Even though Exchange Traded Funds (ETFs)resemble closed-end mutual funds in many respects,ETFs have the unique feature that additional sharescan be created and redeemed by certain institutionalinvestors. The creation–redemption process allowsETF shares to trade continuously during the day asin any publicly traded company at prices determinedby supply and demand rather than at the calculatednet asset value. This article investigates the extentand properties of the resulting premiums (discounts)of ETFs from their fair market value.
Traditional measures of premiums (discounts)for ETFs are misleading because the net asset value isnot accurately represented and/or because the priceof the fund is not accurately recorded. This articleincorporates these features into an errors-in-variablesmodel. The model measures the standard deviationof the remaining pricing errors and investigates thetime variation in this standard deviation.
This article uses data for domestic as well asinternational ETFs from an end-of-day perspectiveand from a minute-by-minute intra-daily frame-work. The overall finding is that, once mismatchesin timing are accounted for, the premiums (discounts)for the domestic ETFs are generally small and highlytransient, typically lasting only several minutes. Thestandard deviation of the premiums (discounts) is15 basis points on average across all domestic ETFs.This standard deviation is substantially smaller thanthe bid-ask spread.
For international ETFs, premiums (discounts)are much larger and more persistent, frequently lastingseveral days. An explanation for this difference mayrest with the higher cost of creation and redemptionfor the international products. The bid-ask spreadsare also much wider but are comparable with thestandard deviation of the premiums. Nonetheless,
when compared with closed-end funds where thereare no opportunities for creation or redemption, theETFs have smaller and less persistent premiums(discounts).
INTRODUCTION
Although in some respects, exchangetraded funds (ETFs) do resembleconventional index mutual funds,they also differ in two important
ways. First, shares of ETFs can be created bysome institutional investors who deposit pre-specified baskets of shares of the companiespresent in the fund portfolio in return forshares in the fund. Second, as a result, ETFsshares trade continuously during the day asin any publicly traded company at prices deter-mined by supply and demand rather than atthe calculated net asset value (NAV).
As closed-end mutual funds exhibit largeand persistent premiums, it is natural to askwhether ETFs also exhibit such premiums.The creation and redemption process for ETFsallows arbitrage opportunities to be exploitedprofitably whenever the share prices deviatefrom the NAV of the underlying portfolio.If the creation–redemption process works effi-ciently, ETF shares should not trade at signif-icant premiums or discounts from the fair valueof the portfolio.
The details of the creation–redemptionprocess differ substantially across funds. Forthe domestic funds, the order to create orredeem is given sometime during the tradingday and is exercised at the end of the day. Fora creation order, shares of stock and cash must
Premiums-Discounts andExchange Traded FundsROBERT ENGLE AND DEBOJYOTI SARKAR
be delivered within a certain number of days, and theETF shares become available at the end of the day. If thereis a discrepancy between the estimated cash account andthe closing cash account, this adjustment is made at thesettlement. For international funds, the process can be farmore complex. In some countries there are substantialtaxes that must be paid when shares are transferred. Inothers, there are prohibitions on transactions made byforeigners so that the account is settled entirely in cash withthe trustee acquiring the shares rather than the creator. Thedelivery period is longer and there is more price risk inassembling the package. The ability to hedge such risksis also much reduced. Altogether, these features make thearbitrage mechanism more complicated, risky and costlyfor the international transactions than for the domestic.Consequently, one might not be surprised if the pricingis not as fast or accurate in these situations.1
This article examines the end-of-day and intra-dailypremiums for a collection of 21 domestic and 16 inter-national ETFs, and measures both the magnitude and thepersistence of the premiums.
LITERATURE REVIEW
As traditional mutual funds guarantee investors theability to buy or sell shares in the fund at the closingNAV, investors who notice any discrepancy between themeasured NAV and the fair market value have theopportunity to buy at a discount and sell at a premium.The importance of this effect has been documented byGoetzmann et al. [2001]; Chalmers et al. [2001]; andBoudoukh et al. [2002]. The highly profitable tradingstrategies proposed in these articles are basically condi-tional schemes as they examine the relation between theofficial NAV and current market variables in choosingwhen to trade. Various solutions have been proposed toprevent such arbitrage trading; one is the ETF solutionthat allows trading at a market price that can differ fromthe measured NAV.
However, premiums that arise because of inaccu-rate recording of the NAV must be treated very differentlyfrom premiums that arise because of trading at bad prices.In the closed-end pricing literature, early attempts toexplain the behavior of fund discounts and premiumsfocused on the mismeasurement of reported NAVs. Thereis also a vast literature examining the comovements ofpairs of asset prices, leading to measures of correlationand predictability. The literature that examines comove-ments of prices of essentially the same asset in different
markets or in different forms of security is of particularinterest here. Two widely studied examples of the sameasset trading in different forms at different prices areclosed-end mutual funds and futures markets.
