`

Mutual Fund Performance Measurement William F. Sharpe Graduate School of Business Stanford University   www-sharpe.stanford.edu

 

Mutual Fund Performance Measures Use statistics from: historic frequency distribution many periods Example: combination of mean and standard deviation for past 36 months To predict statistics for: future probability distribution one period Example: combination of mean and standard deviation for next month

 

Decisions One Fund One Fund plus borrowing or lending A portfolio of potentially many funds   The key question: Which (if any) fund is relevant in each situation?

 

Portfolio Theory

 

Hierarchic Taxonomic Procedures

 

 

Statistics: M Ex Ante: Expected Return Expected geometric return etc. Ex Post: Arithmetic average return Geometric average return Compounded total return over period etc.

 

Statistics: S Ex Ante: Standard Deviation of Return Variance of Return Expected loss etc. Ex Post: Standard deviation of return Variance of Return Average loss etc.

 

Performance Measures Return M Utility-based M - k * S Scale-independent M / S

 

Variables Total Return Fund Return Excess Return Fund Return - Return on a risk-free instrument Differential Return Fund Return - Return on an appropriate benchmark portfolio

 

Absolute and Relative Measures Absolute Use statistics as computed for all funds Relative Each fund assigned to a peer group Performance of funds ranked within each peer group Stars or Ratings assigned based on rankings

 

Frequently-used Measures Relative   Total Return Excess Return Differential Return Return Lipper     Utility-based   Morningstar (form)   Scale-independent   Morningstar (subst.) Micropal Absolute   Total Return Excess Return Differential Return Return     selection mean (alpha) Utility-based       Scale-independent   Sharpe ratio selection Sharpe ratio

 

Scale-independent Measures Variable = Return on A minus return on B Strategy requires zero investment long position in A short position in B Change in value can be doubled by doubling sizes of positions For scale k: Mk = k* M1 SDk = k* SD1 Mk / SDk = M1 / SD1 Therefore, ratio is scale-independent

 

Scale-independent Measures with Positive Expected Returns

 

Scale-independent Measures with Negative Average Returns

 

Morningstar Peer Groups Peer Groups Asset classes Categories Asset Classes Domestic equity International equity Taxable bond Municipal bond Domestic equity categories Diversified (9) Specialty (9) Hybrid Convertible

 

Morningstar Ratings Stars: Rank within asset class (e.g. equity) 3-year, 5 year, 10 year and weighted average of 3,5, and 10 year Net of load charges Category Ratings: Rank within asset category (e.g. Large Growth equity) 3-year Load charges not taken into account Percentages: 1 (worst) 2 3 4 5 (best) 10% 22.5% 35% 22.5% 10%

 

Morningstar Statistics, 3-year Ratings M Compounded return on fund - compounded return on Treasury bills Loss if fund return > Treasury bill return, loss = 0 if fund return < Treasury bill return, loss = - (fund return - bill return) S Average Monthly Loss sum ( monthly loss) takes all 36 months into account

 

Average Monthly Loss versus Standard Deviation of Monthly Returns, Morningstar Diversified Equity Funds, 1994-1996

 

Morningstar Risk-adjusted Rating RARf = Mf / M_ - Sf / S_ M_ if mean ( Mf ) >= compound return on Treasury bills, mean ( Mf ) if mean ( Mf ) < compound return on Treasury bills, compound return on Treasury bills S_ mean ( AMLf )

 

Morningstar Risk-adjusted Ratings as Utility-based Measures RARf = Mf / M_ - Sf / S_ = ( 1/M_ ) * [ Mf - ( M_ / S_ ) * Sf ] Rankings unaffected by initial constant ( 1/M_ ) Rankings depend on: Mf - k * Sf where: k = M_ / S_

Optimal Leverage when Utility = Return - k*Risk

Optimal Leverage when Utility = Return - k*Risk2

 

Indifference Curves and Iso-M/S lines: k = M_ / S_

 

Sharpe Ratio Ranks versus Category Ratings, Morningstar Diversified Equity Funds, 1994-1996

 

 

 

Factor-model Based Analysis

 

An Asset Class Factor Model R~f = [ b1f F~1 + b2f F~2 + ... + bnf F~n ] + e~f R~f Fund return F~1 ,...,F~n Asset class returns b1f ,..., bnf Fund asset class exposures (style) : sum = 1 [ ... ] Fund style return e~f Fund selection return: e~f i uncorrelated with e~f j

 

Benchmark Portfolios and Asset Exposures R~f = [ b1f F~1 + b2f F~2 + ... + bnf F~n ] + e~f R~f Fund return F~1 ,...,F~n Asset class returns b1f ,..., bnf Benchmark portfolio composition [ ... ] Benchmark portfolio return e~f Fund differential return

