Mutual Fund Performance Measures, Factor Models,

and Fund Style and Selection

William F. Sharpe

www-sharpe.stanford.edu

www-leland.stanford.edu/~wfsharpe

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

One fund from a given asset class or category

A portfolio of potentially many funds

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
- Comparisons based on:

- Differences
- Ratios
- Rankings
- Stars

- 5 stars: top 10%
- 4 stars: next 22.5%
- etc.

Frequently-used Measures

Relative

Total ReturnExcess ReturnDifferential ReturnReturnLipperUtility-basedMorningstar (form)Scale-independentMorningstar (subst.)MicropalAbsolute

Total ReturnExcess ReturnDifferential ReturnReturnselection mean (alpha)Utility-basedScale-independentSharpe ratioselection 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:

- M
_{k}= k* M_{1}- SD
_{k}= k* SD_{1}- M
_{k}/ SD_{k}= M_{1}/ SD_{1}Therefore, ratio is scale-independent

Scale-independent Measures with Positive Expected Returns

Scale-independent Measures with Negative Average Returns

Inappropriateness of Total Return M/S Measures

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 Diversified Equity Categories

Based on portfolio composition

- price/earnings, price/book
- market capitalization
Averaged over past three years

Style Boxes

Large Value

Large Blend

Large Growth

Medium Value

Medium Blend

Medium Growth

Small Value

Small Blend

Small Growth

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)2345 (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

Average Monthly loss versus function of Monthly Mean and Std. Deviation

Morningstar Diversified Equity Funds, 1994-1996

Morningstar Risk-adjusted Rating

RAR

_{f}= M_{f}/ M^{_}- S_{f}/ S^{_}M

^{_}

- if mean ( M
_{f }) >= compound return on Treasury bills,

- mean ( M
_{f})

- if mean ( M
_{f }) < compound return on Treasury bills,

- compound return on Treasury bills
S

^{_}

- mean ( AML
_{f})

Morningstar Risk-adjusted Ratings as Utility-based Measures

RAR

_{f}= M_{f}/ M^{_}- S_{f}/ S^{_}= ( 1/M

^{_}) * [ M_{f}- ( M^{_}/ S^{_}) * S_{f}]Rankings unaffected by initial constant ( 1/M

^{_})Rankings depend on:

- M
_{f}- k * S_{f}- where:

- k = M
^{_}/ S^{_}

A bi-linear VnM Utility Function with threshold = 4% and utility ratio = 2.5

Optimal Leverage when Utility = Return - k*Risk

Optimal Leverage when Utility = Return - k*Risk

^{2}

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

^{_}/ S^{_}

Indifference Curves and Iso-M/S lines: k > M

^{_}/ S^{_}

Sharpe Ratio Ranks versus Category Rankings,

Morningstar Diversified Equity Funds, 1994-1996

Three-year Star Ratings and Mean-variance combinations,

Morningstar Diversified Equity Funds, 1994-1996

An Asset Class Factor Model

R

^{~}_{f}= [ b_{1f }F^{~}_{1}+ b_{2f}F^{~}_{2}+ ... + b_{nf }F^{~}_{n}] + e^{~}_{f}

R ^{~}_{f}Fund return F ^{~}_{1 },...,F^{~}_{n}Asset class returns b _{1f},..., b_{nf}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}= [ b_{1f }F^{~}_{1}+ b_{2f }F^{~}_{2}+ ... + b_{nf }F^{~}_{n}] + e^{~}_{f}

R ^{~}_{f}Fund return F ^{~}_{1 },...,F^{~}_{n}Asset class returns b _{1f},..., b_{nf}Benchmark portfolio composition [ ... ] Benchmark portfolio return e ^{~}_{f}Fund differential return

Methods for Selecting a Benchmark

Historic AverageCurrentProjectedCompositionMStar CategoryMStar StyleRegressionActual ReturnsRetrospective ReturnsStyle AnalysisActual ReturnsRetrospective ReturnsProjectionFER Proposal

Taxonomic Factor Models

All conditions for a general asset class factor model hold

plus

For any given fund f

- One b
_{if }= 1

- All other b
_{if}'s = 0Fund expected return = asset class expected return + fund alpha

