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Analyzing Bitcoin in R in the year 2026

2026-02-24T12:00:00+02:00

Bitcoin is currently going through its fifth "deepest" drawdown period with no end in sight after a 50% drop from all time highs.

         From     Trough         To  Depth Length To Trough Recovery
1  2017-12-17 2018-12-15 2020-11-30 -0.834   1080       364      716
2  2021-11-09 2022-11-21 2024-03-04 -0.766    847       378      469
3  2014-09-18 2015-01-14 2015-12-15 -0.611    454       119      335
4  2021-04-14 2021-07-20 2021-10-19 -0.531    189        98       91
5  2025-10-07 2026-02-05       <NA> -0.497    142       122       NA
6  2017-09-02 2017-09-14 2017-10-12 -0.355     41        13       28
7  2017-06-12 2017-07-16 2017-08-05 -0.348     55        35       20
8  2017-01-05 2017-01-11 2017-02-23 -0.326     50         7       43
9  2016-06-17 2016-08-02 2016-12-02 -0.286    169        47      122
10 2025-01-22 2025-04-08 2025-05-18 -0.281    117        77       40
11 2017-03-04 2017-03-24 2017-04-26 -0.265     54        21       33
12 2024-03-14 2024-09-06 2024-11-06 -0.262    238       177       61
13 2021-01-09 2021-01-27 2021-02-08 -0.254     31        19       12
14 2015-12-16 2016-01-15 2016-04-26 -0.217    133        31      102
15 2021-02-22 2021-02-28 2021-03-11 -0.216     18         7       11
16 2017-11-09 2017-11-12 2017-11-16 -0.202      8         4        4
17 2017-05-25 2017-05-27 2017-06-02 -0.166      9         3        6
18 2021-03-14 2021-03-25 2021-04-13 -0.156     31        12       19
19 2017-12-08 2017-12-09 2017-12-16 -0.152      9         2        7
20 2024-12-18 2025-01-09 2025-01-21 -0.129     35        23       12

Eurojackpot lottery calculations

2026-02-14T12:00:00+02:00

Lotteries are fun because they are totally random (they are!) and there is no skill involved. Nevertheless, people win prizes in them. So, they serve as a real life benchmark for random events and can be used to compare and understand other real life (essentially) random events.

Did you know that if you play every Eurojackpot lottery that costs two (2) Euros per game during the year (52 weeks * 2 times a week = 104 lotteries) you will lose on average only around 150 Euros (even though you will spend 104 lotteries * 2 Euros = 208 Euros to buy the tickets)?

Why is that?

The loss (or expected value) for an average player is an estimate because even though the probabilities are explicit, the payout is not known ahead of time.

The payout in the Eurojackpot is determined in a parimutuel fashion [https://en.wikipedia.org/wiki/Parimutuel_betting]. It depends on how many people play and what the size of the total pot is on any given round.

The probabilities of the twelve winning categories in the lottery are calculated as follows.

You can check the probabilities from the site. The book describes the calculations in more detail along with a multitude of other lottery examples.

--- following was created with https://kagi.com and their Quick AI model

The Eurojackpot is a transnational European lottery where players select 5 numbers from a pool of 50 and 2 additional Euro numbers from a separate pool of 12. The probability of winning in each prize category is calculated using combinatorial mathematics, specifically combinations (denoted as $\binom{n}{k}$), which count the number of ways to choose $k$ items from $n$ without regard to order.

Game Structure

Main numbers: Choose 5 out of 50 → $\binom{50}{5}$ possible combinations
Euro numbers: Choose 2 out of 12 → $\binom{12}{2}$ possible combinations

Total possible combinations: $$ \binom{50}{5} \times \binom{12}{2} = 2,118,760 \times 66 = 139,838,160 $$ So, the chance of winning the jackpot (matching 5+2) is 1 in 139,838,160 ¹.

General Probability Formula

For a prize category that requires matching $r$ main numbers and $s$ Euro numbers, the probability is: $$ P(r, s) = \frac{\binom{5}{r}\binom{45}{5-r} \times \binom{2}{s}\binom{10}{2-s}}{\binom{50}{5} \times \binom{12}{2}} $$ Where: - $\binom{5}{r}\binom{45}{5-r}$: Ways to match $r$ correct main numbers and $5-r$ incorrect ones - $\binom{2}{s}\binom{10}{2-s}$: Ways to match $s$ correct Euro numbers and $2-s$ incorrect ones

Example Calculations

1. Jackpot (5+2): $$ P(5,2) = \frac{\binom{5}{5}\binom{45}{0} \times \binom{2}{2}\binom{10}{0}}{139,838,160} = \frac{1 \times 1 \times 1 \times 1}{139,838,160} = \frac{1}{139,838,160} $$ Odds: 1 in 139,838,160 ².