In the early mismeasurement literature, threepotential explanations were mentioned: agency costs(managerial ability is not reflected in the NAV), tax lia-bilities (unrealized capital appreciation is not captured bythe reported NAV) and illiquidity of assets (assets thathave trading restrictions may be overvalued in the NAVcalculation). See Boudreaux [1973]; Rosenfeldt and Tuttle[1973]; and Malkiel [1977]. Today the existence of suchpremiums and discounts is still viewed as a market anomalyin Thaler et al. [1993].
The cash market index futures market literaturelooks at how quickly the cash market responds to market-wide information that has already been transmitted intofutures prices. See MacKinlay and Ramaswamy [1988];Stoll and Whaley [1990]; Atchinson et al. [1992] andChan [1992]. Ahn et al. [2002] compare microstructure-based explanations (i.e., stale prices) to partial adjustment-based explanation for portfolio autocorrelations. Theyconclude that their findings can “most easily be associ-ated” with market microstructure-based explanations.
The source of much of the microstructure noisedocumented above is the observation that the closingtransaction price does not contain all the information onend-of-period value. When there is such staleness in theprice, portfolio autocorrelations may arise. Deviation ofthe observed price from the true price may arise from therandom bouncing of transaction prices between bid andask levels. Roll [1984] shows that the bid-ask price bounceinduces negative first-order autocorrelation in observedprice changes even when price innovations are seriallyindependent. Scholes and Williams [1977] consider asituation where infrequent trading takes place and provideconsistent estimates in such “errors-in-variables” scenarios.For other models of how nonsynchronous trading canexplain portfolio autocorrelations, see Cohen et al. [1986];Atchinson et al. [1987]; Lo and MacKinlay [1988, 1990];Boudoukh et al. [1994]; and Kadlec and Patterson [1999].The general finding is that nonsynchronous trading can,at best, partly explain portfolio autocorrelations. An alter-native to nonsynchronous trading models assumes partialadjustment or slow adjustment to market-wide informa-tion. See Holden and Subrahmanyan [1992]; Brennanet al. [1993]; Foster and Vishwanathan [1993]; Badrinathet al. [1995]; Klibanoff et al. [1998]; Chordia andSwaminathan [2000]; Llorente et al. [2002].
28 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
These studies, however, do not use conditioninginformation and do not assume cointegration.2 As thebasis in these studies closely resembles the ETF premium,especially after corrections for dividends and interest rates,cointegration should be considered as an important partof the analysis. Cointegration is a powerful statisticalconcept that says that, eventually, deviations betweentwo prices must be corrected even when each of theprices is integrated and each may be very close to a ran-dom walk.
DATA AND DESCRIPTIVE STATISTICS
This article examines the premiums of 21 Domesticand 16 International ETFs. The time period covered bythe intra-daily statistics runs from April through September2000 whereas the end-of-day time period has a variablestart date depending on the launch date of a particularETF and ending in September 2000. For the end-of-dayanalysis, the premium is measured by the percentagedifference between the average of the closing bid and askprices (“midquote”) on the ETF and the NAV. For theintra-daily analysis, the premium is measured every minuteas the percentage difference between the midquote andthe IOPV (indicative optimized portfolio value). IOPVis an official estimate of the value of the portfolio postedevery 15 seconds.
The 21 domestic ETFs are divided into three cate-gories: (1) ETFs that close at 4:15 PM and have futuresmarkets for the underlying indexes; (2) ETFs that closeat 4:15 PM but do not have futures markets; and (3) ETFsthat close at 4:00 PM and do not have futures markets.The eight ETFs considered under category 1 representbroad market indexes that have index futures markets.Three of the four ETFs considered under category 2represent Dow Jones subsectors with the fourth onerepresenting the S&P SmallCap. The nine ETFs consideredunder category 3 all represent different subsectors of S&P.
Exhibit 1 shows the end-of-day and intra-dailyaverage premium and standard deviation of premiumfor the three groups of domestic funds. The standarddeviation of last trade-based premium and end-of-daybid-ask spread are also reported.3 Exhibit 2 shows thesame end-of-day and intra-daily statistics for the interna-tional funds.
Exhibit 1 shows that, for domestic funds, the end-of-day average premium is 1.1 basis points (bps) with arange from −0.1 (IWM: Russell 2000) bps to 4.6 bps(DIA: DJIA). The average standard deviation of pre-
mium is 18.3 bps with a range from 10.1 bps (IYF: DJFinancial) to 34 bps (QQQ: Nasdaq 100). The averagestandard deviation of last trade-based premium is42.1 bps with a range from 17.6 bps (IVV: S&P500) to142 bps (IYV: DJ Internet). In each instance, the stan-dard deviation of last trade-based premium is larger thanthe standard deviation of premium. The average bid-askspread (log of Ask/Bid) is 37.7 bps with a range from8.7 bps (SPY: S&P500) to 79.5 bps (XLB: S&P BasicIndustries).
