Monthly Archives: December 2016

AFX Index December Update: Flat Year for Currency Trend Followers …..

Passive currency indices do not reflect any of the money management skills necessary to generate profit out of the Foreign Exchange market. Evidently there is no value in holding a long (or short) position in any currency over the very long term. For this reason passive currency benchmarks fail to adequately describe the performance of currency funds because they do not have an embedded timing process to imitate the short/long currency positions that an active manager would take. For that reason correlation between passive currency indices and currency managers peer group indices tends to be low.

The AFX, aims to replicate the risk/return profile of the average currency manager by using the returns of technical trading rules, namely trend following trading rules. The index was designed by Lequeux & Acar (1998). The timing embedded in the index relies on the buy/sell signals generated by three moving averages. So as to cover a broad spectrum of time horizons the ex-ante statistical properties of technical indicators were used to build the index on the basis of ex-ante measurable criteria of risk reduction and transaction costs. Finally the index uses a weighting scheme derived from the estimated turnover in currency market as reported by the triennial survey on foreign exchange turnover conducted by the Bank for International Settlements. The index is calculated gross of any fee or risk free income and as such express the typical directional market opportunity that was available in G10 FX.

A full description of the index can be found in : Lequeux, P. and Acar, E. (1998) “A Dynamic Benchmark for Managed currencies Funds”, European Journal of Finance Vol. 4.

The historical returns of the AFX Index can be downloaded by through the following link: AFX Historical data

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**Summary Performance Statistics

##                                  AFX
## Annualized Return               2.99
## Annualized Standard Deviation   6.79
## Annualized Sharpe Ratio (Rf=0%) 0.44

Drawdowns Table

##          From              Trough         To  Depth Length To Trough
## 1  2015-04-30 2016-09-30 01:00:00       <NA> -15.21        22     22
## 2  2010-11-30 2014-06-30 00:00:00 2015-01-30 -11.48        51     51
## 3  2004-01-30 2004-09-30 00:00:00 2008-10-31 -10.76        58     58
## 4  1993-05-28 1995-01-31 00:00:00 1996-01-31  -7.86        33     33
## 5  1988-01-29 1988-04-29 00:00:00 1988-11-30  -7.79        11     11
## 6  1991-04-30 1991-08-30 00:00:00 1991-12-31  -7.17         9      9
## 7  2009-01-30 2009-04-30 00:00:00 2010-05-31  -6.26        17     17
## 8  1992-01-31 1992-04-30 00:00:00 1992-07-31  -5.77         7      7
## 9  2002-07-31 2002-11-29 00:00:00 2003-05-30  -5.68        11     11
## 10 1989-06-30 1989-10-31 00:00:00 1990-07-31  -5.58        14     14
##    Recovery
## 1        18
## 2        44
## 3         9
## 4        21
## 5         4
## 6         5
## 7         4
## 8         4
## 9         5
## 10        5

