Financial analysis of Bitcoin and how it is used to optimize the performances of an investment portfolio

1. The use of Bitcoin for
portfolio optimization
Federico Tenga
federico@chainside.net

2. Introduction
For many people Bitcoin is considered the ideal store of value,
but most investor still lack to see the value of this asset due to
its technical complexity.
The scope of this study is to analyze Bitcoin strictly under a
financial point of view and show the benefits it brings to
optimize any investment portfolio and have more awareness of
the risk-reward profile of this new asset.

3. Index
• Bitcoin Supply
• Volatility
• Correlation
• Expected Return
• Daily returns distribution analysis
• Bitcoin for Portfolio Optimization

4. Bitcoin Supply
Figure 1 Bitcoin supply and inflation

5. Gold Supply
Figure 2 Gold supply. Source: Number Sleuth ("All The World's Gold Facts")

6. Volatility
The volatility, measured by the standard deviation, in finance is the degree of variation of
price of an asset, and it can be derived using historical market price data.
To find the volatility of bitcoin, we will compute the standard deviation of daily returns
using the following formula:
𝜎 =
1
𝑛
෍
𝑖=1
𝑛
(𝑥𝑖 − 𝜇)2
Where:
𝜇 = 𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑖𝑙𝑦 𝑟𝑒𝑡𝑢𝑟𝑛𝑠
From April 2016 to May 2017 the average daily volatility of bitcoin is 2.67%

7. Annualized Volatility
To compute the annualized standard deviation, we have to multiply the daily
standard deviation for the square root of the number of trading days.
For traditional financial markets the number of trading days in a year is usually
about 250, but since bitcoin is traded 24/7, every day of the year, we have about
365 trading days, consequentially the formula to compute the annualised
standard deviation will be:
𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝜎 = 𝜎 365
Resulting in an annualized standard deviation of 51.03%.

8. Volatility over time

9. Correlation
The correlation measures the dependence between two variables. The Pearson
correlation, considered to be the “traditional” correlation, is calculated with the
following formula:
𝜌 𝑥,𝑦=
𝐶𝑜𝑣(𝑥, 𝑦)
𝜎 𝑥 𝜎 𝑦
Where:
𝐶𝑜𝑣 𝑥, 𝑦 = 𝐸( 𝑥 − 𝐸 𝑥 𝑦 − 𝐸 𝑦 )
𝜎 𝑥 = 𝐸[𝑥]2− (𝐸 𝑥 )2
If we try to compute the Pearson correlation of bitcoin returns with some major asset
classes we can find some very interesting results

10. p-value
When you perform a hypothesis test in statistics, a p-value helps you determine the significance of
your results. he p-value is a number between 0 and 1 and interpreted in the following way: A small
p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the
null hypothesis.

11. Bitcoin vs S&P 500 Correlation
The Pearson correlation of bitcoin with S&P 500, over the timespan analysed, results to
be 1.57%, which is extremely low. Moreover, the p-value of the Pearson correlation is a
very high 0.582, meaning that we cannot even easily assume that the correlation is
different from zero.
Figure 5 Bitcoin vs S&P 500 daily returns

12. Bitcoin vs MSCI Emerging Markets Index
The Pearson correlation of bitcoin with MSCI Emerging Markets Index, over the timespan
analysed, results to be 2.69%. Still a very low value, and similarly to what we have seen
with S&P 500 the high p-value of 0.345 suggests that the result is not significant so we can
even be sure that the correlation is not actually zero.
Figure 6Bitcoin and MSCI Emerging Markets Index Daily Returns

13. Bitcoin vs Oil Correlation
The Pearson correlation of bitcoin with WTI Crude Oil prices, over the timespan analysed,
results to be 0.8%, and just like in the cases seen above the data cannot be considered
significant due to the high p-value of 0.789.

14. Bitcoin vs Gold Correlation
The Pearson correlation of bitcoin with gold, over the timespan analysed, results to be
just 1.7%, and just like we have seen before the p-value of 0.558 suggests that the results
cannot be considerate significant to assume that the correlation is different from zero.

