Portfolio Optimization with Cvar Budgets

Portfolio optimization with CVaR budgets

∗ Kris Boudt - Peter Carl - Brian G. Peterson

R/Finance 2010

April 16th, 2010

∗ K.U.Leuven and Lessius 1/42 History

Outline Risk budgets: A primer on risk budgets

CVaR budgets as objective or constraint in Standard tool to quantify risk allocation; portfolio allocation

Dynamic portfolio allocation Previous research: non-normality return Conclusion series, CVaR: PerformanceAnalytics. Appendix Instrument to adjust marginally portfolios;

This research: Use of risk budgets as objective and/or constraint in portfolio allocation styles: PortfolioAnalytics;

Collaborative: Peter Carl & Brian Peter- son, David Ardia, Christophe Croux. 2/42 Motivation equity/bonds portfolio

Outline Bender, Briand, Nielsen, Stefek (JPM, Winter 2010): A primer on risk budgets

CVaR budgets as “Traditional approaches to structuring policy portfolios for strategic objective or constraint in portfolio allocation asset allocation have not provided the full potential of diversiﬁcation.

Dynamic portfolio allocation Portfolios based on a 60/40 allocation between equities and bonds

Conclusion remain volatile and dominated by equity risk.”

Appendix Minimum risk portfolios tend to be dominated by bond risk and have lower expected returns.

3/42 Motivation equity/bonds portfolio

Outline Bender, Briand, Nielsen, Stefek (JPM, Winter 2010): A primer on risk budgets

Dynamic portfolio allocation Portfolios based on a 60/40 allocation between equities and bonds

Conclusion remain volatile and dominated by equity risk.”

Appendix Minimum risk portfolios tend to be dominated by bond risk and have lower expected returns.

Optimize the risk allocation directly in the portfolio strategy. Examples: X A 60/40 risk allocation portfolio or an equal-risk portfolio

X The most risk diversiﬁed portfolio subject to return/risk targets. 3/42 Outline

Outline Risk budgets: A primer on risk budgets

CVaR budgets as objective or constraint in Review on risk budgets; portfolio allocation

Dynamic portfolio allocation Use of risk budgets as objective and/or Conclusion constraint in portfolio allocation styles. Appendix Illustrations:

X Static bond-equity portfolio, R code (DEoptim, see also Guy Yollin’s slides RFinance 2009);

X Dynamic 4 assets.

4/42 Outline

A primer on risk budgets

VaR budget

CVaR budget

Min CVaR portfolio

CVaR budgets as objective or constraint in A primer on risk budgets portfolio allocation

Dynamic portfolio allocation

Conclusion

Appendix

5/42 VaR budget on 60/40 portfolio

Outline > library(PortfolioAnalytics) A primer on risk budgets > data(indexes) VaR budget

CVaR budget > head(indexes[,1:2]) Min CVaR portfolio US Bonds US Equities CVaR budgets as objective or constraint in 1980-01-31 -0.027189977 0.0610 portfolio allocation

Dynamic portfolio 1980-02-29 -0.066876898 0.0031 allocation 1980-03-31 0.005275587 -0.0987 Conclusion 1980-04-30 0.099246261 0.0429 Appendix 1980-05-31 0.000000000 0.0562 1980-06-30 0.060511451 0.0296

6/42 VaR budget on 60/40 portfolio

Outline > apply(indexes[,1:2],2,’mean’) A primer on risk budgets US Bonds US Equities VaR budget

CVaR budget 0.006916187 0.008238420 Min CVaR portfolio > apply(indexes[,1:2],2,’sd’) CVaR budgets as objective or constraint in US Bonds US Equities portfolio allocation

Dynamic portfolio 0.01810161 0.04476569 allocation

Conclusion

Appendix

7/42 VaR budget on 60/40 portfolio

Outline > w6040 <- c(0.4,0.6) A primer on risk budgets > library(PerformanceAnalytics) VaR budget

CVaR budget > VaR(R=indexes[,1:2], weights=w6040, Min CVaR portfolio + portfolio_method="component") CVaR budgets as objective or constraint in $MVaR portfolio allocation

Dynamic portfolio [1,] 0.04336715 allocation $contribution Conclusion US Bonds US Equities Appendix -0.0002303964 0.0435975440 $pct_contrib_MVaR US Bonds US Equities -0.005312695 1.005312695

8/42 VaR budgets

Outline ∂VaR(w) VaR A primer on risk budgets Ci = wi ∂wi VaR budget CVaR budget Gourieroux,´ Laurent and Scaillet (2000): Min CVaR portfolio

CVaR budgets as objective or constraint in CiVaR = −E[wiri|rp = −VaR] portfolio allocation

Dynamic portfolio allocation

Conclusion Estimation:

Appendix X Simulation

X Explicit formulae Cornish-Fisher estimator (Boudt, Peterson and Croux, 2008; Peterson and Boudt, 2008).

