SOLUTION NOTES FOR EXERCISES ON FUNCTIONS
4. Code for Standard Deviation of Cash Flows
8.6 CONFIDENCE INTERVALS FOR STYLE WEIGHTS
So far, we have estimated style weights (both on a simple basis and on a rolling basis) but we have no idea of how to find whether the estimated style weights differ signifi- cantly from zero. To do this, in line with longstanding procedures underlying statistical estimation, we need to calculate standard errors to correspond with the estimated style weights.
In an ideal world, the indices used in style analysis would be independent and thus have returns that were uncorrelated with the returns of any of the other indices. In practice, the indices needed to cover the range of possible asset classes will fall short of the ideal and will sometimes have high correlations with other indices. What we would like to do is to eliminate indices from the style analysis if they are too similar to other indices (substitutes) in order that the remaining subset of indices differ as much as possible (i.e. they provide complements). For instance, we might start with eight indices and by dropping, in turn, four of the more closely related indices we would end up with the four most complementary indices for our eventual style analysis.
One way of judging whether indices are substitutes or complements is to generate a correlation matrix for the returns ‘generated’ from the different possible style indices.
Indices that are easiest to replicate using other indices will be highly correlated with other indices. For instance, in Figure 8.9 below, indices 3, 6 and 8 have correlations above 0.5 with four of the seven other indices. Conversely, indices that are most difficult to replicate will have low correlations with other indices. For instance, indices 1 and 2 have correlations below 0.25 with all the other indices.
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O P Q R S T U V W X
Correlation Matrix for Style Index Returns
Index1 Index2 Index3 Index4 Index5 Index6 Index7 Index8
1.00 0.22 0.19 -0.01 0.16 -0.04 -0.06 -0.13 Index1 0.22 1.00 0.11 0.10 -0.09 -0.24 0.15 -0.15 Index2 0.19 0.11 1.00 0.86 0.53 0.63 0.19 0.62 Index3 -0.01 0.10 0.86 1.00 0.48 0.58 0.24 0.61 Index4 0.16 -0.09 0.53 0.48 1.00 0.61 0.34 0.52 Index5 -0.04 -0.24 0.63 0.58 0.61 1.00 0.31 0.63 Index6 -0.06 0.15 0.19 0.24 0.34 0.31 1.00 0.14 Index7 -0.13 -0.15 0.62 0.61 0.52 0.63 0.14 1.00 Index8 Figure 8.9 Correlation matrix of style index returns
Performance Measurement and Attribution 149 Style analysis is a development of constrained linear regression and, as such, it should be possible to find confidence intervals for the style weights. This can be done by estimating the style of each chosen index in terms of the other indices (as first described by Lobosco and DiBartolomeo, 1997). The style is estimated without constraints on individual weights (though the weights must sum to 100%), thus giving an unexplained volatility for each index. This is compared to the active standard deviation of the style model for the fund to give standard errors (and hence confidence intervals) for the estimated style weights.
See Figures 8.10 and 8.11.
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A B C D E F G H I J K L M
Confidence Intervals for Style Weights (see Lobosco & DiBartolomeo, 1997) Solver - Style Model for Confidence Intervals
Target cell M6 EV*10,000 34.1653
Changing cells E14..J14 Error Variance 0.0034
Constraints Active Std Deviation 5.85%
j * 8 Aset Variance 0.0071
Style R-sqd 52.1%
Style Weights
Index1 Index2 Index3 Index4 Index5 Index6 Index7 Index8
Weight 18.6% -29.5% 23.0% 41.4% 22.8% 34.0% -10.3%
Index ASD 2.27% 2.47% 2.69% 2.74% 4.21% 4.21% 5.27% 5.85%
Mth Indexj* Index2 Index3 Index4 Index5 Index6 Index7 Index8 Style Error
1 0.111 0.008 -0.032 0.043 0.032 0.097 0.110 0.026 0.091 0.020
Figure 8.10 Confidence intervals for style coefficients in Style3 sheet
We need to build a spreadsheet that can automatically divide the matrix of index returns into two: returns for the chosen single index,jŁ, in column B; and returns for the remaining indices in columns D to J. The single index is isolated by using the value of jŁin cell B10 in combination with the INDEX command, here used in the array form to return the value of a chosen cell within the array of returns. This is achieved with this formula in cell B21:
=INDEX($P$21:$W$80,A21,$B$9)
The best way to then divide the remaining indices is by writing a user-defined function, StyleSubMatrix, that is described in greater detail in sheet Module1.
The spreadsheet is now set up to use Solver in a fashion similar to the Style1 sheet, though without the constraints on individual weights. This is automated for a single use in the Style3 macro (called by the CtrlCShiftCJ key combination) and for the collection of eight style indices in the Style4 macro (CtrlCShiftCK). These macros are explained in greater detail in the ModuleM sheet.
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AA AB AC AD AE AF AG AH AI
Confidence Intervals for Style Weights Active Std Deviation for Style 2.54%
Observations 60
Non-zero style weights 6
Index1 Index2 Index3 Index4 Index5 Index6 Index7 Index8
Weight 0.0% 15.8% 46.8% 1.4% 13.2% 0.0% 6.3% 16.5%
non-zero 0 1 1 1 1 0 1 1
Index ASD 2.27% 2.47% 2.69% 2.74% 4.21% 4.21% 5.27% 5.85%
Weight (SE) 15.36% 14.10% 12.96% 12.74% 8.29% 8.28% 6.61% 5.96%
T-statistic 0.00 1.12 3.61 0.11 1.59 0.00 0.96 2.76
Figure 8.11 Confidence intervals for style coefficients in Style3 sheet
The resulting active standard deviations for the individual indices are then used to estimate standard errors for the style weights obtained from the original style analysis of the fund. The formula for the standard error in cell AB16 is:
=$AD$5/(AB14∗SQRT($AD$6−$AD$7−1))
The index ASD (from cell AB14) is used as the divisor for the standard error. An index that has a high ASD is very difficult to replicate using the other indices and so should have a lower standard error. This is achieved by dividing the estimated weight by the appropriate index ASD. The formula also depends on the number of observations used for the original style analysis and the number of non-zero style weights.
Using the t-statistics, we can then refine the style analysis by dropping indices from the style analysis with t-statistic values insignificantly different from zero (say below 2 at the 95% confidence interval).
Style analysis is much needed. Since the development of the theory underlying the CAPM (and the associated conventional performance measures) in the 1960s, there has been a continuing avalanche of research into finding anomalies observed in asset pricing markets. Differential performance (good over some periods, bad over other periods) has been ascribed to a whole raft of active strategies at differing times. The 1980s saw the documentation of the small firm effect as well as strategies based on dividend yield or price-earnings ratios. The 1990s followed with the market-to-book ratio and momentum.
Hot on the heels of the discovery of possible anomalies has come the development of passive benchmarks, especially in the US and the UK, to mimic active strategies. Style analysis is currently the most important of these, as it produces multi-index passive benchmarks against which to compare the performance of the current mix of anomaly-led active strategies.
Performance Measurement and Attribution 151