Efficient Portfolio Diversification

according to Stochastic Dominance

Criteria: Applications to Mixed-Asset

Forest Portfolio Management and

Environmentally Responsible Mutual

FundsTimo Kuosmanen

Wageningen University, The Netherlands

Ympäristö ja luonnovarataloustietee kollokvia, Helsinki 15.10.2003

The presentation is based on 3 papers: Kuosmanen, T. (2001): ”Stochastic Dominance Efficient

Diversification ”, Helsinki School of Economics Working Paper W-232?

Heikkinen, V.-P., and T. Kuosmanen (2003): ”Stochastic Dominance Portfolio Analysis of Forestry Assets”, chapter 12 in Wesseler et al. (Eds.): Risk and Uncertainty in Environmental and Resource Economics, Edward Elgar.

Kuosmanen (2003): ”DEA and Stochhastic Dominance Portfolio Analysis: Do Environmentally Responsible Mutual Funds Diversify Efficiently?, paper presented at the 8EWEPA, Oviedo, Spain, 24-26 Sept. 2003.

Stochastic Dominance as a Criterion of

Risk

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Stochastic Dominance as a Criterion of

Risk

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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

Economic interpretation of SD Consider the Expected Utility Theory of von Neumann

& Morgenstern. If portfolio j dominates portfolio k by FSD (SSD, TSD),

then portfolio j is preferred to portfolio k by all investors who are

FSD: non-satiated (u’(x)0).

SSD: non-satiated and risk averse (u’(x)0, u’’(x)0).

TSD: non-satiated and risk averse with decreasing absolute risk aversion (u’(x)0, u’’(x)0, u’’’(x)0).

Second-order Stochastic Dominance

(SSD)

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Setting

N assets T different states of nature (time periods) R(j,t) = rate of return of asset j in state t j = portfolio weight of asset j Rate of return of portfolio in state t is Portfolio can be characterized equivalently in terms

of the return vector R in the state space (primal) or the portfolio weights (dual).

1

( , )N

jj

R j t

Stochastic Dominance (SD) Approach Return is an i.i.d. random variable drawn from an

unknown distribution. Returns in different states are a sample drawn from that distribution.

State independence: investor indifferent between return profiles (x,y) and (y,x).

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

SD criteria applied to the empirical distributions.

Problem of diversification1. Diversification(states / time series)

2. Sorting / Ranking(irreversibility)

3. SD(distribution function)

-20.00%

-15.00%

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0.00%

5.00%

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15.00%

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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

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FSD dominating set Kuosmanen (2001)

Consider R0 = (1,4).

(4,1)

(4,4)(1,4)

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FSD dominating set

SSD dominating set Kuosmanen (2001)

R0 = (1,4).

 

(4,1)

(4,4)(1,4)

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SSD dominating set

SD efficiencyDefinition: Portfolio k is FSD (SSD) inefficient if the

portfolio set includes another feasible portfolio that dominates k by FSD (SSD).

Otherwise k is FSD (SSD) efficient. Typical approach is to apply the basic pairwise

comparisons to a sample of assets/portfolios using the standard crossing algorithms.

However, there are infinite numbers of alternative diversified portfolios! Therefore, even though it is possible to falsify efficiency by pairwise comparisons, it is not possible to verify it.

Testing for SD efficiency: FSD

Is fund A FSD efficient?

C

A

B

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R2

FSD dominating set

C

A

B

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R2

Testing for SD efficiency: SSD

Is fund A SSD efficient?

SSD dominating set

C

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B

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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

Stochastic Dominance Portfolio

Analysis

of Forestry AssetsVeli-Pekka Heikkinen (Varma-Sampo Mutual Pension Insurance Company, Helsinki, Finland)

Timo Kuosmanen (Wageningen University, The Netherlands)

Risk and Uncertainty in Environmental and Resource Economics, June 5-7, 2002 ,Risk and Uncertainty in Environmental and Resource Economics, June 5-7, 2002 ,

Empirical motivation Heikkinen (1999): Cutting Rules for Final Fellings: A Mean-

Variance Portfolio Analysis, J. Forest Econ.

The Faustmann rule can determine the optimal timing of harvest, but the targeting harvest to specific stands can be used for hedging portfolio risk of the land-owner.

