Cryptocurrency Sortino Ratios

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Sortino ratios measure the excess return of an asset per unit of downside risk. The excess return is defined as the return of an asset minus the return of a benchmark asset. The risk is defined as the downside deviation of the returns since we are only concerned with risks related to losses.  Note that this is an attempt to improve on the standard Sharpe ratios. The Sortino ratio $$S_o$$ is defined as $S_o = \frac{\mu}{\sigma_{ds}},$where $\mu = \int_{-\infty}^{+\infty} (r_c-r_b)p(r_c)\text{d}r_c,$ and $\sigma_{ds}=\left(\int_{-\infty}^{+\infty} \text{Min}(r_c-r_b-\mu,0)^2 p(r_c) \text{d}r_c \right)^{1/2},$and $$r_c$$ and $$r_b$$ denote the returns of the cryptocurrency and benchmark asset, respectively, $$p(r_c)$$ is the probability density function of the cryptocurrency returns, and $$\text{Min}(x,0)$$ denotes the minimum of $$x$$ and $$0$$. Shown above are annualized Sortino ratios based on the previous 90, 180, and 365 days of log returns calculated from volume weighted average daily prices.  See this paper for a more detailed description of the Sortino ratio. The benchmark asset was the risk-free LIBOR.

BTC = Bitcoin, ETH = Ethereum, BCH = Bitcoin Cash, XRP = Ripple, LTC = Litecoin, DASH = Dash, XMR = Monero, XEM = NEM, ETC = Ethereum Classic, XLM = Stellar Lumens, ZEC = Zcash, NXT = Nxt, REP = Augur, LSK = Lisk, FCT = Factom, GLD = SPDR Gold Shares.

Raw price data courtesy of Poloniex.