國
立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
16 BBB
corporate bond
-0.05 0.12 0.01 0.05 0.66 1.00
4. Methodology
While it is possible to determine a fund’s investment style from a detailed analysis of the securities held by the fund, but a simpler approach that uses only the realized fund-return is also feasible. We don’t have to understand the knowledge of the actual composition of the portfolio; instead, return-based style analysis requires only easily obtained information.
I. Relation to Multifactor Models
Multiple factor models are commonly used to characterize how industry factors and economy wide pervasive factors affect the return on the individual securities and portfolios of securities. In such models a portfolio of factors is used to replicate the return on a security as closely as possible.
Equation (1) gives a generic n-factor model that decomposes the return on security i into different components:
Ri,t is the return on security i in period t;
F1,t represents the value of factor 1;
F2,t represents the value of factor 2;
Fn,t represents the value of the nth factor and εi,t is the non-factor component of the return.
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different set of industry and economy wide pervasive factors.In factor models the portfolio weights, βi,1 , βi,2 , ...βi,n , need not sum to 1 and a factor, Fk,t, need not necessarily be the return on a portfolio of financial assets.
II. Sharpe’s model for Return-Based Style Analysis
Sharpe’s (1988, 1992) Return-based Style Analysis can be considered a special case of the generic factor model. In Return-based Style Analysis we replicate the performance of a managed portfolio over a specified time period as best as possible by the return on a passively managed portfolio of style benchmark index portfolios.
Asset Class Model brought by Sharpe (1992) is built the foundation of return-based approach. This model is widely-used among the investors, consultants or academic researches because of the minimal data
requirements and convenience to operate.
Based on the return-based analysis, it asserts that a manager’s
investment style can be decided by comparing the returns of the funds to the returns of a string of the selected passive indices. Sharpe proposes the following generic econometric model:
Eq. (1) where
Ri represents the return of the portfolio for t=1,2….T Fi1 ,Fi2…Fin denote the return of index F at time t
bi1,bi2…bin are factor loadings that express the sensitivity of the fund return to the factor-mimicking portfolio return of index F
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et is the excess return at time t, reflecting the idiosyncratic noise, the part that can’t be explained by the return of the N asset classes.
is the particular combination (portfolio) of factors that best
replicates the return Ri.
The slope coefficients bi1,bi2…bin stand for that the managed portfolio average allocation among the different style benchmark index portfolios - or asset classes during the relevant time period. The sum of the terms in the square brackets is that part of the managed portfolio return that can be explained by its exposure to the different style benchmarks and is termed the style of the manger. The residual component of the portfolio return reflects the manager decision to depart from the benchmark composition within each style benchmark class. This is the part of return attributable to the manger stock picking ability and is termed selection.
In order to give the factors the meaning of portfolio weights, asset
allocations or performance benchmarking, the factor loadings are restricted to add-up to one.
A second restriction is the short selling restriction, which means all the holdings should be long positions.
,i=1,2,3……N
In the context of biFnt has the denotation of the return on a passive portfolio with the same style as the fund.
The two important differences when compared to factor models are: (i) Every factor is a return on a particular style benchmark index portfolio, and (ii) The
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weights assigned to the factors sum to unity.
As before, the objective of the analysis is to select a set of coefficients that minimizes the unexplained variation in returns (i.e. the variance of et) subject to the stated constraints. The presence of inequality constraints in (3) requires the use of quadratic programming since standard regression analysis
packages typically do not allow imposing such restriction. Rearranging Equation (1) yields,
where Χ is the T × n matrix of asset classes returns, Rp is the T ×1 vector of portfolio returns and ∆p is the n ×1 vector of slope coefficients bi1,bi2…bin. The term on the left Ep can be interpreted as the T dimensional column vector, [ et , p , t = 1,2..], of differencesbetween the return on the fund and the return on the portfolio of passive benchmark styleindexes corresponding to the n
dimensional vector Δp of style benchmark portfolioweights − also referred to as style-asset class exposures. The goal of Return-based Style Analysis is to find the set of non-negative style-asset class exposures bi1,bi2…bin that sum to 1 and minimize the variance of et , p, referred to as fund’s tracking error over the style benchmark. Note that the objective of this analysis is not to choose style benchmarks that make the fund “look good” or “bad”. Rather the goal is to infer as much as possible about a fund’s exposures to variations in the returns of the given style benchmark asset classes during the period of interest. It is also important to understand that the ‘style’ identified in such an analysis
represents an average of potentially changing styles over the period covered.
