Asset Allocation vs. Security Selection: Their Relative Importance

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1 INVESTMENT PERFORMANCE MEASUREMENT BY RENATO STAUB AND BRIAN SINGER, CFA Asset Allocation vs. Security Selection: Their Relative Importance Various researchers have investigated the importance of asset allocation versus security selection. Although we think this question is conceptually weak because asset allocation and security selection have different missions we address it to ensure appropriate quantitative treatment. We focus on feasibility rather than on what managers actually do. Hence, our approach is free of benchmark thinking and makes no assumptions regarding portfolio positions or potential constraints. This map shows Zurich, the largest city in Switzerland. Zurich s road system comprises all the streets and highways in the city. To get from A to B, you must drive along selected routes of this network, and you must avoid any shortcuts through private property. Overall, a city s road system can have much variety in terms of courts, streets, highways, and interstates. Although you are free to prefer some types of routes to others, you may find that one way or the other is more or less efficient. Next, we conduct an experiment: We randomly choose 20 points in downtown Zurich and then determine their connecting routes to Zurich s main train station. Furthermore, we want to drive as much as possible along primary (yellow) roads. According to the following table, which lists all 20 connections, we drive 83.2 percent of the entire distance on primary roads and only 16.8 percent on secondary (white) roads, such as courts and streets. Thus, by far the biggest portion of each route from some random point to the main train station involves primary roads, despite the fact that the white network is denser than the yellow one by a multiple. Destination On Yellow Street On White Street (%) 83.2% 16.8% cfa institute 1

2 The point is that the yellow streets bundle a multitude of connections. One might almost be tempted to claim that, in essence, traffic could function without the white streets almost. Our point is that we can compare asset allocation to the yellow roads because asset allocation bundles and security selection to the white ones. ISSUE Many authors have investigated the importance of asset allocation versus security selection. The seminal pieces, published by Brinson, Hood, and Beebower (1986) and Brinson, Singer, and Beebower (1991), turned out to be among the most quoted and debated articles in the domain of finance. Brinson and his co-authors figured out that the principal part of a portfolio s return variance is based on the asset allocation decision. Pointing out that the variance based on asset allocation comes mainly from the policy decision, Hensel, Ezra, and Ilkiw (1991) claimed that Brinson et al. had overstated the importance of asset allocation. Consequently, they recommended extracting the portion of return variance arising from policy in order to measure the impact of asset allocation. Increasingly, this kind of benchmark-driven thinking is a misperception in a world where benchmarks are losing relevance. Further, Ibbotson and Kaplan (2000) found that policy explained about 40 percent of the variation of returns across funds. Whether true or not, this issue is not the one that interests us. Our thinking is more along the lines of Kritzman and Page (2003), who suggested that, with respect to judging the importance of asset allocation versus security selection, researchers investigate what managers could do given the return dispersion opportunities rather than what they actually do. In other words, Kritzman and Page proposed a normative rather than a positive (i.e., descriptive) analysis. Although their suggestion makes sense, they heavily constrained asset allocation in their bootstrap simulation, which led to erroneous conclusions. Finally, Staub (2006) observed that asset allocation and security selection have completely different missions with different measures of success, analogous to pitchers and hitters on a baseball team as different as their jobs are, the team needs both. Hence, any comparison of asset allocation and security selection is misguided. Because the question seems unavoidable to some extent, however, we address it despite its conceptual weakness to ensure appropriate quantitative treatment in future research. We focus on what is feasible rather than on what managers actually do. Therefore, our approach is free of benchmark thinking and makes no assumptions regarding how many positions an asset allocation portfolio or a stock selection portfolio should have or the degree to which such positions should be constrained. The debate is about information in the form of return distributions rather than the impact of constrained information on the portfolio. Whether managers ultimately have the skill to exploit this information is their business. RISK AND CORRELATION Like the road system of Zurich, correlation and risk can be portrayed geometrically, as a set of vectors. A vector s length represents the risk of an asset, and the cosine of the angle between two vectors equals the correlation between the two represented assets. Risk vectors can be rescaled without altering the correlations. This rescaling is nothing more than leveraging or deleveraging a security. Analogous to the directions of the streets on a road map, correlations are the unchangeable building blocks of the system. Because of rescalability, therefore, correlation constitutes information but risk does not. In a world of leverage, risk is just a metric. What matters from an information standpoint is the directions of all vectors. We refer to the set of all directions in our investment universe as the available correlation space. Two risky assets can be portrayed as vectors in a plane; three risky assets can be portrayed as vectors in a space. And the entire universe of n risky assets can be represented as a set of vectors in an n-dimensional space. 2