In general, the international funds show largerend-of-day values for each of the statistics. For all inter-national funds except one (EWO: MSCI Austria), thestandard deviation of last trade-based premium is greaterthan the standard deviation of premium. In several casesthe differences are very large suggesting that the last trademay be at a price far from the closing NAV when the fundis infrequently traded. According to Exhibit 2, the inter-national funds often have quite large and positive premiums.EWO (MSCI Austria) and EWN (MSCI Netherlands) areexceptions; they have mean discounts: 2 and 6 bps, respec-tively. Even with these negative values, the average pre-mium for the 16 products is 34.8 bps. These funds havestandard deviations of premium that range from 54 bps(EZU: MSCI EMU) to 117 bps (EWW: MSCI Mexico)with an average of 78 bps. The standard deviation of lasttrade-based premium ranges from 59 bps (EZU) to 211 bps(EWZ: MSCI Brazil) with an average of 100.8 bps. Theaverage end-of-day bid-ask spread is 112.3 bps. Whilethese are larger than for the domestic funds, they are againrather small compared with many traditional costs oftrading. This is especially clear when compared with otherways to invest internationally such as closed-end funds thatoften have persistent discounts of 10 or 20%, or directinvestment with its myriad costs and risks.
For domestic funds, the intra-daily average premiumis 0.25 bps with an average standard deviation of 11.8 bps.See the bottom row of Exhibit 1. For the internationalfunds, the similar numbers are 23.7 bps and 64.8 bps,respectively. See the bottom row of Exhibit 2. Thus, interms of the simple measured premiums, domestic fundsshow small divergences between the prices of the fundsand their estimated NAVs. The premiums for the inter-national funds, on the other hand, exhibit some positivebias, which is, at least in part, due to the greater cost andrisk in the creation and redemption of the internationalfunds. Nonetheless, both the domestic and internationalETFs exhibit smaller standard deviations within the daythan at the end of the day.
Part of the explanation of the smaller end-of-daystandard deviation is simply the diurnal effect of well-known trading patterns. Markets are more volatile at theopen and at the close than in the middle of the day. Thevariability of the premium is closely related to the vari-ability of the underlying index and, therefore, it is notsurprising to see this type of typical effect in the corre-sponding ETF as well. This pattern is illustrated for DIAin Exhibit 3. The plot gives the standard deviation of thepremium for each minute of the day. The typical U-shapedeffect is easily seen along with a small increase in volatilityaround lunchtime.
Some portion of the measured premium is attrib-utable to microstructure effects and the purpose of themethodology proposed in the next section is to correctthese errors. To show that these errors are important,the premiums are reexamined for the funds that haveactive futures contracts. The premium can be measuredrelative to the IOPV, to the cash index or to the futuresprice. The first two will potentially suffer from stale pricesand may, therefore, show delays in pricing. The latter twowill show drift due to dividends, interest rates and portfoliocash balances. Thus a comparison can only be made fordeviations around a slowly moving mean.
30 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
E X H I B I T 1Descriptive Statistics: Domestic ETFs
We calculate the first-order autocorrelation and thestandard deviation of the three premiums (IOPV-based, cashindex-based and futures-based) on a minute-by-minute basisfor each day in the sample and then average over all days.These statistics are calculated around a daily mean that allowsfor the slowly moving components. The results are givenin Exhibit 4. For all funds except IJH, the autocorrelationis lower for the futures-based premium than for either theIOPV- or cash index-based premium. This reduction inpersistence is consistent with the hypothesis that the short-run deviations between price and NAV are due to stalenessin the estimates of NAV. The autocorrelation of the IOPV-and cash index-based premiums are similar in size. Exceptfor IJH, the standard deviations of the three premiums aregenerally of the same size. These autocorrelations are some-what smaller for the ETFs that are most heavily traded.
METHODOLOGY
These results indicate the importance of building astatistical model to correct for these microstructure effectsas most of the funds do not have a futures contract. Thenatural approach for examination of ETF premiums isthe conditional analysis of cointegrated asset prices. The
analysis must recognize the potential staleness of the NAVand the possibility of measurement error in the ETF price.The prices, however, must remain cointegrated even withsuch measurement errors.
To develop the statistical methods it is first necessaryto introduce notation. Let p be the log of the measuredprice of the ETF and let n be the log of the measured NAVat time t. Then
(1)
This premium is the fractional difference betweenthe price and the NAV. A discount is, therefore, a negativepremium. As premium is found to be normally distributed,then the standard deviation of it is a very familiar andeasily quantified measure of the size of the pricing error.
As mentioned earlier, both the price and NAV maybe measured with errors. Consider first the problem ofmeasuring the NAV at the end of the day. The portfolioheld by the fund is known and is evaluated at the closingtransaction prices of each of the assets. This evaluationmethod introduces two potential sources of error.
First, each closing transaction price could haveoccurred as a buy or as a sell order, and therefore, beslightly above or below the closing midquote. Second,
premium t t tp n= −
SUMMER 2006 THE JOURNAL OF DERIVATIVES 31
E X H I B I T 2Descriptive Statistics: International ETFs
the closing transaction could have occurred early in theday, particularly for thinly traded stocks. As a result, thetransaction may not contain information on its end-of-day value. An institutional investor considering creatingor redeeming shares will compare the current value ofthese shares at the end of the day to the fund share priceand will trade regardless of the accounting definitionof NAV at the market close. Even though intra-dailypremium uses IOPV, the estimated value of the portfolioposted every 15 seconds, IOPV suffers from the same stalequote possibility as the NAV at the market end.