Monthly Returns

##       Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct  Nov  Dec YEARLY
## 1984 -0.6 -0.7 -0.7  0.0  0.8  1.4  4.1 -1.7  3.9 -0.9  0.9  1.9    8.5
## 1985  1.5  4.0 -1.5 -2.3 -0.4 -0.3  6.7 -1.5 -1.5  2.1  3.3  0.0   10.1
## 1986  1.4  5.0 -1.3  2.2 -1.1  0.3  3.7  0.1 -1.3  1.0 -0.9  0.8    9.9
## 1987  3.2  0.1  2.3  1.7 -1.6 -1.9  1.7 -0.1 -0.5 -0.2  4.7  5.0   14.3
## 1988 -5.8 -0.3 -0.2 -1.6  1.3  4.2  1.8  0.7 -1.5  0.9  3.1 -2.9   -0.2
## 1989  3.1 -2.5  3.1 -0.3  5.6 -0.5 -1.6 -0.9 -1.6 -1.1  0.8  2.2    6.4
## 1990 -0.4  0.5  0.7 -0.1  0.1 -1.2  3.8  1.1 -0.2  3.3 -0.6 -1.4    5.7
## 1991 -1.8 -1.1  7.9 -1.8 -0.9  2.3 -2.7 -4.2  3.6 -1.6  1.9  5.3    7.0
## 1992 -4.5 -1.0  0.8 -1.1  0.5  3.8  1.7  3.7 -0.6 -0.7  1.7 -0.7    3.6
## 1993 -2.8  2.0  0.5  2.6 -0.1 -1.2 -0.5 -3.3  0.1  0.0 -0.2 -0.4   -3.3
## 1994 -1.5  0.3  1.4 -0.4 -0.9  2.9 -1.1 -1.5 -0.2  1.8 -1.2 -1.1   -1.5
## 1995 -0.8  1.7  6.5  0.2 -3.8 -2.0 -0.5  4.9 -1.1  0.7 -0.9 -0.1    4.7
## 1996  3.3 -1.8  0.6  2.0  0.4  0.3 -0.1 -1.0  0.8  1.8  0.2  1.4    7.9
## 1997  4.0  1.3 -0.3  1.3 -2.6  0.7  2.3 -0.9 -0.4  0.1  1.5  0.1    7.2
## 1998 -0.7 -2.1  2.6 -1.3  1.5 -0.5 -0.8 -1.9  0.9  4.7 -2.5  0.1    0.1
## 1999 -0.4  0.7  0.7  0.6  0.7 -0.5 -0.5 -0.8  0.2 -0.8  2.4 -0.5    1.8
## 2000  1.8  0.8 -0.7  1.4 -1.1 -1.7 -0.2  1.9 -0.7  2.0 -0.6  4.4    7.3
## 2001 -0.3 -1.4  2.6 -2.1  0.8 -0.5 -1.0  1.8 -0.9 -0.6 -0.5  0.3   -2.0
## 2002  1.3 -1.9 -1.3  0.3  2.6  4.7 -0.4 -1.5 -1.7 -1.6 -0.6  2.8    2.8
## 2003  1.2 -0.5 -0.5  0.1  2.9 -1.9 -1.1  0.7  0.7  0.3  0.7  3.0    5.7
## 2004 -0.5 -0.4 -1.0 -0.7 -1.0 -1.8 -1.3 -3.0 -1.5  2.0  3.3  0.3   -5.7
## 2005 -2.7 -0.4  0.0 -0.2  2.6  2.2  0.2 -1.6  0.6  0.3  1.6 -1.7    0.7
## 2006 -1.4 -0.9 -1.6  1.9  1.2 -1.1 -0.6  0.6 -0.1 -0.2  1.9 -0.5   -0.8
## 2007 -0.1 -0.9 -0.8  1.2 -0.4  0.1  0.3 -1.0  1.6  0.6  0.6 -1.3   -0.3
## 2008  0.7  0.9  2.8 -2.5 -1.5 -1.9 -0.7  4.2 -0.9  8.8  1.5  0.3   11.5
## 2009 -1.1 -0.4 -2.2 -2.8  4.5 -0.7 -0.6 -0.7  1.6 -0.8  0.8 -0.7   -3.0
## 2010  0.7  1.3  0.1 -0.3  3.1 -0.7  0.7  0.1  1.1  1.4 -0.7 -1.7    5.0
## 2011 -1.3  0.2  0.3  3.0 -3.2 -1.6 -0.2 -2.9  3.2 -4.1 -0.2  2.0   -4.8
## 2012 -0.9  0.8 -0.6 -1.2  3.3 -3.3  0.8 -1.1  1.2  0.4 -0.1  2.2    1.5
## 2013  2.1 -0.2  0.7 -1.8 -0.5 -2.6 -0.4 -1.5  1.5 -0.5  0.3  1.8   -1.2
## 2014 -2.6  0.2 -0.9 -0.9 -0.2 -0.1  0.5  1.4  3.5  0.7  2.5  1.4    5.5
## 2015  2.5 -0.3  1.9 -1.3 -1.0 -2.5 -0.7 -1.2 -0.5 -1.5  2.3 -3.4   -5.6
## 2016 -0.7  0.6 -0.6  0.6 -1.3 -1.3 -1.1 -0.9 -1.9  1.4  3.6  1.3   -0.3

The AFX is positively correlated to main peer group indices highlighting that currency managers are typically directional in their investment style. The below charts shows the 24-month rolling correlation of the AFX with the BTOP FX Index .

