Visualizing their predictability, profitability, volatility, and correlation

Cryptocurrencies, spearheaded by Bitcoin (BTC), have attracted worldwide attention in recent years. With ever growing interest in this new asset class, it is interesting to compare the basic statistical properties of a cryptocurrency and a traditional financial asset, which will shed light on their predictability, profitability, volatility, and correlation.

In this post, using a range of data visualization methods, I compare a range of statistical properties of Bitcoin (BTC) and S&P500 (SP) index. These assets respectively represent

the most popular of the cryptocurrencies, and

a broad stock portfolio of the U.S. market.

Daily price and return data from the 2nd of January 2014 to the 31st of January 2023 are used (2286 observations). The focus is on visualization, with a minimal use of the mathematical details.

The main points of comparison include

· Profitability of investment,

· Predictability of the return (or price change),

· Volatility and its evolution,

· Correlation of the returns, and

· News impact on volatility.

The results are produced using R and its packages zoo, rugarch, rmgarch, and PerformanceAnalytics. The data and R code are available from here.

#### Time plots

A time plot is used to visualize the variation of a time series over time.

The above figure presents time plots of the prices. The BTC shows wild fluctuations, often with sharp spikes and troughs of enormous scale. The SP index looks quite steady in comparison, but it also has an upward trend with a substantial degree of fluctuations, as clearer from a separate time plot on the right panel.

A $1 invested in BTC in early 2014 would be around $29 by early 2023, while $1 invested in SP in early 2014 would be just over $2, not including the dividends. Hence, the profitability of BTC far exceeds that of SP, if it is a long-term (buy-and-hold) investment. Profitability for risk-adjusted return for medium-term investment will be compared later.

The above are time plots for daily logarithmic returns in percentage. Both returns show frequent spikes up and down, which indicate a high degree of daily volatility. The volatility of BTC returns is extensive, simply overwhelming that of the SP. Most of BTC returns are within ±20% range, while the SP returns are mostly within the range of ±5%. The plots show that the investors of BTC have been facing a disproportionate amount of daily risk, in comparison with the SP investors.

#### Autocorrelation

The autocorrelation of lag order k measures the correlation between time series observations that are k periods apart. The autocorrelation function plots the values of autocorrelation against the values of k, visualizing the degree of linear dependency of a time series on its own past. Strong autocorrelations indicate a high level of in-sample predictability.

We first inspect the autocorrelation function of the returns. All the autocorrelations of BTC return are close to 0, meaning that the degree of linear dependency of the return on its own past is negligible at all lag orders. In contrast, the autocorrelations of the SP returns are much higher, ranging between 0.1 and 0.2 in absolute values. This means a higher level of in-sample predictability for the SP returns.

The autocorrelation functions of the absolute returns are plotted above, which are used to estimate the degree of linear dependency of the volatility in returns. The SP absolute returns show a longer and stronger linear dependency, indicative of presence of a higher degree of volatility clustering. Similar features are evident from the BTC absolute returns, but the degree of volatility clustering is much smaller.

Volatility clustering is a feature of financial return where the current volatility depends on the past values, with a systematic pattern where a period of high volatility is followed by a period of low one, and so on. This feature is also evident in the time plots of the returns presented above, and will further be examined later by fitting a time series model.

From the above autocorrelation functions, we can see that the SP exhibits a higher level of in-sample predictability, in both daily returns and volatility. A higher in-sample predictability means that a profitable technical trading rule may be employed. However, in-sample predictability does not necessarily lead to out-of-sample profitability. In contrast, the price changes of BTC are nearly purely unpredictable, while its volatility shows a moderate degree of predictability.

#### Histogram

Histogram visualizes the general distributional features of the data points, by plotting their frequencies over the intervals covering the data range. A rug on the X-axis is often added to it for a clearer visualization.

The histogram shows bell-shaped distributions centered around 0 for both assets, but there are a large number of extreme values, especially on the negative side. Again, the BTC returns show a much wider variability with more frequent extreme values, indicative of a higher degree of risk associated, on a daily basis.

#### Boxplots

A boxplot presents a box with the 25th percentile, median (50th percentile) and 75th percentile of the data points, respectively as its lower bound, mid-point, and upper bound. The length of the box is the inter-quartile range that contains the middle 50% of the data points. The plot also shows the minimum and maximum of the data points (called whiskers), with the outliers indicated with the dots. The outliers are the values outside three inter-quartile range away from the median.

The boxplots show that the daily returns are concentrated around the median of 0, with the variability measured by the inter-quartile range much higher for the BTC returns. That is, the length of the interval that contains the middle 50% of the returns are nearly three times bigger for the BTC than the SP. Again, the BTC returns are showing a larger number of outliers with bigger sizes. This means that the investors gain virtually zero return on average, but the BTC holders have been facing substantially larger variability and risk than the SP investors.

#### Q-Q plots

The Q-Q (Quantile to Quantile) plot compares the sample quantiles with those of the normal distribution. A sample generated from a normal distribution should show a straight line (like the blue one), indicating a one-to-one correspondence between the sample quantiles and normal quantiles.