3 PRINCIPAL COMPONENT ANALYSIS Principal component analysis (PCA) is the main tool of our investigation. 1 Thus, we will describe how to interpret asset allocation and security selection in the context of PCA. The two arrows in the previous figure are not orthogonal: Each vector points partly in the direction of the other one, and thus each vector includes part of the other vector s information. PCA identifies the uncorrelated factors embedded in a correlation matrix, and it shows the resulting coefficients if these factors are regressed on the matrix items. Let us consider the following correlation matrix: The matrix has two pairs of two mutually highly correlated assets, which we can interpret as four assets from two different asset classes. Using PCA, we find the following coefficients of the most powerful factor versus the four assets of the matrix. Such coefficients are usually referred to as loadings. All coefficients of the first factor have the same size and sign. Thus, we interpret the first factor, which has an indexlike character for all items, as the market factor. For the second-most-powerful factor, we identify the following coefficients versus the four matrix items: Although all the coefficients have the same size, their signs differ; those belonging to the same asset class have identical signs. That is, the second factor resembles a long short portfolio that is long one asset class and short the other asset class. For example, the two asset classes could be equity and bonds, and the second factor thus represents the asset class decision between equity and bonds. Overall, reading the signs of the coefficients when interpreting the factors is key. Now, let us assume that all four assets are perfectly mutually correlated. The first factor would be the only explaining factor, because all assets would be perfectly synchronized; the second and all subsequent factors would have zero explanation power because of their zero volatility. This scenario suggests that the asset class decision is irrelevant: The entire portfolio volatility would result from either investing in risky assets or staying in cash. Nevertheless, the lower the correlation between the two asset classes, the more the resulting portfolio volatility depends on the asset class decision all at the expense of the market decision. We next examine the correlation space as determined by our investment universe. Our objective is to explain how (and how much) standardized dispersion (i.e., correlation) results from investment decisions at various aggregation levels. EXAMINATION As previously mentioned, we want our examination to be independent of any portfolio context. And because risk is rescalable, we think the structure to be analyzed is the available correlation space as provided by the correlation matrix that underlies our universe of securities. Therefore, using PCA, we explore which variety of standardized security return the correlation space offers at which aggregation level. The aggregation levels are assets (as opposed to cash), asset classes, markets, and securities. To that end, we assume an investment universe comprising 20 national equity markets and 20 national bond markets, 2 where each market includes 100 securities cfa institute 3

This construction results in a 4,000 4,000 matrix at the security level. For demonstration purposes, we examine four different matrices representing different degrees of variety.

Matrix 1, Matrix 2, and Matrix 3 are intended exclusively for didactic purposes, whereas Matrix 4 is the one most consistent with reality.

The underlying assumption is that all securities are perfectly correlated. In the case of perfect correlation, we can make only one decision (i.e., invest in risky assets or in cash), regardless of the number of securities, because a single factor explains the entire correlation structure.

4 This construction results in a 4,000 4,000 matrix at the security level. For demonstration purposes, we examine four different matrices representing different degrees of variety. The following figure, reflecting the main results, shows the cumulative explained correlation for the subsequent factors. Matrix 1, Matrix 2, and Matrix 3 are intended exclusively for didactic purposes, whereas Matrix 4 is the one most consistent with reality. The following figures show the share of explained correlation for the four matrices and explain the corresponding composition. Matrix 1. The underlying assumption is that all securities are perfectly correlated. In the case of perfect correlation, we can make only one decision (i.e., invest in risky assets or in cash), regardless of the number of securities, because a single factor explains the entire correlation structure. Because of perfect correlation, the loadings are identical for all securities, whether equity or bond that is, any individual security can be considered the risk factor. Matrix 2. The underlying assumptions are that all stocks are perfectly correlated, all bonds are perfectly correlated, and the stock bond correlations equal Because stocks and bonds are not perfectly correlated, we identify a second factor through PCA. This factor is about the asset class decision in our case, the equity versus bond decision. Although the asset class decision explains 35 percent of the entire correlation space, it would certainly explain an even bigger share were there more than just two asset classes. The second factor loads with opposite signs on equity and bonds, which is why it can be considered the asset class factor (i.e., equity versus bonds). In addition, we identify the following coefficients of the first factor versus the 4,000 securities in our universe, which are lined up along the x-axis of the figure. 4