A statistical model for the premium, therefore, mustincorporate the long run properties of the data: the ETFprice and the underlying value eventually must be thesame. Both prices and NAV are integrated processes asthey are prices for portfolios of traded assets. However,the premium is a stationary process as long as arbitrageopportunities ensure that deviations are self correcting.Thus the system of measured prices, measured NAVs andpremiums is a cointegrated system where the premiumwould represent the error correction term.
We now formulate a novel statistical model ofthis measurement error that preserves the cointegrationproperties of the data. Define as the true value of theunderlying portfolio at t.4 We hypothesize that:
(2)
where x is a set of stationary exogenous or predeterminedvariables that explain differences between measured andtrue NAV.5 When prices change very little, the error issmall but when they change rapidly, the error is large andhas the effect of making the measured price change by lessthan the true price. Thus a natural expectation is that θis negative. This model is roughly consistent with Lo andMacKinlay [1990] where the arrival rates of componentsof the index are constant, see also Le Baron. The expectedtime of the last quote is, therefore, constant over time.Consequently, the expected closing price on a portfoliowould be a fixed proportion of the true change in port-folio value. For funds with futures prices, which do notsuffer from these portfolio problems but which do havenoise from expected dividends and interest rates, thefutures returns can be used as xs.
The goal of the analysis is to measure the size and per-sistence of the true premium that can now be defined as
(3)
where u may be autocorrelated if premiums have somedynamic structure. For example, if the premium follows
p n ut t t− =˜
n n n n xt t t t t t= + −( ) + +−˜ ˜θ φ η1
nt
a first-order autoregression, then Equation (3) can beexpressed as:
(4)
Assume that the growth of NAV has a constant mean,
(5)
and assume that all three shocks are independent andnormally distributed.6
The system of Equations (2), (4) and (5) can thenbe expressed in a state space framework and estimatedwith the Kalman Filter. See for example Harvey [1989]or Hamilton [1994].
(6)
The Kalman filter will provide forecasts of the true NAVand true premium based on past information. Theseestimates can be further refined based on subsequent datato estimate what the true NAV was at any time. The para-meters of this system can be estimated by maximizing thelikelihood with respect to the unknown variance and meanparameters. The standard deviation of the innovation to thetrue premium, ε, is related to the standard deviation of u by
(7)
The methodology, however, is greatly simplified if it turnsout that the errors in the NAV Equation (2) are smallrelative to the others. This would generally be expected,as the magnitude of the stale quote error is likely to besmaller than the rate of change of the price or the devi-ation of the premium. Assuming that Equation (2) has noerror term, it can be solved with Equation (3) to elimi-nate the unobserved true NAV.
(8)
If the first-order autoregressive assumption is sufficient forthe premiums, then Equation (8) will simply require anAR(1) error specification. The unconditional standard devi-ation is estimated by the standard deviation of . Noticethat this model is consistent with the cointegration hypo-thesis as all variables on both sides of the equation are
stationary. Because θ is negative, the coefficient α shouldbe positive. This means that rapid increases in NAV shouldresult in especially large premiums because the measuredNAV will be an underestimate of the true NAV.
If the variance of the measurement error in Equa-tion (2) is not zero, then Equation (8) is only an approx-imation. The disturbance in the equation is
(9)
This additional term has several implications. Becauseη is correlated with ∆nt, the least squares coefficient estimatesof α and β are biased and inconsistent. The estimate of αis biased downward and possibly negative. Thus largeincreases in NAV could be associated with reduced pre-mium if the increase in NAV is mostly measurement error.The standard deviation of Equation (9) exceeds the standarddeviation of u, but the least squares residuals will have asmaller standard deviation as only the part of η orthogonalto the regressors will be left in the residuals. Thus the stan-dard error of the regression Equation (8) will give only a smalloverstatement of the standard deviation of the premium.
The composite error term in Equation (9) has morecomplex time series structure. For example, if u is an AR(1),then the composite error is an ARMA(1,1). Thus, forsmall measurement errors, the standard deviation of theautoregressive error is a conservative estimate of the truepremium standard deviation. If the measurement errors aremore significant than this, then the model in Equation (6)must be used. In the empirical section, we provide someexamples to show the relation between these two estimates.
