Index Tracking Issues

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Index Tracking Issues

Prof. Dr. Marlene M ¨uller This version: December 4, 2009

Plan

Problem Tracking Basket Index Replication Case Study Conclusions

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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

letBdenote the tracking portfolio (”basket”) andI the index (we do not specify if these denote returns or prices for the moment)

consider B= m X i=1 wiXi =w⊤X

wherew = (w1, . . . ,wm)⊤ is a weight vector (unit: no. of stocks) and X = (X1, . . . ,Xm)⊤ the universe of stocks to analyze

notation:

- X =R(returns) or X =S(stock prices) - RI, RB (index and basket returns)

- SI, SB (index and basket values)

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Tracking Error

tracking error (volatility of) return differences

TE=

q

E(RI−RB)2 variants:

tracking error volatility

TE_vol =pVar(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

TEres =σ(R_B) q

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(RI,RB) = Cov(RI,RB) p Var(RI)· p Var(RB)

correlation of portfolio values

Corr=Corr(SI,SB) = Cov(SI,SB) p Var(SI)· p Var(SB) cointegration

SI andSB 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

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Relationships

TE vs. TEvol

E_{(RI−ERI)−(RB−ERB)}2

= (RI−ERI)2 +E(RB −ERB)2 −2E(RI −ERI)(RB−ERB) = Var(RI) +Var(RB)−2 Cov(RI,RB)

= Var(RI−RB)

TEres vs. TEvol

RB = α+βRI+ε

⇒ Var(ε) =Var(RB −βRI) =Var(RB)

n

1₋Corr(RI,RB)2 o

TEres vs. Corr

⇒ TEres leads to a quadratic optimization problem while Corr gives

an non-convex objective function inw (see next slides)

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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 TEvol 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)

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Optimization with TE

vol

and Correlation

assume the following constellation:

- hold a part Bhold of the basket fixed and optimize only w.r.t. a small

stock universe X

- then:

B=Bhold +w⊤X

- optimize

min

w TEvol(w) or maxw Corr(w)

under additional restrictions onw (e.g. w _≥0, . . . )

8

TE

vol

Optimization

we have

TE2_vol(w) = Var(I₋B) = Var(I) +Var(B)₋2 Cov(I,B) = Var(I) +Nw2 −2Dw with Dw = Cov(I,Bhold) + m X i=1 w_iCov(X_i,I) N_w2 =Var(B_hold) +2 m X i=1 w_iCov(B_hold,X_i) + m X i,j=1 w_iw_jCov(X_i,X_j) therefore min w TEvol(w) ⇐⇒ minw n N_w2 ₋2Dw o ⇐⇒ min w n (c+2d⊤w +w⊤Σw)₋2(a+b⊤w)o

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Correlation Optimization

we have now

Corr(w) = _p Cov(I,B)

Var(I)_·pVar(B) =

Cov(I,Bhold) +Cov(I,Bnew) p

Var(I)_·pVar(B)

= _p Dw Var(I)_·Nw with again Dw = Cov(I,B_hold) + m X i=1 w_iCov(X_i,I) N_w2 =Var(B_hold) +2 m X i=1 wiCov(Bhold,X_i) + m X i,j=1 wiwjCov(Xi,X_j) therefore max w Corr(w) ⇐⇒ maxw Dw Nw ⇐⇒ max w a+b⊤_w √ c+2d⊤_w₊_w⊤_Σw 10

Prices or Returns?

simple scenario:

(jointly) log-normal stock prices _⇐⇒ (jointly) normal returns

Rit ∼N(ri, σ2i) iid. over timet

then forreturnsof stocksi,j

ERit =ri, Var(Rit) =σi2, Cov(Rit,Rjt) = ρijσiσj

but forstock prices

ESit = Si0exp(rit+σi2t2/2) Var(Sit) = S2i0exp “ 2rit+σi2t2 ” {exp(σi2t2)−1}

Cov(Sit,Sjt) = Si0Sj0exp

 (ri+rj)t+ 1 2(σ 2 i +σj2)t2 ﬀ n

exp(ρijσiσjt2)₋1o

⇒ optimization over time horizont+ model assumption

⇒ on the other hand: objective function on prices / portfolio values

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Index 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, TEvol, Corr)

- uses a constraint on the maximal numberm 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 indexI

step 1: remove consecutively thosem stocks which give

largest deviations of the objective criterion from its starting value (backward model selection)

step 2: add consecutively up tom stocks which give largest

improvements to the objective criterion from its starting value (forward model selection, test all

combinations of up to mstocks)

remark: stocks removed in step 1 can be added again in step 2 (reweighting!)

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DAX –

Fit: 01–12/2008, Eval: 01–06/2009, Corr Optimization 3500 4000 4500 5000 5500 6000 Eval Period (2009−01−02 to 2009−06−30) Inde x

Jan Mär Mai Jul

Index with Initial and Optimal Baskets

Index Initial Optimal 0.95 1.00 1.05 1.10 1.15 Eval Period (2009−01−02 to 2009−06−30) Ratio Inde x to Bask ets

Jan Mär Mai Jul

Ratio of Index to Initial and Optimal Baskets

Index Initial Optimal

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights D.ALV 0.0294 D.BAS 0.2164 D.BMW 0.1183 D.DAI 0.0495 D.DBK 0.0361 D.MAN 0.0136 D.SIE 0.0911 D.TKA 0.1178 OptWeights D.ALV 0.0294 D.BAYN 0.0771 D.DAI 0.0495 D.DBK 0.0361 D.DTE 0.0427 D.HEN3 0.0290 D.TKA 0.1178 D.VOW 0.0383 Deleted Added Changed 1 D.BAS D.BAYN 2 D.BMW D.DTE 3 D.MAN D.HEN3 4 D.SIE D.VOW StartBasket OptBasket 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 KPSS.p 0.0100 0.0100 14