9/42 Outline

Outline Pearson [2002, p.7]: “Value-at-risk has some well known limitations,

A primer on risk budgets and it may be that some other risk measures eventually supplants VaR budget

CVaR budget value-at-risk in the risk budgeting process.”

Min CVaR portfolio

CVaR: 95% Value at Risk and 95% Conditional VaR CVaR budgets as for a Skewed Student t objective or constraint in portfolio allocation Coherent risk mea- 95% VaR Dynamic portfolio allocation sure (most notably: Conclusion subadditive); Appendix Density

Less incentive to load 95% CVaR on the tail risk below

the VaR used. 0.0 0.1 0.2 0.3 0.4

−10 −5 0 5 10

Portfolio returns

10 / 42 CVaR budgets

Outline ∂CVaR(w) CVaR A primer on risk budgets Ci = wi ∂wi VaR budget CVaR budget Scaillet (2002): Min CVaR portfolio

CVaR budgets as objective or constraint in CiCVaR = −E[wiri|rp ≤−VaR] portfolio allocation

Dynamic portfolio allocation Estimation:

Conclusion X Appendix Simulation

X Explicit formulae Cornish-Fisher estimator (Boudt, Peterson and Croux, 2008).

11 / 42 CVaR budgets

Outline > ES(R=indexes[,1:2], weights=w6040, A primer on risk budgets + portfolio_method="component") VaR budget

CVaR budget $MES Min CVaR portfolio [1,] 0.07725177 CVaR budgets as objective or constraint in $contribution portfolio allocation

Dynamic portfolio US Bonds US Equities allocation -0.001066194 0.078317964 Conclusion $pct_contrib_MES Appendix US Bonds US Equities -0.01380155 1.01380155

12 / 42 CVaR in portfolio allocation

Outline As an objective: Minimum CVaR portfolio (E.g. Rockafellar A primer on risk budgets

VaR budget and Uryasev, 2000)

CVaR budget Min CVaR portfolio As a constraint: Min CVaR/SD portfolio under CVaR CVaR budgets as objective or constraint in constraints (E.g. Alexander and Baptista, 2004, Krokhmal, portfolio allocation

Dynamic portfolio Palmquist and Uryasev, 2002). allocation

Conclusion Why? Better risk measure + convex function of portfolio Appendix weights (easier to optimize).

13 / 42 Min CVaR portfolio

Outline > library(DEoptim) A primer on risk budgets > obj <- function(w) { VaR budget

CVaR budget + if (sum(w) == 0) { w <- w + 1e-2 } Min CVaR portfolio + w <- w / sum(w) CVaR budgets as objective or constraint in + ES(R=indexes[,1:2],weights = w)$MES portfolio allocation

Dynamic portfolio + } allocation > out <- DEoptim(fn = obj, lower = rep(0, 2), Conclusion + upper = rep(1, 2), DEoptim.control(itermax=50)) Appendix > wstar <- out$optim$bestmem > wMinCVaR <- wstar / sum(wstar) > print(wMinCVaR) US Bonds US Equities 0.96443348 0.03556652 14 / 42 CVaR budgets

Outline > ES(R=indexes[,1:2], weights=wMinCVaR, A primer on risk budgets + portfolio_method="component") VaR budget

CVaR budget $MES Min CVaR portfolio [1,] 0.01102894 CVaR budgets as objective or constraint in $contribution portfolio allocation

Dynamic portfolio US Bonds US Equities allocation 0.0106366796 0.0003922610 Conclusion $pct_contrib_MES Appendix US Bonds US Equities 0.96443349 0.03556651

15 / 42 CVaR budgets

Outline Weight allocation Risk allocation A primer on risk budgets style bond equity bond equity VaR budget

CVaR budget 60/40 weight 0.40 0.6 -0.01 1.01 Min CVaR portfolio 60/40 risk alloc 0.84 0.16 0.40 0.60 CVaR budgets as objective or constraint in Min CVaR Conc 0.86 0.14 0.50 0.50 portfolio allocation Min CVaR 0.96 0.04 0.96 0.04 Dynamic portfolio allocation

Conclusion

Appendix

16 / 42 Outline

A primer on risk budgets

CVaR budgets as objective or constraint in portfolio allocation CVaR budgets as objective or Risk budget constraints

Risk budget objective

Efﬁcient Frontier constraint in portfolio

Dynamic portfolio allocation allocation Conclusion

Appendix

17 / 42 Traditional use

Outline ∂CVaR(w) Risk contribution CVaR A primer on risk budgets Ci (w)= wi ∂wi CVaR budgets as objective or constraint in portfolio allocation TM Risk budget constraints Litterman (1999): Hot Spots and hedges. Risk budgets as

Risk budget objective ex post instrument to adjust marginally portfolios. Efﬁcient Frontier

Dynamic portfolio allocation Keel and Ardia (2009):

Conclusion

Appendix 1. Only precise for inﬁnitesimal changes, poor approximations for realistic reallocations.

