paper on PCA_rubik cube_sampling writing

Science Journal of EconomicsISSN: 2276-6286http://www.sjpub.org/sjpsych.html© Author(s) 2011. CC Attribution 3.0 License.

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Research Article

Playing with the Rubik cube: Principal Component Analysis Solving the Close End Funds Puzzle?

Ismael Torres-Pizarro

University of Puerto RicoEmail: [email protected]

Tel:(787) 315-5636

ABSTRACT

A simple PCA model was used to find the direction of mostvariability for the CEF puzzle. Evidence that the MOM factor asdetailed by Carhart (1997) explains this puzzle was found. Datasets used are available for independent verification of results.

KEYWORD:PCA model, CEF puzzle and MOM factor

INTRODUCTION

A CEF is one type of collective investment scheme where thenumber of shares is finite (hence the “close-end” name).The CEF, which is a corporation, is in the business ofinvesting the funds raised from issuing a limited number ofshares (in an initial public offering -or IPO-) in othercorporations and businesses’ securities and managing thoseinvestments for profit. Charrón (2009), define it as “aninvestment company that holds or bundles other publiclytraded securities”. The CEF shares are then traded on anexchange or directly through the fund manager to create asecondary market subject to market forces. New shares arerarely (if ever) issued after the fund is launched; shares arenot normally redeemable for cash or securities until the fundliquidates. This makes them somewhat more illiquid or“restricted” than stocks from open corporations.The market price of a share in a closed-end fund isdetermined partially by the value of the investments inthe fund, and partially by the premium (ordiscount) placed on it by the market. The total value of allthe securities in the fund divided by the number of sharesin the fund is called the net asset value (NAV1) per share.The market price of a close end fund share is often lowerthan the “per share NAV” (that is, it sells at discount).The fact that a closed end funds share usually sells atdiscount is a puzzle considered an anomaly (“the CEFpuzzle”) to the classic finance by challenging the law ofone price2 and by implication, the efficiency markethypothesis (EMH)3. This puzzle has a curious behaviorcommented by Lee, Shleifer and Thaler (1990); for asummary of the complete picture the puzzle behaviorfollows this pattern: (a) The IPO of a CEF starts selling in orabout 10% premium of its NAV; the fund is a shell that justbuys shares, bonds, and other securities from othersfirms;therefore, when first issued, the fund value should be justthe sum of the shares

1 NAV = (Market value of all securities – liabilities)/ number of sharesoutstanding2 The Law of One price states that identical goods (or securities) shouldsell for identical prices in an efficient market; that is, all identical goodsmust have only one price. If two identical assets (identical in the sense theyhave the same risk-return distribution) sell for different prices; sellers willflock to the highest prevailing price, and buyers to the lowest currentmarket price. In an efficient market the convergence on one price shouldbe instant.3 The well-known efficient-market hypothesis (EMH) asserts thatfinancial markets are "informational efficient", or that prices on tradedassets (e.g., stocks, bonds, or property) already reflect all knowninformation, and instantly change to reflect new information. Therefore,according to theory, it is impossible to consistently outperform the marketby using any information that the market already knows, except throughluck. Information or news in the EMH is defined as anything that mayaffect prices that is unknowable in the present and thus appears randomlyin the future.it buys less commissions and other costs. This should implyit should sell at a discount initially; but it does not. (b) In orabout 4 to 6 months after the IPO the CEF's shares startselling at a discount of an average of 10% of its NAV (Notingalso that, the investor pays a premium for the new CEF'sshares when there are existing CEF's shares selling atdiscount); there could be CEF's shares selling at premium,but this is not the norm. In fact, wide variation is theobserved behavior, Lee, Shleifer & Thaler (1991). This addsto the puzzling situation, since it is a well-known fact that,on average, the fund will actually sell at discount about 4months after its IPO. The next question should be: Whywould anybody want to buy today, paying premium prices,something that in all likelihood, will depreciate in about 4months? Not only that, during its lifetime, its market valueis not equal to the net asset value less costs and has a widevariation. Some even sell at premium which is a total riddlebecause when the fund is terminated it does sell at its NAV.Dimson & Minio-Kozerski (1999) reported IPO premiumsfor American CEF of up to 10% and up to 15% for BritishCEF, while Weiss (1994) reported that in 6 monthsAmerican equity CEF were selling at 10% discount onaverage and Levis & Thomas (1995) reported that Britishequity CEF were selling at an average of 5% discount inabout 6.5 months. (c) When the CEF is terminated oropen-ended, its shares value rise to the NAV. See Lee,Shleifer and Thaler (1990) discuss for details in this curiousbehavior found and why the standard response formainstream finance seemed inadequate.The almost 300 funds used as a the original sample in thisstudy represent almost half the total closed ends funds listedwith the SEC; many are bond funds, others are equity ones.We updated the database's information with the onlinedatabases including CRSP and Compustat for monthly data.

