In Search of Diversification by Selecting Strategy Parameters

Diver­si­fi­ca­tion comes in dif­fer­ent shapes and forms. As a sys­tem­at­ic futures trad­er you immense­ly prof­it from the wide range of under­ly­ing assets avail­able. You can not only trade all the major equi­ty indices, but also for­eign exchange rates, bonds, ener­gy, met­als, agri­cul­ture and even volatil­i­ty. Future mar­kets are sim­ply the most diverse mar­kets out there.

This source of instru­ment¹ diver­si­fi­ca­tion is the most pow­er­ful for any futures trad­er. A sec­ond source of diver­si­fi­ca­tion is the use of mul­ti­ple trad­ing strate­gies. To name a few, for exam­ple you can trade trend, mean rever­sion or car­ry.

A third source of diver­si­fi­ca­tion lies in the dif­fer­ent time frames your strat­e­gy trades. Trend plays out over weeks, months or even years.

The same strat­e­gy can be deployed across mul­ti­ple time hori­zons, allow­ing sev­er­al para­me­ter­i­za­tions to trade simul­ta­ne­ous­ly. This source of diver­si­fi­ca­tion is usu­al­ly the weak­est form of diver­si­fi­ca­tion of these three, but nev­er­the­less worth pur­su­ing as we shall see in this arti­cle.

Before open­ing the math tool­box, let’s first ask whether diver­si­fi­ca­tion is real­ly always a good thing.

You may have heard the argu­ment that it is nec­es­sary to run a con­cen­trat­ed port­fo­lio to be suc­cess­ful. There are enough exam­ples of that out there. Look at War­ren Buf­fett: He did not get famous for invest­ing in a broad­ly diver­si­fied index like the S&P 500!

You also may have heard the argu­ment that being as diver­si­fied as pos­si­ble is the best thing ever, because diver­si­fi­ca­tion is the only free lunch in finance.

This argu­ment also has its prob­lems. You just have to look at the stock per­for­mance of con­glom­er­ates. Of course, cor­po­rate diver­si­fi­ca­tion and port­fo­lio diver­si­fi­ca­tion are not the same thing. Nev­er­the­less, the per­sis­tent con­glom­er­ate dis­count illus­trates an impor­tant prin­ci­ple: diver­si­fi­ca­tion is not free when it intro­duces addi­tion­al com­plex­i­ty.

Every­body who has worked for some time in a large organ­i­sa­tion usu­al­ly expe­ri­enced it first hand: Increased com­plex­i­ty and larg­er hier­ar­chies impose an even larg­er toll on oper­a­tive deci­sions.

Diver­si­fi­ca­tion should be viewed more care­ful­ly. It is not always the right answer. Ulti­mate­ly it is a trade-off between focus and com­plex­i­ty.

If you are a sys­tem­at­ic trad­er that devel­ops spe­cial mod­els for inter­est rates the added com­plex­i­ty to do the same for the agri­cul­tur­al mar­kets may be too much. But if you run mod­els on dai­ly data that do not dif­fer­en­ti­ate between asset class­es, more diver­si­fi­ca­tion is a no-brain­er.

Unlike adding entire­ly new strate­gies, trad­ing addi­tion­al instru­ments or addi­tion­al para­me­ters of the same strat­e­gy often increas­es com­plex­i­ty only mar­gin­al­ly. The remain­der of this arti­cle explores whether the result­ing diver­si­fi­ca­tion ben­e­fits jus­ti­fy doing so.

Theory

Let’s look a lit­tle bit deep­er into diver­si­fi­ca­tion from a math­e­mat­i­cal stand­point. Diver­si­fi­ca­tion man­i­fests itself if you com­bine return streams that are not per­fect­ly cor­re­lat­ed. To keep things sim­ple we com­bine the return streams of two strate­gies with the same expect­ed return of 15% and a volatil­i­ty of 20% each. Fur­ther assume that the cor­re­la­tion between the two is 50% and we will put half of our cap­i­tal in each of them.

The volatil­i­ty of a two-strat­e­gy port­fo­lio can be cal­cu­lat­ed by:

\[\mathsf{\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2}}\]

with \(w_i\) denot­ing the weight of each strat­e­gy. In our case of equal weight­ing, \(w_1 = w_2 = 0.5\). The sym­bol \(\sigma_i\) denotes the volatil­i­ty of each strat­e­gy i (each 20% in our case). Final­ly, \(\rho\) is the cor­re­la­tion between the two (50%).