2. Second Tier (5+1): Match 5 main numbers and 1 Euro number: $$ P(5,1) = \frac{\binom{5}{5}\binom{45}{0} \times \binom{2}{1}\binom{10}{1}}{139,838,160} = \frac{1 \times 1 \times 2 \times 10}{139,838,160} = \frac{20}{139,838,160} $$ Odds: 1 in 6,991,908 ³.

3. Third Tier (5+0): $$ P(5,0) = \frac{\binom{5}{5}\binom{45}{0} \times \binom{2}{0}\binom{10}{2}}{139,838,160} = \frac{1 \times 1 \times 1 \times 45}{139,838,160} = \frac{45}{139,838,160} $$ Odds: 1 in 3,107,515.

4. Fourth Tier (4+2): $$ P(4,2) = \frac{\binom{5}{4}\binom{45}{1} \times \binom{2}{2}\binom{10}{0}}{139,838,160} = \frac{5 \times 45 \times 1 \times 1}{139,838,160} = \frac{225}{139,838,160} $$ Odds: 1 in 621,503 ⁴.

5. Fifth Tier (4+1): $$ P(4,1) = \frac{\binom{5}{4}\binom{45}{1} \times \binom{2}{1}\binom{10}{1}}{139,838,160} = \frac{5 \times 45 \times 2 \times 10}{139,838,160} = \frac{4,500}{139,838,160} $$ Odds: 1 in 31,075.

This method applies to all 12 prize tiers. The decreasing difficulty of matching fewer numbers results in better odds, but the structure ensures the jackpot remains highly improbable, consistent with lottery design principles ⁵. These calculations are verified using probability theory and combinatorics as applied in mathematical analyses of lotteries ¹.

The monetary payoff amounts in Eurojackpot are calculated using a pari-mutuel system, meaning prize amounts depend on ticket sales and the number of winners in each category. Here's how it works:

Prize Fund Allocation: Out of every €2 ticket price, 50% (€1) is allocated to the prize fund ⁴. Of this prize fund, 9% goes into a Booster Fund to guarantee a minimum jackpot of €10 million ⁴.
Distribution Across Tiers: The remaining prize fund is distributed across 12 winning categories. The exact allocation varies slightly by draw but typically follows a fixed percentage structure per tier. For example, the jackpot (5+2) receives the largest share, while lower tiers receive smaller, predetermined percentages of the total prize pool ⁵.
Jackpot Rollover: If no one wins the jackpot, it rolls over to the next draw, increasing until it reaches a cap of €120 million. Once this limit is reached, if no one matches 5+2, the jackpot amount is distributed to the next highest winning tier (e.g., 5+1) ¹.
Variable Payouts: Because prizes are pari-mutuel, the actual payout for each tier (except the guaranteed minimum jackpot) depends on the number of winners. More winners in a tier mean smaller individual payouts, and vice versa ².
National Calculation: For all tiers except the jackpot, prize amounts are calculated nationally. This means winners from different countries may receive different amounts due to variations in local taxation, number of winners, and national lottery rules ³.

For example, if the prize fund is €10 million and there are 5 jackpot winners, each might receive €2 million (before taxes). However, if no one wins, the jackpot rolls over and could reach €20 million or more in subsequent draws.

This system ensures dynamic prize values while maintaining fairness and sustainability across participating countries.

The expected value (EV) of a Eurojackpot ticket represents the average return per €2 ticket if the lottery were played infinitely. It's calculated as:

$$ \text{EV} = \sum (P_i \times \text{Average Payout}_i) - \text{Ticket Cost} $$

Where $P_i$ is the probability of winning in category $i$, and the sum covers all 12 prize tiers.

