Rrm Tracking Error
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The Use and Abuse of Tracking Error Jason MacQueen
R-Squared Risk Management
Definition of Tracking Error • Tracking Error is defined as the standard deviation of the differences (the relative returns) between a portfolio’s returns and its benchmark’s returns. • Tracking Variance is the square of the Tracking Error. • It is common practice in the investment management industry to judge risk models by how closely their ex ante tracking error ‘forecast’ for a portfolio corresponds to its ex post observed tracking error. • Typically, the ex ante value will be somewhat lower than the ex post value.
R-Squared
Definition of Tracking Error • Let rt be a vector of the rates of return at time t with mean vector E (rt +1 ) = µ r and covariance matrix Ω r • Let the portfolio weights at time t be given by the vector at and benchmark weights by vector bt. • We may then define the portfolio relative weights as : wt = at - bt.
R-Squared
Ex Ante Tracking Error • Ex ante tracking error is calculated as follows :
TE =
R-Squared
′ ′ var(a t rt − b t rt )
=
(a t − b t )′ var(rt )(a t − b t )
=
(a t − b t )′Ω r (a t − b t ).
=
( w t )′Ω r (w t ).
Ex Post Tracking Error • Ex post tracking error is calculated from the actual portfolio relative returns rpt, where rpt = w′t-1rt • Hence, a time series calculation of ex post tracking error would involve, over a period from t = 1, … T, the terms rp1, rp2, … rpT, or equivalently, w′0r1 , w1′r2 ,..., w′T-1rT • The crucial point is that the portfolio weights wt, which are assumed to be fixed, (or non-stochastic) ex ante, will vary (be stochastic) ex post. • Ex post tracking error is therefore given by : TE =
R-Squared
1 T 2 ( r − r ) ∑ pt p T − 1 t =1
Stochastic Weights • Consider an investment strategy such that at t = 0, we fix the weights w0. • For randomness in the weights w0 not to enter into the calculation, we would require that each wt should be rebalanced back to w0 for all periods from t = 1 to t = T. • However, barring the above special case, all other investment strategies, including buy and hold, or quarterly re-balancing, will involve the weights wt being stochastic. • The same will apply (obviously) to capitalisationweighted strategies, and, of course, to all active strategies.
R-Squared
Ex Ante vs Ex Post • The formulae for calculating ex ante and ex post tracking error are different, and they make different assumptions about whether the weights are fixed or not • Ex Ante and Ex Post Tracking Errors are: THEREFORE NOT DIRECTLY COMPARABLE WITH EACH OTHER.
R-Squared
Active Management & Tracking Error - 1 • Let e be a vector of ones, i.e. e = (1, 1, … 1)’. For conventional portfolio calculations, we have the portfolio budget constraint
e′w t = 1 • However, when we are computing portfolio values relative to a benchmark, we have
e′w t = 0.
• Otherwise, the calculations are the same. • In the conventional calculation, weights wt are given at time t and the portfolio return rpt +1 can therefore be written as:
rpt +1 = w ′t rt +1 R-Squared
Active Management & Tracking Error - 2 • Portfolio tracking variance becomes, by definition :
var( rpt +1 ) = var( w ′t rt +1 ) = w ′t Ω r w t • where Ω r is the conditional (or unconditional) covariance matrix of rt +1 , w t being treated as fixed, and E (rt +1 ) = µ r is again being interpreted conditionally or unconditionally. • We now propose an important theorem describing the fundamental relationship between ex post and ex ante tracking variance (and also tracking error).
R-Squared
Theorem • If a set of portfolio weights wt that satisfies e′w t = 0 is stochastic, i.e., w t = µ w + ν t , where v t ~ (0, Ω w ) , then the ex-post tracking variance of the difference between portfolio returns and benchmark returns, can be decomposed as follows : ∧ 2
TE SD = µ′r Ω wµ r + tr (Ω r Ω w ) + µ′wΩ r µ w .
R-Squared
Proof of Theorem We have: ∧ 2
TE SD = var(rpt +1 ) = var( E (rpt +1 | w t )) + E (var(rpt +1 ) | w t ) = var(w′t µ r ) + E (w′t Ω r w t ) = var(w′t µ r ) + E[(µ w + v t )′Ω r (µ w + v t )] ′ ′ ′ = var(w′t µ r ) + E[ v t Ω r v t + 2 v t Ω r µ w + µ w Ω r µ w ] = µ′r Ω wµ r + tr (Ω r Ω w ) + µ′wΩ r µ w
R-Squared
Remarks • In the special case of fixed (i.e. non-stochastic) weights, we have w t = µ w , Ω w = 0, and var(rpt +1 ) = µ′wΩr µ w • This is the ex post tracking variance with fixed weights, which will be directly comparable to the ex ante forecast tracking variance. • Since e′w t = 0 for all t, then we must have e′µ w = 0 and also var(e’wt) = e′Ω we = 0 • Lawton-Browne (2000) establishes empirically that µ′r Ω wµ r is, indeed, very small in the cases she examines • (see Tables below).