5 Practically speaking, if we standardized all 4,000 return series and regressed them on the first and second factors, we would find that the first factor explains 65 percent of the standardized movement of all securities and the second factor explains the remaining 35 percent. Again, the first factor is constructed in such a way that it explains the highest percentage possible. Matrix 3. The underlying assumptions are that stocks in a national market are perfectly correlated, bonds in a national market are perfectly correlated, stocks of different national markets have a correlation of 0.40, bonds of different national markets have a correlation of 0.60, stocks and bonds of the same national market have a correlation of 0.30, and stocks and bonds of different national markets have a correlation of Matrix 3 portrays stocks of different national markets and bonds of different national markets as less than perfectly correlated. As a result, additional factors that is, the various (national) market decisions come into play. Overall, we find 38 additional factors 38 because they must add up to 40 factors (including the asset versus cash factor and the asset class factor), which is the number of national markets in our matrix. Within each of the 40 national markets, there is no distinction among securities because they are perfectly correlated. The third factor has identical loadings for 100 subsequent securities (i.e., the securities of the same market) because they are perfectly correlated. Thus, the third factor can be considered a market factor, like the following 37 factors: Again, if we standardized all 4,000 return series and regressed them on the 1st, 2nd, and 3rd 40th factors, we would find that the 1st factor explains 38 percent of the standardized movement of all securities, the 2nd factor explains 15 percent, and the 3rd 40th factors the remaining 47 percent. Matrix 4. The underlying assumptions are that stocks in a national market have a correlation of 0.50, bonds in a national market have a correlation of 0.80, stocks of different national markets have a correlation of 0.40, bonds of different national markets have a correlation of 0.60, stocks and bonds of the same national market have a correlation of 0.30, and stocks and bonds of different national markets have a correlation of cfa institute 5

6 In Matrix 4, each security has partial independence that is, a single security cannot be replicated by any combination of other securities. Consequently, Matrix 4 has 4,000 factors, albeit of varying strengths (Matrix 4 is depicted in a figure at the end of this article). Accordingly, 35 percent of the dispersion is the result of security selection, and 65 percent is the result of the asset/cash decision, the asset class decision, and market selection, which can be subsumed under asset allocation decisions because they are aggregated decisions inferred with instruments provided by asset allocation. Ultimately, the 41st and all subsequent factors represent bets at the security level. For instance, the 41st factor involves long short positions in the equity space exclusively. The meaning of the first 40 factors is the same as previously discussed. Although slightly different quantitatively in this case vis-à-vis the previous cases because of the correlation matrices further refinement, the factors are unchanged in qualitative terms. Finally, if we again standardized all 4,000 return series and regressed them on the 1st, 2nd, 3rd 40th, and 41st 4,000th factors, we would find that the 1st factor explains 37 percent of the standardized movement of all securities, the 2nd factor explains 14 percent, the 3rd 40th factors 14 percent, and the 41st 4,000th factors the remaining 35 percent. EXAMINATION OF COVARIANCE STRUCTURE The strongest claim of our examination is that risk is merely a metric because we can always rescale it. If this perception is challenged on the ground that focusing on the correlation matrix is an unacceptable route, we can run a PCA on the covariance matrix instead of the correlation matrix. Assigning total risks of 40 percent to all stocks and 6 percent to all bonds and then running a covariancebased PCA, we find the following shares of explained covariance in the covariance space. 6