Volume 2012 (2012), Issue 2

Accepted 28�� March, 2012

Corresponding Author:Ismael Torres-PizarroUniversity of Puerto RicoEmail:[email protected]

http://www.sjpub.org/sjmb.html

Science Journals Publication(ISSN:2276-6286) Page 2

We want to know if any of the different models havepredictable value on the CEF's discounts and returns. Wewould also like to have a better understanding of thepeculiar financial behavior of these companies. II. Explanations that might be as puzzling as the cube:There have been many attempts to explain the CEF puzzle,but so far they have not been completely satisfactory. Forexample, the "tax liability theory" states that the NAV doesconsiderate the capital tax liability that will be imposed onthe fund if all its assets are sold. Therefore, the NAV itself isoverestimated (meaning there is no puzzle at all, but it is amiscalculation of the NAV), Malkiel (1977, 1995) found apositive relationship between discounts and unrealizedcapital appreciation and restricted (illiquid) stocks4. Theproblem with this attempt is that is does not explain thecases where the CEF shares are sold at premium. The theorypredicts there will be not a premium ever as the actual NAVwill always be less than the calculated value. What is evenmore puzzling is that the observed behavior of a CEF whenit is open-ended (that is, when converted from a CEF to anOpen-End Fund) is to converge to its NAV which is preciselythe opposite behavior predicted by this theory. Datar (2001)found out that funds with higher liquidity, measured byproxies of trading activity, have higher premiums or lowerdiscounts than funds with lower liquidity.4 Malkiel’s idea is that funds with high unrealized capital appreciationshould sell at discounts from NAV because the holder of such a fund wouldbe assuming a potential tax liability that depends on the holding period ofthe investor.Therefore, these authors have proposed that the liquidityof the restricted5 securities could be part of the puzzle. Thatis, highly illiquid restricted securities not having a readilyavailable market to price them in an efficient manner willcause the NAV to be overstated6, as the managers willerr on the side of overpricing them. However, the manyCEF's that actually have large well- diversified portfoliosand therefore, their portfolio consist of liquid securities thatare efficiently priced and are still affected by the observedbehavior of being sold at discount or at premium; cannot beexplained by this theory, Lee, Shleifer & Thaler (1991). Thatis, Lee et al contend illiquidity could be part of the answerif the NAV consisted of special machinery or equipment ofdifficult valorization but, when it consists of highly liquidand readily available market prized assets it makes no sense.A more plausible explanation to the puzzle is the "agencycost theory". The "entrenchment" of the CEF's managerscauses its share to sell at discount because they canconsume CEF benefits without the fear of being fired, thus,the shares will sell at discount; however, the theory cannotexplain why any rational investor would ever buy the shareat premium when initially issued. Malkiel (1995) is a largesupporter of this position as he proposed the larger theinsider ownership position, the larger the windfall gain forthose insiders when the large5A restricted security is one that generally cannot be sold in thesecondary market or, for those which can be sold, in order to sell it certainvery restrictive criteria must be met before the selling can occur.6When you have a property which worth cannot be ascertain in aobjectively manner, it is likely you will think it is worth more than it reallydoes; after all, you bought it and that means you thought it was worth it.discount stock is sold at the higher NAV value. He found outthat unrealized gains and turnover were possible candidatesto explain the puzzle. This idea is supported by Barclay,Holderness, and Pontiff (1993).