R-Squared
Lawton-Browne Research • We now digress briefly to discuss Carola LawtonBrowne’s research. • She provides two estimates of Ω r , one based on individual stock returns, and the other based on industry returns. • Her universe is an active UK portfolio of 187 stocks, with a benchmark of the FT-SE 100. • In her paper, Lawton-Browne reports the following tables.
R-Squared
Table 1 : TE results with stock returns Component Term
Actual Variance
% Variance contribution
% Standard deviation
µ 'r Ω w µ r
0.168
1.6
0.41
tr (Ω r Ω w )
4.847
45.8
2.20
µ 'w Ω r µ w
5.566
52.6
2.36
10.581
100.0
3.25
Total
R-Squared
Table 2 : TE results with industry returns Component Term
Actual Variance
% Variance contribution
% Standard deviation
µ 'r Ω w µ r
0.091
2.9
0.30
tr (Ω r Ω w )
1.971
63.0
1.40
µ 'w Ω r µ w
1.068
34.1
1.03
Total
3.130
100.0
1.77
R-Squared
Results of Lawton-Browne • These results show that, in this case at least, the overall contribution from stochastic weights accounts for between 45% to 65% of the ex post tracking error. • Furthermore, the contribution of both cases, as anticipated.
R-Squared
µ 'r Ω w µ r
is tiny in
Conclusions - 1 • Tracking error is becoming increasingly influential in the investment management business. Sponsors of defined benefit plans increasingly pay attention to ‘risk budgeting’ that purports to allocate an allowable ‘budget’ of tracking error for the whole plan across managers of different asset classes; see Gupta, Prajogi, & Stubbs (1999). • Another example is that last year Barclays Global Investors agreed to return back a portion of its management fee to the Sainsbury’s Pension Scheme if the portfolio exceeded its agreed tracking error limits.
R-Squared
Conclusions - 2 • If tracking error is used to help judge fund performance, the crucial difference between ex ante and ex post tracking errors described in this study must be taken into account. • More generally, both portfolio managers and their clients should take particular care to avoid confusing the two different kinds of tracking error. • Finally, it should now be clear that risk models cannot be judged on a spurious comparison of the ex ante tracking error – based on the initial relative weights - with the observed ex post tracking error, which reflects, inter alia, the changing relative weights through time. • Ex ante and Ex post tracking error are subtly different
R-Squared
References - 1 • Baierl, G. T. and P. Chen, 2000, “ChoosingManagers and Funds,” Journal of Portfolio Management 26(2), 47-53. • Gardner, D. and D. Bowie, M. Brooks, M. Cumberworth, 2000, “Predicted Tracking Errors : Fact or Fantasy?” Faculty and Institute of Actuaries, Investment Conference paper, 25-27 June 2000. • Grinold, R. and R. Kahn, 1995, Active Portfolio Management, Irwin. • Gupta, F., R. Prajogi, and E. Stubbs, 1999, “The Information Ratio and Performance,” Journal of Portfolio Management (Q3 1999), 33-39.
R-Squared
References - 2 • Larsen, G. A. and B. G. Resnick,, 1998, “Empirical Insights on Indexing,” Journal of Portfolio Management 25(1), 51-60. • Lawton-Browne, C. L. 2000, Masters dissertation, Dept. of Economics, Birkbeck College, London University • Markowitz, H. M., 1959, Portfolio Selection, 1st Edition, New York: John Wiley & Sons. • Pope, P. and P. K. Yadav, 1994, “Discovering Error in Tracking Error,” Journal of Portfolio Management (Q4 1994), 27-32. • Roll, R., 1992, “A Mean/Variance Analysis of Tracking Error,” Journal of Portfolio Management, 13-22. • Sharpe, W, 1964, “Capital Asset Prices : A Theory of Capital Market Equilibrium under Conditions of Risk,” Journal of Finance 19, 425-442.
R-Squared
Summary • “The essence of investment management is the management of risks, not the management of returns” - Ben Graham • It is also often said that you can manage risks, but you can’t manage returns • Portfolio managers obviously need to come up with expected returns, or forecasts of what will do well or badly, or just ‘pick stocks’ • But this is just the first step; the true skill in portfolio management lies in managing its risk
R-Squared
20
Contact Information R-Squared Risk Management Limited The Nexus Building, Broadway, Letchworth Garden City, Hertfordshire, SG6 3TA, United Kingdom +44 1462 688 325 +44 7768 068 333 455 Lakeland Street, Grosse Pointe, MI 48230, U. S. A. +1 313 469 9960 +1 646 280 9598 Email: [email protected]
R-Squared Risk Management
R-Squared Risk Management
Definition of Tracking Error • Tracking Error is defined as the standard deviation of the differences (the relative returns) between a portfolio’s returns and its benchmark’s returns. • Tracking Variance is the square of the Tracking Error. • It is common practice in the investment management industry to judge risk models by how closely their ex ante tracking error ‘forecast’ for a portfolio corresponds to its ex post observed tracking error. • Typically, the ex ante value will be somewhat lower than the ex post value.