2. Assume changing a single position keeping ﬁxed all other positions >< full investment constraint.

18 / 42 Proposal I: Risk budget constraints

Outline Avoid downside risk concentration by: A primer on risk budgets

CVaR budgets as CiCVaR objective or constraint in li ≤ %CiCVaR ≡ ≤ ui portfolio allocation CVaR Risk budget constraints

Risk budget objective Efﬁcient Frontier Min CVaR portfolio with

Dynamic portfolio allocation X 60/40 risk allocation constraint Conclusion X Appendix equal risk allocation constraint.

19 / 42 60/40 risk allocation portfolio

Outline > obj <- function(w) { A primer on risk budgets + if (sum(w) == 0) { w <- w + 1e-2 } CVaR budgets as objective or constraint in + w <- w / sum(w) portfolio allocation

Risk budget constraints + CVaR <- ES(R=indexes[,1:2],weights = w) Risk budget objective + tmp1 <- CVaR$MES Efﬁcient Frontier

Dynamic portfolio + tmp2 <- max(CVaR$pct_contrib_ES allocation + - c(0.405, 0.605) , 0) Conclusion + tmp3 <- max(c(0.395, 0.595) - Appendix + CVaR$pct_contrib_ES , 0) + out <- tmp1 + 1e3 * tmp2 + 1e3 * tmp3 + }

20 / 42 60/40 risk allocation portfolio

Outline > out <- DEoptim(fn = obj, lower = rep(0, 2), A primer on risk budgets + upper = rep(1, 2), DEoptim.control(itermax=50)) CVaR budgets as objective or constraint in > wstar <- out$optim$bestmem portfolio allocation

Risk budget constraints > w6040riskalloc <- wstar / sum(wstar) Risk budget objective > print(w6040riskalloc) Efﬁcient Frontier

Dynamic portfolio US Bonds US Equities allocation 0.8382035 0.1617965 Conclusion

Appendix

21 / 42 60/40 risk allocation portfolio

Outline > ES(R=indexes[,1:2], weights=w6040riskalloc, A primer on risk budgets + portfolio_method="component") CVaR budgets as objective or constraint in $MES portfolio allocation

Risk budget constraints [1,] 0.01400341 Risk budget objective $contribution Efﬁcient Frontier

Dynamic portfolio US Bonds US Equities allocation 0.005671224 0.008332185 Conclusion $pct_contrib_MES Appendix US Bonds US Equities 0.4049888 0.5950112

22 / 42 Special case: Equal-risk portfolio

Outline Min CVaR with equal-risk constraint A primer on risk budgets

CVaR budgets as objective or constraint in %CiCVaR(w)=1/N (i =1,...,N) portfolio allocation

Risk budget constraints

Risk budget objective

Efﬁcient Frontier Dynamic portfolio In this portfolio: allocation wi ∂CVaR/∂wj Conclusion = . wj ∂CVaR/∂wi Appendix Downweights “hot spots”: positions for which a marginal

decrease in weight leads to a large reduction in CVaR.

23 / 42 Proposal II: Risk budget objective

Outline Avoid downside risk concentration by: A primer on risk budgets

CVaR budgets as objective or constraint in min max CiCVaR(w) portfolio allocation w i

Risk budget constraints

Risk budget objective

Efﬁcient Frontier

Dynamic portfolio allocation

Conclusion

Appendix

24 / 42 Proposal II: Risk budget objective

Outline Avoid downside risk concentration by: A primer on risk budgets

CVaR budgets as objective or constraint in min max CiCVaR(w) portfolio allocation w i

Risk budget constraints

Risk budget objective Efﬁcient Frontier Objective trades off Risk Minimization and Risk Diversiﬁcation, Dynamic portfolio allocation since:

Conclusion

Appendix max CiCVaR = CVaR max{%C1CVaR,..., %CN CVaR} i

24 / 42 Relation with equal-risk portfolio

Outline If the set of equal-risk portfolios is non-empty and the A primer on risk budgets

CVaR budgets as minimum CVaR concentration portfolio has a unique optimum. objective or constraint in portfolio allocation Then the minimum CVaR concentration portfolio criterion is Risk budget constraints equivalent to: Risk budget objective Efﬁcient Frontier min CVaR(w) w Dynamic portfolio allocation s.t. %C1CVaR = ... =%CN CVaR Conclusion

Appendix But computationally more simple and has also a solution if there is no equal-risk portfolio.