There are others authors (see, for example, Ferguson andLeistikow (2004) or Berk and Stanton (2007)) whoconjecture that discounts are related to market'sassessment of anticipated future performance in terms ofmanagerial ability. Yet, some other authors (see, forexample, Woan and Kline (2003) who compareMunicipal funds (i.e., munis) vs. equity and non-munisfunds concluding there are no differences) think partialexplanations might come from "market segmentation" suchas country funds or bonds vs. equity funds, etc. Datar (2001)did find some discounts differences between bonds andequity funds.Country funds could be different based on direct or indirectforeign restrictions on ownership and investments, Bonser-Neal, Brauer, Neal, and Wheatley (1990) or country'smarkets integration to global economy (Chang, Eun, andKolodny (1995)).Another proposed explanation is the "dividend yieldhypothesis" which states closed-end bond funds areprimarily held by individual investors who are short-termindividuals who seek a high current yield. They found suchyield in CEF's investing in bonds because they are lessvolatile in price than equity funds and they pay monthlydividends (equity funds pay annual dividends); that is, it isa monthly income vehicle with a short investment horizonthat can easily get in and outs; therefore, the idealinvestment vehicle for them. Lee and Moore (2003) founda very strong negative relationship between dividend yieldand discounts. There are many other explanations as thereare researchers but those just mentioned are by far themost common ones. There is, however, one newapproach in finance dealing primarily with violations to thestandard efficiency theory. It is called Behavioral Finance.The best-known example7, (Thaler, 1999) of an apparentviolation of the law of one price was Royal Dutch / Shellshares. Royal Dutch Petroleum and Shell Transport wereindependently incorporated in, respectively, theNetherlands and England. After merging in 1907, holders ofRoyal Dutch Petroleum (traded in Amsterdam and US andare part of the S&P 500 Index) and Shell Transport shares(traded in London and was part of the Financial Times StockExchange Index) were entitled to 60% and 40% respectivelyof all future profits.Royal Dutch shares should, therefore, automatically havebeen priced at 50% more than Shell shares; that is,according to any rational model, the shares of these twocomponents (after adjusting for foreign exchange) shouldtrade in a 60-40 ratio (if the 40% of Shell is taken as 100%;then, Royal Dutch 60% is 150%). However, they divergedfrom this amounts by up to 15%. This discrepancydisappeared with their final merger in 2005. Simpleexplanations, such as taxes and transaction costs, could notexplain the disparity. Shleifer and Vishny (1997)7 Another good example of violation of this “law” can be found inLamont and Thaler (2001) where the author studied the case of equitycarve outs and concluded that even if the EMH could be incorrect (i.e., themarket forces got the price wrong), it does not necessary imply profitablearbitrage opportunities since the associate shorting costs could make theoperation unfeasible. An equity carve out is occurs when a parentcompany sells a minority (usually 20% or less) stake in a subsidiary for anIPO or rights offering.foresaw this exact setting. They argue that arbitrageurs8'behavior will be limited, even in the face of price disparitiesas this case. In any case, some of the flaws of EMH were

Page 3 Science Journals Publication(ISSN:2276-6286)

pinpointed in Shleifer and Summers (1990); in particular,they foretold about the limits to arbitrage as early as thatyear.So, how is this related to CEF? The new set of theories justmentioned, loosely fit together under the name of behavioralfinance9 (BF) have been since the 1990's10 trying todevelop a new school of thought primarily based on the factthat humans are not rational beings all the time, muchless when taking decisions with strong emotional content.This is a departure from standard economic theory whereone of the pillars is that human beings make rationaldecisions. For example, Thaler (1999) suggests that even ifthe decision maker is rational, rational decisions might notbe possible to execute and the investor will behaveirrationally or just not as rational as the theory predict. Thatis one possible explanation to the Royal Dutch/Shell sharespuzzle and, by induction, to the CEF conundrum as well.Thaler argues that although hedge funds do makeinvestments based on disparities such as the RoyalDutch/Shell case, an8 the practice of taking advantage of a price differential between twoor more markets: striking a combination of matching deals thatcapitalize upon the imbalance, the profit being the difference betweenthe market prices; in simple terms, a risk-free profit9Sometimes referred as “behavioral economics” (BE) although it can beargued that “BE” is a more ample term contained the research interest of“BF” and others topics. A detailed and updated account of BF history,methodology and principles could be found in Shiller (2003), Barberis(2003) and Ritter (2003).

