Measuring the tracking error of exchange traded funds: an unobserved components approach

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1 Measuring the tracking error of exchange traded funds: an unobserved components approach Giuliano De Rossi Quantitative analyst UBS Investment Research June 2012 Analyst Certification and Required Disclosures Begin on Slide 30

2 Motivation We address an empirical question: How should the tracking error (TE) of an exchange traded fund (ETF) be calculated? This is a crucial issue given the growth in ETF size and the importance of TE as a selection criterion Reported TEs for Emerging Markets (EM) ETFs are typically overestimated in the existing literature We propose a solution based on an unobserved components approach

3 Outline Introduction: ETFs TE estimation The problem with EM ETFs Our approach: An unobserved components model Inference Results Conclusion

4 What is an (equity) ETF? Listed fund which tracks a chosen benchmark Exchange traded: provides liquidity and transparency Low management fees, usually deducted when dividends are distributed Academic studies: Gastineau (2001), Poterba and Shoven (2002)

5 How do ETFs replicate the benchmark? Two main alternative methods are available: Physically-backed: holds the underlying in a ring fenced account (i.e. no exposure counterparty risk) Complete replication Partial replication (holds a subset of the constituents of the index, selected through an optimisation procedure) Swap-backed: ETFs are collateralised

6 Estimating the TE of a fund The TE measures the dispersion of a portfolio s returns relative to the returns of the manager s benchmark Several approaches in the academic literature, see Pope and Yadav (1994), Shin and Soydemir (2010) and references therein One can use ETF price returns or the return to the net asset value (NAV) of the fund, which is less noisy The NAV is not accurately represented due to agency costs, tax liabilities, asset illiquidity, asynchronous prices

7 Estimating the TE of a fund A common definition of TE (e.g. Vardharaj, Fabozzi and Jones, 2004) is the standard deviation of the active return: TE 0 = 1 T ( ) 2 rf,t r I,t T 1 where r I,t the logarithmic return of the index and r F,t the log return to the NAV of an ETF. Shin and Soydemir (2010) use: TE 1 = 1 T T r F,t r I,t t=1 TE 2 = St. error in the regression r F,t = a + br I,t + u t TE 3 = 1 T ( RDt RD ) 2 where RDt = rf,t r I,t T 1 t=1 t=1

8 Typical levels of TE 0 According to Vardharaj, Fabozzi and Jones (2004) the typical levels of tracking error, measured as an annualised standard deviation, should be: zero for an index fund below 2% for an enhanced index fund between 5 and 10% for an actively managed fund They also note that TEs tend to increase when the volatility of the benchmark increases

9 Sources of TE in an ETF The performance of an ETF in tracking the index is affected by Management Fee and administrative/operating expenses Trading Costs Rebalance Costs Dividend Treatment Portfolio Optimisation Tax Considerations Premiums/Discounts Non-Concomitant Trading Hours As a result, the TE is not zero in practice.

10 Example: Two Emerging Markets ETFs Source: Bloomberg, UBS estimates. NDUEEGF is the total return to the MSCI Emerging Markets index. CSEM VWO St Dev TE TE TE Source: UBS estimates. Daily data between 6/10/2009 and 6/10/2010.

11 Errors in variables Source: UBS estimates. Daily data between 6/10/2009 and 6/10/2010 used to compute the ACF of active returns. The active returns are measured with error due to the use of asynchronous stock prices and asynchronous exchange rates This results in the MA(1) structure of autocorrelations typically found in the data

12 Potential solutions Using low frequency data: Blitz and Huij (2012) use quarterly or even annual data to mitigate the problem Filtering the data or using instrumental variables in order to take the autocorrelation into account. We show in How to pick the right ETF (5 November 2010) that a simple MA(1) model yields an estimated tracking error of 0.88% for the Vanguard fund! Engle and Sarkar (2006), in a paper on ETF premia, model the true NAV of an ETF as a latent variable and use the Kalman filter to estimate its value Moise and Russell (2012) use high frequency data to decompose the intraday price returns of SPDR into a fundamental return and a price deviation component

13 An unobserved components model Call p I,t the log of the value of the index, p F,t the log of the true, unobservable NAV. [ ] [ ] [ ] [ ] pi,t µ = I pi,t 1 η + + I µ I + µ F p F,t 1 η I,t + η F,t p F,t where µ I, µ F are constants while η I and η F are mutually independent random variables with zero mean and variances σi 2 andσf 2. While we do observe the index, we can only obtain a noisy realisation p F of the NAV p F : [ ] [ ] [ ] pi,t pi,t 0 = + p F,t σ t ɛ t where ɛ t is standard normal. p F,t

14 An unobserved components model (2) The variable σ t introduces stochastic volatility: σ t = exp (h t /2) h t = ω + φh t 1 + η h,t where ω, φ are constant and η h,t is normally distributed with mean zero and variance q.

15 An unobserved components model (3) The variables of interest are the cumulated active return of the fund λ t and its observable counterpart λ t λ t p F,t p I,t λ t p F,t p I,t The observed one period active return is just the difference λ t. Over an interval of τ periods we obtain, conditional on h 1,..., h t, ) E ( λ t λ t τ = τµ F ) Var ( λ t λ t τ = τσf 2 + exp (h t τ ) + exp (h t )

16 An unobserved components model (4) From the equations in the previous slides we obtain a nonlinear state space model: λ t = λ t + exp (h t /2) ɛ t λ t = µ F + λ t 1 + η F,t h t = ω + φh t 1 + η h,t We need to estimate the parameter vector ψ = {µ F, σ 2 F, ω, φ, q}.