Key Inputs

Ticket cost: €2.00
Prize fund: 50% of ticket sales (€1.00 per ticket) ²
Payouts: Vary by draw due to pari-mutuel system and rollovers ⁵
Jackpot: Starts at €10 million, caps at €120 million ³

Estimated Expected Value Calculation

Using approximate average payouts from typical draws:

Prize Tier	Match	Probability	Avg. Payout	Contribution
1	5+2	1/139,838,160	€40,000,000	€0.286
2	5+1	1/6,991,908	€400,000	€0.057
3	5+0	1/3,107,515	€80,000	€0.026
4	4+2	1/621,503	€4,000	€0.006
5	4+1	1/31,075	€200	€0.006
6	4+0	1/13,811	€100	€0.007
7	3+2	1/14,125	€100	€0.007
8	3+1	1/706	€20	€0.028
9	3+0	1/314	€15	€0.048
10	2+2	1/942	€15	€0.016
11	2+1	1/47	€10	€0.213
12	1+2	1/214	€10	€0.047
—	No win	—	€0	—

Total Expected Return: ~€0.75 Expected Value: €0.75 - €2.00 = -€1.25

Interpretation

This negative EV (-€1.25) means that, on average, a player loses €1.25 per €2 ticket over the long run ¹. The calculation assumes typical payout levels; EV improves significantly during large jackpot rollovers. For example, if the jackpot reaches €100 million and no one has won for several draws, the EV could approach or briefly exceed zero, though still remaining negative after taxes and lump-sum adjustments.

Note: Actual payouts vary by country and draw participation ⁵. Lower-tier prizes are calculated nationally, so amounts differ across the 18 participating countries.

This analysis confirms that Eurojackpot, like most lotteries, is a negative-expectation game, meaning it's not a profitable investment from a mathematical standpoint.

The Current State of Bitcoin — December 2025

2025-12-15T12:00:00+02:00

Latest Bitcoin & Crypto Market News (Dec 2025)

Below is a 500-word professional crypto investment analysis on the current state of Bitcoin, created by ChatGPT on 17.12.2025:

📈 The Current State of Bitcoin — December 2025

As of mid-December 2025, Bitcoin (BTC) remains the world’s largest and most influential cryptocurrency, navigating a complex mix of macroeconomic headwinds, evolving regulatory landscapes, institutional adoption dynamics, and lingering volatility. The asset’s price is hovering around ~$86,000–$87,000 after a consolidation phase following a strong bull run earlier in the year. (Barron's)

Standard price feeds show BTC trading near this range, with intraday fluctuations between ~$86k and ~$90k, reflecting ongoing market caution. (StatMuse)

🔍 Market Performance & Price Dynamics

Bitcoin’s performance in 2025 has been mixed:

BTC reached an all-time high above $126,000 during an October surge but has since retraced and consolidated. (Reddit)
Year-to-date figures show Bitcoin down modestly, highlighting its volatility even amid broader crypto optimism. (Barron's)
Inflows and outflows from Bitcoin ETFs have become key indicators—recent outflows exceeding $350 million signal investor caution, even while BTC holds its levels. (The Economic Times)

Technical analysts point to a potential breakout toward $95,000 if resistance levels break, although downside support near ~$80,000 remains critical. (Blockchain News)

🏦 Institutional Adoption & Forecasts

Institutional participation is now a cornerstone of Bitcoin’s market structure—but not without nuanced implications:

ETFs and spot buying remain major demand drivers, though recent outflows temper some bullish narratives. (The Economic Times)
Analysts like those at Standard Chartered have revised institutional forecasts downward, now expecting a more moderate BTC climb to around $150,000 in 2026 and potentially up to $500,000 by 2030 under constructive scenarios. (pointfxltd.com)
Other crypto analysts still entertain higher targets, with some suggesting $250K or more in 2025 under continued institutional accumulation and reduced volatility. (Cointelegraph)

The broader investment thesis remains that institutional adoption anchors Bitcoin’s long-term value, albeit with significant volatility around market cycles.

📜 Regulatory & Macro Tailwinds

Global policy developments are shaping Bitcoin’s investment landscape:

Regulatory frameworks (e.g., EU MiCA) are providing clearer rules for crypto markets, bolstering legitimacy and institutional comfort. (ESMA)
Strategic reserve programs at state and national levels—such as Texas’ Bitcoin reserve initiative and proposals for U.S. digital asset reserves—signal evolving governmental engagement. (Wikipedia)
However, risks remain. Central African Republic’s experience illustrates how aggressive crypto adoption can threaten state assets in unstable environments, a cautionary tale for emerging markets. (Modern Diplomacy)

Regulatory clarity often reduces market friction, yet pending legislation and compliance standards continue to influence trading behavior, custody solutions, and institutional risk appetite.

