Ulcer Index, a practical application

In previous articles, we have seen how gold has been able to transfer value over time (even centuries or millennia), and how Bitcoin is emerging as the best performing asset ever, thanks in part to scarcity characteristics quite similar to gold, but in the digital sphere. 

From the analysis of these characteristics, the research department of Diaman Partners came up with a strategy to have in a single index both of these wonderful assets weighted with an intelligent and dynamic logic. 

Many will be familiar with the Risk Parity approach, proposed in 2005 by Eward Qian in a fine paper entitled 'Risk Parity portfolios'. 

In fact, a risk-parity strategy weights two or more financial instruments within an investment portfolio according to a risk indicator, traditionally the inverse of volatility, so as to overweight the more conservative securities and allocate a smaller risk budget for the more aggressive instruments. 

Volatility, however, has mathematical characteristics that do not lend themselves well to these types of approaches, because according to its calculation formula, squaring the returns actually eliminates the sign (plus or minus) and therefore a historical series that tends to lose, can have the same volatility as a historical series that tends to gain, without any distinction; whereas it is clear that I would prefer to have the one that gains and not the one that loses in my portfolio. 

Figure 1: formula for calculating Volatility

Moreover, volatility as a statistical indicator does not take the sequence of returns into account at all, nor does VaR (Value at Risk), so they are in fact not indicators that can determine whether risk is increasing before the real risk becomes apparent, but only afterwards. 

A striking example, although there are many others, was in September 2008, when Lehman Brothers went bankrupt, until mid-September, perhaps even until early November, both volatility and VaR had given no clear signals that risk was increasing in the financial markets. One had to wait until early November, with the stock market doing -10% and +10% one day to the next, to see VaR and volatility change to an appreciable level. 

To make these dynamics understood, I have assumed a random time series, which has mean = 0 and volatility = 20.7 

Figure 2: daily returns by volatility calculation 

To make the perverse mechanism of the volatility calculation clear, the Xi returns of the formula are the vertical bars below the graph. 

Volatility does not change if I change the sequence of returns, so paradoxically if I sorted the returns from lowest to highest, I would get such a result as a time series: 

Figure 3: Yields ordered from smallest to largest 

If, on the other hand, I sorted the returns from highest to lowest, I would obtain a completely different historical series

Figure 4: yields ordered from largest to smallest 

It is clear that the three time series have completely different characteristics, and that in all likelihood an investor's feeling and perhaps behaviour in experiencing these time series would be completely different. 

Figure 5: possible time series with the same volatility and average 

So if we graph the three time series, which we see below, we can also conclude that in the middle of the two extremes there are a very large number of time series (n!) that vary according to the number of days of observation that have the same mean and same variance, but not necessarily the same perception of trends by investors. 

From this awareness of the limits of volatility as a risk indicator, we came up with the idea of using another, lesser known, but certainly more efficient risk indicator, the Ulcer Index. 

Few are yet familiar with this indicator, created in 1987 by Peter Martin, because it was published in a Fidelity book, but has never been officially presented in an academic paper. 

Diaman brought it to Italy in 2004, thanks to the scientific committee that at the time also included Paolo Sassetti, an American hedge fund enthusiast. 

The ulcer index has a similar formula to the volatility index, only the returns are not the daily returns, but the ratio of the day's price to the previous high. 

Figure 6: Ulcer Index formula 

Thus the ulcer index is able to measure the relationship between loss and recovery time, but more importantly it is sensitive to the sequence of returns, so unlike volatility, if I change a sequence between two returns, I get a different Ulcer Index. 

When is this indicator useful? When I need to understand whether a historical series is going badly or going well, a detail that volatility cannot provide me with. 

Figure 7: example of the area covered by the Ulcer Index 

It is obvious that if I have to create a portfolio consisting of two financial instruments, using a risk indicator to determine the weight of each, the ulcer index allows me to weigh the components more dynamically and promptly, taking into account the current trend. 

Figure 8: example of yields used for Ulcer Index calculation 

As can be seen in the example, in the first part the ulcer index was very low, so the indication would have been to overweight this historical series compared to the other one which may have been doing badly, but in the second part of the graph the ulcer index continues to rise, indicating that it is better to reduce the exposure of this asset. 

Figure 9: limiting example with returns ordered from worst to best 

In the extreme example above, where volatility remained the same, the Ulcer Index practically doubles, clearly indicating that this price trend situation, especially in the first part, is not good, while it improves in the second part, because the average value gradually decreases. 

Figure 10: boundary example from best to worst performance 

In the other example, on the other hand, the Ulcer index is practically nil in the first part of the rise of the graph, and then gradually increases in the second part. 

Again, the Ulcer Index has a different value than the other two examples, demonstrating its superiority to volatility. 

Furthermore, to definitively realise the validity of this instrument as a dynamic indicator of increasing or decreasing risk, we took the 90-day return on gold, compared with the 90-day Ulcer Index. 

Figure 11: 90-day return and 90-day Ulcer Index of Gold 

As you can see, the ulcer index moves in perfect complementarity with the performance of Gold, so if I use the ulcer index as an indicator that determines the weight of Gold to put in the portfolio, the higher the ulcer index is, the less I will put in the portfolio, and vice versa, if the ulcer index is low, I will be more confident in putting more Gold in the portfolio. 

Figure 12: 90-day Ulcer Index and 90-day tolling yield of Bitcoin 

The same is true for Bitcoin, with the difference that the Ulcer Index values on average are higher than the Gold values, clearly indicating that the predominant weight in the portfolio is Gold, as opposed to Bitcoin, even though the changes are very sudden over time, as evidenced by the graph below

Figure 13: weight of Gold and Bitcoin over time 

The end result however is surprising, because the portfolio mix between Gold and Bitcoin, if started in 2015, would have led to an average annual return of 27%, with a volatility of 22%, a better result than the Nasdaq over the last 10 years, but above all with a lower drawdown. 

Figure 14: Back Test of the Diaman Bitcoin / Gold Index

In fact, this strategy would return more than the S&P500 for 8 years out of the last 10, with slightly higher volatility and very low correlation, which highlights its potential in terms of improving the risk-adjusted return in an investment portfolio. 

But more on this in another article.