 

Methods for Selecting a Benchmark       Historic Average Current Projected Composition MStar Category MStar Style   Regression Actual Returns Retrospective Returns   Style Analysis Actual Returns Retrospective Returns   Projection     FER Proposal

 

Overall Portfolio Return R~p = [ b1p F~1 + b2p F~2 + ... + bnp F~n ] + e~p where: bjp = X1 b1j + X2 b2j + ... + Xn bnj e~p = X1 e~1 + X2 e~2 + ... + Xn e~m [...] = (style) return on assets ( R~A ) e~p = selection return

 

Selection Return Statistics Ex post mean ( e~f ) Average selection return ( alpha ) stddev ( e~f ) Selection return variability Ex ante expected ( e~f ) Expected selection return ( alpha ) stddev ( e~f ) Selection return risk

 

Factor-model Based Analysis: Optimization Inputs Asset Classes Expected Returns Standard Deviations Correlations Funds Styles ( Benchmark portfolios) Expected selection returns (alphas) Selection risks Investor Risk tolerance: t other constraints, assets, liabilities, etc

 

Optimization with Unlimited Short Positions in Assets Creating a hedge fund Long: fund Short: fund's benchmark asset mix Zero investment required Return is scale-independent Asset allocation unaffected by scale of investment Select Xi to maximize: Xi expected (ei ) - ( Xi 2 Var ( ei ) ) / t

 

Optimal Position in a Fund with Unlimited Short Positions in Assets Xi = [ expected (ei ) / Var ( ei ) ) ] * ( t / 2 ) Amount of risk taken: Xi * stdev ( ei ) = [ expected (ei ) / stdev ( ei ) ] * ( t / 2 ) = [ selection Sharpe ratio ] * ( t / 2 ) Relative values independent of investor preferences

 

 

Taxonomic Factor Models All conditions for a general asset class factor model hold plus For any given fund f One bif = 1 All other bif's = 0 Fund expected return = asset class expected return + fund alpha Fund Variance = asset class variance + fund selection variance

 

 

The Optimal Fund for an Asset Class with a Taxonomic Factor Model Assume that Xj is to be invested in a fund in a given asset class From the funds in the asset class, select the fund with the largest value of: expected ( ef ) - ( Xj / t ) * variance ( ef ) A utility-based differential return measure with k a function of: the amount to be invested in the asset class ( Xj ) the investor's risk tolerance (t) The appropriate measure will thus depend on an investor's portfolio and degrees of risk tolerance -- no single measure will be appropriate for everyone

 

 

The Optimal Fund for a Small Portion of a Portfolio The preferred fund for an investment of Xj in asset class j has maximum: z = expected ( ef ) - ( Xj / t ) * variance ( ef ) If Xj is small: ( Xj / t ) * variance ( ef ) is small z is approximately equal to expected ( ef ) = alpha Hence the best fund is the one with the largest alpha This may not be the case if a significant amount is to be invested in the fund

 

 

Why There is no Universally Relevant Single Performance Measure Measures such as Morningstar's risk-adjusted ratings or excess return Sharpe ratios are inappropriate for choosing funds for a multi-fund portfolio since they are based on total risk or excess return risk rather than the contribution of the investment in a fund to overall portfolio risk Most investors do not or cannot take short positions in asset classes, hence measures such as selection Sharpe ratios which are based on differential return risk may be of limited value Most investors place enough money in a fund to make alpha an insufficient measure of performance since it does not take risk into account In most cases, no universal single measure can provide a sufficient statistic for choosing funds for a multi-fund portfolio

 

An Alternative to the Hierarchic Taxonomic Approach with Single Measures of Mutual Fund Performance Asset Classes as Factors Funds as Decision Variables All fund exposures to asset classes taken into account All components of risk and return considered Unbiased estimates of future risks, returns and correlations obtained using all relevant information (for example, fund expense ratios) Efficient combinations of funds obtained using Markowitz' optimization procedures

Needed Inputs (1) For asset classes: future expected returns future risks future correlations For funds: future asset exposures ( appropriate benchmark portfolios or styles) future fund selection risks future fund selection expected returns

 

Needed Inputs (2) For the investor: Current amount saved Future savings rate Horizon Liabilities Other assets Degree of aggressiveness

 

 

Outputs For a given set of investor inputs: The associated efficient combination of funds Range of outcomes (in terms relevant for the investor) For different sets of investor inputs, the associated: Efficient combinations of funds Ranges of outcomes For the set of inputs ultimately chosen by the investor: The optimal combination of funds The preferred range of outcomes

 

 

The Bottom Line   We all have computers Why not use them?