Your Portfolio is Not as Diversified as You Think

Financial investments are based on future expectations regarding returns and risks.  Historical data provides guidelines for estimating both.  While expected returns often are then adjusted up or down based on contemporaneous information not captured by past data, the measures of risk invariably rely on historical data and vary primarily due to the time horizon used in the calculation.  The risk for a single asset is measured by its variance, or the square root ov variance (the standard deviation).  The risk for a portfolio is measured by the collection of standard deviations and their covariances, and the correlation between any two assets is measured by the ratio of their covariance to the product of their standard deviations.  When moving from two assets to a portfolio of multiple assets, corresponding formulae are applied by expressing these statistical terms in vectors and matrices and applying linear algebra.  Neat, tidy, and easy to program a computer to do.  Alas, this occurs so routinely that it is done without thought.

However, a standard deviation is a proper measure of variability only when the underlying variable can be assumed to follow a symmetric distribution, and correlation as invariably measured using the classic Pearson formula is a proper measure of statistical association only when the underlying association is linear.  While there is theory to support return (or log-return) symmetry, there is no theory to support linearity.

Pearson correlation is the measure of association taught in every introductory course.  Sometime later one might be introduced to the Spearman rank-order correlation, although within investment circles even respected websites like Investopedia (https://www.investopedia.com/terms/c/correlationcoefficient.asp) does not suggest alternatives to classic Pearson, and claim: “By adding a low or negatively correlated mutual fund to an existing portfolio, the investor gains diversification benefits”.

But do they?  Can classic Pearson estimates by trusted without considering alternatives — and without at least visually inspecting the pairwise association over time of a pair of asset returns?  There are several alternatives, each with potential to better flush out the true underlying association between and among variables that deviate from an underlying linear pattern.  The R software has functions to measure several alternatives.  “R” is a freely-available open-source statistical computing and visualization environment that you can download or access from the cloud, and it is accompanied by about 60,000 “packages” that implement ideas across disciplines (to access “R” or learn more about it, click the “Computing Resources” link within the “Resource Links” tab at our website (https://retirementfinance.org/resources/).  Below is a list of five correlation measures that are implemented in “R” and well-documented for ease of use.  

Five Measures of Correlation R Package Package Function

  • Pearson Correlation stats cor()
  • Spearman Correlation pspearman pspearman()
  • Hoeffding Correlation Hmisc foeffd()
  • Maximal Correlation acepack ace()
  • Distance Correlation energy dcor()

Following are examples where reliance on classic Pearson correlation may mislead.  For each, three correlation measures are provided: classic Pearson (with which you likely are familiar); Spearman correlation (which substitutes rank-orders for the raw data, thereby tempering the impact of outliers), and; Maximal correlation (which considers various transformations separately for each variable and then selects the two transformations that provide the highest correlation between the two transformed variables.  Maximal correlation equals 0.00 when the variables are independent, and equals 1.00 when the Pearson correlation between the optimally transformed pair of variables equals 1.00.  Maximal correlation measures strength of relationship, not direction (since patterns can be multidirectional).  So, for any pair of asset returns measured over time, Pearson correlation captures linear trends, Spearman correlation captures monotone trends, and Maximal correlation captures complex trends.

Example A: The Anscombe Quartet

Anscombe (https://garstats.files.wordpress.com/2016/08/anscombe-as-1973.pdf) used simulated data to demonstrate the power of visual observation.  He create four variants of dt pairings, each with an outcome (y) and a predictor (x).  All sets have 11 observations, the same mean of x (9) and y (7.5), the same fitted regression line (y = 3 + 0.5 x), the same regression and residual sum of squares and therefore the same r-squared (0.67) and same classic Pearson correlation (0.816).  The summary measures are the same across the four pairings, but the situations vary greatly.  Pairing #1 confirms to a pure linear relationship. #2 is nonlinear, #3 has an outlier, and #4 an influential point.  The four displays below make clear these are four different scenarios.

Example B: A Messy Arc

An example of a perfect downward bending arc can be formed by letting x range from -100 to +100, and setting y equal to the square root of 100 squared minus x squared.

In this example, both the classic Pearson and the Spearman rank correlation measures equal 0.00, whereas the maximal correlation coefficien equals 1.00 (actually 0.999817). 

A more plausible relationship between asset returns might add noise to y, adding an unbiased random normal element to each pairing with standard deviation set to 10.  In this modified example, shown below, it is clear that the relationship between returns for Asset x and returns for Asset y is nonlinear, better captured by the arc than straight line.

Mechanical application of modern portfolio theory will include classic Pearson correlation as a key ingredient in estimation portfolio risk, clearly understating degree of association.