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If you need more information or have questions about the above, feel free to contact me at pollux@argonautae.com

Growing Randomness of Currency Markets…

Back in 20114 I wrote a chapter for The Role of Currency in Institutional Portfolio by Professor Levich and M. Pojarliev.In my research I debated about the growing efficiency in foreign exchange markets potentially making a more arid ground for active managers to generate alpha. Clearly some could argue about the timing of my publication on the subject as since then some strong trends have occurred in US$ crosses. In fact the last quarter of 2014 proved to be a significant localised alpha bonanza for many currency managers. This fed into much enthusiasm from managers and a regain of interest for active currency management. However since then those trends have abated and it is lean times again for currency managers who use single factor strategies. Anyhow, my study focus on long term dynamics and the secular growing efficiency of market which I suggest is driven by a cocktail of world globalisation and advances in information technology. This has enhanced in an unprecedented way the availability of information, access to market and provided a level field market pricing to market participant.

In a seminal paper Emmanuel Acar laid the theoretical background demonstrating that the expected return of directional trading rules can be attributed mainly to autocorrelations (i.e. how the daily returns of an asset are correlated from one period to another) and drift (i.e. the absolute percentage deviation of the price series). In my paper I proposed a methodology based on his finding to classify financial time price series. The below shows what was the drift, autocorrelation and volatilities of the 45 G10 FX cross exchange rates over the period 1996 to 2015.

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Using significance tests for the drift and first order autocorrelation of the time series over a rolling windows of 125 days it is possible to classify each of the 45 G10 FX crosses into 6 specific behaviours, namely: Strong trend, strong mean reversion, short term trend, Long term trend + short term reversion, Long term trend + white noise, random walk. More details on this can be found in my paper. In the below I have aggregated the time dimension (i.e. long and short term) so as to end up only with three states: Trending, Mean Reverting and Random walk. The bar chart shows the percentage of time that each currency pair spent in each of those state. It is quite apparent that some currencies have had a greater propensity to trend than other (i.e. US$ and JPY crosses) and also that currencies spent most of their time in a random walk state. It is still possible to generate value in the later as long as the risk is compensated by a high level of carry. Clearly this has not been the case over recent time and may explain why so many currency managers had poor perfpormace.

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The following chart shows the number of US$ crosses that have been in trending regime over the previous 750 days. It is quite clear that aside the last quarter of 2014, trends have been seldom.

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Finally, the last charts shows the number of currencies that would have been classify as trending, mean reverting or random on a rolling basis since the seventies. It is quite clear that currencies have become more random over the last few decades. This in turn means that currency manager performance has become far much more dependent on the level of carry and volatility. I am always happy to have a natter about what I produce so feel free to contact me at Pierre@Argonautae.co.uk.

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G10 FX Implied Volatilities: Cheap or Expensive ?

The following report provides a granular analysis of implied volatilities within G10 FX. I use primarily the same formatting than for my G10FX positioning report to estimate how extended the 1-month FX implied volatilities are over various time horizon.

The first set of charts shows the historical T-stat of the 1-day changes in 1-month implied volatilities over a rolling period of 61-days. This is my statistical metric to quantify how stretched the implied volatilities are, but clearly other time period could be used as shown further down on in that report. The purple line represents the median value since 1996 and the red lines represent the 95% confidence intervals. Therefore if the value is above or below those the deviation of the given implied volatility should be deemed as atypical relative to what would be expected under a normal distribution (I am not saying that implied volatilities have a normal behaviour to be clear….) and therefore overbought/oversold.

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The below charts shows the current implied volatilities relative to their historical distributions since 1996. Once again the red lines delimit the 95% confidence intervals and the purple line the median value. The blue line indicates the most current level of 1-month implied volatility.