DAX –

Fit: 01–12/2008, Eval: 01–06/2009, TE Optimization

3500 4000 4500 5000 5500 6000 Eval Period (2009−01−02 to 2009−06−30) Inde x

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights D.ALV 0.0294 D.BAS 0.2164 D.BMW 0.1183 D.DAI 0.0495 D.DBK 0.0361 D.MAN 0.0136 D.SIE 0.0911 D.TKA 0.1178 OptWeights D.ALV 0.0294 D.BAS 0.2402 D.BMW 0.0937 D.DAI 0.0495 D.DBK 0.0361 D.DPW 0.1302 D.DTE 0.1067 D.SIE 0.0911 Deleted Added Changed 1 D.MAN D.DPW D.BAS 2 D.TKA D.DTE D.BMW StartBasket OptBasket 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 KPSS.p 0.0100 0.0100 15

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DAX –

Fit: 01–12/2008, Eval: 01–06/2009, TE_vol Optimization 3500 4000 4500 5000 5500 6000 Eval Period (2009−01−02 to 2009−06−30) Inde x

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights D.ALV 0.0294 D.BAS 0.2164 D.BMW 0.1183 D.DAI 0.0495 D.DBK 0.0361 D.MAN 0.0136 D.SIE 0.0911 D.TKA 0.1178 OptWeights D.ALV 0.0294 D.BAS 0.2070 D.BMW 0.1183 D.DAI 0.0495 D.DBK 0.0361 D.DTE 0.1156 D.SIE 0.0911 D.TKA 0.0944 Deleted Added Changed 1 D.MAN D.DTE D.BAS

2 D.TKA StartBasket OptBasket 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 KPSS.p 0.0100 0.0100 16

DAX –

Fit: 01–12/2008, Eval: 01–06/2009, TEres Optimization

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights D.ALV 0.0294 D.BAS 0.2164 D.BMW 0.1183 D.DAI 0.0495 D.DBK 0.0361 D.MAN 0.0136 D.SIE 0.0911 D.TKA 0.1178 OptWeights D.BMW 0.1183 D.DAI 0.0495 D.FME 0.0092 D.FRE3 0.0029 D.MRK 0.0005 D.SIE 0.0911 D.TKA 0.1178 D.VOW 0.0370 Deleted Added Changed 1 D.ALV D.FME 2 D.BAS D.FRE3 3 D.DBK D.MRK 4 D.MAN D.VOW StartBasket OptBasket 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 KPSS.p 0.0100 0.0100

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DAX –

Fit: 01–12/2009, Eval: 01–06/2009, TE_mad Optimization 3500 4000 4500 5000 5500 6000 Eval Period (2009−01−02 to 2009−06−30) Inde x

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights D.ALV 0.0294 D.BAS 0.2164 D.BMW 0.1183 D.DAI 0.0495 D.DBK 0.0361 D.MAN 0.0136 D.SIE 0.0911 D.TKA 0.1178 OptWeights D.ALV 0.0294 D.DAI 0.0495 D.DBK 0.0361 D.EOAN 0.1176 D.MAN 0.0136 D.MEO 0.1261 D.MUV2 0.0474 D.TKA 0.1832 Deleted Added Changed 1 D.BAS D.EOAN D.TKA 2 D.BMW D.MEO 3 D.SIE D.MUV2 StartBasket OptBasket 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

Jan Mär Mai Jul

Index Initial Optimal 0.95 1.00 1.05 1.10 Eval Period (2009−01−02 to 2009−06−30) Ratio Inde x to Bask ets

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights 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 OptWeights 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 Deleted Added Changed 1 E.BBVA F.GSZ H.RDSA 2 F.TAL S.ABB StartBasket OptBasket 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 KPSS.p 0.0100 0.0100 19

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SX5E –

Fit: 01–12/2008, Eval: 01–06/2009, TE Optimization 160 180 200 220 240 Eval Period (2009−01−02 to 2009−06−30) Inde x

Jan Mär Mai Jul

Index Initial Optimal 0.95 1.00 1.05 1.10 Eval Period (2009−01−02 to 2009−06−30) Ratio Inde x to Bask ets

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights 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 OptWeights 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 Deleted Added Changed 1 E.BBVA S.ABB F.MIDI 2 S.CSGN S.UBSN H.RDSA StartBasket OptBasket 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 KPSS.p 0.0100 0.0100 20

SX5E –

Fit: 01–12/2008, Eval: 01–06/2009, TE_madOptimization

Jan Mär Mai Jul

Index Initial Optimal 0.95 1.00 1.05 1.10 1.15 1.20 Eval Period (2009−01−02 to 2009−06−30) Ratio Inde x to Bask ets

Jan Mär Mai Jul

- initial basket: regression (w≥0) - maximal changes: 4 StartWeights 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 OptWeights 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 Deleted Added Changed 1 D.DAI D.EON F.MIDI 2 S.CSGN M.NOKP StartBasket OptBasket 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 KPSS.p 0.0100 0.0100

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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, TEvol, 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

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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 Managementpp. 13–22. Rudolf, M., Wolter, H.-J. and Zimmermann, H. (1999). A linear model for tracking error

minimization,Journal of Banking & Finance23(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, M ¨unchen. 4. ¨uberarb. Aufl.