Fund Variance = asset class variance + fund selection variance

Overall Portfolio Return

R

^{~}_{p}= [ b_{1p }F^{~}_{1}+ b_{2p }F^{~}_{2}+ ... + b_{np }F^{~}_{n}] + e^{~}_{p}where:

b

_{jp}= X_{1}b_{1j}+ X_{2 }b_{2j}+ ... + X_{n}b_{nj}e

^{~}_{p}= X_{1}e^{~}_{1}+ X_{2 }e^{~}_{2}+ ... + X_{n}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

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 X

_{i}to maximize:X

_{i}expected (e_{i})_{ }- ( X_{i}^{2}Var ( e_{i }) ) / t

Optimal Position in a Fund with Unlimited Short Positions in Assets

X

_{i}= [ expected (e_{i})_{ }/ Var ( e_{i }) ) ] * ( t / 2 )Amount of risk taken:

X

_{i }* stdev ( e_{i })= [ expected (e

_{i})_{ }/ stdev ( e_{i }) ] * ( t / 2 )= [ selection Sharpe ratio ] * ( t / 2 )

Relative values independent of investor preferences

Choosing a Fund for an Asset Class Position with a Taxonomic Factor Model

Assume asset allocation is fixed

Then:

E

_{p}= E_{A}+ X_{1}expected ( e_{1 }) + . . . + X_{m }expected ( e_{m })V

_{p}= V_{A}+ X_{1}^{2}variance ( e_{1 }) + .... + X_{m}^{2}variance ( e_{m })Utility:

[ E

_{A }- V_{A }/ t ] +[ X

_{1}expected ( e_{1 }) - X_{1}^{2}variance ( e_{1 }) / t ] +. . . +

[ X

_{m}expected ( e_{m }) - X_{m}^{2}variance ( e_{m }) / t ]

The Optimal Fund for an Asset Class with a Taxonomic Factor Model

X

_{j}is a given constantFrom the funds in the asset class, select the fund for which

[ X

_{j}expected ( e_{f }) - X_{j}^{2}variance ( e_{f }) / t ] is the largestEquivalently, select the fund with the largest value of:

expected ( e

_{f }) - ( X_{j}/ t ) * variance ( e_{f })A utility-based differential return measure with k a function of:

- the amount to be invested in the asset class ( X
_{j})- the investor's risk tolerance (t)

The Optimal Fund for a Small Portion of a Portfolio

The preferred fund for an investment of X

_{j }in asset class j has maximum:z = expected ( e

_{f }) - ( X_{j}/ t ) * variance ( e_{f })If X

_{j }is small:( X

_{j}/ t ) * variance ( e_{f }) is smallz is approximately equal to expected ( e

_{f }) = alphaHence best fund is the one with the largest alpha relative to an appropriate benchmark

Correlations of Percentiles within Categories

SR Cat. Star Alpha SSR Sharpe Ratio 1.000 0.986 0.945 0.831 0.744 Category Rating 0.986 1.000 0.957 0.829 0.735 Star Rating 0.945 0.957 1.000 0.790 0.694 Selection Mean (Alpha) 0.831 0.829 0.790 1.000 0.940 Selection Sharpe Ratio 0.744 0.735 0.694 0.940 1.000

Style Analysis Alpha Ranks versus Category Rankings,

Morningstar Diversified Equity Funds, 1994-1996

Style Analysis Selection Sharpe Ratios versus Category Rankings,

Morningstar Diversified Equity Funds, 1994-1996

Conclusions (1)

Hierarchic taxonomic approaches will generally be suboptimal

lower-level characteristics not taken into account when making decisions

asset category characteristics not taken into account when allocating among asset classes

fund characteristics not taken into account when allocating among asset classes and categories

No universal single measure can provide a sufficient statistic for choosing

one fund in each category, or

multiple funds in each category

Conclusions (2)

Need good estimates of:

future asset exposures

appropriate benchmark portfolio (fund style)

future fund selection risk

future fund selection expected return

This information should be combined optimally with estimates of

future asset risks, expected returns and correlations

investor risk tolerance and other characteristics

All useful predictors of future performance should be taken into account

include fund expense ratios, turnover, etc..