Investment Style of Portfolio Investment Style of Portfolio Management: Excel Management: Excel

ApplicationsApplications

Journal of Applied Finance, Journal of Applied Finance, 20012001

Stan Atkinson and Yoon ChoiStan Atkinson and Yoon Choi

IntroductionIntroduction

To investigate Sharpe's (1988, 1992) To investigate Sharpe's (1988, 1992) investment style model of managed investment style model of managed portfolios in terms of asset allocation (style), portfolios in terms of asset allocation (style), using the Solver function. using the Solver function.

Style analysis (or asset allocation) models are a Style analysis (or asset allocation) models are a valuable tool for investors, plan sponsors, and valuable tool for investors, plan sponsors, and consultants. consultants.

Investors want to know the investment style so they Investors want to know the investment style so they can create an effective mix of assets that fits their can create an effective mix of assets that fits their tastes. Plan sponsors and consultants are interested tastes. Plan sponsors and consultants are interested in how well the portfolio manager meets the in how well the portfolio manager meets the investment objectives.investment objectives.

Sharpe’s Style ModelSharpe’s Style Model

Ri = bRi = bi1i1F1 + bF1 + bi2i2F2 + ….+ bF2 + ….+ bininFn+ ei,Fn+ ei,

Ri ?Ri ? F = ?F = ? b ? ei?b ? ei?

Target Cell?Target Cell?

Constraints?Constraints?

Sharpe’s Style ModelSharpe’s Style Model

Sharpe introduces an objective styleSharpe introduces an objective style

model based on asset classes (or factors). model based on asset classes (or factors).

Ri = bRi = bi1i1F1 + bF1 + bi2i2F2 + ….+ bF2 + ….+ bininFn+ ei,Fn+ ei,

He assumes that a mutual fund’s return is He assumes that a mutual fund’s return is assumed to be a function of several factor assumed to be a function of several factor exposures and firm-specific risks. The exposures and firm-specific risks. The factor exposures or sensitivities determine factor exposures or sensitivities determine the style or asset allocation of the fund.the style or asset allocation of the fund.

Empirical SupportsEmpirical Supports

Trzcinka (1995) defends Sharpe’s style Trzcinka (1995) defends Sharpe’s style model, model, – Easy to implement and very objective.Easy to implement and very objective.– The assumptions and data are clear to the The assumptions and data are clear to the

analyst, so the results can be replicated. analyst, so the results can be replicated. – In spite of some debates about its merits, In spite of some debates about its merits,

the Sharpe style analysis has been the Sharpe style analysis has been popular and applied to actual portfolios. popular and applied to actual portfolios.

Style AnalysisStyle Analysis

Sharpe (1992) defines the asset allocation Sharpe (1992) defines the asset allocation of a mutual fund as the way in which the of a mutual fund as the way in which the fund manager allocates his assets across a fund manager allocates his assets across a number of major asset classes. number of major asset classes.

Consider the following equation:Consider the following equation:

Ri = bRi = bi1i1F1 + bF1 + bi2i2F2 + ….+ bF2 + ….+ bininFn+ ei,Fn+ ei, (1) (1) where Ri is the mutual fund return, Fn where Ri is the mutual fund return, Fn

is the value of the nth factor, bis the value of the nth factor, binin is the factor is the factor sensitivities, and ei is the unsystematic sensitivities, and ei is the unsystematic residual. residual.

InterpretationInterpretation

In a way, we estimate bs so that the In a way, we estimate bs so that the model fits the mutual fund return model fits the mutual fund return behavior best. behavior best.

What is the best representation of What is the best representation of the fund in terms of its exposure (bthe fund in terms of its exposure (bss) ) to different asset classes?to different asset classes?

Ri = bRi = bi1i1F1 + bF1 + bi2i2F2 + ….+ bF2 + ….+ bininFn+ ei,Fn+ ei,

EstimationEstimationMathematically, It is the one for which the Mathematically, It is the one for which the variance of ei in Equation (1) is the least. Thus, variance of ei in Equation (1) is the least. Thus, we rearrange Equation (1) as follows:we rearrange Equation (1) as follows:

ei = Ri – [biei = Ri – [bi11F1 + biF1 + bi22F2 + ….+ biF2 + ….+ binnFn]Fn] (2) (2)

The objective is to choose the style (set of asset The objective is to choose the style (set of asset class exposures) that minimizes the variance of class exposures) that minimizes the variance of this difference (or the sum of the residual this difference (or the sum of the residual squared with all constraints). squared with all constraints).

IIthe investment style problem is to obtain a set of the investment style problem is to obtain a set of exposure coefficients that minimizes the exposure coefficients that minimizes the variance of the residual, var(ei) with the variance of the residual, var(ei) with the constraints that each of the factor sensitivities constraints that each of the factor sensitivities (or exposures), bi, lies between zero and one, (or exposures), bi, lies between zero and one, and that the factor sensitivities should add up to and that the factor sensitivities should add up to one (e.g., bi1 + bi2 + ….+ bin = 1). one (e.g., bi1 + bi2 + ….+ bin = 1).