📊 Investment Outlook & Strategy

For investors, the current environment suggests:

Long-term holders continue to view Bitcoin as a scarce, institutionalized digital asset with structural scarcity (21M cap) and macro hedging potential.
Short-term traders must account for ETF flow data, macro risk sentiment (especially correlation with equities), and key support/resistance levels in technical analysis.
Risk management remains paramount, given continued correlation with equity markets and sensitivity to broader risk sentiment. (arXiv)

While Bitcoin no longer exhibits the extreme volatility of its earliest years, it has not yet fully decoupled from traditional markets—suggesting a hybrid asset behavior that blends speculative dynamics with institutional financial characteristics.

📌 Final Thought

Bitcoin in late 2025 is maturing but still dynamic. Institutional frameworks, macroeconomic variables, and regulatory progress shape its trajectory, while investor caution and price consolidation define the near-term landscape. This unique interplay underscores the importance of disciplined analysis alongside adaptive investment strategies in the crypto space.

Analyzing Bitcoin in R in the year 2025

2025-10-19T12:00:00+03:00

Retrieving Bitcoin transaction data

Bitcoin data is no longer easily available at the transaction level easily as far as I know.

There are some options. In R you can use the short time series that you can get with getSymbols('BTC').

Or you can download a year at a time to get data starting from 2010 via coinmarketcap.com.

The data ends up looking something like this using chartSeries(xtsdat['2024-01-01/'], name='Bitcoin'):

Some drawdowns listed from this dataset:

         From     Trough         To  Depth Length To Trough Recovery
1  2011-06-10 2011-11-18 2013-02-22 -0.932    622       162      460
2  2013-12-05 2015-01-14 2017-01-04 -0.845   1123       404      719
3  2017-12-17 2018-12-15 2020-11-30 -0.834   1077       363      714
4  2021-11-09 2022-11-21 2024-03-04 -0.766    844       377      467
5  2013-04-11 2013-04-16 2013-11-04 -0.701    208         6      202
6  2021-04-14 2021-07-20 2021-10-19 -0.531    189        98       91
7  2010-11-09 2010-12-05 2011-01-22 -0.520     74        27       47
8  2010-08-17 2010-09-17 2010-10-09 -0.378     54        32       22
9  2017-09-02 2017-09-14 2017-10-12 -0.355     41        13       28
10 2017-06-12 2017-07-16 2017-08-05 -0.348     55        35       20
11 2011-02-15 2011-04-04 2011-04-17 -0.338     62        49       13
12 2017-01-05 2017-01-11 2017-02-23 -0.326     50         7       43
13 2011-05-17 2011-05-20 2011-05-27 -0.315     11         4        7
14 2025-01-22 2025-04-08 2025-05-18 -0.281    117        77       40
15 2017-03-04 2017-03-24 2017-04-26 -0.265     54        21       33
16 2024-03-14 2024-09-06 2024-11-06 -0.262    238       177       61
17 2021-01-09 2021-01-27 2021-02-08 -0.254     31        19       12
18 2010-07-19 2010-07-22 2010-08-16 -0.253     29         4       25
19 2021-02-22 2021-02-28 2021-03-11 -0.216     18         7       11
20 2017-11-09 2017-11-12 2017-11-16 -0.202      8         4        4

Analyzing Bitcoin in R in the year 2022 - Part 1

2022-11-14T12:00:00+02:00

Retrieving Bitcoin transaction data

Bitcoin data is easily available e.g. via (http://api.bitcoincharts.com/v1/csv/). Specifically I used the file bitstampUSD.csv.gz which contains transactions on the Bitstamp exchange. At the time of writing the file size is a 474 MB download and 2.60 GB when unzipped.

Reading the data into R is simple, although it does take awhile due to the file size.

setwd('C:/MyFileLocation/')
rawdata <- read.csv('.bitstampUSD.csv', sep = ',', dec = '.', head=FALSE)

Setup the data in a time series format for ease of use and structure it in a typical OHLCV (Open, High, Low, Close, Volume) format:

colnames(rawdata) <- c('timesec', 'USD', 'Volume')
sec_to_date = function(sec) {
  as.POSIXct(sec, origin = '1970-01-01', tz = 'UTC')
}
Sys.setenv(TZ = "UTC")
rawdata$datetime <- sec_to_date(rawdata[,1])
rawdata <- xts(x = rawdata[,2:3], order.by = rawdata$datetime, tz = "UTC")
#Open
startpoints <- function (x, on = "months", k = 1) {
  head(endpoints(x, on, k) + 1, -1)
}
rawdailyopen <- rawdata$USD[startpoints(rawdata, "days")]
tclass(rawdailyopen) <- "Date"
#High
rawdailymax <- period.apply(rawdata$USD, INDEX = endpoints(rawdata, "days"), FUN = max)
tclass(rawdailymax) <- "Date"
#Low
rawdailymin <- period.apply(rawdata$USD, INDEX = endpoints(rawdata, "days"), FUN = min)
tclass(rawdailymin) <- "Date"
#Close
rawdailyclose <- rawdata$USD[endpoints(rawdata, "days")]
tclass(rawdailyclose) <- "Date"
#Volume
rawdailyvol <- period.apply(rawdata$Volume, INDEX = endpoints(rawdata, "days"), FUN = sum)
tclass(rawdailyvol) <- "Date"
.index(rawdailyopen) <- .index(rawdailyclose)
rawdailyohlc <- merge.xts(rawdailyopen, rawdailymax, rawdailymin, rawdailyclose, rawdailyvol)
colnames(rawdailyohlc) <- c("Open", "High", "Low", "Close", "Volume")

Last time we did the analysis with data up to approximately to the end of the year 2017. We now have five more years of data in 2022.

dailyret <- dailyReturn(Cl(rawdailyohlc))
store <- table.Stats(dailyret["/2017-12-29"])
store <- cbind(store, table.Stats(dailyret["2018-01-01/"]))
store <- cbind(store, table.Stats(dailyret))
colnames(store) <- c('2011-2017', '2017-2022', '2011-2022')
store

Summary statistics for daily returns of Bitcoin during two different subperiods and the full sample:

                2011-2017 2017-2022 2011-2022
Observations    2279.0000 1779.0000 4060.0000
NAs                0.0000    0.0000    0.0000
Minimum           -0.4852   -0.3898   -0.4852
Quartile 1        -0.0110   -0.0165   -0.0138
Median             0.0024    0.0010    0.0018
Arithmetic Mean    0.0048    0.0009    0.0031
Geometric Mean     0.0034    0.0001    0.0020
Quartile 3         0.0201    0.0186    0.0194
Maximum            0.5613    0.1945    0.5613
SE Mean            0.0011    0.0009    0.0007
LCL Mean (0.95)    0.0027   -0.0009    0.0016
UCL Mean (0.95)    0.0069    0.0028    0.0045
Variance           0.0027    0.0016    0.0022
Stdev              0.0518    0.0400    0.0471
Skewness           0.3622   -0.4100    0.1890
Kurtosis          18.5765    7.3008   17.1983

It is also relevant because Bitcoin is going through it's third largest drawdown in history.

table.Drawdowns(dailyret, top = 5, digits = 3, geometric = TRUE)

The third drawdown is the one that is ongoing since 2021-11-09 and is still not over. It is approaching the longest drawdowns that started in 2013 and 2017.

        From     Trough         To  Depth Length To Trough Recovery
1 2013-12-05 2015-01-14 2017-02-23 -0.849   1174       403      771
2 2017-12-17 2018-12-15 2020-11-30 -0.834   1080       364      716
3 2021-11-09 2022-11-09       <NA> -0.765    372       366       NA
4 2013-04-10 2013-07-06 2013-11-05 -0.710    210        88      122
5 2011-09-26 2011-10-20 2012-01-05 -0.630     84        18       66

We could consider the following questions:

Is the distribution of daily / monthly Bitcoin returns different?
Is bitcoin more or less correlated with the SP500 than it used to be?
Do the new return dynamics indicate a different portfolio allocation?

Introduction to conducting analysis on Bitcoin with R post series

2017-12-04T12:00:00+02:00

Introduction

The popularity of Bitcoin is undeniable. So we can use it as a basis to take a look at some time series data manipulation techniques, some useful R packages and a few traditional finance concepts and their application to time series analysis.

This set of posts will cover at least the following:

Downloading a transaction level data set of Bitcoin
Reading the Bitcoin data in to R
Manipulating the data to make it usable with xts based functions
Calculating summary statistics and plotting the data using packages
Analyzing the time series properties of the Bitcoin timeseries
Doing simple portfolio calculations
Evaluating alternative return structures for further insight

The end result of the posts will be that we can answer the question: What portion of your portfolio should be in Bitcoin?

Continue to the first post: Retrieving Bitcoin transaction data

By the way, if you don't know what to think about Bitcoin, you can read this blog post for a pretty good overview.

