, but the thresholds analyzed here can serve as an important starting
point for forming such a view.
CONCLUSION
The main takeaways from this article are as follows:
1. Directly using realized average historical bitcoin returns and risk character-
2.
3.
4.
5.
6.
istics in portfolio optimization leads to unrealistic—and arguably “too good
to be true”—portfolio allocations and improvements. The naively optimal
bitcoin allocation is around 25%, and just a 5% allocation would have raised
the Sharpe ratio of a traditional balanced portfolio from 1.08 to 1.46.
However, the brevity of the available historical bitcoin return sample, combined
with its extraordinary volatility, makes it difficult to assess its true mean
return with the degree of accuracy required for sensible portfolio analysis.
We showed that the naively “optimal” bitcoin allocations are greatly impacted
by innocent tweaks to the return sampling approach underlying the exercise.
A better approach is to take bitcoin’s risk characteristics as given, and then
back into the level of return that would be required to rationalize a 1% or 5%
bitcoin allocation given those risk characteristics. Readers can decide independently whether their bitcoin investment thesis can support that kind of
return expectation.
The return assumptions needed to motivate small bitcoin allocations in this
setting are strikingly low considering its volatility. Depending on assumptions
for traditional assets, Sharpe ratios of 0.1 or slightly above can be sufficient
to motivate a small bitcoin allocation.
In most cases, the implied geometric average bitcoin return at that threshold
level is negative, suggesting that a small allocation can be justified even if
one believes that the value of bitcoin may trend toward zero over time.
The holding period and rebalancing frequency assumed play important roles
when injecting bitcoin into a multi-asset portfolio analysis.
APPENDIX
DATA
Data are sourced from Bloomberg: S&P 500 Index (SPX INDEX) for Equity, Bloomberg
US Aggregate Index (LBUSTRUU INDEX) for Bonds, Bloomberg US T-bills 2–5 Months
Index (I31982 INDEX), the US three-month T-bill yield (GB03 GOVT) for cash, and bitcoin
(XBTUSD CURNCY). For the indexes we directly source total return, and for bitcoin we
source the price and calculate returns from price changes. We use the 2–5 month T-bill
index for cash returns where available, and otherwise use the three-month T-bill yield to
create a synthetic cash return. We then express all returns in excess of the cash before
proceeding with the analysis.
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76 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
PORTFOLIO OPTIMIZATION
The study utilizes two types of optimization. For the “naïve” optimization we solve
the problem in Equation 1 below:
max u ′w −
w
g
w ′ Sw
2
(1)
s.t. 0 ≤ wi ≤ 1 and Siwi ≤ 1 for i ∈ {E,B,BTC}
where E = equity, B = bonds, and BTC = bitcoin. u are the arithmetic average sample
returns for the three assets expressed as a vector, and S is their sample covariance
matrix. We obtain the risk aversion parameter g by first posing a 60% equity/40% bond
portfolio (w = (0.6,0.4,0)) and solving for the value of g that makes this portfolio optimal
among all portfolios that are not permitted to own BTC. We then permit BTC and directly
re-solve Equation 1, using the risk aversion g obtained in the first step.
The utility-based optimization in the second part of the article uses the so-called
Constant Relative Risk Aversion (CRRA) utility function. It is defined as
U(1 + r ) =
(1 + r )1− g − 1
1− g
for g > 1
(2)
Utility U(.) is a function of consumption or ending wealth, and the advantage of the
CRRA formulation is that scale does not matter. Hence for the single-period optimizations
employed here, we can standardize starting wealth to 1 and obtain ending wealth as 1 +
r, with r being the realized portfolio return. In this setting, investors choose the portfolio
that maximizes the expected utility E[U(1 + r)], recognizing that for a given allocation w,
r is a random variable. E[U] can then be understood as a function of the shape of that
distribution as represented by its moments. Equation 1 then can be interpreted as a
second-order approximation of Equation 2, where the higher-order terms that pick up
skewness and kurtosis are dropped. It is in this sense that traditional mean-variance
optimization is ultimately rooted in such a utility-based approach. The possible justifications for relying simply on Equation 1 to construct portfolios are either that the investor
truly does not “care” about higher moments (in the sense that Equation 1 rather than
Equation 2 represents true preferences), or that r itself is normally distributed, in which
case the third and fourth moment are zero. The view expressed in this article is that both
assumptions are inappropriate when dealing with bitcoin.