10The beginnings of behavioral finance could be traced back toKahneman and Tversky (1979).

example of what could stop the investor to behave rationallyis the situation of the Long-Term Capital Management(LTCM) in the summer of 1998 when they hold investmentsin the Royal Dutch/Shell disparity but in August 1998, whenthings started to unravel for LTCM, the disparity wasrelatively large, so at a time when LTCM might have chosento increase the money it was willing to bet on this anomaly,it had to cut back instead.This example shows that even when the investor is rationalit might have to behave irrationally11, thus validating theneed to further research in this area. Another interestingpoint of view supporting behavioral finance is the one byOwen (2002), who contends that the investment done inhigh technology forms in the late 20th century cannot beexplained by investors rational behavior given the chancethose firms have to make money; however, his papercontains the same ad-hoc line of arguments that have madebehavioral finance little good. For example, Titman (2002)critiques this ad-hoc approach by stating he is notconvinced of the explanations given by behavioral theoristsfor the persistence of investor's under-reaction to stocksplits and other events because the implicit assumption ofinhibit learning is not explicitly modeled in Daniel, Hirsleiferand Subrahmanyam (1998) -i.e., DHS- one of the two papershe analyzed. Shiller (1990) displayed an overwhelming casefor simple fundamentals feedback models such wherefactors such as demographics and GNP deflators couldexplain much of the variation observed in speculative prices.11 Perhaps we can say that the investor did behave rationallyunder the irrational circumstances being faced. If so, behavioral financeis not a contrarian hypothesis to the EMH, but a theory that includes itas a subset. That is, maybe the EMH is just a special case of BF.Barbieris and Huang (2001) avoided much of the ad hocapproach and developed a working mathematical model forloss aversion that suggests narrow framing and lossaversion affect investor decision making; however, they also

admit other explanations to their observation are possible.In the discussion that followed for the same paper Brennan(2001) prescribes that caution in the relaxation of therational criteria will lead to the adoption of different(and perhaps even conflicting) postulates for behavioralfinance.As for the CEF puzzle, the behavioral finance frameworkprovides for yet another plausible explanation: individualinvestor sentiment, Lee, Shleifer and Thaler, (1991). Theseauthors argue that most CEF shares are initially sold byprofessionals to less informed (i.e., irrational) individualsin a "marketing hypothesis" and in a "hot"' thread mode,therefore, the less informed investors pay the initialpremium prices that rational investors are not willing topay; but in a secondary "wave", after the initial fad is over,rational investors are willing to buy the CEF shares if theirprices are discounted enough to make it for the additionalrisk. The same investor sentiment theory would explain thewide variation observed for the share prices during theirlifetime; depending on the stochastic (and therefore,unpredictable and highly risky) mood of the irrationalinvestors or the new fad they make their decision or on thepiece of informational noise they form their trading strategy,etc, they will tend to bid on the CEF's shares. Last, when theCEF is to be open-ended the risk from noise traderdisappears, as the rational investor knows exactly the endprice of the CEF share at the time of termination, andthus, can short sell without fear of any arbitrageur risk.Ross (2005) developed a neoclassical approach. He modelsa management fee based simple approach that seems toprovide a very good answer to the puzzle. As themanagement fee is correlated to the funds' distributions;Ross' states the initial high price for the funds is due to theinvestors' high expectation on management ability; if suchexpectations are not met, the funds will trade at discount;properly accounting for the management's fees that now donot cover themselves. When close to end, the funds investorsexpect the elimination of management's fees; therefore, itsvalue will come closer to the NAV. Funds distribution is,therefore, one important variable to watch in CEF modelstrying to explain the puzzle.Therefore, there could be at least two distinct schools ofthought regarding this puzzling situation: one that says thecurrent view is not enough to explain it (behavioral finance)and another that says there is practically no puzzle at all(neoclassical). Finding the factors that amount to thisbehavior might shed some light toward one or anotherpoint of view. If, for example, the sentiment momentumfactor results a relevant input and the distribution does not;that might support behavioral finance claims the oppositeresults will call neoclassical finance claims as victorious.Maybe neither or both factors results important calling foran integrated approach or a whole new one.We intend to use a PCA simple model to get evidence ofrelevant factors in this puzzle.