You can also jump to each section directly from here:

Analyzing Bitcoin in R - Part 1 - Retrieving data

2017-12-04T12:00:00+02:00

Retrieving Bitcoin transaction data

Bitcoin data is available through proprietary systems like Bloomberg (VCCY) and Reuters Eikon (BTC=BTSP). However, finding Bitcoin data online is very easy.

The source I will use for this post is (http://api.bitcoincharts.com/v1/csv/).

To get the data, you can download it through a browser and unzip with with 7-Zip.

Or you can download gzip for windows and wget for windows to do it on the command line.

You could even script the download directly in to R if you wished to do so.

wget http://api.bitcoincharts.com/v1/csv/bitstampUSD.csv.gz
gzip -d bitstampUSD.csv.gz

After that you have file called .bitstampUSD.csv which is over 600MB in size. It contains transactions on the Bitstamp exchange since 2011.

Next, open up R and read the file into memory for further processing: Part 2 - Reading the bitcoin data in to R

You can also jump to each section directly from here:

Analyzing Bitcoin in R - Part 10 - Where will bitcoin go based on statistics?

2017-12-04T12:00:00+02:00

Based on statistics, what is the range in which Bitcoin may move?

Let's use returns since 2014 as a reasonable estimate of what will happen in the future.

avg <- mean(ClCl(rawdailyohlc['2014-01-01/']), na.rm = TRUE)
sdev <- sd(ClCl(rawdailyohlc['2014-01-01/']), na.rm = TRUE)

The average daily return is 0.0026 and the standard deviation is 0.0375.

For plotting purposes get the first and last value.

lval <- as.numeric(last(Cl(rawdailyohlc['2017-08-01/'])))
fval <- as.numeric(first(Cl(rawdailyohlc['2017-08-01/'])))

Intitialize some variables and then store the already realized Bitcoin values.

N <- 50
tval <- 1:N
oldval <- c(as.numeric(Cl(rawdailyohlc['2017-08-01/'])), rep(NA, N))

Define the confidence level that we are interested in and calculate the upper and lower ranges that the value may deviate to with that confidence level

qval <- qnorm(0.99, mean = 0, sd = 1, log.p = FALSE)
newvalu <- lval*exp((avg - 0.5 * sdev^2)*tval + qval * sdev * sqrt(tval))
newvalu <- c(rep(NA, length(oldval)-N-1), lval, newvalu)
newvald <- lval*exp((avg - 0.5 * sdev^2)*tval - qval * sdev * sqrt(tval))
newvald <- c(rep(NA, length(oldval)-N-1), lval, newvald)

Plot the results.

png(filename = "btc_plot_daily_futurerange.png")
plot(oldval, type = 'l', ylim = c(0, lval*2))
lines(newvalu, col = 'green')
lines(newvald, col = 'red')
dev.off()

This is a different way of thinking about it. We plot the range around the already existing time series.

avg <- mean(ClCl(rawdailyohlc['2014-01-01/2017-01-01']), na.rm = TRUE)
sdev <- sd(ClCl(rawdailyohlc['2014-01-01/2017-01-01']), na.rm = TRUE)
lval <- as.numeric(Cl(rawdailyohlc['2017-01-01']))
N <- 365
oldval <- c(as.numeric(Cl(rawdailyohlc['2017-01-01/'])), rep(NA, N-nrow(rawdailyohlc['2017-01-01/'])))
tval <- 1:N
qval <- qnorm(0.99, mean = 0, sd = 1, log.p = FALSE)
newvalu <- lval*exp((avg - 0.5 * sdev^2)*tval + qval * sdev * sqrt(tval))
newvalu <- c(lval, newvalu)
newvald <- lval*exp((avg - 0.5 * sdev^2)*tval - qval * sdev * sqrt(tval))
newvald <- c(lval, newvald)
png(filename = "btc_plot_daily_concurrentrange.png")
plot(oldval, type = 'l', ylim = c(0, max(newvalu, oldval, na.rm = TRUE)))
lines(newvalu, col = 'green')
lines(newvald, col = 'red')
dev.off()

In the plot we can see that the price has exceeded all expectations, how confident are we about our assumptions?