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Finally the below shows a stretch map of the T-Stats to help visualise how much implied volatilities have departed from their equilibrium levels over time horizons ranging from 1-month to 6-month. The bigger the square the most significant the observed upside (Green) or downside (Red) of the implied volatility over the given period.

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Chinese Stock Market Risk Report Update

The following report provides an update on some of the metrics I use to classify market risk. The word classify is more appropriate as I think that in essence you cannot forecast risk but rather attempt to adjust to it into a timely fashion. Clearly risk would not be a risk if you could forecast it accurately. However as there is generally some degree of persistence in risk regimes, using a dynamic classification may be a useful approach for portfolio rebalancing and hedging. In this report I use the CBOE China ETF Volatility Index (VIX China) as a measure of stock market risk for China . The same methodology can be successfully applied to other inputs. Feel free to contact me at Pierre@argonautae.com for more information on the subject.

In my approach I recognise that the nominal level of implied volatility is a crude metric of risk therefore I also use two other measures. The VIX Volga, a measure of uncertainty of risk and the ShockIndex a measure of market dislocation. VIX Volga is simply the volatility of the VIX China over a given period. This measure highlights how uncertain and unstable the level of risk has become. Though positively correlated to the level of the VIX China the VIX China Volga is not necessarily dependent on it. You can have a high level of volga whilst the VIX China is trading at rather innocuous levels. This is not a trivial observation as the leverage undertaken by market participants tends to be an inverse function of market volatility which implies a greater vulnerability when volatility becomes uncertain at low levels and therefore cannot be accurately budgeted fo r. The ShockIndex is the ratio between the Volga and VIX at the beginning the historical window chosen to evaluate the Volga. It quantifies sharp changes and acceleration in risk levels. Historically it has proven to be a good classifying measure for market event risks.

The below charts shows those three measures both relative to a time axis and their historical distribution. The red lines are the 95% confidence intervals, the purple line the median. The blue line highlight the current level. The VIX Volga and ShockIndex in this report are evaluated over a period of 14 days. The medians and 95% confidence intervals are calculated over the full history going back to 1990 though the charts shows only the recent years.

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At close of business the 2016-12-30 the VIX China was trading at 24.6 at the 41 percentile. The 14-day VIX China Volga was estimated at 13.1 its 46.6 percentile and the China shockindex at 0.6 or its 56.1 percentile.

The above charts are useful, however their visualisation is quite limiting. On the one hand we need quite a few charts to present the data on the other hand it is difficult to show the full VIX history going back to 1990 as this would make the charts unreadable. Therefore clustering and aggregating the whole data into a single chart should be useful to the end user. To answer this I use a mapping technique developed by Kohonen in the 1980′. It uses an unsupervised neural network to re-arrange data around meaningful clusters. Though computationally complex is a practical way to summarise multidimensional data into a low (usually 2) dimensional system.

The below chart shows how the VIX price history was split into 4 distinct clusters. Those clusters where computed not only as a function of the VIX level but also as a function of the other variables, namely VIX volga and Shockindex.

Since 03/2011 the VIX China traded 53 % of the time in Cluster 1, 30 % in Cluster 2, 13 % in Cluster 3 and 4 % in Cluster 4. Overall the layering provided seems quite intuitive as the increase in risk and time spent in each cluster points toward what would generally be expected from market risk regimes ranging from low to high risk.

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In the chart below we zoom on the various regimes within which the VIX has been trading for the current year. so far it traded 80 % of the time in Cluster 1, 20 % in Cluster 2, 0 % in Cluster 3 and 0 % in Cluster 4.

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Finally the below chart shows a Self Organising Map of the above mentioned risk metrics. The data has been grouped and colored as a function of four clusters of increasing market risk regimes. Obviously as shown on the map, the minimum level of volatility pertains to cluster 1 and the highest to cluster4. The current regime and its progression from 21 days ago is also highlighted on the map.

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Always happy to discuss any of the above, feel free to reach me at: Pierre@argonautae.co.uk

Trade Weighted Currency Indices Stretch Map

Trade Weighted Currency Indices Report

Sat Dec 31 09:06:54 2016

The following report aims to provide a gauge to the current strenght of major currencies. For doing so I use the Bank of England Trade weighted Exchange rate indices and a standardised statistical measures of price deviation to provide an estimate of how stretched major currencies are on a trade weighted perspective.