R-Squared
Definition of Tracking Error • Let rt be a vector of the rates of return at time t with mean vector E (rt +1 ) = µ r and covariance matrix Ω r • Let the portfolio weights at time t be given by the vector at and benchmark weights by vector bt. • We may then define the portfolio relative weights as : wt = at - bt.
R-Squared
Ex Ante Tracking Error • Ex ante tracking error is calculated as follows :
TE =
R-Squared
′ ′ var(a t rt − b t rt )
=
(a t − b t )′ var(rt )(a t − b t )
=
(a t − b t )′Ω r (a t − b t ).
=
( w t )′Ω r (w t ).
Ex Post Tracking Error • Ex post tracking error is calculated from the actual portfolio relative returns rpt, where rpt = w′t-1rt • Hence, a time series calculation of ex post tracking error would involve, over a period from t = 1, … T, the terms rp1, rp2, … rpT, or equivalently, w′0r1 , w1′r2 ,..., w′T-1rT • The crucial point is that the portfolio weights wt, which are assumed to be fixed, (or non-stochastic) ex ante, will vary (be stochastic) ex post. • Ex post tracking error is therefore given by : TE =
R-Squared
1 T 2 ( r − r ) ∑ pt p T − 1 t =1
Stochastic Weights • Consider an investment strategy such that at t = 0, we fix the weights w0. • For randomness in the weights w0 not to enter into the calculation, we would require that each wt should be rebalanced back to w0 for all periods from t = 1 to t = T. • However, barring the above special case, all other investment strategies, including buy and hold, or quarterly re-balancing, will involve the weights wt being stochastic. • The same will apply (obviously) to capitalisationweighted strategies, and, of course, to all active strategies.
R-Squared
Ex Ante vs Ex Post • The formulae for calculating ex ante and ex post tracking error are different, and they make different assumptions about whether the weights are fixed or not • Ex Ante and Ex Post Tracking Errors are: THEREFORE NOT DIRECTLY COMPARABLE WITH EACH OTHER.
R-Squared
Active Management & Tracking Error - 1 • Let e be a vector of ones, i.e. e = (1, 1, … 1)’. For conventional portfolio calculations, we have the portfolio budget constraint
e′w t = 1 • However, when we are computing portfolio values relative to a benchmark, we have
e′w t = 0.
• Otherwise, the calculations are the same. • In the conventional calculation, weights wt are given at time t and the portfolio return rpt +1 can therefore be written as:
rpt +1 = w ′t rt +1 R-Squared
Active Management & Tracking Error - 2 • Portfolio tracking variance becomes, by definition :
var( rpt +1 ) = var( w ′t rt +1 ) = w ′t Ω r w t • where Ω r is the conditional (or unconditional) covariance matrix of rt +1 , w t being treated as fixed, and E (rt +1 ) = µ r is again being interpreted conditionally or unconditionally. • We now propose an important theorem describing the fundamental relationship between ex post and ex ante tracking variance (and also tracking error).
R-Squared
Theorem • If a set of portfolio weights wt that satisfies e′w t = 0 is stochastic, i.e., w t = µ w + ν t , where v t ~ (0, Ω w ) , then the ex-post tracking variance of the difference between portfolio returns and benchmark returns, can be decomposed as follows : ∧ 2
TE SD = µ′r Ω wµ r + tr (Ω r Ω w ) + µ′wΩ r µ w .
R-Squared
Proof of Theorem We have: ∧ 2
TE SD = var(rpt +1 ) = var( E (rpt +1 | w t )) + E (var(rpt +1 ) | w t ) = var(w′t µ r ) + E (w′t Ω r w t ) = var(w′t µ r ) + E[(µ w + v t )′Ω r (µ w + v t )] ′ ′ ′ = var(w′t µ r ) + E[ v t Ω r v t + 2 v t Ω r µ w + µ w Ω r µ w ] = µ′r Ω wµ r + tr (Ω r Ω w ) + µ′wΩ r µ w
R-Squared
Remarks • In the special case of fixed (i.e. non-stochastic) weights, we have w t = µ w , Ω w = 0, and var(rpt +1 ) = µ′wΩr µ w • This is the ex post tracking variance with fixed weights, which will be directly comparable to the ex ante forecast tracking variance. • Since e′w t = 0 for all t, then we must have e′µ w = 0 and also var(e’wt) = e′Ω we = 0 • Lawton-Browne (2000) establishes empirically that µ′r Ω wµ r is, indeed, very small in the cases she examines • (see Tables below).