III. Setting the game:

In his well-known undergraduate textbook, Madura(2003) states that stock prices are affected by economicfactors (such as market yields, bond yields which are aproxy measurements for market risk and changes in thebond markets that might cause investors to switch frombonds to stocks and viceverse), firm specific factors (such


as dividend policy, acquisitions, expectations, etc) andmarket related factors (such as investor sentiment, etc.).Fama & French (1992, 1993) discussed the use of twofactors in addition to the firm beta to model stock returns12:a) SMB stands for "small (market capitalization) minus big"b) HML for "high (book-to-market ratio) minus low"; theymeasure the historic excess returns of small caps over bigcaps and of value stocks over growth stocks. These factorsare calculated with combinations of portfolios composed byranked stocks and available historical market data.Historical values were downloaded from French's web page.Carhart (1997) extended the Fama-French model with anadditional momentum factor (MOM), which is long prior-month winners and short prior- month. We have monthlydata points for all the variables (namely, inputs and outputsfactors in the PCA approach) for each of the close to 300funds starting since 1987 in some cases.

1. Average Monthly Market Price Discount

2. Monthly NAV

3. Fund Monthly Market Return

4. Dividend Distribution per share

5. Fama & French Rm-Rf

6. Fama & French SMB (small [cap] minus big: a measure the historic excess returns of small caps over the market as a whole)12Should be noted the most basic definition of a stock return = {(Valueof investment at the end of the year – Value of investment at beginningof the year) + Dividends} / Value of investment at beginning of the year= Total Return. That is, it is just another way to see prices changes.7. Fama & French HML (high [book/price] minus low: historic excess returns of "value" stocks over the

market as a whole)8. Fama & French MOM

9. Market Yield 1 year

10. Market Yield 10 years

11. Corporate Bond Yields

The response (dependent) variable Discount is definedhere as the difference between the Funds’ AverageMonthly Market Price and its Monthly NAV. All the modelshere will hypothesize the response (output or dependent)variable is a function of the other inputs (independent)variables (factors), namely: distribution, monthly return,Rm-Rf, SMB, HML, MOM, market yields for: 1 year and 10years length and corporate bond yields for 1 year and 10years.Neoclassical finance might expect that the input variable“distribution” has a great weight in both response modelswhile the others variables should not be significant (Rm-Rf,SMB, HML, MOM) or its significance be related to the fundnature of equity or bond (that should give a positive ornegative relationship to market and corporate yields) in aregression analysis. That is, the neoclassical finance schoolwould hypothesize the first and most important principalcomponent in a PCA analysis for the output variable wouldbe the “distribution” of dividends variable, while theothers input variables should not have any componentat all (or perhaps be insignificant for all practical purposes)..Behavioral finance would expect the MOM factor to be ofgreat significance; to be either the first principal componentor one the most important among a few other variables ortheir combinations.The process has a similarity to the well-known Rubik’scube toy. A numerical example clarifies it:

Let us say we have two variables13 named such as: f(x,y)T =

10.0 10.4 9.7 9.7 11.7 11.0 8.7 9.5 10.1 9.6 10.5 9.2 11.3 10.1 8.510.7 9.8 10.0 10.1 11.5 10.8 8.8 9.3 9.4 9.6 10.4 9.0 11.6 9.8 9.2

FIGURE 1. Scatter Plot of X vs. Y for the PCA Example.