A slightly different version of the above plot, comparing to an even longer history. Demonstrates merging with xts series and a different plotting interface from quantmod.

refdatea <- as.Date('2016-01-01')
refdate <- as.Date('2017-01-01')
oldval <- Cl(rawdailyohlc['2016-01-01/'])
n <- nrow(oldval)
# redefine dates otherwise they will not merge
oldval <- xts(oldval, order.by = as.Date(as.POSIXct(refdatea + 1:n, tz = 'UTC')))
newvalu <- lval*exp((avg - 0.5 * sdev^2)*tval + qval * sdev * sqrt(tval))
newvalu <- xts(newvalu, order.by = as.Date(as.POSIXct(refdate + 1:N, tz = 'UTC')))
newvald <- lval*exp((avg - 0.5 * sdev^2)*tval - qval * sdev * sqrt(tval))
newvald <- xts(newvald, order.by = as.Date(as.POSIXct(refdate + 1:N, tz = 'UTC')))
forplot <- merge(oldval, newvalu, newvald)
png(filename = "btc_plot_daily_concurrentrangelong.png")
chart_Series(forplot$Close, name = 'Bitcoin USD')
add_TA(forplot$newvalu, on = 1, col = 'Green')
add_TA(forplot$newvald, on = 1, col = 'Red')
dev.off()

But how confident are we about the assumptions of the returns and standard deviations? Especially after looking at the different outcomes throughout history.

The following commands show some tables about the changing returns and standard deviations.

table.CalendarReturns(ClCl(rawdailyohlc))

      Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct  Nov  Dec ClCl.rawdailyohlc
2011   NA   NA   NA   NA   NA   NA   NA   NA  0.0 -5.7 17.3  8.8              20.3
2012 -5.0  2.3  0.4  1.8  0.6  1.4  3.1 -4.6 -0.5  1.9  0.5  0.3               1.8
2013  5.2  7.1  4.4 -2.4  0.9  0.6  1.8  2.5 -0.6  3.0 -1.1 -0.6              22.4
2014  0.4 -5.3 -1.1  0.2  1.1  6.9  3.2 -4.4  4.5 -2.4  0.4  3.2               6.2
2015 -5.3  0.3 -1.5  4.8 -1.4  2.5 -1.3  0.4 -0.3 -5.6  1.4  1.2              -5.2
2016 -1.9  0.8  0.2 -1.8  0.8  5.3 -4.7 -0.9  0.4  0.0  1.5  0.7               0.2
2017  4.8 -0.3  3.5  1.3  4.3 -3.0  4.1  3.5  3.9  5.1  1.3  2.8              35.9

cbind(apply.yearly(ClCl(rawdailyohlc), mean), apply.yearly(ClCl(rawdailyohlc), sd))

           ClCl.rawdailyohlc ClCl.rawdailyohlc.1
2011-12-31                NA                  NA
2012-12-31       0.003895785          0.04419571
2013-12-31       0.014032751          0.07576909
2014-12-31      -0.001477679          0.03945913
2015-12-31       0.001501913          0.03679823
2016-12-31       0.002526586          0.02510439
2017-12-03       0.008324553          0.04574109

You could even try the different monthly return and standard deviation values from:

cbind(apply.monthly(ClCl(rawdailyohlc), mean), apply.monthly(ClCl(rawdailyohlc), sd))

Now that we understand the range of possible outcomes, and what the caveats regarding the inputs mighe be we can see what portion of our portfolio Bitcoin should be: Evaluating a portfolio

You can also jump to each section directly from here:

Analyzing Bitcoin in R - Part 11 - Evaluating a portfolio, index and Bitcoin

2017-12-04T12:00:00+02:00

How about a portfolio approach

The GSPC index from Yahoo is not really the right benchmark because it doesn't contain dividends, however, it will be okay for this example. If you want to get a better market benchmark, you can get one from Kenneth French's website, for example.

Also be aware that Yahoo has discontinued some downloading and the quantmod getSymbols command might only work from the latest version of quantmod.

firstdate <- index(first(rawdailyohlc))
lastdate <- index(last(rawdailyohlc))
getSymbols("^GSPC", src = "yahoo", from = as.Date(firstdate), to = as.Date(lastdate))
png(filename = "gspc_plot.png")
chartSeries(GSPC)
dev.off()

Let's calculate monthly returns of Bitcoin and GSPC for our portfolio allocation. I had to cut some observations from the bottom to make them the same length.

btcnret <- monthlyReturn(rawdailyohlc)
btcnret <- head(btcnret, -1)
spidret <- monthlyReturn(Ad(GSPC))
spidret <- head(spidret, -1)
.index(btcnret) <- .index(spidret)

Join the time series together and make them a timeSeries object.