For both BTC and SP returns, the departure from the normality is dramatic, with the former showing a substantially larger departure from normality than the latter, especially at the tail areas of the distribution. These plots highlight the fat-tailed (excess kurtosis) property, which is extremely strong on the BTC returns, again indicative of extensively high degree of risk associated.

#### Volatility

The time-varying (conditional) volatility can be estimated using the GARCH(1,1) (generalized autoregressive conditional heteroskedasticity) model of the following form, proposed by Bollerslev (1986):

where Yt is the asset return at time t. The variance of the return (σt²) depends on its own past value with the coefficient β and the past observed volatility with the coefficient of α1.

The above figure plots the daily conditional standard deviation from the estimated GARCH(1,1) models. As might be expected, the standard deviation is much higher for the BTC than the SP over time. The median of the standard deviations for the BTC is 4.15, while that of SP is 0.80, meaning that the risk of BTC is more than 5 times higher. This is again clear evidence that BTC has been showing a much higher volatility than SP.

The estimated coefficients of the GARCH(1,1) model are presented below:

The SP returns show a slightly higher value of (α1 + β1), which is a sign of a higher degree of volatility clustering. The unconditional variance [α0 /(1-α1-β1)] is much higher for BTC, which again show a substantially higher risk involved with it. The unconditional standard deviation is 5.21, which is more than 4 times that of the SP return which is 1.28.

#### News Impact Curve

The leverage effect is the tendency that a negative return (bad news) has a higher impact on the variance of the return, more than a positive return (good news) of the same magnitude. This is reflection that, investors over-react to negative news more than positive news in financial markets.

The GJR-GARCH(1,1) model is an extension of the GARCH(1,1) model, which is designed to capture the leverage effect. It has the same structure as the GARCH(1,1) as above, except that its variance equation takes the following form:

where the coefficient ϕ > 0 captures the degree of the leverage effect. If ϕ = 0, the model reduces to a GARCH(1,1) that shows a symmetric volatility. The news impact curve plots the variance of the return (σt²) against the value of the shock (ut).

**News Impact Curve for SP**

**News Impact Curve for BTC**

The news impact curves above are obtained from the GJR-GARCH(1,1) model. An asymmetric effect of news on volatility is clearly shown in the SP news impact curve. In contrast, BTC shows nearly symmetric news impact, which means that the good news and bad news have nearly identical impact on its volatility. The estimated value of ϕ is 0.250 for SP return, which is statistically significant; while that of BTC is 0.059 and statistically insignificant. These results are also consistent with the shape of the new impact curves plotted above.

The evidence indicates that the BTC returns show symmetric volatility in response to good and bad news, which is different from what is usually expected from a financial return.

#### Correlation

Are BTC returns correlated with SP returns over time? If so, how? The question has strong implications to asset management and portfolio selection. Here we measure the time-varying correlation between the two, using

historical correlation calculated from the returns with one-year moving window (220 daily observations), and

dynamic conditional correlation of Engel (2002) in the similar spirit of the GARCH(1,1) model.

The figure above plots the daily conditional and historical correlations between BTC and SP returns. The two alternatives show a similar pattern over time. The correlation is low and close to 0 up until early 2020, but it gets positive and stronger from then on. By 2023, the correlation gets as larger as 0.5.

That is, a positive linear association between the two returns has been identified, which is getting stronger from the end of 2020. This is an interesting feature, which means that a positive (negative) price change in one market is more and more likely to lead to the positive (negative) price change in the other.

#### Profitability

In finance, return or profit from an investment is a compensation for taking risk. That is, a higher return is expected from an investment with a higher risk. To compare the profitability, we consider both unstandardized and standardized returns (return per unit of risk) with 3-year holding period. The latter is the risk-adjusted return, which is the daily return divided by daily conditional standard deviation from the GARCH(1,1) models.

The above figure plots the log return summed over a moving sub-sample window of 3 years (660 observations). The unstandardized return has been much higher for BTC, which represents a big compensation for taking huge risk much higher than that of SP. In terms of the risk-adjusted or standardized returns, BTC also has been much more profitable until 2020, but its risk-adjusted returns become quite close to those of SP afterwards. Similar results are obtained under different holding periods or lengths of sub-sample windows. The evidence shows that the two assets have shown a similar degree of risk-adjusted profitability in recent times (from 2021), for medium-term buy-and-hold investors.

From the results discussed above, the main findings are summarized as follows:

Both BTC and SP returns overall have been showing the properties consistent with the stylized facts of financial return: little predictability, a clear departure from normality, and a time-varying risk.

Investment in BTC has been much riskier than investment in SP, about 5 times more when the risk is measure by the standard deviation of returns.

The BTC volatility shows nearly symmetric news impact, which is different from other financial assets such as the SP.

The correlation of BTC return with SP return gets positively stronger from 2021, while it was negligible before.

The risk-adjusted return (from a medium-term buy-and-hold investment) of BTC has been much higher in the past, but it has become close to that of SP from around 2021.

While it is a considerably risky asset, the BTC in recent years has become much more in line with the traditional stock investment such as SP, in terms of its risk-adjusted profitability and correlation.

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