Index Tracking Issues

Tracking Issues Prof. Dr. Marlene Müller This version: December 4, 2009 Plan Problem Tracking Basket Replication Case Study Conclusions 1

Problem index tracking issues - tracking an index (e.g. to construct an ETF) - determining a strategy to (regularly) re-adjust tracking basket - limit transaction costs by restricting the number of adjusted weights - implement other restrictions on weights (e.g. no shortselling) - implement budget constraints (e.g. self-financing sell buy) this talk - which optimization criterion to use? - case study for comparing different optimization criteria 2 Tracking Criteria let B denote the tracking portfolio ( basket ) and I the index (we do not specify if these denote returns or prices for the moment) consider B = w i X i = w X i=1 where w = (w 1,..., w m ) is a weight vector (unit: no. of stocks) and X = (X 1,..., X m ) the universe of stocks to analyze notation: - X = R (returns) or X = S (stock prices) - R I, R B (index and basket returns) - S I, S B (index and basket values) 3

Tracking Error tracking error (volatility of) return differences q TE = E(R I R B ) 2 variants: tracking error volatility TE vol = p Var(R I R B ) volatility of portfolio value instead of return differences TE vol,prices = p Var(S I γ S B ) (see e.g. Meucci; 2005) residual standard deviations q TE res = σ(r B ) 1 Corr(I, B) 2 (see e.g. Spremann; 2008) mean absolute deviation TE mad = E R I R B (Rudolf, Wolter and Zimmermann; 1999; Satchell and Hwang; 2001) 4 Correlation and Other Concepts correlation Corr = Corr(R I, R B ) = Cov(R I, R B ) p Var(RI ) pvar(r B ) correlation of portfolio values Corr = Corr(S I, S B ) = Cov(S I, S B ) p Var(SI ) pvar(s B ) cointegration S I and S B are cointegrated (see e.g. Alexander; 2001) i.e. S I γ S B is stationary, can be tested by e.g.: - Dickey-Fuller (DF), Augmented Dickey Fuller (ADF) tests - Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test market capitalization weight method (see e.g. Alighanbari and Mougeot; 2009) 5

Relationships TE vs. TE vol E{(R I ER I ) (R B ER B )} 2 = (R I ER I ) 2 + E(R B ER B ) 2 2E(R I ER I )(R B ER B ) = Var(R I ) + Var(R B ) 2 Cov(R I, R B ) = Var(R I R B ) TE res vs. TE vol R B = α + βr I + ε Var(ε) = Var(R B βr I ) = Var(R B ) n1 Corr(R I, R B ) 2o TE res vs. Corr TE res leads to a quadratic optimization problem while Corr gives an non-convex objective function in w (see next slides) 6 Which Criterion to Use? all criteria are,,average deviations between index and basket no difference between over- and under-performance in particular, important to note: - a TE vol of 0 does not imply that index and basket (returns) are proportional - a correlation of 1 does not imply that index and basket (returns) are proportional (as this may happen if both have constant difference) 7

Optimization with TE vol and Correlation assume the following constellation: - hold a part B hold of the basket fixed and optimize only w.r.t. a small stock universe X - then: B = B hold + w X - optimize min TE vol (w) or max Corr(w) w under additional restrictions on w (e.g. w 0,... ) w 8 TE vol Optimization we have TE 2 vol(w) = Var(I B) = Var(I) + Var(B) 2 Cov(I, B) = Var(I) + Nw 2 2D w with D w = Cov(I, B hold ) + w i Cov(X i, I) Nw 2 = Var(B hold) + 2 w i Cov(B hold, X i ) + w i w j Cov(X i, X j ) i=1 i=1 i,j=1 therefore n o min TE vol (w) min Nw 2 2D w w w min w n (c + 2d w + w Σw) 2(a + b w) o 9

Correlation Optimization we have now Corr(w) = = Cov(I, B) p p = Cov(I, B hold) + Cov(I, B new ) p p Var(I) Var(B) Var(I) Var(B) D w p Var(I) Nw with again therefore N 2 w = Var(B hold) + 2 D w = Cov(I, B hold ) + w i Cov(X i, I) i=1 w i Cov(B hold, X i ) + i=1 w i w j Cov(X i, X j ) i,j=1 D w a + b w max Corr(w) max max w w N w w c + 2d w + w Σw 10 Prices or Returns? simple scenario: (jointly) log-normal stock prices (jointly) normal returns R it N(r i, σ 2 i ) iid. over time t then for returns of stocks i, j ER it = r i, Var(R it ) = σ 2 i, Cov(R it, R jt ) = ρ ij σ i σ j but for stock prices ES it = S i0 exp(r i t + σi 2 t 2 /2) Var(S it ) = Si0 2 exp 2r i t + σi 2 t 2 {exp(σi 2 t 2 ) 1} j Cov(S it, S jt ) = S i0 S j0 exp (r i + r j )t + 1 ff n o 2 (σ2 i + σj 2 )t 2 exp(ρ ij σ i σ j t 2 ) 1 optimization over time horizon t + model assumption on the other hand: objective function on prices / portfolio values 11