In some markets, it is possible to improve the mea-surement of n using futures prices. As, under standardarbitrage assumptions, the futures are priced as:
(10)
with F as the futures price, S as the spot price, T as theremaining time to expiration of the futures contract, q asthe continuously compounded dividend rate and r as thecontinuously compounded interest rate, a futures priceimplicitly estimates the cash price just by resolving thisequation. To incorporate this into the measurementequation for NAV, define
(11)
Then Equation (2) becomes
(12)
where one might anticipate a value of φ = −1.
n n n n At t t t t t= + −( ) + +−˜ ˜θ φ η1
A F r q T nt t t= ( ) − −( ) −log
F S et tr q T= −( )
ut t− +( )η θ/ 1
Further measurement errors are introduced throughthe timing of market closing. For many of the domesticbroad-based ETFs, the market closes at 4:15 Eastern timewhile the NAV is calculated at 4:00, when the equitymarkets close. As a consequence, in daily data thereis another important measurement error in the NAV.Calculation of the 4:15 NAV for funds with futurescontracts simply requires the change in the futures pricebetween the close of the two markets. Calling this post-market change in futures, Fpm, Equation (12) now can bewritten as,
(13)
and the premium Equation (8) becomes:
(14)
We refer to Equation (14) as the dyna model. The regres-sion of premium, therefore, includes the change in NAV,the futures-based cash adjustment and the future returnsfrom 4:00 PM to 4:15 PM. For some ETFs, only a subsetof these variables will be available or relevant.
Serial correlation corrections will be needed if auto-correlation remains in the premium in Equation (14).Allowing for a first-order autoregressive error structure asin Equation (4), the estimating equation, then, is
(15)
The coefficients are estimated more efficiently in Equa-tion (15) but the residuals measure only the unpredictableportion of the premium, not the entire premium. Whenthese differ, it is the entire premium that reflects theimportance of premiums and discounts. The uncondi-tional error in the premium can be calculated by exam-ining the sum of squared residuals of Equation (14) usingthe coefficients estimated in Equation (15).
While there may still be errors in the premium dueto noisy measurement of p due to bid-ask spread orstaleness, these price effects can be almost eliminated byusing closing midquotes rather than last trade prices. Infact, we will show later that the standard deviation ofmidquote premium regression is smaller than that oftransaction premium regression.
Once the effects of the independent variables aretaken out, the residuals reflect the remaining premiumand discount. Thus, the standard deviation of the residualsis a good measure of the size of the pricing errors that
p n p n n n
A A Fpm Fpm
t t t t t t
t t t t t
− = −( ) + −( )+ −( ) + −( ) +
− − −
− −
ρ α ρ
β ρ β ρ ε1 1 1
1 1 2 1
∆ ∆
p n n n A Fpm ut t t t t t t− = −( ) + + +−α β β1 1 2
n n n n A Fpmt t t t t t t= + −( ) + + +−˜ ˜θ φ β η1
34 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
actually occur. If the residual variance changes over time,as it is likely to do from the model presented above,heteroskedasticity corrections can measure when it is largeand when it is small. We assume the error variance to beproportional to the volatility of the underlying asset.Suppose the residual variance is modeled as
, (16)
where z reflects a vector of variables measuring thevolatility of the underlying asset. The simplest versiontakes zt′ = (log(hight/lowt),c) where c is an intercept. Amore flexible GARCH model sets:
(17)
where e are the residuals from the model. With eitherof these formulations of the heteroskedasticity—Equation(16) or Equation (17)—the model is estimated by max-imum likelihood with a conventional conditional Gaussianlikelihood function given by
(18)
END-OF-DAY PREMIUM:DIA, XLK, AND EWA
Descriptive Statistics
The three models derived in the section Method-ology are: the Kalman Filter State Space model, the dynamodel, and the GARCH model. The Kalman Filter StateSpace model is represented by Equation (6) that providesforecasts of the true NAV and true premium based onpast information. The dyna model of Equation (14)regresses premium on the change in NAV, the future-based cash adjustment and the future returns from 4:00PM to 4:15 PM. The GARCH model is representedby Equation (15) with the residual variance defined asEquation (17). The GARCH model corrects for auto-correlation if it is present in the premium.
To compare the performance of these models, weconsider end-of-day data for three ETFs: DIA (based onbroad-based market index DJIA), XLK (based on sectorindex S&P Technology) and EWA (based on internationalindex MSCI Australia).
The ETF trading time and the underlying indextrading time pose very different problems for these threeseries. DIA trades until 4:15 PM but the NAV is calculated
Le
tt
tt
= − +
∑1
22
2
2log( ) .σ
σ
σ δ δt t t t t t th z h a b ae z bh21
211 2= = − −( ) + −( ) +− −exp( ), exp
σ δt tz2 = exp( ),
at 4:00 PM. The DJIA futures contract that trades until4:15 PM can be used to correct the NAV in DIA bothfor stale quotes and for the timing discrepancies. XLKcloses at 4:00 PM and it has no futures contract on it.However, as it is a sector of the S&P500 index, it is pos-sible that staleness in its NAV would be related to theS&P500 measures. EWA closes at 4:00 PM but it tradesentirely while the underlying market is closed. As a result,it probably contains a very stale value for NAV. Therecorded value of NAV in this case is simply the closingprice of the basket in Australia, adjusted for changes in cur-rency values until 4:00 PM Eastern time.