Figure 1 shows both variables. Its mean vector and covariance matrix are:

Mean(f(x,y)T)= [10,10]

Covariance(f(x,y)T)=[ 0.79857142857143 0.67928571428571 0.67928571428571 0.7342857142857]

13 Numerical example taken and modified from Jackson (1991).The correspondent eigenvalues and eigenvectors from thatcovariance matrix are:

eigenvalues=[1.44647433819575 0.08638280466139]

eigenvectors=[-0.72362480830445 0.69019355024975 -0.69019355024975 -0.72362480830445]

The eigenvectors values are just the arccosines of a new“principal” rotation of the original variables about theirmeans over the original set of axis (X,Y). That is, moving“upward” and to the “left” the original axis to align a newset of axis with the data set that also pinpoint on thedirection of the highest variation. In this case, we move14:

arccosine(eigenvectors)=[43.645432° 133.645432°46.354568° 43.645432°]

That is, the new abscissa, E1, moved “up” and to the left43.65° measured from the old abscissa, X (or moved “down”and to the right 46.35° measured from the old ordinate, Y).As the new ordinate, E2, must be orthogonal to E1, we havecompleted the process15 for this simple case.14 It should be noted that the eigenvector could also be represented as=[0.72362480830445 -0.69019355024975; 0.69019355024975;0.72362480830445]; that is, the negative sign just shows the fact the linecross over to the other quadrant.15 Just by adding 90° to the angles; that is, the new ordinate is 43.65°+90° = 133.65° and 46.35°+ 90°= 136.35, which are nothing morethan the angles from the second eigenvector.

Now, the first set of values from the old set (X,Y) was(10,10.7) taking the mean from each variable we have now(0.0,0.7) which is the same as moving the originto(10,10)from the old (0,0). Now, performing thecalculation:

X*Eigenvector(1,1) + Y* Eigenvector(1,2)= 0.0*0.7236 + 0.7*0.6902= 0.4831;X*Eigenvector(2,1) + Y* Eigenvector(2,2)= 0.0*(-0.6902)+ 0.7*0.7236 = 0.5065;

Therefore, we have mapped (10.0, 10.7) into (0.48, 0.51) inthe new set of ordinates. The process is just the inverse16

if you need to get from the new set of variables (E1, E2) to(X, Y):

E1*Eigenvector(1,1)+E2*Eigenvector(2,1)=0.4831*0.7236+0.5065*(-0.6902)= 0.0E1*Eigenvector(1,2)+E2*Eigenvector(2,2)=0.4831*0.6902+0.5065*0.7236 = 0.7

Note that we just normalized the original variables17 andthat the sample variance for each normalized variable is notother than its eigenvalue. We can observe from the fact thatthe axis has moved almost 45° (also, from the comparisonof the eigenvalues; which are close to one another; that is,E1 ? E2 = - 0.72362480830445 ? -0.69019355024975) thatthe pattern between the observed data might be modeledby the linear equation: Y = X + Intercept. This is furthersupported by estimating the simple linear regression

methodology in an Excel spreadsheet, which gives us thefollowing summary output:16 Usually this would require to calculate the inverse matrix; however,the eigenvector matrix inverse its transpose matrix. This should dismissany claim that the original variables cannot be recovered, although itwould be a cumbersome process for a multivariable space.17 From the statistics world, this is the reason the numbers in theeigenvectors are also known as "scores", short for z scores.

FIGURE 2. Summary Output of X vs. Y for the PCA Example


From Figure 2 it can be seen, a very simple modelfor this could be conjectured to be: Y=0.85X+1.49; or whentaking the p-values (or confidence intervals) intoconsideration we can also use Y = X (a 45° degree angleline). This is what we referred when we talked about thedata reduction feature of PCA and its pattern discoveredfeature.