Pearson (and Spearman) correlation is measured in the [-1, +1] range,  with the signage signalling direction of the monotonic association.  In contrast, maximal correlation is expressed within the [0, +1] range, expressing strength of association without regard for direction.  As seen in this example, the direction is positive for low x, negative for high x.

Example C: Pennington Fund v. S&P500
The Pennington Fund (https://fund.pennington-trading.com/) features a managed trading platform (MTP) with, in their words, returns that “have historically had a near-zero relation to the S&P500”.  Their portfolio is a mix of blue chip stocks, initial public offerings (IPOs), options, real estate investment trusts (REITs), a commodities fund, and a client investment margin account.  Over a recent 55-month period both the classic Pearson and the Spearman rank correlation measures indeed were near zero — but the maximal correlation coefficient (0.352) indicates potentially far less diversification than presumed.

Example D: Asset and Sector ETFs

Clearly, mechanical or robotic application of classic Pearson correlation measures to assess degree of diversification among assets or asset classes can be misleading.  It nonetheless is done routinely by financial advisors and professionals, and reported without recognition of caveats such as the potential for curvatures, outliers, influential points or other empirical possibilities that are assumed away by Pearson correlations. 

The Portfolio Visualizer website (https://www.portfoliovisualizer.com/asset-class-correlations

provides a correlation matrix for 14 common exchange-traded funds (ETFs), representing typical asset classes and subclasses.  Asset class Pearson correlations are for the time period 05/01/2009 – 08/31/2020, based on monthly returns and assuming linear relations.

Ticker Name

VTI Vanguard Total Stock Market ETF

VO Vanguard Mid-Cap ETF

VB Vanguard Small-Cap ETF

SHY iShares 1-3 Year Treasury Bond ETF

BND Vanguard Total Bond Market ETF

TLT iShares 20+ Year Treasury Bond ETF

TIP iShares TIPS Bond ETF

MUB iShares National Muni Bond ETF

VEU Vanguard FTSE All-Wld ex-US ETF

VSS Vanguard FTSE All-Wld ex-US SmCp ETF

VWO Vanguard FTSE Emerging Markets ETF

VNQ Vanguard Real Estate ETF

DBC Invesco DB Commodity Tracking

GLD SPDR Gold Shares

Conclusion

Before investing on the basis of this information, compute maximal correlations; while the truth may well lie somewhere between the two correlation estimates, collecting and plotting data pairs likely will be informative, and may suggest less diversifaction value.

Especially for those nearing or in retirement, a conservative approach to investing in risky portfolios likely is prudent, as there may not be opportunities to recover losses.  In such cases a conservative approach would be to measure risk using Maximal correlations.

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  163. จริงๆเล่นมาหลายเว็บไซต์เลยจ้ะครับผม ส่วนใหญ่ก็จ่ายจริงครับ แต่ที่ต่างเป็นบางเว็บเวลาฝาก-ถอนจึงควรผ่านผู้แทนนะครับ จำเป็นที่จะต้องแคปจอส่งให้บุคลากร ผมว่าเจ็บป่วยเลย มี UFABET นี่แหละขอรับ ผมว่าระบบดีสุดเลย ฝากถอนอัตโนมัติทำเองได้เลยขอรับ

  164. บาคาร่าออนไลน์เป็นเกมพนันออนไลน์ที่ต้องใจคนอีกจำนวนไม่น้อยเลยครับ เพราะนอกจากจะเล่นง่ายยังทำเงินได้จริง UFABET นอกเหนือจากที่จะมีบาคาร่า ยังมีเกมอื่นๆมีมากมายไม่ว่าจะเป็นแทงบอล สล็อต ยิงปลา เรียกว่ายกคาสิโนมาไว้บนเว็บไซต์เลยนะครับ สมัครง่าย ได้เงินเร็ว ระบบฝากถอนแบบอัตโนมัติขอรับ

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  170. เเล่นบาคาร่าออนไลน์ได้อย่างไม่ยากเย็นเพียงแต่เข้ามาที่ UFABET กดเข้าหน้าสมัครก็สารมารถทำเงินได้ในทันทีทันใด พวกเราเป็นคาสิโนออนไลน์ที่ดีที่สุด ระบบใช้งานง่ายเป็นมีคณะทำงานคอยให้คำแนะนำและก็เสนอแนะถ้าหากเกิดปัญหา บริการดีกันเอง จะเล่นพันออนไลน์ทั้งทีจำเป็นที่จะต้องUFABET

  171. ถ้าเกิดจะถามหาเกมออนไลน์ที่เล่นง่ายและทำเงินได้ง่ายที่สุดปะทุงหนีไม่พ้นบาคาร่าออนไลน์ UFABET ก็เลยเอาใจสมาชิกโดยการรวมรวมเกมบาคาร่าออนไลน์ไว้มากไม่น้อยเลยทีเดียวเพื่อสามาชิกได้เลือกเล่นอย่างเต็มเปี่ยมนอกจากนั้นยังมีเกมฯลฯ