Exhibit 5 reports the end-of-day estimates for thethree funds for the three models. Panel A gives the pre-mium, and standard deviations of premium and last trade-based premium. All figures are expressed in percentageterms. For example, the premium of DIA is 4.6 bps,1.9 bps for XLK, and 47.1 bps for EWA. The standarddeviation of last trade-based premium for DIA is 22 bpswhereas the standard deviation of premium is 20 bps. Theuse of the midquote reduces the standard deviation foreach of these products particularly for the less activelytraded XLK and EWA.
The dyna Model
The regression results with the midquote-based pre-mium based on dyna model are given in Panel B ofExhibit 5. The futures price change from 4:00 PM to4:15 PM (“FutPM”) has a very large and significant effecton DIA. The correction to the NAV is estimated to be 70%of the change in the futures price. The adjustment to theestimated cash value at 4:00 PM (“CashAdj”) is only 10%of the prediction based on the futures price. The coeffi-cient of the change in the NAV from one day to the next(“dNAV”) is found to be significantly positive. RisingNAV implies that the measured NAV is too low becausesome quotes are stale and, consequently, the premium istoo high. The autocorrelation in the errors is estimated tobe 0.13, which is quite small. Therefore, the estimatedstandard deviation of the true premium (11.8 bps, calcu-lated by simply ignoring the autocorrelation) is almostidentical to the standard error of the regression or 11.7 bps.The adjustments to NAV based on the futures prices, cor-recting for the timing discrepancy and for the estimatedcash value, are effective in bringing the standard deviationfrom 20 bps to 12 bps, almost a 40% reduction.
For XLK, there is no timing mismatch and no futurescontract. Hence the cash adjustment for the S&P500 futures
is used in the regression. The adjustment to the estimatedcash value is 3.8%. The change in NAV is significant, but nowit has the negative sign associated with errors-in-variables:increasing NAV reduces the premium. In other words,when the NAV increases, the premium is now measuredrelative to a typically overstated estimate of NAV. Whilethe stale quote feature may still be important, it is domi-nated by the measurement error in NAV. There remainslittle serial correlation and the final estimate of the standard
deviation of the premium is 15.3 bps, which is not dif-ferent from the standard deviation of the unadjusted pre-mium. Notice that the standard deviation is now higherthan the one in the DIA. This is expected because of thereduced transaction volume and narrower sector coverage.
EWA has no futures contract traded in the UnitedStates and, therefore, is priced only with reference to themeasured NAV. The coefficient on the change in NAVis significantly negative reinforcing the prior expectation
36 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
E X H I B I T 5Results for Three ETFs: End-of-Day Analysis
that the NAV is measured with large errors. The auto-correlation is estimated to be large and significant. Thestandard deviation of the true premium is, therefore, biggerthan the standard error of the regression and is, in fact,about the same size as the unconditional standard devia-tion. This reflects the finding that only a small part of thepremium can be attributed to the hypothesized forms ofmeasurement error. The mean premium is now notice-ably positive at 47 basis points. The finding of a large pos-itive mean premium is characteristic of the internationalETFs and will be discussed later.
Kalman Filter and Errors-in-VariablesModel
For two of these series (XLK and EWA), there isevidence that the measurement errors on the NAV areimportant in that changes negatively affect the premium.Therefore, it may be important to estimate the KalmanFilter State Space version of the model given in Equation (6).This estimation procedure identifies the measurementerrors in NAV and the premium separately from the timeseries data. The results are given in Panel C of Exhibit 5.
For DIA and XLK, the estimated standard devia-tion of the measurement error is much smaller than thestandard deviation of the premium, and the premiumautocorrelation—measured by rho—is nearly zero. Hence,the model gives practically the same estimated standarddeviation of premium as the dyna model. However, forEWA, the standard deviation of the measurement errorin NAV is much bigger and there is autocorrelation of0.92 in the premium. The estimate of the standard devi-ation of the premium adjusted for measurement errorsaccording to Equation (7), is now 49 bps. This suggests asubstantially better performance of this model for thisfund over other models. This model attributes much ofthe measured premium to errors in the NAV. The modelalso estimates the persistence of premiums of the errorfree true premium to be greater than that estimatedpreviously indicating that the correctly measured premiumis smaller but lasts longer.
The estimates given by the dyna model are conser-vative as argued in the development of the model but give auseful upper bound on the standard deviation of premium.
GARCH Model
The volatility of the premium changes over time. InPanel D of Exhibit 5, Equation (15) is estimated with the
GARCH heteroskedasticity correction given by Equa-tion (17). While the parameter values are rather similarto those in the upper panels, the graphs of conditionalvariance are quite interesting. In Exhibit 6, the standarddeviation of the DIA premium is graphed from the basicmodel with no adjustment for measurement errors inNAV and from the dyna model. The time variation inthe standard deviation is partly a result of variation in thevolatility of the DJIA index itself as measured by the dailyhigh/low ratio. It also is due in part to persistent swingsin standard deviations that are modeled by GARCH. Thereduction in standard deviation is more or less uniformacross time, but is particularly effective at times whenthe standard deviations are greatest.
