“No Pain, No Gain!” The Ulcer Index for Cryptos…


An Alternative Approach for Measuring Investment  Risk-Adjusted Return


Martin Ratio for Cryptos
A question that is often asked is: “Given the very high volatility in cryptocurrencies ( 30 day standard deviation ranges have historically been between 60~600%!) which one is a better investment on a relative risk-return basis?”

But first some background:


Whats wrong with using Standard Deviation?
Standard Deviation is a statistical measure of variability of an investments return but as a measure of risk is suffers from several drawbacks including (1) the calculation of SD involves both upside and downside variance whereas real investors would actually enjoy upside risk but would want to avoid/minimize any downside risk in their investments.  And (2) the calculation of SD does not recognize the sequence of gains or losses.

The chart below shows 3 total different price lines with the same SD but clearly a normal investor would not see them as having the same “risk”. 


The Ulcer Index – as a measure of  investor “Pain”

Ulcer Index (UI) is a method was originally developed by Peter Martin back in 1987 for measuring investment risk that addresses the real concerns of investors, unlike the widely used Standard Deviation of return (SD). UI is a measure of the depth and duration of drawdowns in prices from earlier peaks.

Using UI instead of SD as a volatility measure can lead to very different conclusions about investment risk and risk-adjusted return, especially when evaluating strategies that seek to avoid major declines in portfolio value (market timing, dynamic asset allocation, hedge funds strategies etc).

The Ulcer index formula is defined as:

ulcer index formula.png
and is designed to measure (and thus penalize) downside variance – in particular the depth and duration of drawdowns in price from its highs. A higher UI implies a greater drawdown and a longer recovery time to previous peak equity highs.

Risk-adjusted Return: Sharpe Ratio vs Ulcer/Martin Ratio

The UI can be used to replace SD in calculating risk adjusted return, as in the case of the Sharpe Ratio which uses SD in its denominator as its “risk/volatility” measurement.

Sharpe Ratio = (Investment Return – Risk free return)/ SD

Replacing SD with UI becomes:

Martin Ratio = (Investment Return – Risk free return)/ UI

Martin Ratio for Cryptos
Now with the theory and background behind us – back to the question: Given the very high volatility in cryptocurrencies (30 day standard deviations range have historically between between 60~300%!) which one is a better investment on a relative risk-return basis?  We can have a stab at measuring and ranking risk adjusted returns using the Martin Ratio.

Just looking a snapshot of Cumulative Returns for a selection of 10 cryptocurrencies over the last 360 days ( Oct 15 2016 to Oct 15 2017) is clear from the chart which ones have visibly outperformed but it is NOT clear what risk you took in getting those returns (or if indeed the risk was worth taking to get those returns!) but more specifically in the case of suffering big drawdowns in achieving it.


And over this time period the drawdowns were deep and many ! As seen in the chart below, anywhere between a -20~80% drawdown from a given peak equity in just the last 1 year!

For example, in the last 360 days Bitcoin suffered 32 drawdowns with the top 4 drawdowns suffering a drop of between -28~35% and all lasting between 41~55 days each.  The most recent Bictoin drawdown was caused by the infamous “China ICO ban” which lasted 41 days (Sept 2 ~ Oct 12) with a drop of -34% and taking only 13 days to hit a trough but the recovery back to its previous peak took twice as long at 28 days! With these type of trading patterns in cryptoworld, every investor should be focused on potential drawdowns and volatility and not just the “blue sky” upside.

The Table below shows the annualized returns (360 days) of the above 10 cryptos with the corresponding SD’s and their Sharpe ratios (Risk free rate assumed to be 0).


BTC does pull in an impressive 773% return with a relatively modest 81% SD compared to the other cryptos.  But even using an standard Sharpe ratio it ranks only 6th on a risk-adjusted return basis.  Dash comes out #1 and beats ETH given its slightly better return for a similar SD.

The above exercise is repeated again using the Ulcer Index instead of SD to give the following Martin Ratios:

Martin Ratio_top 10

Surprisingly Dash wins again, and BTC drops a rank to #7 with LTC, XRP remaining the same. What’s more interesting is that 6 other coins (including smaller cap coins like VTC and BTCD) move UP in rankings despite their higher SD measurements. The most likely reasons being due to having a lower number and/or more shallower drawdowns and possibly that their overall upside variance was greater than their downside deviation (and something that is masked behind a generic volatility measurement like SD)

And why did Dash rank #1? Closer analysis shows it had only 23 drawdowns vs BTC’s 32 of between -1~50% (still quite deep) but it’s impressive gains of 2635% were enough to justify the downside risk and more so than other cryptos which achieved even more impressive gains (eg VTC +3867% and BTCD +2912%).

Whilst clearly historical returns and volatility may not be representative of future gains, a solid understanding of historical downside risk is a good starting point when deciding to add any cryptos to a portfolio. The Martin Ratio via the Ulcer index is a valuable metric.

And what does a “good portfolio” look like? Well that another question. But based on historical risk and returns it involves mean-variance optimization to find an optimal portfolio on the efficient frontier. That is, a maximizng a portfolio return for a given risk or minimizing portfolio risk for a given level of expected return.

Good luck out there!


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