Using Stochastic Dominance criteria in

Data Envelopment Analysis of mutual

funds

Timo Kuosmanen Wageningen University, The Netherlands

EURO / INFORMS joint meeting, Istanbul 6-10 July 2003

DEA and Mutual Fund Performance Murthi, Choi, Desai (1997), EJOR.

transaction costs. Morey & Morey (1999), Omega.

multiple investment horizons. Basso & Funari (2001), EJOR

multiple risk measures, sub-period dominance Joro & Na (2001), w.p.

skewness preferences

Stochastic Dominance portfolio

analysis Kuosmanen (2001), w.p. SD efficiency tests and measures that account for

portfolio diversification Post (2003), J. of Finance (to appear)

dual approach, statistical properties, bootstrapping Heikkinen and Kuosmanen (2003), book chapter

application to the management of a mixed asset forest portfolio

Setting

N mutual funds T different time periods R(j,t) return for fund j in period t

Return possibilities frontier: 2-periods

Funds A, B, C; returns RA=(1,4), RB=(3.5,1.6), RC=(2,2).

C

A

B

0

1

2

3

4

5

0 1 2 3 4 5 R1

R2

Elementary DEA-model

1

1

max

. .

(0, ) ( , ) 1, 2,...,

1

0

N

jj

N

jj

j

s t

R t R j t j T

Returns as outputs, no inputs

Properties - elementary DEA model

The previous approach takes into account diversification opportunities risk: entire distribution of returns considered,

not just the first moments (mean, variance).

Can we do better? Preference information

Stochastic Dominance (SD) Approach Return is a random variable drawn from an unknown

distribution. Returns in different time periods are a sample drawn from that distribution.

State independence: timing of returns does not matter.

Empirical distribution function gives a nonparametric minimum variance unbiased estimator of the underlying distribution function.

SD criteria applied to the empirical distributions.

Stochastic Dominance as a criterion of

Risk

0

0.2

0.4

0.6

0.8

1

-10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00%

A

B

Definition of SD Risky portfolios j and k, return distributions Gj and Gk.

Portfolio j dominates portfolio k by FSD (SSD, TSD) if and only if

FSD:

SSD:

TSD:

with strict inequality for some z.

( ) ( ) 0k jG z G z

z ( ) ( ) 0

z

k jG t G t dt

( ) ( ) 0z u

k jG t G t dtdu

Problem of diversification

1. Diversification(time series)

2. Sorting / Ranking(irreversibility)

3. SD(distribution function)

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129 133

HEX PineLog

0

0.2

0.4

0.6

0.8

1

-30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00%

HEX

ST3

FSD dominating set Kuosmanen (2001)

Consider R0 = (1,4).

(4,1)

(4,4)(1,4)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

FSD dominating set

SSD dominating set Kuosmanen (2001)

R0 = (1,4).

 

(4,1)

(4,4)(1,4)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

SSD dominating set

Combining SD with DEA

Is fund A FSD efficient?

C

A

B

0

1

2

3

4

5

0 1 2 3 4 5 R1

R2

FSD dominating set

C

A

B

0

1

2

3

4

5

0 1 2 3 4 5 R1

R2

Combining SD with DEA

Is fund A SSD efficient?

SSD dominating set

C

A

B

0

1

2

3

4

5

0 1 2 3 4 5 R1

R2

Measuring efficiency

How much higher return should be obtained in all periods to make A efficient?

FSD efficiency measure

Return profile R0 is FSD efficient if and only if

1 0,

1

1 1

1 1

( ) max /

. .

( , ) (0, ) =0 1,...,

1 , 1,...,

0,1 , 1,...,

ti

ti t

T

tP

t

N T

j tj i

T T

i

ti

i t

R s T

s t

R j t R i s t T

t i T

t

P

P

i T

P

P

1 0( ) 0R

SSD efficiency measure

Return profile R0 is SSD efficient only if 2 0( ) 0R

1 0,

1

1 1

1 1

( ) max /

. .

( , ) (0, ) =0 1,...,

1 , 1,...,

0,1 , 1,...,

ti

ti t

T

tP

t

N T

j tj i

T T

i

ti

i t

R s T

s t

R j t R i s t T

t i T

t

W

W

i T

W

W

Efficiency of env. resp. mutual funds

Part of Socially Responsive Investing (SRI) US SRI funds amounted to $2.34 trillion in 2001

Methods: screening (positive/negative) shareholder advocacy community investing

Does portfolio efficiency of environmentally responsible mutual funds differ traditional large blend funds?

Return possibilities frontier

175 stocks traded in NYSE and included in the DJSI sustainability index

Weekly returns for 26/11/2001 - 26/11/2002 Constraints on portfolio weights

no shortsales weight of any single stock should not exceed

5.8% total weight of the US stocks at least 65%

Results: Green funds

SSD: Inefficiency premium (% per annum)Fund % p.a.Calvert A 0.35Calvert C 0.36Women's 0.36Neuberger 0.43Devcap 0.43Advocacy 0.45Green Century 0.48Domini 0.51

Results: Traditional fundsFund % p.a. Fund % p.a.NPPAX 0.00 AFEAX 0.44ASECX 0.28 EVSBX 0.45SSLGX 0.32 HFFYX 0.45WFDMX 0.39 HIGCX 0.45MMLAX 0.39 HGRZX 0.45MDLRX 0.40 FGIBX 0.46OTRYX 0.40 FBLVX 0.46STVDX 0.42 PWSPX 0.47PRFMX 0.43 FLCIX 0.49PRACX 0.43 WCEBX 0.50GESPX 0.43 FRMVX 0.50ACQAX 0.43 IGSCX 0.51

IBCCX 0.44 EGRCX 0.51

Dominating distribution

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-5 -4 -3 -2 -1 0 1 2 3 4return

cum

Further details... A full paper with an application to environmentally

responsible mutual funds available soon.

Send an e-mail to Timo.Kuosmanen@wur.nl

The paper will be uploaded shortly on my homepage:http://www.sls.wau.nl/enr/staff/kuosmanen