17 Bayesian inference We adopt a MCMC approach that is a slight modification of Kim, Shephard and Chib (1998) and Chib, Nardari and Shephard (2002). The object of interest is the augmented posterior π (ψ, h, λ) π (ψ) T f t=1 ( λ t h t, λ t, ψ) f (λ t λ t 1, ψ) f (h t h t 1, ψ) We take draws from the posterior in two blocks: {µ F, σ 2 F, λ} and {ω, φ, q, h}.

18 Bayesian inference: µ F, σ 2 F In order to draw the first block of variables, ψ (1) = {µ F, σf 2 }, we use the Metropolis-Hastings algorithm. First we run the Kalman filter on the linear system λ t = λ t + exp (h t /2) ɛ t (1) λ t = µ F + λ t 1 + η F,t conditional on h, treating λ t as the state variable. We then find the value m of {µ F, σ 2 F } that maximises the (log) density g ( λ t ) in the linear Gaussian state space (1). V is obtained by inverting the Hessian of the same function at the optimum.

19 Bayesian inference: µ F, σ 2 F The proposal draw ψ (1) is then sampled from N(m, V ) and accepted with probability { ( g ψ (1) ) π ( ψ (1) ) ( f N ψ (1) ; m, V ) } min g ( ψ (1)) π ( ψ (1)) f N ( ψ (1) ; m, V ), 1 We then draw the state λ by running the simulation smoother of de Jong and Shephard (1995) in (1).

20 Bayesian inference: ω, φ, q, h A similar procedure is used to draw {ω, φ, q, h}. Here we use the linearisation proposed e.g. in Harvey, Ruiz and Shephard (1994) ( ) ) 2 y t = log ( λ t λ t = h t + z t (2) h t = ω + φh t 1 + η h,t where z t log(ɛ 2 ). The non Gaussian density of z is approximated by a mixture of seven normal distributions. This introduces a new state variable s t {1,..., 7}. Conditional on s t, z t is assumed to be Gaussian with mean and variance depending on s t. We then draw the state h by running the simulation smoother in (2).

21 Bayesian inference: Summary Summary of the MCMC algorithm: 1. Initialise ψ (1) = {µ F, σf 2 }, λ, ψ(2) = {ω, φ, q}, h 2. Sample ψ (1) and λ from ψ (1), λ λ, h 3. Sample s from s y, h, ψ (2) 4. Sample ψ (2) and h from ψ (2), h λ, s 5. Go back to 2.

22 Prior distribution 1. µ F and ω have normal priors 2. σf 2 and q have lognormal priors 3. φ has a rescaled beta prior with support on ( 1, 1). We have chosen the prior on φ so as to concentrate the probability mass between 0.75 and 1. The prior mean of ω implies an equilibrium value of σ t of 50bp when φ = 0.9. The prior mean of σf 2 implies an annualised contribution of 1.5% to the TE. The prior mean of q is chosen so as to set the standard deviation of the stationary distribution of h to 4 when φ = 0.9. As a result, if h t were one standard deviation away from its equilibrium value then σ t would equal 3.69%.

23 Data Source: Bloomberg, UBS estimates. Daily NAVs and dividends, total return index from Bloomberg Two funds tracking the MSCI EM index: Vanguard (full replication) and ishares (partial replication until 2009, moved to full replication from early 2010)

24 Posterior distributions Source: UBS estimates. We used 8000 Gibbs iterations and discarded the first 1000.

25 Posterior distributions (2) Source: UBS estimates. The long term standard deviation is exp(0.5ω/(1 φ)), while the long term dispersion of h is q/(1 φ 2 ).

26 Posterior distributions of the state h t Source: UBS estimates. The chart shows the median of the distribution of the MCMC draws.

27 Posterior distributions (3) Source: UBS estimates. The chart shows the median and 10% and 90% quantiles of the distribution of the MCMC draws.

28 Conclusion We developed a method to decompose the TE of an ETF into a temporary and a permanent component Our approach allows for stochastic volatility in the transitory component of the return differences The results seem to provide realistic values of the TE and estimate accurately its changes over time The proposed methodology also expresses TE as a function of the investment horizon

29 References D. Blitz and J. Huij (2012): Evaluating the performance of global emerging markets equity exchange-traded funds, Emerging Markets Review 13, S. Chib, F. Nardari and N. Shephard (2002): Markov chain Monte Carlo methods for stochastic volatility models, Journal of Econometrics 108, P. de Jong and N. Shephard (1995): The simulation smoother for time series models, Biometrika 82, R. Engle and D. Sarkar (2006): Premiums-discounts and exchange traded funds, Journal of Derivatives 13, G.L. Gastineau (2001): Exchange-traded funds: an introduction, Journal of Portfolio Management 27, A.C. Harvey, E. Ruiz and N. Shephard (1994): Multivariate stochastic variance models, Review of Economic Studies 61, S. Kim, N. Shephard and S. Chib (1998): Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models, Review of Economic Studies 65, C.E. Moise and J.R. Russell (2012): The joint pricing of volatility and liquidity, Working paper. P.F. Pope and P.K. Yadav (1994): Discovering errors in tracking error, Journal of Portfolio Management 20, J.M. Poterba and J.B. Shoven (2002): Exchange-traded funds: A new investment option for taxable investors, American Economic Review 92, S. Shin and G. Soydemir (2010): Exchange-traded funds, persistence in tracking errors and information dissemination, Journal of Multinational Financial Management 20, R. Vardharaj, F.J. Fabozzi and F.J. Jones (2004): Determinants of tracking error for equity portfolios, Journal of Investing 13,

30 Analyst certification Analyst Certification

31 Required Disclosures

32 Required Disclosures (continued)

33 Compendium Disclosure

34 Global disclaimer

35 Global disclaimer (continued)

36 Contact Information

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