Bitcoin for every Moment: part 2
In the prior article of this two part series we examined portfolio optimization when Bitcoin is included. There were three general conclusions: (1) Bitcoin price series should be detrended by the fundamental power law long-term trend in order to compute higher moments, (2) the positive skew (third moment) is favorable to Bitcoin allocations, (3) Bitcoin should receive a very large allocation, up to 100% relative to the S&P 500 when skew and kurtosis are incorporated in the analysis.
Typically, skew and kurtosis (fourth moment) calculations are based on a series of log price returns, as suitable for long-term exponential growth (compound annual growth) assets. Stocks also generally have negative skew, with left tail risk. Bitcoin is different, it is not an exponential growth equation asset, but adheres to a power law equation of its age, with price proportional to T^5.9, approximately, where T is Bitcoin’s age. Bitcoin’s occasional large bubbles provide significant positive skew to Bitcoin.
In this article we include GLD, the gold ETF and this changes the results considerably from a two asset portfolio. Their can be substantial variation depending on how much weight one gives to the skew and kurtosis.
The MVSK method has skew, kurtosis included
For an 8-year daily price series, power law detrending yielded +1 normalized skew for Bitcoin and -2 for the SPY ETF (S&P 500). Bitcoin’s current expected return based on the best fit power law (median quantile regression) is 42%, about 4 times higher than the S&P, and with 3 x the volatility (second moment). Given those values and the skew advantage, an MVSK (mean-variance-skew-kurtosis) optimization calculation yielded 100% allocation to Bitcoin for a two asset BTC and SPY portfolio. MVSK enhances typical Sharpe and Markowitz mean-variance analyses that only use the mean and variance parameters (first two moments).
The MVSK method maximizes the utility function: mean / variance + λ skew - β kurtosis where λ is an adjustable weight parameter for skew and β is the weight parameter for kurtosis. Note that positive skew (right tail bias) increases an allocation, and positive kurtosis (which is the width of the residual distribution) decreases it.
Three Asset Portfolio: BTC, SPY, GLD
A reasonable neutral choice is 1.0 for each parameter. In this case MVSK optimization for the four moments of mean expected return, variance (square of standard deviation), skew, and kurtosis for the three assets yields the current portfolio allocation shown in Table 1, which is 58% Bitcoin, and with 27% in the gold ETF as the second largest weighting.
Gold has had a similar return to the SPY but with lower volatility in recent years, and acts as ballast in the overall portfolio, it seems.
The New 60/40?
I ran 8 different cases with various values for Lambda (λ) and Beta (β). The results are summarized in Figure 1, which shows the Bitcoin allocations as a function of the skew and kurtosis weights; the allocation is color-coded and each box has the value as a label. For zero kurtosis weight the Bitcoin allocation is about 50%. For kurtosis weight 2, the allocation is 75 to 80%. The allocations vary mildly with skew weight, increasing moderately as one increases λ.
There is no single answer from the MVSK analysis for Bitcoin allocation in this three asset portfolio; it is a matter of how much one wants to weight the skew and kurtosis terms.
But the general takeaway is that the New 60 / 40 has Bitcoin in the 60 slot relative to a broad stock index and gold in a three asset portfolio.