15. Correlation and portfolio volatility
The non-existent correlation means that bitcoin does not share systematic risk with other
asset classes, making it a great tool for diversification of a portfolio. Indeed, this is
evident looking at the formula of a portfolio volatility:
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 = 𝑆𝐷 𝑎
2
∗ 𝑊𝑎
2
+ 𝑆𝐷 𝑏
2
∗ 𝑊𝑏
2
+ 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑒𝑐𝑡
Where:
SD = standard deviation of the asset
W = weight of the asset in the portfolio

16. Expected Returns (1/2)
A common method to attempt to calculate the assets appropriate return is the Capital
Asset Pricing Model (CAPM), which describes the relationship between systematic risk and
expected return of an asset. The CAPM formula for calculating the expected return of an
asset given its risk is as follows:
ra = rf + βa (rm – rf)
Where:
rf = risk-free rate
βa = beta of the asset
rm = expected return of the market
Unfortunately, due to the lack of correlation with a benchmark portfolio, it is not possible
to calculate the β of bitcoin

17. We can use instead historical data to estimate future trends, but it is not easy to decide
which timespan can be useful for our analysis. Old trading data are less significant due to
the high market manipulation of the early days, a good option to use post-MtGox data.
We can find the mean daily return of the period April 2014 – May 2017 using the following
formula:
𝜇 =
1
𝑛
෍
𝑖=1
𝑛
𝑋𝑖
Which gives us a mean daily return of 0.13%
Expected Returns (2/2)

18. Annualized expected returns
We can than derive the annualized expected return raising 𝜇 to the power of number of
trading days in a year, using the following formula:
𝜇 𝑎𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 = (1 + 𝜇 )365−1
Once again since bitcoin is traded 24/7, differently from any other asset we will 365
trading days. Which gives us an annualized expected return of 60.35%

19. Daily Returns Distribution
During the period from April 2014 to June 2017, we can see an average daily return of
0.2% with a standard deviation on 3.06%
However, it is necessary to proceed with a normality test to discover if the standard
returns can be described by a Gaussian distribution.

20. Shapiro–Wilk test
In statistics, the Shapiro–Wilk test tests the null hypothesis that a sample x1,...,xn originated
from a normally distributed population. The test is:
𝑊 =
σ𝑖=1
𝑛
aixi 2
σ𝑖=1
𝑛
(xi − lj𝑥)
2
𝑎1, … , 𝑎 𝑛 =
𝑚T 𝑉−1
(𝑚 𝑡 𝑉−1 𝑉−1 𝑚)1/2
Where:
𝑚 = (𝑚1, … , 𝑚 𝑛)T
and m1, ..., mn are the expected values of the order statistics of independent and identically
distributed random variables sampled from the standard normal distribution, and V is the
covariance matrix of those order statistics.

21. Shapiro–Wilk test
Doing the Shapiro-Wink test on the bitcoin daily returns for the period April 2014 – April 2017
we find the following results:
As the computed p-value is lower than the significance level alpha=0.05, one should reject the
null hypothesis (the sample follows a Normal distribution), and accept the alternative
hypothesis (the sample does not follow a Normal distribution).
The risk to reject the null hypothesis while it is true is lower than 0.01%
W 0.868
p-value (Two-tailed) < 0.0001
alpha 0.05

22. Skewness analysis
The Skewness is a measure to analyse the symmetry of the distribution. If the coefficient of
skewness is positive the distribution is skewed right, if it is negative the distribution is skewed
left.
The coefficient of skewness of a data set is:
skewness: 𝑔1 = 𝑚3/𝑚2
3/2
Where:
𝑚3 =
σ x − തx 3
𝑛
𝑎𝑛𝑑 𝑚2 =
σ x − തx 2
𝑛
A normal distribution has a skewness coefficient of zero (perfect symmetry). The distribution
of bitcoin daily returns from April 2014 to April 2017 has skewness coefficient of 0.024, which
means that the distribution is almost perfectly symmetrical, indicating a good market
efficiency.

23. Kurtosis Analysis
Another common measure of shape is the Kurtosis which provides information on how the
data are spread among the peak and the tails of the distribution.
Higher kurtosis also means that more of the variance is the result of infrequent extreme
deviations, as opposed to frequent modestly sized deviations. The reference standard is a
normal distribution, which has a kurtosis of 3.
The coefficient of kurtosis of a dataset is:
kurtosis: 𝑎4 = 𝑚4/𝑚2
2
Where:
𝑚4 =
σ x − തx 4
𝑛
𝑎𝑛𝑑 𝑚2 =
σ x − തx 2
𝑛
The distribution of bitcoin daily returns from April 2014 to April 2017 has kurtosis
coefficient of 12.439, meaning that most of the variance is caused by infrequent extreme
deviations. This can also be seen as volatility on volatility.