Month-to-month deviations of the fund’s return from that of style itself can arise from selection of specific securities within one or more asset classes, rotating
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among asset classes, or both.
III. A Six Asset Class Model
Based on Sharpe(1992)’s twelve asset class model, the return of each asset index is represented by a market capitalization weighted index of the returns on a large number of securities. These indices imply a very important idea that each index can be realized as an investment strategy at lower cost by constructing an index fund. In other word, assumed that one mutual fund is classified by its mutual fund company as a growth fund, we can examine this statement by this model through the magnitude of its beta coefficient. Even it has a large beta coefficient, if the return of the mutual fund is inferior to the passive index fund, then the investor can think about just invest in passive fund which equally means this manager doesn’t have superior marketing ability and stocks selected ability. This method also enables an investor to track the returns with little error.
In order to make this model to accommodate the situations against our sample data – domestic mutual funds, we replacing and excluding some indices in Asset Class Model. We remove the inappropriate indices such as Mortgaged-Related Securities Index, Non-U.S. Government Bond Index, FTA Euro-Pacific EX Japan Index, FTA Japan Index and so on; transforming to a six asset class model rather than a twelve asset class model.
Table 3 descries six indices used for the associated return analysis.
Table 3 Bills
Cash-equivalents with less than 3 months to maturity
Index: commercial paper rate 90 days in the secondary market
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Corporate bonds with ratings of at least BBB Index: tw BBB
Small-capitalization stocks
Index: FTSE TAIWN small-price index Medium- capitalization stocks
Index: FTSE TAIWN medium-price index Large-capitalization value stocks
Index: MSCI TAIWAN Large-Value - PRICE INDEX Large-capitalization growth stocks
Index: MSCI TAIWAN Large-growth - PRICE INDEX This results in a 6-factor model.
IV. A Rolling Window Method
Dybvig and Ross (1985), for instant, show how linear risk models fail to properly rank fund managers when they change their asset weights through time. Connor and Korajczyk (1991) consider how to risk-adjust for nonlinear portfolio strategies by mutual fund managers. Grinblatt and Titman (1993) avoid problems posed by nonlinearities by explicitly considering active strategies as the basis for a benchmark-free approach to performance measurement. While such nonlinearities present problems for style
identification as well, our procedure accommodates nonlinear strategies by allowing factor loadings to change on a month-to-month basis. This is critical in the light of the fact that many fund managers actively change their exposure to market. Brown & Goetzmann (1997) find some evidence, in the form of
time-varying factor loadings, that this is due to the presence of dynamic
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management styles in the mutual fund.
V. STYLE DRIFT SCORE
Style drift happens when the holdings of a mutual fund "drift" from one asset class to another class or a manager’s investment style “drift” from one particular attitude to another. In other words, style drift is the evolution of the asset class coefficients over time.(see Idzorek and Bertsch [2004])
Style drift can produce the result in changing the risk-to-rewarding characteristics of a portfolio because manager may overweight in one
investment class and underweight in another one. This will mislead investor to evaluate the overall optimal portfolio and misallocate their assets when they make decisions.
In other words, the effect of diversification may be destroyed.
Generally speaking, if we have more quantity of data to calculate the asset coefficients, the smoother the charges can see. We will use a‘rolling window’
technique to ascertain the extent of style variation over time: taking 36 months as an initial time window, then the window will be moved forward by 1 month and drop the oldest data point, so the number of the time periods used is fixed.
Idzorek and Bertsch (2004) developed a quantitative measure to measure the variability of a fund’s asset mix over time called style drift score(SDS). The SDS is calculated as the square root of the sum of the variances of the asset class coefficients derived from Eq. (1) as demonstrated by
SDS =
where b1,t, b2,t……bn,t represent the time series style weights obtained from the style analysis process. The SDS is the average Euclidian distance of the T
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rolling-window asset class coefficients from the center of gravity in K-dimensional space. A fund with a high SDS will represent greater style inconsistency than a fund with a low SDS.
Idzorek and Bertsch stand for the point that it is an effective, time-efficient way that comparing style consistency and eliminating the need to examine rolling window style graphs via SDS.