For two equal­ly weight­ed strate­gies with the same volatil­i­ty, above for­mu­la can be sim­pli­fied to:

\[\mathsf{\sigma_p = \sigma\sqrt{\frac{1+\rho}{2}}}\]

The rela­tion­ship between the port­fo­lio volatil­i­ty and the cor­re­la­tion is visu­al­ized in fig­ure 1. As can also be seen from the for­mu­la, the result­ing port­fo­lio volatil­i­ty is always low­er than the volatil­i­ty of the com­po­nent strate­gies if they are not iden­ti­cal.

Espe­cial­ly for futures traders this is the mag­ic of diver­si­fi­ca­tion. In the above exam­ple you get the same return — 50% of cap­i­tal in strate­gies yield­ing 15% each results in a port­fo­lio with a return of also 15%. But the volatil­i­ty (or risk) of the port­fo­lio drops to 17%. The Sharpe ratio of com­bin­ing these two strat­e­gy there­fore ris­es from 0.75 to 0.87.

Because futures are lever­aged instru­ments it is way eas­i­er for futures traders to con­trol their port­fo­lio volatil­i­ty. You are able to crank up the return if you pre­fer to have a risk lev­el of 20% instead of hav­ing the same return but a low­er risk of 17%.

The diver­si­fi­ca­tion you get is rough­ly lin­ear for strate­gies with a cor­re­la­tion between ‑50% up to +100%. In real­i­ty find­ing strate­gies that are cor­re­lat­ed below ‑50% and have pos­i­tive expectan­cy is quite uncom­mon.

Nev­er­the­less this qua­si lin­ear rela­tion­ship is quite a dri­ver of per­for­mance. With two uncor­re­lat­ed strate­gies (cor­re­la­tion 0%) you get a risk reduc­tion of 30%. But even with a cor­re­la­tion of 50% you still get a risk reduc­tion (or per­for­mance increase) of still 13%. That’s an impres­sive rela­tion­ship!

Before you get too excit­ed, let us dis­cuss the for­mu­la in prac­tice. Unfor­tu­nate­ly none of the para­me­ters in the for­mu­la are sta­ble in prac­tice. Espe­cial­ly cor­re­la­tions of under­ly­ing assets tend to increase togeth­er in finan­cial sit­u­a­tions of stress. In such a sit­u­a­tion, every­body tries to sell and all assets fall togeth­er and your assumed diver­si­fi­ca­tion effect breaks down.

Math­e­mat­i­cal­ly the cor­re­la­tion also only cap­tures lin­ear rela­tion­ships between assets. That’s some­thing to keep in mind — think about it more as a rough mea­sure­ment about how much two time series move togeth­er. The real rela­tion­ship between them is also most like­ly not lin­ear.

There are oth­er mod­els out there that try to cap­ture rela­tion­ships between time series like cop­u­las or coin­te­gra­tion but they also have assump­tions that are ques­tion­able in real­i­ty and use more para­me­ters — and that is some­thing to avoid unless there are very strong incen­tives in doing it.

But even the volatil­i­ty of a strat­e­gy is an unknown quan­ti­ty that may exhib­it wild swings. So please don’t take this for­mu­la too lit­er­al­ly and crank up the lever­age like there is no tomor­row! This even hap­pened to the very smart peo­ple of Long-Term Cap­i­tal Man­age­ment (one of them even a Nobel lau­re­ate) — you can read about this in the excel­lent book of Roger Lowen­stein When Genius Failed.

Take this con­cept more as a guide to improve your trad­ing sys­tem and know about dif­fer­ent sources of diver­si­fi­ca­tion. This way you can prof­it from the con­cept with­out lulling your­self into a false sense of secu­ri­ty.

Let’s meet Reality

So far we have learned that diver­si­fi­ca­tion works in the­o­ry.

The inter­est­ing ques­tion is whether para­me­ter diver­si­fi­ca­tion behaves the same way in a real trad­ing sys­tem.

We will do so by apply­ing it to a real-world strat­e­gy: Cross-sec­tion­al momen­tum from Robert Carver’s book Advanced Futures Trad­ing Strate­gies.

What is cross-sec­tion­al momen­tum? It cap­tures a sig­nal from an asset that out- or under­per­forms oth­er assets in its group.

Let’s say you grouped your trade­able equi­ty indices by coun­try and the US-list­ed tech-heavy Nas­daq index out­per­forms the broad­er S&P 500 over a longer peri­od due to some tech rev­o­lu­tion espe­cial­ly tech com­pa­nies prof­it from.

That’s an exam­ple of cross-sec­tion­al momen­tum. Some under­ly­ing fac­tor plays out bet­ter over a time peri­od for one spe­cif­ic asset than oth­er assets in the same group — in this case bet­ter com­pared to oth­er broad­er US indices.