5 assets: 4 harvestable mixed stands of borealis forest Stock market (index) represents investment alternatives

Forest stands offer physical growth (assumed certain) but involve a risk in stumpage prices. The composition of species and thickness influences the price risk.

Research questions Are the current portfolio weights of stands and

the stocks SD efficient?

Does risk aversion (FSD vs SSD) play a role?

Do additional constraints on acquiring additional growing stock with characteristic similar to existing stands influence the result?

Overview of the 4 forest standsStand#162

Stand#163

Stand#165

Stand#173

Total PricesFIM/m3

1996:12Pine sawlogs 16 m3 17 m3 248.1Spruce sawlogs 67 m3 1 m3 267 m3 206.1Birch sawlogs 98.1Pine pulpwood 9 m3 16 m3 123.8Spruce pulpwood 25 m3 2 m3 191 m3 246.9Birch pulpwood 3 m3 14 m3 94.2

Total 92 m3 31 m3 47 m3 485 m3 628 m3

Growth (%) 3.2 4.1 4.1 3.7Value (FIM) 16904 5589 7106 78675 108273Portfolio weight 0.16 0.05 0.07 0.72Area (ha) 0.7 0.5 0.6 2.2 4

r = - 0.016Price volatility

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1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131

HEX PineLog

The MV assumptions

All asset Returns are normally distributed the higher moments of the distribution

(skewness, etc) equal to zero.

OR Forest owners expected utility function is

of quadratic form, U(x) = a + bx + cx2

the higher moments do not matter.

r = 0.442r = 0.442

Empirical fit of Normal distribution:

Stand 165

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r = 0.442r = 0.442

Empirical fit of Normal distribution:

Stand 165

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Results

1 2n

1) Unconstrained case0.00080.00082) Constrained case 0.00000.0000

Conclusions

Original portfolio slightly inefficient (0.08 % points p.a. inefficiency premium).

Risk preferences did not play a role. If new identical timber stock cannot be acquired,

the current portfolio is actually efficient. The MV model suggests very similar reference

portfolios. Offers 1.8 percent decrease in portfolio variance in the constrained case.

Stochastic Dominance Efficiency Analysis of

Investment Portfolios:Do Evironmentally Responsible Mutual Funds

Diversify Efficiently?

Timo Kuosmanen Wageningen University, The Netherlands

Lunch presentation 6 October 2003

Environmentally responsible mutual

funds Part of Socially Responsive Investing (SRI) or Ethical Investing

”Green” funds with special focus on the environment

Most ethical/religious funds also have environmental criteria in their investment strategy

Methods of SRI funds screening of corporate securities

positive screens (invest in clean firms)negative (avoid polluting firms)

shareholder advocacy

community investing

Screening of corporate securities Common screens

AlcoholTobacco GamblingWeapons/DefenceAnimal testingHuman RightsLabor relationsEqual opportunitiesEnvironment

Shareholder advocacy Influence the CEOs and the board of

directors as shareholder Proxy voting in annual general meetings of

the companies Present resolutions Vote to resolutions presented by other

shareholders in accordance with the values of the fund

Community investing Support development initiatives in low-income

communities and get responsible businesses get started. Help people who may not be able to obtain financing through traditional lenders.

Channeled through: Community Banks, Community Credit Unions, Community Loan Funds Microenterprise lenders

Are ”green” funds efficient? Constraints on fund managers => cannot

hedge risk as efficiently as normal funds => higher risk/lower return.

Focus on best practice within each industry. If environmental performance is correlated with profitability (Porter hypothesis), environmental indicators contain useful information => higher return/lower risk

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%

Shapiro-Wilks normality test

Reject normality at significance level Total1% 5% 10%

SRI funds 0 1 1 8Securities[DJSI &SP500]

13 17 22 175

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

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Dominating distribution

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Conclusions Stochastic Dominance criteria applicable

for measuring portfolio efficiency and finding efficient diversification strategies.

Dominating reference portfolios can be composed directly from stocks rather than peer funds

No notable differences in the efficiency distribution of green funds and traditional funds

Questions & comments

The first two papers are available by request, the third one is work in progress.

Coordinates: E-mail: Timo.Kuosmanen@wur.nl homepage:

http://www.sls.wau.nl/enr/staff/kuosmanen/