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I first calculate the T-stat of the mean price deviations over a rolling period of 61 days. The charts below show the results for each currency over the last 500 days. The purple line represents the median value since 1990-01-03 and the red lines represent the 95% confidence intervals. Therefore if the value is above or below those the deviation of the given currency would be deemed as atypical relative to what #would be expected under a normal distribution and therefore overbought/oversold.

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The following Map chart shows how stretched the currencies are over time horizons ranging from 1-month to 1-year. The bigger the square the most significant the upside (green) or downside (red) of currencies over the given period.

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The charts below show how the daily changes in the Trade weighted indices have correlated since January 1990 and since the begining of 2015.

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Finally, the following provide an ARIMA forecast for each of the trade weighted indices. My script selects the best ARIMA fit over the previous 250-day to generate a forecast for the next 21 days.
It also shows the forecast confidence intervals.

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GBP TWI Break Analysis…

In the following I us an R package BFAST designed to detect strucutural breaks in time series.The script Iteratively detects breaks in the seasonal and trend component of a time series. The first chart shows the various break and fitted regressions. The second chart shows the deviations from the regression lines and 95% interval of confidence. This could be used as an overbought/oversold indicator. Anyway, just work in progress…so any input / suggestions are always welcome as usual. Feel free to contact me at:Pierre@argonautae.com

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GBP TWI update….

Whatever the market being traded, there always will be a a question being asked at one moment: How far can this thing go ? Clearly not an easy question to answer as this will invariably depends on factors that are partly unknown or difficult to estimate, such as fundamentals, market positioning or market risk amongst others. The first part is obviously to assess how atypical the move experienced in the given instrument is. This report aims to contribute to this.

The below chart shows the GBP TWI over the period of January 1990 to December 2016 . On the 29 December 2016 it was trading around 76.924.

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In the below I plot the previous 125 days against other similar historical periods that would have closely matched the recent history. The data has been normalised so as to be on the same scale. The chart shows the latest 125 days in black, and overlay similar historical patterns in grey. It Also shows what has been the price path for the following 125 days as well as the observed quartiles.

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Finally I plot the last 125 days and a trend forecast derived from an ARIMA(2,1,1) model as well as the 95% confidence intervals. The ARIMA model is fitted to the past 625 historical values whilst ignoring the last 125 days, therefore we can look at the recent price path against the trend forecast and its confidence intervals to gauge how (a)typical the recent move has been.

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G10 FX Position Report Update

G10 FX POSITIONING REPORT

Sat Dec 31 09:07:55 2016

The following report aims to provide a gauge to the current market positioning in G10 FX. It focuses on US$ crosses and uses a standardised statistical measures of price deviation as well as a regression methodology to produce an estimate of how stretched currency exchange rates are and also to evaluate how currency managers are likely to be positioned and leveraged in G10 Currency. I use the BTOPFX in the report but can do the computations for any other peer group benchmark.

G10 FX STRETCH MAP

The stretch indicator looks at how much exchange rates are extended by calculating the T-stat of the mean price deviation over a rolling period of 61 days. The charts below shows the results for each currency pairs over the last 500 days. The spot prices are expressed as 1 unit of foreign currency versus the USD. The purple line represent the median value since 2005 and the red lines represent the 95% confidence intervals. Therefore if the value is above or below those the deviation of the given exchange rate would be deemed as atypical relative to what would be expected under a normal distribution and therefore overbought/oversold.

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The below shows the above calculated T-stats but this time relative to their historical distributions. Once again the red lines delimit the 95% confidence intervals and the purple line the median value. The blue line indicates the most current value of the T-stat.

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The following Map chart shows how stretched G10 FX exchange rates are over time horizons ranging from 1-month to 6-month. The bigger the square the most significant the upside (green) or downside (red) of the exchange rate over the given period. All the exchange rates are quoted on CCY-US$ basis so red indicate a depreciation of a given CCY against US$ and green an appreciation versus the US$.