24. Autocorrelation Analysis
In order to understand if the bitcoin market is efficient, it is useful to test the autocorrelation
of the daily returns. If the returns result to be autocorrelated, it indicates that the price
movements can be predicted and the market is not efficient.
To test the autocorrelation, we will use the Durbin-Watson statistic, which always has a value
d between 0 and 4, where 2 means there is no autocorrelation, 0 means that there is positive
correlation and 4 means that there is negative correlation.
The value d of the Durbin-Watson statistic is calculated with the following formula:
d =
σ 𝑡=2
𝑇
(𝑒𝑡 − 𝑒𝑡−1)2
σ 𝑡=1
𝑇
𝑒𝑡
2
Where:
et = residual associated with the observation at time t
Running the Durbin-Watson test on the bitcoin daily returns from April 2014 to April 2017 we
obtain a d value of 2.094, meaning that there is almost no autocorrelation and the market is
efficient.

25. Bitcoin Price Resilience
In the history of Bitcoin there have been many big events that caused high volatility periods and
the collapse of the price. It is interesting to notice that even if big events that can potentially
damage the Bitcoin ecosystem keep happening, the impact is becoming less significant.
Figure 13 Bitcoin price change comparison after exchanges' losses

26. Portfolio Optimization

27. Portfolio Optimization
A general solution to portfolio selection problem was proposed by Markowitz in
1952, building the foundation of the Modern Portfolio Theory (MPT) in the coming
decades. The MPT is based on the utility maximization concept, emphasizing the
trade-off between risk and return.
One of the assumption of the MPT is that investors are risk-adverse, giving two
portfolios with similar returns a rational investor will always choose the one with
lower risk, and any extra risk will have to be compensated.
With the Modern Portfolio Theory, it is possible to keep a certain level of risk while
maximising the returns, or keeping the returns fixed while minimising the risk.
An investor can then generate a so called efficient frontier where the optimal asset
allocation is achieved.

28. Portfolio Returns
An asset return can be easily defined as the price difference over a time period divided
the price of the asset at the beginning of the period, as described by the following
formula:
𝑅𝑖,𝑡 =
𝑃𝑖,𝑡 − 𝑃𝑖,𝑡−1
𝑃𝑖,𝑡−1
Where:
𝑃𝑖,𝑡 = 𝑝𝑟𝑖𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
𝑃𝑖,𝑡−1 = 𝑝𝑟𝑖𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
In a portfolio, the return is simply the weighted average of the return of the assets in the
portfolio, as described by the following formula:
𝑅 𝑝,𝑡 = ෍
𝑛=1
𝑁
𝑤𝑖 𝑅𝑖,𝑡
Where:
𝑤𝑖 = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 𝑖

29. Portfolio Risk
We can measure the risk of the portfolio using its standard deviation. For an asset,
the standard deviation can be found, as we have previously seen, using the
following formula:
𝜎 =
1
𝑛
෍
𝑖=1
𝑛
(𝑥𝑖 − 𝜇)2

30. Portfolio Risk
In a portfolio, we have to consider the standard deviation and the correlation coefficient of all
the assets that compose the portfolio, we can then compute the standard deviation of the
portfolio using the following general formula:
𝜎 𝑝 = ෍
𝑖=1
𝑁
෍
𝑗=1
𝑁
𝑤𝑖 𝑤𝑗 𝜎𝑖 𝜎𝑗ρ𝑖𝑗
Where:
w = weight of the asset
σ = standard deviation of the asset
ρ = correlation coefficients of the assets
Considering a portfolio with two assets, the resulting standard deviation will be:
𝜎 𝑝 = 𝑤1
2
𝜎1
2
+ 𝑤2
2
𝜎2
2
+ 2𝑤1 𝜎2ρ1,2

31. The Sharpe Ratio
The Sharpe ratio is a tool to examine the performance of an investment, by measuring
how well the return of an asset compensates for its risk, making also possible to compare
assets with very different levels of risk.
The Sharpe ratio is calculated as follows:
𝑆 𝑎 =
𝐸[𝑅 𝑎 − 𝑅 𝑏]
𝜎 𝑎
Where:
Ra = return of asset
Rb = return of benchmark asset, usually the risk-free rate
σa = standard deviation of the asset
The Sharpe ratio can be used to identify the optimal asset allocation in a portfolio, the
one that provides the highest return per unit of deviation.

