**From Guinness to GARCH**, and kindly contributed to R-bloggers)

This is a ‘do over’ of a project I started while at my former employer in the fall of 2012. I presented part 1 of this framework at the FX Invest West Coast conference on September 11, 2012. I have made some changes and expanded the analysis since then. Part 2 is complete and will follow this post in the week or so.

Adam Duncan September, 2012

Also avilable on R-bloggers.com

## Part 1: Constructing a Regime Framework

### Motivation

People often aske me, “What trades should we be doing right now?” or “What trades do you like right now?” I have often answered these questions with a combination of personal preference and an informal sampling of trades other people have been talking about. Rather than continue offering such subjective responses, I decided to embark on a small quest to apply some discipline answering those questions. What if we had a roadmap that told us: “When the world looks like this, do that.” What if we could map out all states of the world and test all our favorite strategies in each of those states? We could then assess where we are and apply the strategies that do the best. The following is the first in a series of pieces that will develop this framework. It will:

- Give you a framework for assessing the current state of the world.
- Give you a way to test and rank your favorite strategies in each state or regime.
- Give you a way to estimate the conditional probability of migtrating to another state.
- Give you a way to form expectations about how long each state might persist.

In addition, we will try to estimate a more formal regime swithcing model to see if the transition probabilities can give us any foresight into regime transitions. You should come away from this reading with a useable tool for deciding which stategies you should be employing, given the state of the world as identified by the FX vol surface. In short, we want to know if conditioning our strategy selection based on the volatility regime leads to better performance. But, first things first. Let’s map the world…

### Mapping the World

Volatiltiy regimes are nothing more than states of the world defined by volatility characteristics. What sort of characteristics? Well, in this framework, we consider 2 factors: (1) level of volatility and (2) the term structure or slope. These 2 factors give us a lot of information about what state the world is currently in.

In a simple 4 state world, we might let each of our 2 factors take on 2 levels each. For example, vol levels might be (1) High or (2) Low and our term structure of vol might be (1) Steep or (2) Flat. One can imagine looking at the world in 2008 when vol levels were quite high and term structures were assively inverted (short dated vols traded above longer dated vols). We might call this state 1: HIGH and FLAT. As the policy makers introduced measures and time passed, short dated vols came down, but overall vol levels remained quite high. This might be called state 2: HIGH and STEEP. As more time passed and market participants became more comfortable with the fallout fromt he crisis, we experienced a parallel shift lower in all vols, but long date risk premiums remained. That is, long dated vols still traded expensive to short dated vols, albeit at much lower levels. We might call this state 3: LOW and STEEP. One can imagine coming full circle back to the risk seeking months and years before the crisis where long dated risk premiums collapse and the term structure flattens out. This is similar to the 2005-2006 experience. Vol levels settle in a low levels and the slope of the term structure is relatively flat. We can call this state 4: LOW and FLAT.

#### The Data

Source: Bloomberg

Frequency: Daily

Vol Tenors:

- 1 month
- 1 year

Start: 30-Dec-1994

End: 11-Jan-2013

Currency Pair: EURUSD

Note that you could average together, say, all of the G-10 and EM vols together and use some composite vol index as your term structure and level variables. At least as far as G-10 vols go, the results do not differ significantly from just using EURUSD vol.

#### A Visual Inspection

Before I get too far in any analysis, I like to make a visual inspection of the data. This is always useful and can reveal problems or new ideas before too much work has been done. Let’s do that here:

The visual inspection of the data reveals that our simple 2 levels per factor assumption is probably not the right formulation. Looking left to right along the x-axis it’s pretty clear that there are 3 distinct clusters in the vol level data. There also appears to be 3 reasonable levels across the term structure variable, with points above, below, and well below the horizontal zero-line.

#### A Shallow Dive into the Realm of Cluster Analysis

The first thing we’ll want to do is to standardize our variables by transforming them into z-score measures. This is always a useful thing to do when dealing with variables of vastly different bases. It also makes us level agnostic, which is a good thing.

Rather than going through each day in history and deciding whether that day was STEEP or FLAT and HIGH or LOW, we can employ a simple clustering algorithm to split up our standardized variables into an arbitrary number of groups. In this case we have 2 groupings per factor. So, we can use a k-means clustering algorithm with 2 means per factor to split each factor into groups.

#### Is our 2 Levels per Factor (4-State World) Appropriate?

One natural question is,”Have we selected the right number of levels for each factor?” What if some other number of groupings is more appropriate for each of our factors? There is a way to determine the appropriate number of groupings for a given factor. It’s very similar to using a scree plot in PCA to determine the number of principal components you want to use. By iteratively running the k-means clustering algorithm with a variable number of means, we can examine the sum of squared distances between the elements of each group.If we plot the results, we’ll get something like this:

The scree plots suggest that 3 or 4 groups is defintely more appropriate. It might be more appropriate to allow each factor (term structure and level) to take on 3 levels (high, medium, and low) and (steep, normal, and flat), for example. 4 clusters might also work, but we start to lose some intuition about the construction. For now, let’s just stick with our 4 state world and see if we can map the historical observations.

After running the k-means clustering algorithm with 2 means per factor, we should have each standardized factor assigned to one of 2 groups. We can inspect the results of this assignment:

Now that both factors have been assigned a grouping, we can plot the results. A trellis plot is a nice way to view the data by factor grouping:

The plot is nice in some respects. We can clearly see the crisis periods in the upper left quadrant “High/Flat”. And, each of the other quadrants seems to have reasonable representation. Though, there is a tremendous amount of overplotting (dots on top of each other). This suggests that our 4 state grouping might not be sufficient. We knew this, from our scree plot. Let’s re-run the analysis and allow the k-means algorithm to bucket each factor into 3 groupings.