The standard deviation estimator for XLK is plottedwith the premium in Exhibit 7. On the graph, ±1.96standard deviations form an approximate 95% confidenceinterval. Clearly this is highly variable but pretty reliableas an indicator of the possible movements. In Exhibit 8,the standard deviation of the EWA premium is plotted.The scale on this plot is noticeably greater with someperiods having a standard deviation greater than 2%.
END-OF-DAY PREMIUM:A COMPREHENSIVE VIEW
The dyna model from Equation (14) was estimatedfor end-of-day data for all funds and the results for thedomestic and international funds are reported in Exhibits 9and 10, respectively. The base model standard deviationof premium is reported as well. For completeness, wereport the DIA, XLK, and EWA results once again.
Exhibit 9 shows that for all the domestic funds, thedyna estimated standard deviation of premium is smallerthan the standard deviation of the base premium. Theseare dramatically smaller for the ETFs that close at 4:15 andthose with futures contracts on the index. For the otherdomestic ETFs, there is little difference between the twostandard deviations. The dyna numbers range from 9 bps(IJR: S&P SmallCap) to 24 bps (XLB: S&P Basic Indus-tries). According to Exhibit 10, the international funds, onthe other hand, show little change from the base modelstandard deviation of premium. In fact, in many cases, thebase model standard deviation of premium is smaller thanthe corresponding dyna standard deviation.
Exhibits 9 and 10 show the persistence of premiumsand discounts for both domestic and international funds.The autocorrelation parameters indicate whether a pre-mium on one day has predictability for the premium on
the next day. For the domestic funds, the average auto-correlation estimate is a little over 0.1 indicating that about10% of today’s premium can be expected to remain by theclose tomorrow. As the premiums are small to start with,this is quite a small effect. For the international funds, theaverage autocorrelation is about 30% suggesting that carryover from one day to the next is potentially important.
INTRA-DAILY PREMIUM:A COMPREHENSIVE VIEW
Even though the intra-daily IOPV is not designedfor trading, it does give a quick snapshot of underlyingindex value on a high frequency basis. Nevertheless, thesame analysis performed on a daily basis can be performedon an intra-daily basis allowing for the possibility thatthere are biases in the IOPV due to stale quotes anddynamic adjustment of the true premium to its equilib-rium level.
The results of the analysis of intra-daily data aregiven in Exhibits 11 and 12. Each table also reports thebase model standard deviation of premium. The tablesreport the results for 16 domestic and 16 internationalETFs. The other five domestic ETFs do not have com-plete data.
For the domestic ETFs, Exhibit 11 shows that, exceptfor SPY, the dIOPV coefficient is large negative (rangesfrom −11 bps to −55 bps) and significant indicating theexistence of measurement errors in IOPV. For interna-tional ETFs too, the coefficient is large negative (rangesfrom −11 bps to −45 bps) and significant. Measurementerrors seem to be present in international IOPVs as well.
The minute-to-minute change in the S&P futuresand an ARMA(1,1) are used in each regression.7 Gener-ally, the futures coefficient is positive and significant. TheAR term is large, close to 1 in many cases, and the MAterm is generally negative, when significant. It is clear thatthe autocorrelation is much greater for these high
38 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
E X H I B I T 6Standard Deviation of DIA Premium from Base and Dyna Models with GARCH
frequency data sets than it was for the daily data. For thedomestic funds, the average autocorrelation is 0.90 whilefor the international funds it is 0.99. Although the hypoth-esis that the price and the IOPV are cointegrated mightappear tenuous for the international funds, a direct testconcludes that these series are cointegrated in every case.While there is some explanatory power in the regressorsintroduced into these regressions, the estimated standarddeviation of the premium is reduced imperceptibly inalmost all cases.
It is also of interest to examine the persistence ofthe intra-daily premiums. For the domestic funds, thefirst-order autocorrelation is on average 0.90 indicatingthat the half-life of a premium is about 5.6 minutes. Thehalf-life is the expected time it takes for the premium todrop to half of its value and is given by log(0.5)/log(ρ)−1,where log is the natural logarithm. For the internationalfunds, the first-order autocorrelation is 0.99 with a half-life of 68 minutes.
The lengths of the lags can be examined in moredetail with these high frequency data. We calculate thelength of time that a large premium takes to revert to themean value. For each asset we define an upper and a lowerthreshold. We measure the duration of a large premiumstarting when it first exceeds the upper threshold and con-tinuing until it first crosses the lower threshold. This isdone separately for premiums and discounts, and fordomestic and international funds. These durations arepresented in Exhibits 13 and 14.
From these results it is clear that the typical largeduration episode lasts only a few minutes for the majorityof the domestic funds. For SPY and DIA, the medianduration is 5-7 minutes. The average overall domesticfunds is 10 minutes and the distribution is more or lesssymmetric between premium events and discount events.For the international funds, the average premiums anddiscounts are 176 minutes with some lasting more thanone day.