IV. Playing with the “Rubik’s cube”:

By simple transforming the original data with MatlabÒwe obtained the most important eigenvalues and theiraccumulative weight:

TABLE 1. Eigenvalues relative weight in this caseFrom Table 1 we can affirm that with only five (5)eingenvalues about99.54% of the variability of the data could be well modeled.Thus a significant data compression has been achieved. ThePCA transformation matrix is now a (15,5) matrix where theconvention is (variable, eingenvalue) where the order of

the variable is: 1) hi_price; 2)lo_price; 3) N A V ; 4) Monthly_Return; 5) AVE_Discount; 6) DPS; 7)Rm_Rf; 8) SMB; 9) HML; 10) MOM; 11) Rf; 12)Yield_OneYear; 13) Yield_TenYear; 14) AAA; 15) BAA andlook as follows:

TABLE 2. Details of the five most principal Eigenvalues and the fifteen variables

We know this eigenvalue matrix is also the arccosine matrix.That is, an original datapoint must moved as directed fromits original 15 dimensional word to a new mapping with only5 dimensions. We can observe fom Table 2 that some of the

angles have cosines so little that will only cause thedatapoint to move in almost right angles. We set such anglesto 90° which is the same as to have set their arccosines tozero. This is shown in Table 3 below.


From Table 3, it is quite obvious the most importantvariables are: 1 (high price), 2 (low price), 3 (NAV), 5(average discount) and 10 (MOM) for the first eigenvaluewhich happens to be the one where the most variability isassociated. Most important variable for the 2nd eigenvalueare: 3 (NAV), 8 (SMB) ; 9 (HML) and 10 (MOM); for thethird eigenvalue: : 3 (NAV), 9 (HML) and 10 (MOM); forthe fourth eigenvalue: 5 (average discount), 8 (SMB) ; 9(HML) and 10 (MOM); and for the fifth and last eigenvalue:1 (high price), 2 (low price), 3 (NAV), 5 (average discount)and 9 (HML).The first eigenvalue variable list is illustrative as we positthe average discount = average price – NAV. It pops out thatMOM variable has a substantial effect in all five eigenvalues;in particular, MOM is the only other variable that is not partof the linear equation that defines the puzzling discount inthe first eigenvalue which is the principal direction of thesystem variability; therefore, it should be an importantvariable affecting the discount.As a validation exercise we used fifty-six (56) funds wereused in the validation of the models with 11,327 data

points. This represent an 18.98% sample size (56 out of295 funds; a mix of new and previously used funds) and a21.59% (11,327/52,462 datapoints) close to the general20% normally used as guideline for model validationpurposes. The interpolations18 dates range from18

Interpolation: A method of constructing new data points within therange of a discrete set of known data points. It could be use as a validationof the model as the model will predict values inside its range that shouldbe in close agreement with the actual values observed. Greatdepartures from such values indicate a poor model.

August 1987 to December 2010 and the extrapolations19date from January 2011 to October 2011.The Euclidean distance between the validating sample sizeof 11,327 discount data values and their estimates asobtained by the transformation PCA matrix was a total of18.26358994273340 separation units; equivalent to a MSE≈ 0.02944810784818.V. I think I can solve the puzzle now:The puzzling behavior of the CEF discount seems to becaused by the MOM. The linear equation: average discount=average price –NAV becomes for the first eigenvalue:

TABLE 3. Details of the five most principal Eigenvalues and the fifteen variables where non-significant valueswere set to zero

Abs{[arccosine(0.59227071437245) + arccosine(0.53641494107392)]/2 –arccosine(0.59955661177596)} = arccosine(0.03521378405278)

=

[33.93461226273970° + 30.73431219129440°]/2 - 34.35206343392610°

= abs{ -2.01760120690908°}

≈

2.01760120690938°

19Extrapolation: The process of constructing new data points outside

the range of a discrete set of known data points. It is similar to theprocess of interpolation, but the results of extrapolations are subject togreater uncertainty. It could be use as a validation of the model as themodel will predict values outside its range that should be in closeagreement with the actual values observed. Close agreement from suchvalues indicate a good model.

Any difference in the actual discount seen not coveredby this equation must come from the MOMangle’s arccosine(-0.02408744663071) = -

1.38010903118630°.


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