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Estimated Currency Managers Positioning in G10 FX

To determine the sensitivity of currency managers to exchange rates and therefore their current positioning we regress the daily returns of the BTOPFX index against the daily logarithmic returns of G10 FX rates. We then calculate the T-stat for each of the regression’s slope coefficients. The higher the T-stat the higher the sensitivity to a given currency and therefore likely positioning. Using the regression weights as well as the variance of the independent and explanatory variables as input we can then easily deduce an estimation of the current risk utilisation of the typical currency manager as inferred by the values of the BTOPFX.

The below shows the T-stat of the regression’s slope coefficients over the last 500 days. The purple line represents the median value since 2005 and the red lines represent the 95% confidence intervals. Therefore if the value is above or below the red lines the positioning in a currency would be deemed as extreme and therefore the risk of unwinding would be greater since the market inventory would likely be close to its highest. Probably highlighting a good environment to enter a contrarian trade.

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The sensitivity of currency managers returns to changes in G10 FX rates relative to their historical distribution is shown below. Once again the red lines are the 95% confidence intervals and the purple line the median value. The blue line indicates the most current value of the T-stat. If this one is either side of the intervals of confidence it indicates a potentially overextended market positioning in the given currency.

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The exposure to the US$ is derived from the combined sensitivities to the other currencies and is shown in the same fashion than for the other currencies. Namely against an axis of time and relative to its historical distribution.

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The below Map chart shows the currency managers sensitivity to G10 FX exchange rates over time horizons ranging from 1-month to 6-month. The bigger the square the most significant the sensitivity to a currency the exchange rate over the given period. Long positioning is shown in green and short in red.

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Estimated Leverage

As explained previously the level of risk utilisation of currency managers and therefore their gearing can easily be derived by using the regression coefficients and the variances of both the independent and explanatory variables. The chart below shows the rolling estimation of risk utilisation as well putting it in respect of its historical distribution. Average Risk utilisation over the last 61 days is estimated at 41.68 % of maximum.

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Minimum Spanning Trees and G10FX implied volatilities…

I have always been keen on clustering methods as they are a practical way to visualise meaningful relationships that may exist in the somehow chaotic financial markets…..Following my previous post on the subject I decided to extend this to FX Implied volatilies…

The following charts show how major 1-month FX volatilities have been trading over the last 20-years and for 2016.

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The folowing charts shows the correlations of daily changes since 1996 and for 2016.

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The below plot the minimum spanning tree for G10FX implied vols. The distance between the nodes being a function of the above correlations. Some groupings are quite intuitive…some other less so…I would say the recent period seems to be at odd with the period 2010-2015 where we had two specific group: one for European currencies the other for commodity currencies….

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If you want a natter about this or just to exchange some ideas on the subject or other concepts presented in my blog, contact me at Pierre@argonautae.com

UK Assets Only Investor Dynamic ETF Allocation Portfolio Update

The following report provides analyticals in respect of an investible ETF multi-asset dynamic portfolio for UK assets only investors (I am clearly not saying nor advising that you should invest in such porfolio, I am just producing this for general information). For my allocation exercise I used Ishares ETF. My choice for the Ishares was purely driven by the fact that they have the longest price history. However, bearing in mind that Ishare Equity ETF have a total expense ratio of 0.40% , I therefore would rather use Vanguard or State street ETFs when available for implementation as they have a far much more reasonable TER (close to 10 bps). So my choice of IShares ETF probably affects negatively the numbers shown in the below.

I used the FTSE 100 , FTSE 250, FTSE high Div. ,UK Property , Corporate Bonds, Inflation Linked bonds and Gilts ETFs as my investible universe. The description of each ETF can be accessed by clicking on the assets and following the web link.

The below charts shows the rolling 36-month return, volatility and risk-adjusted return for each of the assets considered. Clearly equities and property have a higher volatility than bonds but also higher/lower localised returns highliting that timing is key in unlocking those higher returns.

