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The Mathematics of Diversification

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In a pre­vi­ous blog post on diver­si­fi­ca­tion, we dis­cussed the key role that cor­re­la­tions play in the suc­cess or fail­ure of a diver­si­fi­ca­tion strat­e­gy. In this post, we dig a lit­tle bit deep­er to see how the num­bers play out. Although it is cer­tain­ly pos­si­ble to go over­board with the math­e­mat­ics when it comes to port­fo­lio opti­miza­tion, we will keep it sim­ple and straight­for­ward in this dis­cus­sion.

When Har­ry Markowitz start­ed an intel­lec­tu­al rev­o­lu­tion with his sem­i­nal paper “Port­fo­lio Selec­tion” in 1952, one of his key con­tri­bu­tions was quan­ti­fy­ing the impact that cor­re­la­tions between pairs of invest­ments had on reduc­ing a portfolio’s over­all risk. The idea was that the con­tri­bu­tion to over­all port­fo­lio risk that a sin­gle invest­ment brought to a port­fo­lio was not just its weight in the port­fo­lio, but how it inter­act­ed with the oth­er con­stituents of the port­fo­lio.

Calculating the Risk and Return of Portfolio Components

This inter­ac­tion is mea­sured by cor­re­la­tion. Tech­ni­cal­ly speak­ing, cor­re­la­tion is a scaled ver­sion of covari­ance. Covari­ance mea­sures how two vari­ables change in rela­tion­ship to one anoth­er. Cor­re­la­tion nor­mal­izes covari­ance to a scale rang­ing from -1.0 to +1.0.

Cal­cu­lat­ing how much return the var­i­ous invest­ments con­tribute to a port­fo­lio return is quite straight­for­ward. It is sim­ply the weight­ed aver­age of the returns of each indi­vid­ual invest­ment. The gen­er­al­ized for­mu­la looks like this:

Equation calculating investment's contribution to return |Swan Blog - Math of Diversification

But for simplicity’s sake let’s assume a two-asset port­fo­lio, in which case the for­mu­la is this:

Formula for calculating return contribution of two asset portfolio |Swan Blog - Math of Diversification

Pret­ty sim­ple, right? The weight of one asset (), mul­ti­plied by its return () plus the weight of the oth­er asset (), mul­ti­plied by its return ().

The for­mu­la for cal­cu­lat­ing an investment’s con­tri­bu­tion of risk to a port­fo­lio is more com­pli­cat­ed, but that’s actu­al­ly a good thing. Cor­re­la­tion is the “X-fac­tor” that has the poten­tial to low­er a portfolio’s over­all volatil­i­ty. As before, “w” is the weight and the new terms “σ” and “ρ” sig­ni­fy the stan­dard devi­a­tion and covari­ance, respec­tive­ly.  The for­mu­la for a portfolio’s vari­ance is:

Formual for calculating investment's contribution to risk in a portfolio|Swan Blog - Math of Diversification


Again, the gen­er­al­ized for­mu­la looks intim­i­dat­ing, but break­ing it out in to a sim­ple two-asset port­fo­lio makes it eas­i­er to under­stand.

Formual for calculating investment's contribution to risk in a two asset portfolio|Swan Blog - Math of Diversification

The first two terms in the equa­tion rep­re­sent the stand-alone con­tri­bu­tion of the risk of assets A and B to the port­fo­lio. Because the weights will be rep­re­sent­ed by a dec­i­mal and because they are being squared, the impact of the stand­alone risk will always be less than the weight. For exam­ple, an 80% weight squares out to .64, and a 40% weight squared would be .16.

This makes the third term, the com­bined risk of assets A and B, pret­ty impor­tant. The sym­bol for cor­re­la­tion is high­light­ed in red.

Boiling Down the Diversification Math

By work­ing through the for­mu­la, it becomes evi­dent just how impor­tant that piece of the puz­zle is.

  • If the cor­re­la­tion between A and B is 1.0, then over­all vari­ance is not reduced at all.
  • If the cor­re­la­tion between A and B is 0.0, the whole third term of the equa­tion will be erased and the vari­ance of the port­fo­lio will sim­ply be the first two terms — a big improve­ment.
  • If cor­re­la­tions are neg­a­tive, then the whole third term becomes neg­a­tive, and we actu­al­ly get to sub­tract the third term from the first two.

That’s when risk reduc­tion real­ly comes in to play.

By walk­ing through the for­mu­la above, we see how impor­tant cor­re­la­tion is to the over­all risk equa­tion.

Unfor­tu­nate­ly, this is where most diver­si­fi­ca­tion strate­gies fall apart.

Although this post is based on math­e­mat­ics, the old say­ing “a pic­ture is worth a thou­sand words” holds true.  In the three illus­tra­tions below, we attempt to show the how high cor­re­la­tions fail to bal­ance risk and how low cor­re­la­tions help to even the scales.

See Saw illustration - Two assets with correlations near 1.0 |Swan Blog - Math of Diversification

See Saw illustration - Two assets with correlations near 0.0 |Swan Blog - Math of Diversification

See Saw illustration - Two assets with correlations near -1.0 |Swan Blog - Math of Diversification


Be Wary of False Diversification

As dis­cussed in our pre­vi­ous blog post “Serv­ing Up Diver­si­fi­ca­tion — The Impor­tance of Cor­re­la­tions to Diver­si­fi­ca­tion”, many investors pur­sued what we call “false diver­si­fi­ca­tion” strate­gies, where the cor­re­la­tions might have been 0.80, 0.90, or high­er. Plug­ging such num­bers into our for­mu­la above does lit­tle to address over­all port­fo­lio risk.