At first glance, this sounds like a mean­ing­ful strat­e­gy we can work with, because the eco­nom­ic ratio­nale makes sense. This is a very impor­tant first step in strat­e­gy design and should nev­er be skipped. Don’t just run back­tests for some for­mu­las and pick the strat­e­gy, that per­formed best in a back­test. That is a recipe for dis­as­ter!

Cross-sectional Momentum

Let’s define a trad­ing strat­e­gy around this eco­nom­ic intu­ition:

This con­tin­u­ous trad­ing sig­nal has to be con­vert­ed to actu­al trad­ing posi­tions and each of the instru­ments can be trad­ed sep­a­rate­ly. Let’s not bog down in fur­ther details like only trad­ing instru­ments that are cheap enough (com­mis­sion or spread wise) and trade just all of them.

The more instru­ments you trade, the more diver­si­fi­ca­tion you get. Fig­ure 2 shows my cur­rent uni­verse of trade­able instru­ments over time. The his­toric data of the uni­verse starts in June 1975. Back then there were only a few clas­sic futures like soy­beans or US bonds.

Over time the futures mar­ket got more and more diverse, new instru­ments entered the mar­ket, some of them suc­cess­ful oth­ers have been dis­con­tin­ued (like the famous pork bel­lies). In aggre­gate, more and more instru­ments have been trade­able over time.

Now let’s plug the cables direct­ly in the wall and trade all of them avail­able at each point in time. To do that, we actu­al­ly need to decide which look­back peri­od s to use for our cross-sec­tion­al momen­tum strat­e­gy.

Which look­back peri­od should we choose? Twen­ty days? One hun­dred days? Are there time frames that do not work at all? This turns out to be sur­pris­ing­ly dif­fi­cult.

The Parameter Surface

With­out look­ing at any back­tests, it is clear by the def­i­n­i­tion of the sig­nal that short­er time frames will trade more often because \(\mathsf{R_t}\) will be more volatile due to a short­er rel­e­vant his­to­ry. It will there­fore vary more from day to day and may switch between long and short more often.

This will incur high­er trad­ing costs and most prob­a­bly impact per­for­mance. To get a bet­ter feel­ing for the sig­nal, let’s run back­tests for a range of dif­fer­ent look­back peri­ods s.

If para­me­ter diver­si­fi­ca­tion is going to work, we first need a strat­e­gy whose per­for­mance is rea­son­ably sta­ble across para­me­ters. Fig­ure 3 shows the Sharpe ratios for a range of look­backs between 20 and 350.

Now that is quite some­thing! It is a trad­ing sig­nal that works pret­ty well: The sig­nal has a Sharpe ratio of about 0.5 or high­er for a para­me­ter range from about 30 to 280. That is actu­al­ly a very impor­tant prop­er­ty of any worth­while strat­e­gy. If the para­me­ter sur­face is very noisy, you either have not enough instru­ments in your uni­verse, the sig­nal is not defined in a sta­ble way or it is not good in gen­er­al.

Con­sid­er one par­tic­u­lar strat­e­gy of the whole flock: Look­back peri­od s=215. This is the best strat­e­gy with a Sharpe ratio of 0.93. Its equi­ty curve and draw­down is shown in fig­ure 4.

There are sev­er­al impor­tant obser­va­tions of this graph. The per­for­mance seems to dete­ri­o­rate after about the year 2010. Bad draw­downs are about ‑40% and espe­cial­ly pro­nounced in dura­tion after 2010.

Let’s have a look at some per­for­mance sta­tis­tics of this para­me­ter from table 1:

Espe­cial­ly the large draw­down dura­tion of the strat­e­gy is of some con­cern. A whop­ping 1837 days, that’s over 5 years! The small pos­i­tive skew­ness of the strat­e­gy is an inter­est­ing prop­er­ty. On the neg­a­tive side, the return stream of this look­back peri­od has fat­ter tails on the down­side than the upside.

Selecting Parameters

Should you trade this par­tic­u­lar para­me­ter, because it is the best? This is pre­cise­ly the point where inex­pe­ri­enced quants get it wrong. The look­back of s=215 is the best look­back peri­od in a finite sam­ple of his­tor­i­cal data. If you do this opti­miza­tion over a sub­set of instru­ments or only parts of your his­to­ry, it will not be the best any more.

Nev­er­the­less there seems to be some­thing there. The per­for­mance increas­es steadi­ly from a look­back of around 30 up to the max­i­mum of 215 and then decreas­es again. Cross-sec­tion­al momen­tum seems to play out on a big­ger range of time frames.

A sen­si­ble approach is to trade sev­er­al para­me­ters of this strat­e­gy. But which ones? We need a bet­ter frame­work to make a deci­sion.