Instead, for a given portfolio w, we create the historical distribution of portfolio returns
r t using the historical returns of the three underlying assets (after rescaling equity and
bond returns to conform to the postulated long-term assumptions in Scenarios B). Each
single period return r t conforms to the rebalancing horizon studied (so monthly, quarterly,
or semi-annual returns). We then separately evaluate each individual historical realization
of r t under (2) and obtain the realized utility as the average of those values across all r t.
The threshold results for bitcoin returns in Exhibits 12 and 13 are then generated using a
grid search. We create a fine grid of possible portfolios w and evaluate their realized utility
under the risk aversion parameter g that motivated the original 60/40 portfolio containing
no bitcoin. Directly using historical bitcoin returns, the optimal portfolio (maximizing
realized utility) has very large bitcoin allocations, similar to that in Exhibit 6. We then
progressively reduce the average level of historical bitcoin returns (by applying a “haircut”
to its return history) until the optimal bitcoin allocation falls to 1% or 5%. Exhibits 12 and
13 then describe these reduced historical return levels that motivated this exact outcome.
When scaling returns downward, we divide by a scaling factor rather than subtracting a
fixed haircut from the return:
1 + rB*,t =
1 + rB,t
1+f
for f > 0
(3)
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The Journal of Alternative Investments | 77
Winter 2023
Here, rB is the historical bitcoin return (to be distinguished from portfolio return r), rB* is
the downscaled bitcoin return, and f is a constant scaling factor applied to all periods.
Since the rB are given by the historical data, this approach ensures that rB* is always above
-100%, and that geometric means remain above -100% as well. The haircuts needed to
motivate a small bitcoin allocation are so large that if, alternatively, an absolute haircut
were taken, some constellations would have bankruptcy events with rB* = -100%. To be
clear, by avoiding this, we do not mean to express the view that bitcoin cannot “go to
zero.” What we are trying to avoid is the artifact of bitcoin “going to zero” in our simulations because of its volatility. This conforms with the general practice of modeling security
distributions in terms of log returns (ln(1 + r) rather than the raw return r directly. This
scaling approach does preserve the volatility of bitcoin’s log returns, albeit with the side
effect of reducing its raw volatility.
CHANGING THE CALENDAR CONVENTION
Most studies, including the initial analysis in the main section of this article, use
calendar monthly returns. But there are other ways of defining discrete periods that are
monthly in length. For the analysis underlying Exhibit A1 we use daily data to construct
discrete return periods that are 21 trading days in length (roughly equivalent to a calendar
month), but that begin on different days. Naturally there are 21 different ways of doing
this. The bars in Exhibit A1 show the “optimal” bitcoin allocation (equivalent to Exhibit 6)
obtained when using these 21 different ways of defining the “calendar.” They range from
18%–30%.
THE IMPACT OF INDIVIDUAL OBSERVATIONS
For Exhibit A2 we end the dataset during any of the 12 calendar months preceding
and including October 2021.
EXHIBIT A1
Optimal Bitcoin Allocation Based on 21-Trading Day Rebalancing Periods Starting on Different Days
35%
30%
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21
NOTES: Based on rolling 21-day trading data covering January 2015 to October 2021and ending on different days intra-month, annu-
alized. Monthly rebalancing. Equity and bonds are represented by S&P 500 and Bloomberg US Aggregate index, respectively. All data
are sourced from Bloomberg. See the appendix for details on data sources and calculations.
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78 | Bitcoin in a Multi-Asset Portfolio
Winter 2023
EXHIBIT A2
Optimal Bitcoin Allocation Based on Latest Calendar Month in Sample
35%
30%
25%
20%
15%
10%
5%
1
t-2
Oc
21
pSe
21
gAu
1
l-2
Ju
21
nJu
ay
M
r-2
Ap
-2
1
1
1
M
ar
-2
21
bFe
21
nJa
0
c-2
De
No
v-2
0
0%
NOTES: Based on calendar monthly data beginning in January 2015 and ending in November 2020 to October 2021, annualized.