Even if port­fo­lios were built using asset class­es or styles where the long-term cor­re­la­tions were low, the unfor­tu­nate real­i­ty is that cor­re­la­tions spiked in the midst of a cri­sis.  Dur­ing a major mar­ket sell-off, the car­nage tends to be wide­spread as pan­icked investors hit the “sell” but­ton indis­crim­i­nate­ly. The ben­e­fits of diver­si­fi­ca­tion tend to evap­o­rate right when they are most need­ed. In the exam­ple below, we see how some asset class­es used for diver­si­fi­ca­tion pur­pos­es actu­al­ly per­formed worse than the core S&P 500 dur­ing major mar­ket down­turns.

Investors, there­fore, need a way to deal with risk direct­ly.

Performance of Various Assets During Credit Crises |Swan Blog - Math of Diversification



Diversification by Design

The Swan Defined Risk Strat­e­gy was designed to have dif­fer­ent com­po­nents con­tribut­ing to per­for­mance at dif­fer­ent points in time.

  1. Pri­mar­i­ly the DRS is always invest­ed in an under­ly­ing asset, com­prised of all 9 Select Sec­tor SPDRs.
  2. Then the DRS applies a hedge posi­tion on the full notion­al val­ue of that under­ly­ing asset, in the form of long-term put LEAPS. The put option has a neg­a­tive cor­re­la­tion to the equi­ty por­tion.
  3. The pre­mi­um-sell­ing, income trades are designed to be mar­ket neu­tral, with the intent of hav­ing a zero cor­re­la­tion to mar­ket direc­tion.

By com­bin­ing our equi­ty under­ly­ing, hedge, and income com­po­nents, we believe the DRS is a bet­ter way of imple­ment­ing a tru­ly diver­si­fied strat­e­gy.

To learn more about Swan’s DRS invest­ment approach and how this approach has fared in the past, please con­tact Swan Glob­al Invest­ments at 970–382-8901, or vis­it our Con­tact page.


Marc Odo, Marc Odo, CFA®, CAIA®, CIPM®, CFP®, Director of Investment Solutions - Swan Global InvestmentsAbout the author: Marc Odo, CFA®, CAIA®, CIPM®, CFP®, Direc­tor of Invest­ment Solu­tions, is respon­si­ble for help­ing clients and prospects gain a detailed under­stand­ing of Swan’s Defined Risk Strat­e­gy, includ­ing how it fits into an over­all invest­ment strat­e­gy. For­mer­ly Marc was the Direc­tor of Research for 11 years at Zephyr Asso­ciates.



Impor­tant Notes and Dis­clo­sures:

Swan Glob­al Invest­ments, LLC is a SEC reg­is­tered Invest­ment Advi­sor that spe­cial­izes in man­ag­ing mon­ey using the pro­pri­etary Defined Risk Strat­e­gy (“DRS”). SEC reg­is­tra­tion does not denote any spe­cial train­ing or qual­i­fi­ca­tion con­ferred by the SEC. Swan offers and man­ages the DRS for investors includ­ing indi­vid­u­als, insti­tu­tions and oth­er invest­ment advi­sor firms. Any his­tor­i­cal num­bers, awards and recog­ni­tions pre­sent­ed are based on the per­for­mance of a (GIPS®) com­pos­ite, Swan’s DRS Select Com­pos­ite, which includes non-qual­i­fied dis­cre­tionary accounts invest­ed in since incep­tion, July 1997, and are net of fees and expens­es. Swan claims com­pli­ance with the Glob­al Invest­ment Per­for­mance Stan­dards (GIPS®). All data used here­in; includ­ing the sta­tis­ti­cal infor­ma­tion, ver­i­fi­ca­tion and per­for­mance reports are avail­able upon request. The S&P 500 Index is a mar­ket cap weight­ed index of 500 wide­ly held stocks often used as a proxy for the over­all U.S. equi­ty mar­ket. Index­es are unman­aged and have no fees or expens­es. An invest­ment can­not be made direct­ly in an index. Swan’s invest­ments may con­sist of secu­ri­ties which vary sig­nif­i­cant­ly from those in the bench­mark index­es list­ed above and per­for­mance cal­cu­la­tion meth­ods may not be entire­ly com­pa­ra­ble. Accord­ing­ly, com­par­ing results shown to those of such index­es may be of lim­it­ed use. The adviser’s depen­dence on its DRS process and judg­ments about the attrac­tive­ness, val­ue and poten­tial appre­ci­a­tion of par­tic­u­lar ETFs and options in which the advis­er invests or writes may prove to be incor­rect and may not pro­duce the desired results. There is no guar­an­tee any invest­ment or the DRS will meet its objec­tives. All invest­ments involve the risk of poten­tial invest­ment loss­es as well as the poten­tial for invest­ment gains. Pri­or per­for­mance is not a guar­an­tee of future results and there can be no assur­ance, and investors should not assume, that future per­for­mance will be com­pa­ra­ble to past per­for­mance. All invest­ment strate­gies have the poten­tial for prof­it or loss. Fur­ther infor­ma­tion is avail­able upon request by con­tact­ing the com­pa­ny direct­ly at 970–382-8901   065-SGI-121715

By |2018-10-09T16:32:17+00:00December 15th, 2015|Blog|Comments Off on The Mathematics of Diversification