DEA and Stochastic Dominance Efficiency
Analysis of Investment Portfolios:Do Evironmentally Responsible Mutual Funds
Diversify Efficiently?
Timo Kuosmanen Wageningen University, The Netherlands
8EWEPA Oviedo 24-26 September 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 t 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 an i.i.d. 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
( ) ( ) 0z
k jG t G t dt
( ) ( ) 0z u
k jG t G t dtdu
zR
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 advocacycommunity investing
Do environmentally responsible mutual funds differ from traditional large blend funds in their portfolio efficiency?
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
Return distributions: 8 green funds
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
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
Conclusions Stochastic Dominance criteria applicable
for measuring portfolio efficiency and finding efficient diversification strategies.
Direct analogy with DEA Elementary DEA can be enhanced by
accounting for permutations composing dominating portfolios directly from
stocks rather than peer funds
Further details... The theory and the LP efficiency measures
available in working paper ”Efficient Diversification According to Stochastic Dominance Criteria”.
A DEA oriented paper with an application to environmentally responsible mutual funds is still work in progress.
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