32. Bitcoin – S&P 500 portfolio
To efficiently build a two assets portfolio using S&P 500 and bitcoin, it is first needed to
quickly analyse the two separate assets:
We can already notice that bitcoin has a higher Sharpe ratio than S&P 500, considering
that we have also previously seen that the two assets have no correlation we can already
deduce that bitcoin will significantly improve the returns of the portfolio per unit of
deviation.
S&P 500 bitcoin
Daily mean SD 0.80% 2.67%
Annualized SD 12.66% 51.03%
Daily mean return 0.04% 0.13%
Annualized return 9.46% 60.35%
Sharpe ratio 0.68 1.15

33. Bitcoin – S&P 500 portfolio
Btc% S&P 500% SD Return Sharpe
0% 100% 12.66% 9.46% 0.6838
30% 70% 17.69% 24.73% 1.3532
100% 0% 51.03% 60.35% 1.1673
In the table below we can see the return and the risk for each possible allocation of
the two assets:
As we can see, the portfolio with the highest Sharpe ratio would be the one with
30% allocated in bitcoin an 70% allocated in S&P 500.

35. Gold – S&P 500 portfolio
The fact that bitcoin could improve the performance of the S&P 500 portfolio was easily
predictable due to its low correlation coefficient, so it is more interesting to see how the
performance improvment provided by bitcoin is compared to the improvement provided
by gold, an asset that bitcoin has the ambition to replace, at least partially, in its safe
heaven role for investors.
Gold seems to be very good for diversification purposes due to its low correlation with
S&P 500 (only 0.03), but as we can see in the table below, its poor performances in the
past years greatly affect its Sharpe ratio, which is significantly lower compared to
bitcoin’s ratio.
S&P 500 Gold
Daily mean SD 0.80% 0.79%
Annualized SD 12.66% 12.57%
Daily mean return 0.04% 0.013%
Annualized return 9.46% 3.37%
Sharpe ratio 0.68 0.20

36. Gold – S&P 500 portfolio
After trying to plot every possible portfolio combing the two assets we can find that the
allocation that maximise the Sharpe ratio has 21% of gold. However, the contribution to
improve the performance of the portfolio is modest, the Sharpe ratio of the optimal
portfolio is just slightly higher compared to the S&P ratio of 0.68, while bitcoin provided
a much more significant improvement, almost doubling it.
Gold% S&P 500% SD Return Sharpe
0% 100% 12.66% 9.46% 0.6838
21% 79% 10.42% 8.18% 0.7081
100% 0% 12.57% 3.37% 0.2044

38. Too much risk?
Choosing an asset allocation that maximise the Sharpe ratio does not allow an investor to
set its desired level of risk. This problem can be solved introducing a risk-free asset,
which let an investor maximise the Sharpe ratio and set the portfolio risk level.
Introducing the risk-free asset, the expected return and the standard deviation of a
portfolio are:
𝐸(𝑅 𝑝) = 𝑤1 𝐸(𝑅1) + (1 − 𝑤1) 𝑅 𝑓
σ 𝑝 = 𝑤1
2
σ1
2
Where:
𝑅1= return of the portfolio with only risky assets
𝑅 𝑓= return of the risk-free asset
𝑤1= weight of the risky assets
𝜎1=standard deviation of the portfolio with only risky assets

39. Capital Allocation Line
After having found the portfolio that maximise the Sharpe ratio, an investor
with a lower risk preference can allocate part of his capital in a risk-free asset,
reducing his exposure, and an investor with a higher risk preference can borrow
at the risk-free rate and invest his capital with leverage in the optimal
portfolio. These practices generate the Capital Allocation Line (CAL), defined
by the following formula:
𝐸 𝑅 𝑝 = 𝑅𝑓 +
𝐸(𝑅1) − 𝑅𝑓
σ1
σ 𝑝

41. Conclusion
Whatever is your risk preference, Bitcoin can help you
to optimize your portfolio

42. Federico Tenga
COO at Chainside
federico@chainside.net