In Excel Solver

II

MIN Var (ei) MIN Var (ei)

Changing Cell: exposure coefficients Changing Cell: exposure coefficients Or biOr bi

Constraints: Constraints:

bi >= 0,bi >= 0,

bi <= 1, bi <= 1,

bi1 + bi2 + bi1 + bi2 + …….+ bin = 1. .+ bin = 1.

Excel Solver

Guidelines in choosing the asset Guidelines in choosing the asset class factorsclass factors

such asset classes should be such asset classes should be mutually exclusive and exhaustive.mutually exclusive and exhaustive.

By containing exhaustive classes of By containing exhaustive classes of assets, the asset class factors (F1 assets, the asset class factors (F1 through Fn) as a whole can mirror through Fn) as a whole can mirror the market portfolio as closely as the market portfolio as closely as possible.possible.

Sharpe’s 12 asset classesSharpe’s 12 asset classesTreasury bills, Treasury bills,

Intermediate government bonds, Intermediate government bonds,

long-term government bonds, long-term government bonds,

corporate bonds, corporate bonds,

mortgage-related securities, mortgage-related securities,

large-cap value (growth) stockslarge-cap value (growth) stocks

Medium (small) -cap stocks, Medium (small) -cap stocks,

non-U.S. bonds, non-U.S. bonds,

European stocks, and European stocks, and

Japanese stocks.Japanese stocks.

DATADATA

We obtained the asset classes from various We obtained the asset classes from various sources, including BARRA investment data sources, including BARRA investment data (available through the Internet, (available through the Internet, www.barra.com). Mutual fund data from finance.yahoo.com.). Mutual fund data from finance.yahoo.com.

MSCI acquired Barra in 2004 and these data are MSCI acquired Barra in 2004 and these data are not updated any longer. not updated any longer.

http://www.barra.com/Research/DownloadMonthlyReturns.aspx up to November 2005. up to November 2005.

Or, use www.Russell.com for the index (factor) Or, use www.Russell.com for the index (factor) data.data.

DATADATA

We concentrated on the domestic asset world:We concentrated on the domestic asset world:S&P 500/Barra Growth (Value) Index, S&P 500/Barra Growth (Value) Index, S&P MidCap 400/Barra Growth (Value) IndexS&P MidCap 400/Barra Growth (Value) IndexS&P SmallCap 600/Barra Growth (Value) Index IBBS Corporate S&P SmallCap 600/Barra Growth (Value) Index IBBS Corporate Bond Index, Bond Index, IBBS Government Bond Index, and IBBS Government Bond Index, and IBBS Treasury Bill Index.IBBS Treasury Bill Index.

In the paper, we used previous 36 monthly returns each In the paper, we used previous 36 monthly returns each month for the estimation. For example, month for the estimation. For example,

In Dec. 1996 est., from Jan. 1994 to Dec. 1996. In Jan. In Dec. 1996 est., from Jan. 1994 to Dec. 1996. In Jan. 1997 est., from Feb. 1994 to Jan. 1997. 1997 est., from Feb. 1994 to Jan. 1997.

Style DriftStyle DriftIt could be helpful for investors to know how the It could be helpful for investors to know how the

style changes over time so that they can style changes over time so that they can rebalance or reallocate their portfolios of rebalance or reallocate their portfolios of mutual funds.mutual funds.

Sharpe also shows how to estimate the style Sharpe also shows how to estimate the style drift by performing a series of style analyses, drift by performing a series of style analyses, rolling the window used for the analysis over rolling the window used for the analysis over time. time.

Since he uses the past returns in this Since he uses the past returns in this procedure, the Sharpe model only detect procedure, the Sharpe model only detect style drift with a lag.style drift with a lag.

Interesting Findings on Style Drift Interesting Findings on Style Drift

Israelsen (1999) finds that style drift Israelsen (1999) finds that style drift happens to quite a number of funds. happens to quite a number of funds.

He also finds that funds with the greatest He also finds that funds with the greatest style drift have managers style drift have managers

– with less tenure, more fluctuations in annual with less tenure, more fluctuations in annual returns, lower tax efficiency, higher expense returns, lower tax efficiency, higher expense ratios, higher turnover ratios, and fewer assets. ratios, higher turnover ratios, and fewer assets.

Interesting Findings on Style Drift Interesting Findings on Style Drift

Tergesen (1999) finds that, Tergesen (1999) finds that,

those managers that stick to their those managers that stick to their styles have higher risk-adjusted styles have higher risk-adjusted returns than do their more eclectic returns than do their more eclectic peers. These results held across all peers. These results held across all types of funds. types of funds.

II

MIN Var (ei) MIN Var (ei)

Changing Cell: exposure coefficients Changing Cell: exposure coefficients Or biOr bi

Constraints: Constraints:

bi >= 0,bi >= 0,

bi <= 1, bi <= 1,

bi1 + bi2 + bi1 + bi2 + …….+ bin = 1. .+ bin = 1.

Excel Solver