If you remem­ber the intro­duc­tion, we may also kill sev­er­al birds with one stone. As the return streams of dif­fer­ent para­me­ters are not per­fect­ly cor­re­lat­ed, we get some diver­si­fi­ca­tion and can trade a range of cross-sec­tion­al momen­tum with­out com­mit­ting us to just one time frame.

We can answer this by com­put­ing a cor­re­la­tion matrix of the return streams of the dif­fer­ent para­me­ters. Fig­ure 5 shows just that. You may notice that espe­cial­ly larg­er val­ues of the look­back peri­od s are high­er cor­re­lat­ed than low­er val­ues.

The expla­na­tion for this lies in the design of the trad­ing sig­nal. The sig­nal for the look­back peri­od s=345 shares 345 (98%) data points with the sig­nal for s=350. There­fore all the out­per­for­mances of these neigh­bor­ing para­me­ters will be quite sim­i­lar. Some short­er look­backs like s=20 on the oth­er hand share only 15 (75%) of their data with their neigh­bor s=15. The expo­nen­tial weight­ning has anoth­er decay for each of the para­me­ters but will not trans­form shared data in some­thing total­ly dif­fer­ent.

Because of this it is not advis­able to trade all of the para­me­ters. Doing that results in over­weight­ing the larg­er para­me­ters.

One approach is to trade para­me­ters that have a max­i­mum cor­re­la­tion between them. Which thresh­old you choose is not that impor­tant. Each has the prop­er­ty that you will trade less of the high­er cor­re­lat­ed large para­me­ters. I will use a max­i­mum cor­re­la­tion of 95% between neigh­bor­ing para­me­ters and want to cap­ture strate­gies with a Sharpe ratio of 0.6 or high­er.

The rel­e­vant para­me­ter range comes down to look­backs between 40 and 270. Start­ing with 270 and then search­ing for the next para­me­ter with a cor­re­la­tion of 95% or less and repeat­ing this pro­ce­dure you end up with the para­me­ter set of 270, 230, 195, 165, 135, 110, 90, 75, 60 and 45.

As you are read­ing an arti­cle from Sys­tem­at­ic Trading—Made in Ger­many, I will step into my country’s tra­di­tion of over-engi­neer­ing and also pro­vide you with the algo­rithm to select para­me­ters to trade from a giv­en range.

params <- c(lower, ... , upper)
result <- c(upper)
param <- upper
while (param >= lower) {
 if (cor(param, result[1]) < 95%) {
  selected <- c(param, result)
 }
 next param
} 

The equi­ty curve and draw­down of the par­tic­u­lar set of para­me­ters is shown in fig­ure 6 and table 2 shows some sta­tis­tics of the strat­e­gy.

If you com­pare table 2 with the sta­tis­tics of the best strat­e­gy of table 1, you will find that the strat­e­gy with a mix of look­back peri­ods is worse than the best strat­e­gy: The Sharpe ratio drops from 0.93 to 0.83 and the max­i­mum draw­down increas­es to 50%. The skew and tail prop­er­ties are about the same, only the max­i­mum draw­down dura­tion decreased by over a year.

So where is the ben­e­fit of diver­si­fi­ca­tion?

Above com­par­i­son was made to show you how dan­ger­ous it is to com­pare any­thing against some data mined best strat­e­gy. The cor­rect com­par­i­son is to com­pare the aver­age Sharpe ratio of our para­me­ter range (0.72) to the para­me­ter mix with­in this range (0.83).

The cor­re­la­tions of the select­ed para­me­ters are shown in fig­ure 7. The Sharpe ratio increase of 0.1 cor­re­sponds to an improve­ment of 15%. As can be seen from fig­ure 1, such an improve­ment can be expect­ed from a com­bi­na­tion of return streams that are cor­re­lat­ed by 44%. All in all, not bad!

Of course you may argue that such com­par­isons are not very help­ful, because the strate­gies also have dif­fer­ent expect­ed returns, which also play a role in the Sharpe ratio. Diver­si­fi­ca­tion in prac­tice is a much more messy con­cept than in the­o­ry.

Nev­er­the­less, the cen­tral idea sur­vives con­tact with real­i­ty. Diver­si­fy­ing across care­ful­ly select­ed para­me­ter­i­za­tions does not cre­ate mir­a­cles, but it reduces depen­dence on any sin­gle choice.

Instead of bet­ting every­thing on one look­back peri­od, we allow sev­er­al rea­son­able vari­ants to work togeth­er. In sys­tem­at­ic trad­ing, that is often the clos­est thing to a free lunch that actu­al­ly exists.