It’s helpful to visualize how our term structure and level z-score variables were grouped by the clustering algorithm. The following plots show the time series of the z-scores colored by the group they were assigned to. You can easily see the “Steep”, “Normal” (for lack of a better word), and “Flat” groupings. Similarly for the level z-score variable.

Now we can plot our 9 states as we did for our 4 state mapping:

### The *State* of Play

Now that we have some nice states of the world mapped out, we might like to know something about the frequency of occupying a particular state and other questions like:

- Conditional on being in state s, what is the probability of transitioning to state r?
- Over the next n periods, how long are we expected to be in any one of the states?

A state table is a very good way to represent multi-state data. What follows is adapted from excellent work done by Christopher Jackson at the Department of Epidemiology and Public Health, Imperial College, London. The interested reader is referred here: Multi-state modelling with R: the msm package

The state table shows the number of times each state has followed another state in succession in the data Here is the state table for our 9 state representation of the world:

```
to
from FH FM NH NL NM SH SL SM
FH 63 3 4 0 0 0 0 0
FM 4 25 0 0 5 0 0 0
NH 3 0 157 0 9 10 0 0
NL 0 1 0 1004 20 0 70 0
NM 0 5 10 21 521 0 1 37
SH 0 0 5 0 0 69 0 10
SL 0 0 0 69 3 0 717 4
SM 0 0 3 1 37 5 6 755
```

This table shows the number of times that one state has followed another state in successive observation times. So, conditional on being in the “SM” state (the bottom row of the table), we have migrated to “SL” on *n* occassions, to “SH” on *m* occassions, to “NM” on *p* occassions, and so on. This is informative. It is also worth noting the zero entries in the table. This means that from some states we have NEVER transitioned to some other states. For example, we have never transitioned from “SM” to “FH”. Nor from “SM” to “FH”. Note that the count of observed states is a function of the grouping algorithm. Each time we re-run the clustering, we’ll get slightly different answers. This is expected and you should run the algorithm many times to establish confidence intervals about the estimates.

While *counts* of historical occurrences is helpful, let’s see if we can estimate actual transition probabilities. We’ll use this state table to seed the estimation. First we need to estimate an initial transisiton intensity matrix. Once we have the initial intensity matrix constructed (easily done with the msm package), we can then estimate the transition probabilities. Here are the results of that estimation:

```
FH FM NH NL NM SH SL SM
FH 0.907 0.036 0.051 0.000 0.004 0.001 0.000 0.000
FM 0.098 0.770 0.004 0.002 0.121 0.000 0.000 0.004
NH 0.015 0.000 0.887 0.001 0.045 0.048 0.000 0.004
NL 0.000 0.001 0.000 0.923 0.017 0.000 0.059 0.001
NM 0.001 0.007 0.015 0.032 0.886 0.001 0.003 0.057
SH 0.000 0.000 0.051 0.000 0.004 0.838 0.000 0.106
SL 0.000 0.000 0.000 0.080 0.004 0.000 0.911 0.005
SM 0.000 0.000 0.004 0.002 0.042 0.006 0.007 0.939
```

Excellent. Now we can say something about the 1-day probability of transitioning from one state to another You can see the persistence present in each state. That makes sense. It’s sort of like asking, “What is the probability that firm XYZ will be downgraded *TOMORROW*?” Well, there is a very high liklihood that whatever rating firm XYZ has today, it will have it tomorrow, too. But, there is always some small chance of a downgrade event. Similarly here, there is a very high liklihood that we will maintain the current state tomorrow. It might be more useful for us to estimate, say, 1 month ahead probabilities. Here are the 1-month transition probabilites:

```
FH FM NH NL NM SH SL SM
FH 0.138 0.035 0.164 0.100 0.207 0.064 0.056 0.236
FM 0.091 0.026 0.117 0.155 0.210 0.046 0.096 0.258
NH 0.051 0.016 0.127 0.131 0.213 0.063 0.081 0.317
NL 0.009 0.006 0.023 0.414 0.120 0.009 0.306 0.112
NM 0.025 0.012 0.067 0.221 0.203 0.031 0.148 0.293
SH 0.029 0.012 0.096 0.133 0.215 0.052 0.088 0.376
SL 0.007 0.006 0.020 0.420 0.111 0.008 0.325 0.104
SM 0.018 0.010 0.063 0.173 0.214 0.033 0.118 0.370
```

Now, instead of a 94% chance of being in, say, “SM” tomorrow, we can say that there is approximately 30% chance of still being in “SM” 1 month from now, having started in “SM” today. Similarly, starting in “SM”, we can see that “SL”, “NM”, and “NL” are the 3 most likely transitions from “SM”. Those are the states we should be focusing on as portfolio managers trying to anticipate the next most likely states of the world. Confidence intervals of the probability estimates are available, but not shown here.

The last thing we’ll do before moving on is to estimate the “total length of stay” for each of our designated states of the world. That is, over the next, say, 1yr, how many days do we expect to spend in each state? Here is a table that shows the expected length of stay in each state (in days):

```
FH FM NH NL NM SH SL SM
18 5 20 60 43 8 43 56
```

This concludes part 1 of Volatility Regimes: Identification and the Impact on Strategy Selection. In the next part of the analysis, we’ll select a number of strategies and run them without selection based on regime and then with conditioning on Volatility Regime. We’ll show how the conditioning information improves performance. We’ll also draw some conclusions about which strategies do best in which regimes. Lastly, we’ll estimate a regime switching model and see how well it does in picking up changes in regimes.

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**From Guinness to GARCH**.

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