  172. ชี้แนะเลยขอรับที่นี่เลย UFABET มีเกมพนันออนไลน์มาก มีทุกเกมมีทั้งหมดทุกอย่าง ที่ถูกใจสุดเห็นจะคือเรื่องของระบบครับผม ฝากถอนอัตโนมัติและก็มีข้างบริการลูกค้ารอช่วยเหลือ โปรโมชั่นเยอะแยะครับผม ที่สำคัญไปกว่านั้นเป็น เค้าจ่ายจริงนะครับไม่มีโกง ผมพึ่งเบิกเงินออกมาเลยครับ

  173. UFABET คาสิโนออนไลน์เล่นง่ายได้เงินง่ายเพียงแต่ปลายนิ้วสัมผัส ใช่แล้วครับพวกเรามาอยู่ในยุคที่เรียกว่าใช้นิ้วเขี่ยๆโทรศัพท์ก็ได้เงิน เพราะเหตุว่าคาสิโนออนไลน์เป็นศูนย์รวมของเกมออนไลน์ต้องการเล่นแบบไหนมีหมด ทำเงินได้อย่างง่ายดายสบายรวดเร็วปลอดภัยสุดๆครับผม

  174. คงไม่ยอมรับไม่ลงถ้าจะกล่าวว่าคาสิโนออนไลน์เป็นแหล่งวางเดิมพันที่มากมายที่สุด สร้างเงินทำเงินได้ง่ายที่สุดและที่สำคัญไม่มีอันตรายที่สุด UFABET คาสิโนออนไลน์ที่เหมาะสมที่สุดในยุคนี้ ตอบโจทย์ทุกความชอบของเหล่าสมาชิก

  175. คาสิโนออนไลน์ถือได้ว่าเป็นหนทางที่ดีเยี่ยมๆในการนักเสี่ยงโชคเพราะเหตุว่าทั้งสบายรวมถึงไม่เป็นอันตราย เล่นตรงไหนตอนไหนก็ได้ซึ่งก็กล่าวได้ว่าคุณสามารถทำเงินได้ตลอดระยะเวลา UFABETได้สะสมทุกเกมไว้รอคุณแล้วแค่เพียงสมัครเข้ามาก็ทำเงินได้ในทันทีทันใด

  176. เกมบาคาร่าก็ยังคงเป็นเกมที่ครองใจคนอีกหลายๆคนในตลอดระยะเวลาที่ผ่านมา ยิ่งถูกพัฒนาเป็นรูปแบบออนไลน์ยิ่งง่ายต่อการเข้าถึงไม่จำเป็นที่จะต้องไปเล่นที่คาสิโนเพียงมาที่ UFABET คุณก็สามารถร่วมเล่นเกมแล้วก็ทำเงินได้อย่างง่ายๆกับบาคาร่าออนไลน์แล้ว พร้อมระบบฝากถอนอัตโนมัติ

  177. UFABET เว็บไซต์พนันบอลออนไล์ที่สามารถเรียกได้เลยว่าแจ๋วสุดปัจจุบันนี้เลยก็ได้เนื่องจากว่าเป็นเว็บที่ถูกกล่าวถึงมากไม่น้อยเลยทีเดียวบนโลกออนไลน์ ในด้านการเงินได้รับการยอมรับแล้วว่ามีความมั่นคงสูง และก็ยังมีการพัฒนาระบบอยู่ต่อเนื่องเพื่อให้สมาชิกได้เข้าใช้งานอย่างไม่มีสะดุดเลยขอรับ

  178. พนันบอลออนไลน์ กับ UFABET มีดีเช่นไรหนะหรอ วันนี้มาทำความรู้จักกับเว็บ แทงบอลออนไลน์ ที่มีเกมคาสิโนออนไลน์ ไว้ให้บริการอย่้างครบถ้วนบริบูรณ์ อย่าง UFABET เป็นเว็บที่มีความน่าไว้ใจสูง มีความยั่งยืนมั่นคงทางด้านการเงิน การบริการเสมือนเพื่อนซี้ การเอาใจใส่ราวกับแฟน

  179. หากจะเอ๋ยถึงบาคาร่าออนไลน์ หรือมีการค้นหาบาคาร่าออนไลน์ UFABET จะเป็นชื่อแรกๆเลยที่คุณจะเจอ โน่นด้วยเหตุว่าพวกเราคือเว็บไซต์ให้บริการบาคาร่าระดับแนวหน้า ซึ่งได้รับการยอมรับว่าจ่ายจริง จ่ายเต็มไม่มีกลั้น ไม่มีกั๊ก ยิ่งทำให้สมาชิกถูกใจอย่างมากครับผม