R-Squared
Lawton-Browne Research • We now digress briefly to discuss Carola LawtonBrowne’s research. • She provides two estimates of Ω r , one based on individual stock returns, and the other based on industry returns. • Her universe is an active UK portfolio of 187 stocks, with a benchmark of the FT-SE 100. • In her paper, Lawton-Browne reports the following tables.
R-Squared
Table 1 : TE results with stock returns Component Term
Actual Variance
% Variance contribution
% Standard deviation
µ 'r Ω w µ r
0.168
1.6
0.41
tr (Ω r Ω w )
4.847
45.8
2.20
µ 'w Ω r µ w
5.566
52.6
2.36
10.581
100.0
3.25
Total
R-Squared
Table 2 : TE results with industry returns Component Term
Actual Variance
% Variance contribution
% Standard deviation
µ 'r Ω w µ r
0.091
2.9
0.30
tr (Ω r Ω w )
1.971
63.0
1.40
µ 'w Ω r µ w
1.068
34.1
1.03
Total
3.130
100.0
1.77
R-Squared
Results of Lawton-Browne • These results show that, in this case at least, the overall contribution from stochastic weights accounts for between 45% to 65% of the ex post tracking error. • Furthermore, the contribution of both cases, as anticipated.
R-Squared
µ 'r Ω w µ r
is tiny in
Conclusions - 1 • Tracking error is becoming increasingly influential in the investment management business. Sponsors of defined benefit plans increasingly pay attention to ‘risk budgeting’ that purports to allocate an allowable ‘budget’ of tracking error for the whole plan across managers of different asset classes; see Gupta, Prajogi, & Stubbs (1999). • Another example is that last year Barclays Global Investors agreed to return back a portion of its management fee to the Sainsbury’s Pension Scheme if the portfolio exceeded its agreed tracking error limits.
R-Squared
Conclusions - 2 • If tracking error is used to help judge fund performance, the crucial difference between ex ante and ex post tracking errors described in this study must be taken into account. • More generally, both portfolio managers and their clients should take particular care to avoid confusing the two different kinds of tracking error. • Finally, it should now be clear that risk models cannot be judged on a spurious comparison of the ex ante tracking error – based on the initial relative weights - with the observed ex post tracking error, which reflects, inter alia, the changing relative weights through time. • Ex ante and Ex post tracking error are subtly different
R-Squared
References - 1 • Baierl, G. T. and P. Chen, 2000, “ChoosingManagers and Funds,” Journal of Portfolio Management 26(2), 47-53. • Gardner, D. and D. Bowie, M. Brooks, M. Cumberworth, 2000, “Predicted Tracking Errors : Fact or Fantasy?” Faculty and Institute of Actuaries, Investment Conference paper, 25-27 June 2000. • Grinold, R. and R. Kahn, 1995, Active Portfolio Management, Irwin. • Gupta, F., R. Prajogi, and E. Stubbs, 1999, “The Information Ratio and Performance,” Journal of Portfolio Management (Q3 1999), 33-39.
R-Squared
References - 2 • Larsen, G. A. and B. G. Resnick,, 1998, “Empirical Insights on Indexing,” Journal of Portfolio Management 25(1), 51-60. • Lawton-Browne, C. L. 2000, Masters dissertation, Dept. of Economics, Birkbeck College, London University • Markowitz, H. M., 1959, Portfolio Selection, 1st Edition, New York: John Wiley & Sons. • Pope, P. and P. K. Yadav, 1994, “Discovering Error in Tracking Error,” Journal of Portfolio Management (Q4 1994), 27-32. • Roll, R., 1992, “A Mean/Variance Analysis of Tracking Error,” Journal of Portfolio Management, 13-22. • Sharpe, W, 1964, “Capital Asset Prices : A Theory of Capital Market Equilibrium under Conditions of Risk,” Journal of Finance 19, 425-442.
R-Squared
Summary • “The essence of investment management is the management of risks, not the management of returns” - Ben Graham • It is also often said that you can manage risks, but you can’t manage returns • Portfolio managers obviously need to come up with expected returns, or forecasts of what will do well or badly, or just ‘pick stocks’ • But this is just the first step; the true skill in portfolio management lies in managing its risk
R-Squared
20
Contact Information R-Squared Risk Management Limited The Nexus Building, Broadway, Letchworth Garden City, Hertfordshire, SG6 3TA, United Kingdom +44 1462 688 325 +44 7768 068 333 455 Lakeland Street, Grosse Pointe, MI 48230, U. S. A. +1 313 469 9960 +1 646 280 9598 Email: [email protected]
R-Squared Risk Management
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