SUMMER 2006 THE JOURNAL OF DERIVATIVES 39
E X H I B I T 7Confidence Bands of XLK Premium from Dyna Model with GARCH
One reason these episodes last so long is the infre-quency of trades and quote revisions for the internationalfunds. In Exhibit 15, it can be seen that trades occur onaverage about every 25 minutes and quotes are revisedless frequently, sometimes more than 2 hours apart. Thissluggish response to information is only consistent withthe absence of arbitrage when the spreads are large. Thisis indeed the case with these international funds.
The story of the international fund pricing is thatthe prices move slowly in response to economic news,but that bid-ask spreads are apparently wide enough toprevent arbitrage. Exhibit 1 shows that for domestic fundsthe spread is 37.7 bps on average, although the average isdriven by a few of the sector funds. The broad indiceshave spreads under 20 basis points. The international fundshave spreads that average 112 bps. See Exhibit 2. How-ever, these spreads are small compared with the persistentpremiums of closed-end country funds and are smaller
than one typically finds for ADRs and other internationalreplication instruments.
CONCLUSIONS
This study has examined the magnitude of premiumsand discounts for a wide range of Exchange Traded Funds.These include domestic funds with and without futurescontracts, and closing at 4:00 PM or 4:15 PM. Theseinclude broad market indices and narrow sector funds.The sector funds range from utilities and basic industriesto technology and internet sectors. In almost all cases, themean premium was less than 5 bps and the standard devi-ation was less than 20 bps.
We develop a statistical approach to measuring thetrue premium by correcting some of the measurementerrors in net asset value. This reduces further the observedstandard deviation. We examine how the standard
40 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
E X H I B I T 8Standard Deviation of EWA Premium from Dyna Model with GARCH
deviation moves over time. The resulting standard devi-ation of the premium is 9 bps for some funds and averages14 bps. For the international funds the estimate of thestandard deviation averages 77 bps.
From a minute-by-minute point of view, the stan-dard deviations are even smaller. It now becomes possibleto see how long episodes of premium or discount last.The domestic episodes generally last only a few minuteswith an average across funds of 10 minutes. The interna-tional episodes last typically almost 3 hours with someeven slower to recover.
The overall impression of the domestic ETFs is ofa set of products that are priced very close to their trueNAVs with only brief excursions any significant distanceaway. The international ETFs are less actively traded andless accurately priced; yet they operate in a more strin-gent environment and may still be performing accordingto expectations.
ENDNOTES
The American Stock Exchange provided support for thisresearch. The opinions and conclusions expressed here, how-ever, do not necessarily reflect those of the American StockExchange. The authors would like to thank Chia Hsun Chanand We Chen Foo for excellent research support.
1Bonser-Neal et al. [1990] argue that segmentation ofthe international capital market from the US capital marketcan raise a closed-end country fund’s price-NAV ratio. In thatcase, the country fund can trade at a premium. Each fundprovides two distinct market-determined prices: the countryfund’s share price quoted on the domestic market and its NAV,determined by the prices of the underlying shares traded onthe foreign market. Barriers to international investment cancause the expected returns on assets of equal risk to differacross countries. If capital markets are integrated interna-tionally, a closed-end country fund’s shares and its underlyingassets should have similar risk. International investment restric-tions can affect the ratio of a country fund’s price to NAV ifthey are binding. All other things constant, binding investmentrestrictions will raise the price of a fund’s shares relative to its
SUMMER 2006 THE JOURNAL OF DERIVATIVES 41
E X H I B I T 9End-of-Day dyna Model Regression Results: Domestic Funds
Significant coefficients (at 5%) are presented in bold.
E X H I B I T 1 2Intra-Daily dyna Model Regression Results: International Funds
Significant coefficients (at 5%) are presented in bold.
E X H I B I T 1 3Persistence of Large Intra-day Premium (in Minutes): Domestic Funds
Products with smaller standard deviations have an upper bound of 25 bps and a lower bound of 10 bps (from the mean) and correspondingly (−25 bps, −10 bps) fordiscounts. These are not starred.*Products with large standard deviations. These have an upper bound of 50 bps and a lower bound of 20 bps (from the mean) and correspondingly (−50 bps,−20 bps) for discounts.
NAV by approximately the amount the marginal domesticinvestor is willing to pay to avoid these restrictions.
2See Engle and Granger [1987] for this theory andspecification.
3Even though we use last trade-based premium forcomparison purposes, “premium” would refer tomidquote-based premium throughout the article unlessotherwise stated.
4Potentially this would be a slightly different numberfor an investor considering creation from one consideringredemption because of the difference between the buyingand selling prices of the underlying securities.
5Miller et al. [1994] define a price process where themeasured price this period is a weighted function of mea-sured price last period and true price this period.
6The normality assumption can be weakened whenthe Kalman Filter is interpreted as the linear projectionrather than the conditional distribution.
7Bailey and Lim [1992] show that returns on closed-end fund stock prices are more highly correlated with theU.S. market returns than the corresponding foreign indices.
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44 PREMIUMS-DISCOUNTS AND EXCHANGE TRADED FUNDS SUMMER 2006
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