Send a Message

[contact-form-7 id="2362" title="Send a Message"]

The Importance and Value of a Non-Normal Distribution of Returns

Down­load PDF


The Shape of Things to Come

In this, the fourth and final post in our Math Mat­ters series, we dis­cuss dis­tri­b­u­tion of returns and how investors react to “tail” events.

Every­thing we dis­cussed pre­vi­ous­ly in this series, “Com­pound Growth”, “The Impor­tance of Avoid­ing Large Loss­es”, and “Volatil­i­ty is a Drag”, cul­mi­nates with this post on the shape of the dis­tri­b­u­tion of returns.

In the image below we see the text­book def­i­n­i­tion of a nor­mal Gauss­ian dis­tri­b­u­tion. Most of the occur­rences fall near the mean val­ue, while a few data points occur far from the mean. If a set of data over a long peri­od of time falls into this pat­tern, then we can make assump­tions about future occur­rences with a high degree of accu­ra­cy.

Gaussian Distribution Curve | Blog

How­ev­er, in the real world two unfor­tu­nate facts col­lide with this the­o­ry and are imped­i­ments to an investor’s long-term suc­cess:

  1. When these tail events do occur, investors tend to make poor choic­es.
  2. Actu­al mar­ket returns do not fit the nice, clean, sym­met­ric lay­out of the nor­mal dis­tri­b­u­tion.

It is often said that two things dri­ve the mar­ket: fear and greed. That might not be true all of the time, but it does appear to be true when mar­kets are at their extremes.

A recent study by DALBAR found that the aver­age investor did much worse than the broad mar­ket. More­over, the biggest gaps in under­per­for­mance tend­ed to hap­pen when mar­kets were expe­ri­enc­ing “tail” events.


Top 10 Months of Most Acute Investor Underperformance – Dalbar, Inc. | Swan Blog

Any­one with mar­ket expe­ri­ence will rec­og­nize the cul­prits: pan­ic sell­ing in a bear mar­ket, chas­ing after “hot” sto­ries in a bull mar­ket, sell­ing low and buy­ing high…all of these are quan­ti­fied in the above table. And in the pre­vi­ous posts on the math­e­mat­ics of invest­ing, we’ve seen how hard it is to over­come these mis­takes once they hap­pen. Sad­ly, investors are often their own worst ene­my.

Anoth­er issue is that actu­al mar­ket returns don’t fit the ide­al­ized nor­mal dis­tri­b­u­tion.

Normal vs. Non-Normal Distribution of Returns

In this illus­tra­tion below, we see the return dis­tri­b­u­tion for the Swan Defined Risk Strat­e­gy, as rep­re­sent­ed by the Select Com­pos­ite, super­im­posed over that of the S&P 500 for the peri­od July 1997 to Decem­ber 2015.  It has two char­ac­ter­is­tics that mark its devi­a­tion from the nor­mal dis­tri­b­u­tion:

  1. the S&P 500 has neg­a­tive skew­ness, mean­ing the left-hand, neg­a­tive tail is more extreme than the right-hand, pos­i­tive tail; and
  2. the dis­tri­b­u­tion has pos­i­tive excess kur­to­sis, mean­ing that the volatil­i­ty that does occur tends to be dri­ven by those extreme mar­ket events.


Distribution of Returns – Swan DRS vs SP 500 | Swan Blog



We see how the DRS has effec­tive­ly min­i­mized the impact of the tail events by push­ing more of the obser­va­tions towards the mean. While a dis­tri­b­u­tion like this might be called “safe and bor­ing” by some, we believe these types of results are best suit­ed for long-term suc­cess.

In this post and the pre­vi­ous posts in this series, we have estab­lished that extreme loss­es and high volatil­i­ty are harm­ful to investors from both a math­e­mat­i­cal as well as a behav­ioral stand­point. And yet in the graph above we see that the actu­al mar­ket con­di­tions are worse than what would have been pre­dict­ed by a “nor­mal” dis­tri­b­u­tion.

How to Seek a Non-Normal Distribution of Returns

Faced with these chal­lenges, how should investors pro­ceed?

At Swan Glob­al Invest­ments, we would argue that these prob­lems are best avoid­ed by imple­ment­ing a strat­e­gy that seeks to reduce the impact of the tails and volatil­i­ty as much as pos­si­ble. If returns could be moved from the tails of the dis­tri­b­u­tion to the mid­dle range of the dis­tri­b­u­tion, the investor would be bet­ter off.

Across these four blog posts and our white paper “Math Mat­ters: Rethink­ing the Math Behind Invest­ment Returns” we estab­lished four key points to achiev­ing bet­ter invest­ment results over time. They are:

  1. The impor­tance and pow­er of com­pound­ing
  2. The val­ue of avoid­ing large loss­es
  3. The impor­tance of vari­ance drain
  4. The val­ue of a non-nor­mal dis­tri­b­u­tion of returns

More­over, as an invest­ment strat­e­gy, the Defined Risk Strat­e­gy was meant to be a man­i­fes­ta­tion of these prin­ci­ples.

The DRS should be viewed as a long-term, strate­gic solu­tion and one that we believe has the best chance of help­ing investors stay invest­ed and reach their goals over time.

To learn more about Swan’s Defined Risk invest­ment approach or for more details regard­ing his­tor­i­cal per­for­mance uti­liz­ing put options to hedge a port­fo­lio since 1997, please con­tact Swan at 970–382-8901.


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 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 non­qual­i­fied 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. All Swan prod­ucts uti­lize the Defined Risk Strat­e­gy (“DRS”), but may vary by asset class, reg­u­la­to­ry offer­ing type, etc. Accord­ing­ly, all Swan DRS prod­uct offer­ings will have dif­fer­ent per­for­mance results, and com­par­ing results among the Swan prod­ucts and com­pos­ites may be of lim­it­ed use. Eco­nom­ic fac­tors, mar­ket con­di­tions, and invest­ment strate­gies will affect the per­for­mance of any port­fo­lio and there are no assur­ances that it will match or out­per­form any par­tic­u­lar bench­mark. There­fore, 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.  This analy­sis is not a guar­an­tee or indi­ca­tion of future per­for­mance. 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 or vis­it  289-SGI-111016[/fusion_builder_column][/fusion_builder_row][/fusion_builder_container]

By |2018-10-02T11:27:43+00:00December 1st, 2016|Blog|Comments Off on The Importance of Distribution of Returns