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The summary performance statistics show that over the period April 2007 to date a UK investor would have had the best risk adjusted return by holding a broad basket of Inflation linked bonds and the worse by investing in the Property index which suffered hugely during the financial crisis.

##                                 FTSE100 FTSE250 FTSE HIGH Div. Property
## Annualized Return                  1.03    4.19          -4.00    -5.84
## Annualized Standard Deviation     14.31   17.35          17.06    22.61
## Annualized Sharpe Ratio (Rf=0%)    0.07    0.24          -0.23    -0.26
## Worst Drawdown                    45.25   53.05          66.41    79.38
##                                 Corporate Bds Inflation Linked Gilts
## Annualized Return                        0.60             6.08  2.97
## Annualized Standard Deviation            9.81             9.77  6.94
## Annualized Sharpe Ratio (Rf=0%)          0.06             0.62  0.43
## Worst Drawdown                          32.18            14.86  8.49

Below I show the Markowitz Efficient Frontier based on the past 36-month return series. Each investible asset, the minimum variance and tangent portfolio are shown on the plot as well as the in-sample 36-month annualised returns. The Green line is just the risk free line (I assumed zero risk free).

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I then used a mean-variance model to compute the weights of the portfolio that maximises the risk return ratio on the efficient frontier.The model is optimised for “long only” and weights adding to one constraints. I used a rolling window of 36-month to estimate the returns, volatility and correlation input fed into the Markovitz model. The use of a rolling window implies that the momentum effect in the input is captured by the optimisation. Therefore if an asset becomes more attractive through time in terms of its risk adjusted return and/or diversification potential its participation into the final portfolio should increase and vice versae as time goes. The two charts below show how the optimised portfolio weights have changed throughout time and also what were the weights at the end of the last month.

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Using the above weights I then calculate the return of the portfolio for the folowing period assuming a transaction cost of 0.15% of adjusted notional for each monthly rebalancement so as to factor in bid-ask spreads. The performance is compared to the return of a portfolio composed of 40% Gilts and 60% UK equities.

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**Summary Performance Statistics

##                                 Benchmark 40Eq./60Bds Optimal Portfolio
## Annualized Return                                3.40              3.42
## Annualized Standard Deviation                    5.62              6.89
## Annualized Sharpe Ratio (Rf=0%)                  0.60              0.50
## Worst Drawdown                                   6.90             12.90

Drawdowns Table

##         From     Trough         To Depth Length To Trough Recovery
## 1 2015-06-30 2016-02-29       <NA> -12.9        20     20        9
## 2 2013-05-31 2013-06-28 2014-02-28 -4.59        10     10        2
## 3 2010-09-30 2011-01-31 2011-09-30 -4.52        13     13        5
## 4 2012-04-30 2012-06-29 2013-02-28 -2.22        11     11        3
## 5 2014-03-31 2014-06-30 2014-08-29 -1.75         6      6        4

Monthly Returns

##       Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct  Nov  Dec YEARLY
## 2010   NA   NA   NA -0.2  0.4  1.0 -0.9  4.4 -0.1 -2.6 -0.4  0.9    2.3
## 2011 -2.3  1.0  0.5  0.5  0.9 -0.6  2.0 -0.8  1.9  0.7  1.5  1.4    6.6
## 2012  1.7 -0.1  0.1 -1.9  0.2 -0.5  1.3  0.4 -0.7 -0.8  0.4  0.3    0.5
## 2013 -0.3  1.7  2.1  0.1 -1.4 -3.2  3.0 -1.4  1.1  1.1 -0.9  0.1    2.1
## 2014  0.7  1.9 -0.6 -1.0  0.7 -0.9  0.2  3.0 -1.7  1.5  4.6  0.4    8.8
## 2015  5.4  1.1  0.7 -1.0  2.7 -3.8  2.7 -3.3 -0.5  2.9 -2.2 -1.6    3.0
## 2016 -5.7 -1.8  0.9 -0.1  1.6 -0.5  3.0  5.3 -0.7 -1.6 -3.5  3.0   -0.2

If you need more information or have questions about the above, feel free to contact me at pollux@argonautae.com