Monthly rebalancing. Equity and bonds are represented by S&P 500 and Bloomberg US Aggregate index, respectively. All data are
sourced from Bloomberg. See the Appendix for details on data sources and calculations.
EXHIBIT A3
P-values for Historical Bitcoin Mean under Threshold
Assumptions
Bitcoin
Allocation
Scenario
1M Holding
Period
3M Holding
Period
6M Holding
Period
1%
A
B
0.005
0.001
0.021
0.008
0.059
0.023
5%
A
B
0.020
0.002
0.053
0.011
0.174
0.030
NOTES: The exhibit shows the p-values for the realized historical
bitcoin mean return, using 10,000 simulated bitcoin return histories where the simulated mean corresponds to the threshold
arithmetic means in Exhibits 12 and 13. See surrounding text
for technical details.
The bars indicate the “optimal” bitcoin allocation had
the dataset been finalized that month. Similar to Exhibit
A1, we note meaningful variation, with results spanning
a 25%–33% allocation to bitcoin.
DOES THE HISTORICAL AVERAGE RETURN
SUPPORT THE THRESHOLDS?
We fitted an ARMA (10,6) process to daily BTC
returns.21 Modeling the bitcoin return structure in more
detail was necessary in order to obtain scenarios with
longer holding period returns that match the moments of
the historical bitcoin returns at those holding periods. We
then created 10,000 daily scenarios (of the same length
as our historical sample) by bootstrapping innovations
from this process. For a given rebalancing horizon we
then proceeded as follows:
1. Rescale all returns so that as a group (for the entire panel, across all scenarios)
they exactly match the sample volatility at that rebalancing horizon22
2. Adjust the mean of all scenarios as a group to match the threshold arithmetic
mean required by the case in question
3. Calculate sample arithmetic means separately within each scenario
4. Obtain P-value as the ranking of the historical sample mean in that vector of
simulated means
21
We selected the ARMA parameters from a grid search allowing up to 12 lags for each parameter,
selecting the model that maximized the Akaike Information Criterion (AIC).
22
The ARMA process itself comes close but fails to exactly match historical volatility.
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The Journal of Alternative Investments | 79
Winter 2023
Based on this construction, low p-values imply a low likelihood that true historical
mean was lower than the threshold required for investment. For example, a p-value 0.08
means that only 8% of the 10,000 scenarios have higher means than the historical mean.
In other words, if the true mean were at our necessary threshold, only 8% of the scenarios would have produced a higher mean than what we have seen historically. Exhibit A3
below shows p-values obtained from this simulation.
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Butterfill, J., and C. Bendiksen. 2020. “A Little Bitcoin Goes a Long Way.” White paper, CoinShares.
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Disclosure
My employer, T. Rowe Price Associates, Inc., is a provider of multi-asset investment strategies.
Further, in addition to being a T. Rowe Price shareholder and an employee of T. Rowe Price Associates, Inc.,
I am individually involved in the management of multi-asset investment strategies on behalf of T. Rowe Price
(none of which however have allocations to digital assets like bitcoin today).
This research has not been supported by any dedicated funding or grants from third parties other than my
employer.
The views expressed are the author’s, are subject to change without notice, and may differ from those of
other T. Rowe Price associates. Information and opinions are derived from proprietary and nonproprietary
sources deemed to be reliable; the accuracy of those sources is not guaranteed. This material does not constitute a distribution, offer, invitation, recommendation, or solicitation to sell or buy any securities; it does
not constitute investment advice and should not be relied upon as such. Investors should seek independent
legal and financial advice, including advice as to tax consequences, before making any investment decision.
Past performance is not a reliable indicator of future performance. All investments involve risk. The charts
and tables are shown for illustrative purposes only.
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It is illegal to make unauthorized copies, forward to an unauthorized user, post electronically, or store on shared cloud or hard drive without Publisher permission.