Replication two essential steps - selection - allocation variety of methods (+ constraints!) - regression type - least squares regression - principal components regression - partial least squares -... - optimization - correlation - tracking error -... - cointegration analysis -... 12 Case Study - aiming at comparing different criteria (TE, TE vol, Corr) - uses a constraint on the maximal number m of re-adjustments (take care of transaction costs) - uses a budget constraint (self-financing) - selection simple search algorithm step 0: find an initial basket by (positive) regression on the 8 stocks which are most correlated with the index I step 1: remove consecutively those m stocks which give largest deviations of the objective criterion from its starting value (backward model selection) step 2: add consecutively up to m stocks which give largest improvements to the objective criterion from its starting value (forward model selection, test all combinations of up to m stocks) remark: stocks removed in step 1 can be added again in step 2 (reweighting!) - allocation automatically by optimizing TE, TE vol, Corr over w 13

DAX Fit: 01 12/2008, Eval: 01 06/2009, Corr Optimization with and Baskets 3500 4000 4500 5000 5500 6000 D.BAS 0.2164 D.MAN 0.0136 D.BAYN 0.0771 D.DTE 0.0427 D.HEN3 0.0290 D.VOW 0.0383 Ratio of to and Baskets 1 D.BAS D.BAYN 2 D.BMW D.DTE 3 D.MAN D.HEN3 4 D.SIE D.VOW Ratio to Baskets 0.95 1.00 1.05 1.10 1.15 Value 20.6832 20.6832 Corr 0.8523 0.9185 TE 0.0001 0.0002 TE.vol 0.0118 0.0140 TE.res 0.0100 0.0040 TE.mad 0.0006 0.0014 DF.p 0.4162 0.1234 ADF.p 0.8102 0.4796 14 DAX Fit: 01 12/2008, Eval: 01 06/2009, TE Optimization with and Baskets 3500 4000 4500 5000 5500 6000 D.BAS 0.2164 D.MAN 0.0136 D.BAS 0.2402 D.BMW 0.0937 D.DPW 0.1302 D.DTE 0.1067 Ratio of to and Baskets 1 D.MAN D.DPW D.BAS 2 D.TKA D.DTE D.BMW Ratio to Baskets 0.95 1.00 1.05 1.10 1.15 Value 20.6832 20.6832 Corr 0.8523 0.8596 TE 0.0001 0.0001 TE.vol 0.0118 0.0116 TE.res 0.0100 0.0099 TE.mad 0.0006 0.0002 DF.p 0.4162 0.1456 ADF.p 0.8102 0.5297 15

DAX Fit: 01 12/2008, Eval: 01 06/2009, TE vol Optimization with and Baskets 3500 4000 4500 5000 5500 6000 D.BAS 0.2164 D.MAN 0.0136 D.BAS 0.2070 D.DTE 0.1156 D.TKA 0.0944 Ratio of to and Baskets 1 D.MAN D.DTE D.BAS 2 D.TKA Ratio to Baskets 0.95 1.00 1.05 1.10 1.15 Value 20.6832 20.6832 Corr 0.8523 0.8612 TE 0.0001 0.0001 TE.vol 0.0118 0.0115 TE.res 0.0100 0.0098 TE.mad 0.0006 0.0005 DF.p 0.4162 0.1142 ADF.p 0.8102 0.4703 16 DAX Fit: 01 12/2008, Eval: 01 06/2009, TE res Optimization with and Baskets 3500 4000 4500 5000 5500 6000 D.BAS 0.2164 D.MAN 0.0136 D.FME 0.0092 D.FRE3 0.0029 D.MRK 0.0005 D.VOW 0.0370 Ratio of to and Baskets 1 D.ALV D.FME 2 D.BAS D.FRE3 3 D.DBK D.MRK 4 D.MAN D.VOW Ratio to Baskets 0.95 1.00 1.05 1.10 1.15 Value 20.6832 20.6832 Corr 0.8523 0.8882 TE 0.0001 0.0002 TE.vol 0.0118 0.0138 TE.res 0.0100 0.0051 TE.mad 0.0006 0.0010 DF.p 0.4162 0.0461 ADF.p 0.8102 0.3132 17

