Summary
Cryptocurrencies are extremely volatile. This creates an interesting opportunity to extract extra returns from rebalancing a portfolio and extract a “diversification dividend”.
Part 1
In Part 1: Geometric and Arithmetic Returns, I explored the difference b/w geometric and arithmetic returns and linked risk and returns for an investment. Normally these effects are tiny (for standard, less volatile investments) and hence are exploited only by the very large pension funds and sovereign wealth funds. But given the high volatility of cryptocurrencies I believe they may be interesting for all crypto-investors.
In this post I will quantify the effect.
In Part 3: how to make 8% more per year I will calculate how much it can mean for the average portfolio
Portfolio variance
The variance of a portfolio of assets is lower than we weighted average variance if the assets have a correlation lower than 1.
The variance of a portfolio, in matrix form is
Where
is the covariance matrix
- w is the weight vector
- T is the transpose operator
For an equal weight portfolio this general formula reduces to:
Where
- n is the number of assets
is the average variance
is average covariance
If we now further assume that all variances are the same and all correlations are the same (this is just to see a nicer formula but the principles will still apply if we relax these assumptions), then
where 
This make sense: if n=1 or
otherwise in all other cases portfolio variance is reduced.
Benefit to Geometric Returns
Given the Geometric Returns are reduced by half the variance, if we can reduce the variance we can improve our returns.
From the equation above we can see that the reduction in portfolio variance (and hence half of it would be a direct positive impact on returns) is:
This is interesting because we can now separate 3 effects:
the higher the number of assets n the closer (asymptotically) to 1 this first component becomes
the higher the average variance the higher the effect is
the lower the average correlation the higher the effect is