7 The share of security selection increases to 49 percent versus 35 percent in the correlation matrix. The asset class decision diminishes dramatically, from 15 percent to 1 percent, but this drop is predictable because equity risk also at the asset class level is a multiple of bond risk. This example demonstrates how pointless covariance is as the driving parameter in today s world because risk can be scaled. Risk scaling is something we have been doing for quite some time (e.g., in portfolio design along all markets) and is nothing out of the ordinary. Even if we accept covariance as the driving parameter, however, asset allocation decisions and security selection are still at par. SUMMARY AND CONCLUSIONS We examined the dispersion arising from asset allocation decisions versus security selection. We performed our analysis independent of any perception as to how a portfolio should be structured. We examined the extent to which returns are systematic versus each other and on what aggregation level. To conduct this examination, we needed only the correlation matrix of the underlying investment universe. Our strongest claim is that correlation constitutes information but risk does not. On the basis of a correlation analysis, we found that two-thirds of variation is due to asset allocation and one-third to security selection. In that regard, asset allocation is comparable to the yellow roads on the map because asset allocation bundles assets as the yellow roads bundle connections. If one does not buy into the concept of free scalability, one can replace correlation with covariance. In that case, we found that half the variation is due to asset allocation and the other half to security selection. Stock pickers can always claim that our correlation assumptions regarding individual securities are too high, that we assign too much importance to aggregate decisions. They may even claim the existence of many unexplored and uncorrelated securities. This approach is fooling by measurement, however, as the various financial crises over the past 15 years have taught us. Claiming uncorrelated stocks is as heroic as declaring private equity low risk and uncorrelated with public markets. Asset allocation decisions are very important, the more so when considering the costs of investigating 40 equity and bond markets versus the costs of investigating 4,000 individual securities. Appendix A. Eigenvalues and Eigenvectors Assume the (n n) correlation matrix, R. Then [X, D] = eig(r) (A1) provides the (n n) diagonal matrix (D) of eigenvalues and the (n n) matrix (X) of eigenvectors. The n eigenvectors are defined by the columns of X and scaled to a length of 1. In addition, D and X are inferred such that XDX = R. (A2) The point is to transform R such that the resulting factors are uncorrelated. Hence, D is a diagonal matrix. Further (and this property is also important), the first factor of D explains the most dispersion possible for R, the second factor explains the most dispersion possible for the remainder, the third factor the most dispersion possible for the remainder s remainder, and so on. The structure of Equation A2 is the well-known structure of the standard factor-based risk modeling that is, D assumes the role of the factor matrix, and X is the loadings matrix involving the exposures of the assets to be modeled versus the factors, D. In this brief example, we have the following correlation matrix, R. (A.I) Equation A1 provides the following eigenvalues, D, 3 (A.II) cfa institute 7

and the following eigenvectors, X. (A.III) 0.46 0.65 0.53 0.27 0.53 0.27 0.46 0.65 0.53 0.27 0.46 0.65 0.46 0.65 0.53 0.27 That is, the first asset, defined by the first row of X, loads with 0.

8 and the following eigenvectors, X. (A.III) That is, the first asset, defined by the first row of X, loads with 0.46 on the first factor, 0.65 on the second factor, 0.53 on the third factor, and 0.27 on the fourth factor. Note that, by definition, an asset s sum of squared loadings times the corresponding eigenvalues equals 1 because it equals the calculation of a diagonal element of R according to Equation A2. Thus, for the first asset, ( 0.53) = 1. (A3) Underlying Correlation Matrix 8

9 NOTES 1. We assume that readers understand the fundamentals of PCA. For a brief introduction to PCA, see Appendix A. 2. Bond markets include Treasuries and investment-grade bonds. 3. By definition, the sum of D s elements equals the sum of R s diagonal elements in our case, 4. REFERENCES Brinson, G.P., L.R. Hood, and G.L. Beebower Determinants of Portfolio Performance. Financial Analysts Journal, vol. 42, no. 4 (July/August): Brinson, Gary P., Brian D Singer, and Gilbert L. Beebower Determinants of Portfolio Performance II: An Update. Financial Analysts Journal, vol. 47, no. 3 (May/June): Hensel, Chris R., D. Don Ezra, and John H. Ilkiw The Importance of the Asset Allocation Decision. Financial Analysts Journal, vol. 47, no. 4 (July/August): Ibbotson, Roger G., and Paul D. Kaplan Does Asset Allocation Policy Explain 40, 90, or 100 Percent of Performance? Financial Analysts Journal, vol. 56, no. 1 (January/February): Kritzman, Mark, and Sébastien Page The Hierarchy of Investment Choice. Journal of Portfolio Management, vol. 29, no. 4 (Summer): Staub, Renato Asset Allocation vs. Security Selection Baseball with Pitchers Only? Journal of Investing, vol. 15, no. 3 (Fall): Renato Staub is with Investments, Risk Capital Allocation, Singer Partners LLC. Brian Singer, CFA, is CIO of Singer Partners LLC cfa institute 9

Does Asset Allocation Policy Explain 40, 90, or 100 Percent of Performance?

Does Asset Allocation Policy Explain 40, 90, or 100 Percent of Performance? Roger G. Ibbotson and Paul D. Kaplan Disagreement over the importance of asset allocation policy stems from asking different