DAX Fit: 01 12/2009, Eval: 01 06/2009, TE mad Optimization with and Baskets 3500 4000 4500 5000 5500 6000 Ratio of to and Baskets D.BAS 0.2164 D.MAN 0.0136 D.EOAN 0.1176 D.MAN 0.0136 D.MEO 0.1261 D.MUV2 0.0474 D.TKA 0.1832 1 D.BAS D.EOAN D.TKA 2 D.BMW D.MEO 3 D.SIE D.MUV2 Ratio to Baskets 0.95 1.00 1.05 1.10 1.15 Value 20.6832 20.6832 Corr 0.8523 0.8084 TE 0.0001 0.0002 TE.vol 0.0118 0.0135 TE.res 0.0100 0.0096 TE.mad 0.0006 0.0000 DF.p 0.4162 0.1682 ADF.p 0.8102 0.4913 KPSS.p 0.0100 0.0445 18 SX5E Fit: 01 12/2008, Eval: 01 06/2009, Corr Optimization 160 180 200 220 240 260 with and Baskets D.DAI 0.0547 E.BBVA 0.0310 E.SCH 0.0786 E.TEF 0.1386 F.MIDI 0.1043 F.TAL 0.0773 H.RDSA 0.1675 S.CSGN 0.0852 D.DAI 0.0547 E.SCH 0.0786 E.TEF 0.1386 F.GSZ 0.0587 F.MIDI 0.1043 H.RDSA 0.1793 S.ABB 0.0642 S.CSGN 0.0852 Ratio of to and Baskets 1 E.BBVA F.GSZ H.RDSA 2 F.TAL S.ABB Ratio to Baskets 0.95 1.00 1.05 1.10 Value 14.6749 14.6749 Corr 0.9666 0.9718 TE 0.0000 0.0000 TE.vol 0.0059 0.0053 TE.res 0.0053 0.0052 TE.mad 0.0003 0.0006 DF.p 0.0100 0.0402 ADF.p 0.0369 0.0632 19

SX5E Fit: 01 12/2008, Eval: 01 06/2009, TE Optimization with and Baskets 160 180 200 220 240 D.DAI 0.0547 E.BBVA 0.0310 E.SCH 0.0786 E.TEF 0.1386 F.MIDI 0.1043 F.TAL 0.0773 H.RDSA 0.1675 S.CSGN 0.0852 D.DAI 0.0547 E.SCH 0.0786 E.TEF 0.1386 F.MIDI 0.0895 F.TAL 0.0773 H.RDSA 0.2061 S.ABB 0.0728 S.UBSN 0.0739 Ratio of to and Baskets 1 E.BBVA S.ABB F.MIDI 2 S.CSGN S.UBSN H.RDSA Ratio to Baskets 0.95 1.00 1.05 1.10 Value 14.6749 14.6749 Corr 0.9666 0.9729 TE 0.0000 0.0000 TE.vol 0.0059 0.0052 TE.res 0.0053 0.0052 TE.mad 0.0003 0.0009 DF.p 0.0100 0.3859 ADF.p 0.0369 0.0946 20 SX5E Fit: 01 12/2008, Eval: 01 06/2009, TE mad Optimization 160 180 200 220 240 260 with and Baskets D.DAI 0.0547 E.BBVA 0.0310 E.SCH 0.0786 E.TEF 0.1386 F.MIDI 0.1043 F.TAL 0.0773 H.RDSA 0.1675 S.CSGN 0.0852 D.EON 0.0705 E.BBVA 0.0310 E.SCH 0.0786 E.TEF 0.1386 F.MIDI 0.1555 F.TAL 0.0773 H.RDSA 0.1675 M.NOKP 0.0966 Ratio of to and Baskets 1 D.DAI D.EON F.MIDI 2 S.CSGN M.NOKP Ratio to Baskets 0.95 1.00 1.05 1.10 1.15 1.20 Value 14.6749 14.6749 Corr 0.9666 0.9600 TE 0.0000 0.0000 TE.vol 0.0059 0.0063 TE.res 0.0053 0.0062 TE.mad 0.0003 0.0000 DF.p 0.0100 0.1427 ADF.p 0.0369 0.5016 21

Conclusions (discrete-time) index tracking is closely related to Markowitz portfolio optimization, requires similar algorithms (quadratic optimization), see also Roll (1992) the optimization task gets more involved as soon as there are constraints like e.g. the number of stocks to be reweigted when re-adjusting the basket criteria like TE, TE vol, Corr seem to be similar on a first glance, but their choice may be sensitive for the optimization results... more analyses needed other concepts are cointegration approaches and market capitalization weight methods 22 References Alexander, C. (2001). Market Models: A Guide to Financial Data Analysis, Wiley. Alighanbari, M. and Mougeot, N. (2009). On optimal tracking error, Working paper, Deutsche Bank, Quantitative Research. Meucci, A. (2005). Risk and Asset Allocation, Springer-Verlag, Berlin, Heidelberg. Roll, R. (1992). A mean-variance analysis of tracking error, Portfolio Management pp. 13 22. Rudolf, M., Wolter, H.-J. and Zimmermann, H. (1999). A linear model for tracking error minimization, Journal of Banking & Finance 23(1): 85 103. Satchell, S. and Hwang, S. (2001). Tracking error: Ex-ante versus ex-post measures, Working Paper wp01-15, Warwick Business School, Financial Econometrics Research Centre. Spremann, K. (2008). Portfoliomanagement, Oldenbourg, München. 4. überarb. Aufl. 23