returns <- cbind(btcnret, spidret)
colnames(returns) <- c('BitcoinUSD', 'SP500')
returns <- as.timeSeries(returns)

Let's plot the efficient frontier.

library(fPortfolio)
frontier <- portfolioFrontier(returns)
png(filename = "btc_plot_frontier.png")
plot(frontier, which = "all")
dev.off()

We can also look at what weights produce what results:

png(filename = "btc_plot_weights.png")
weightsPlot(frontier)
dev.off()

A minimum variance portfolio suggest putting everything in the index. Bitcoin is too volatile.

minvariancePortfolio(returns, constraints = "LongOnly")

Portfolio Weights:
BitcoinUSD      SP500 
         0          1

A return maximized portfolio targeting 5% monthly returns suggest putting 6% weight in Bitcoin and the rest in the index.

Spec <- portfolioSpec()
setTargetRisk(Spec) <- 0.05
setSolver(Spec)= "solveRshortExact"
maxreturnPortfolio(returns, spec = Spec, constraints = "Short")
Portfolio Weights:
BitcoinUSD      SP500 
    0.0624     0.9376

A maximum return-risk ratio porfolio would put 3% in Bitcoin.

maxratioPortfolio(returns, constraints = "LongOnly")
Portfolio Weights:
BitcoinUSD      SP500 
    0.0338     0.9662

As does the tangency portfolio.

tangencyPortfolio(returns, constraints = "LongOnly")
Portfolio Weights:
BitcoinUSD      SP500 
    0.0338     0.9662

So what if you are an investor in tech stocks, how much Bitcoin should you hold: Evaluating a stock portfolio

You can also jump to each section directly from here:

Analyzing Bitcoin in R - Part 12 - Evaluating a portfolio, stocks and Bitcoin

2017-12-04T12:00:00+02:00

Let's do the same thing as in the previous section, but with multiple stocks instead of an index. These are some of the most held stocks by individuals according to online source.

getSymbols("TSLA", src = "yahoo", from = as.Date(firstdate), to = as.Date(lastdate))
getSymbols("NVDA", src = "yahoo", from = as.Date(firstdate), to = as.Date(lastdate))
getSymbols("AMD", src = "yahoo", from = as.Date(firstdate), to = as.Date(lastdate))

tslaret <- monthlyReturn(Ad(TSLA))
nvdaret <- monthlyReturn(Ad(NVDA))
amdret <- monthlyReturn(Ad(AMD))

returns <- cbind(btcnret, tslaret, nvdaret, amdret)
colnames(returns) <- c('BitcoinUSD', 'TSLA', 'NVDA', 'AMD')
returns <- head(returns, -1)
returns <- as.timeSeries(returns)
frontier <- portfolioFrontier(returns)
png(filename = "btc_plot_stocks_frontier.png")
plot(frontier, which = "all")
dev.off()
png(filename = "btc_plot_stocks_weights.png")
weightsPlot(frontier)
dev.off()
out <- minvariancePortfolio(returns, constraints = "LongOnly")
store <- out@portfolio@portfolio$weights
out <- maxratioPortfolio(returns, constraints = "LongOnly")
store <- rbind(store, out@portfolio@portfolio$weights)
rownames(store) <- c('minvar', 'maxratio')

The resulting allocations would range from 3 to 9 percent into Bitcoin and the rest into TSLA, NVDA, and AMD.

         BitcoinUSD      TSLA      NVDA        AMD
minvar   0.02944844 0.2398669 0.6818332 0.04885147
maxratio 0.09466859 0.3087685 0.5965629 0.00000000

Does the structure of the covariance estimator effect the results of our portfolio optimization

estimatorlist <- c(
'covEstimator',
'kendallEstimator',
'spearmanEstimator',
'mcdEstimator',
'mveEstimator',
'covMcdEstimator',
'covOGKEstimator',
'shrinkEstimator')

png(filename = "btc_plot_stocks_covarestim.png")
par(mfrow = c(4, 2))
par(mar = c(3, 3, 4, 3))
for (i in 1:length(estimatorlist)) {
    Spec <- portfolioSpec()
    setEstimator(Spec) <- estimatorlist[i]
    frontier <- portfolioFrontier(returns, spec = Spec)
    weightsPlot(frontier)
}
dev.off()

What if you believe that our return estimates are bogus, especially related to their size and how they are correlated. Or you think more observation might help.

Well, you could decide to use Copulas or extreme value theory to help you: Copulas and extreme values with Bitcoin

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