25 / 42 Min CVaR Concentration portfolio

Outline > obj <- function(w) { A primer on risk budgets + if (sum(w) == 0) { w <- w + 1e-2 } CVaR budgets as objective or constraint in + w <- w / sum(w) portfolio allocation

Risk budget constraints + CVaR <- ES(R=indexes[,1:2],weights = w) Risk budget objective + out <- max(CVaR$contribution) Efﬁcient Frontier

Dynamic portfolio + } allocation > out <- DEoptim(fn = obj, lower = rep(0, 2), Conclusion + upper = rep(1, 2),DEoptim.control(itermax=50)) Appendix > wstar <- out$optim$bestmem > wMinCVaRConc <- wstar / sum(wstar) > print(wMinCVaRConc) US Bonds US Equities 0.8584465 0.1415535 26 / 42 Min CVaR Concentration portfolio

Outline > ES(R=indexes[,1:2], weights=wMinCVaRConc, A primer on risk budgets + portfolio_method="component") CVaR budgets as objective or constraint in $MES portfolio allocation

Risk budget constraints [1,] 0.01315665 Risk budget objective $contribution Efﬁcient Frontier

Dynamic portfolio US Bonds US Equities allocation 0.006578325 0.006578323 Conclusion $pct_contrib_MES Appendix US Bonds US Equities 0.5000001 0.4999999

27 / 42 Overview portfolio allocations

Outline Weight allocation Risk allocation A primer on risk budgets style bond equity bond equity CVaR budgets as objective or constraint in 60/40 weight 0.40 0.6 -0.01 1.01 portfolio allocation

Risk budget constraints 60/40 risk alloc 0.84 0.16 0.40 0.60

Risk budget objective Min CVaR Conc 0.86 0.14 0.50 0.50 Efﬁcient Frontier Min CVaR 0.96 0.04 0.96 0.04 Dynamic portfolio allocation

Conclusion

Appendix

28 / 42 Adding a return target: Efﬁcient frontier

Outline Min CVaR Min CVaR Concentration Efficient frontier Efficient frontier A primer on risk budgets

0.0085 S&P 500 0.0085 S&P 500 CVaR budgets as objective or constraint in 0.0080 0.0080 portfolio allocation 0.0075 0.0075 Risk budget constraints

0.0070Monthly return 0.0070Monthly return Risk budget objective

0.0065 US bond 0.0065 US bond Efﬁcient Frontier

Dynamic portfolio 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 allocation Monthly CVaR (full) Monthly CVaR (full) Monthly CVaR concentration(dashed) Monthly CVaR concentration(dashed)

Conclusion Weight allocation Weight allocation 1.0 1.0 Appendix 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0

0.0065 0.0068 0.0071 0.0074 0.0077 0.008 0.0083 0.0067 0.007 0.0073 0.0076 0.0079 0.0082

Monthly return Monthly return US bond S&P 500 29 / 42 Outline

A primer on risk budgets

CVaR budgets as objective or constraint in portfolio allocation

Dynamic portfolio allocation

Dynamic strategies Dynamic portfolio allocation

Data

Conclusion

Appendix

30 / 42 Dynamic investment strategies

We consider the following investment strategies, quarterly Outline

A primer on risk budgets rebalancing, 4 assets:

CVaR budgets as objective or constraint in 1. Benchmark strategies: portfolio allocation Dynamic portfolio X allocation “Equal Weight”

Dynamic strategies X “Min CVaR” Data

Conclusion X “Min CVaR + 40% Position Limit”

Appendix X “Min CVaR + EW return target”

2. Strategies that use CVaR budgets:

X “Min CVaR + 40% Perc CVaR Alloc Limit”

X “Min CVaR Conc”

X “Min CVaR Conc + EW return target”. 31 / 42 Summary statistics data:

4 assets: Merrill Lynch US bond, S&P 500, MSCI EAFE and S&P GSCI Real monthly returns Jan 1976-December 2009, total return indices

US bond S&P 500 MSCI EAFE S&P GSCI Mean (in %) 0.32 0.52 0.39 0.10 StdDev (in %) 1.86 4.46 4.98 5.50 Historical 95% CVaR (in %) 3.64 10.64 12.46 13.58

Quarterly rebalanced based on time-varying conditional moment estimates (EWMA mean, GARCH volatility, rolling 8 year correlation, coskewness and cokurtosis). Weight allocation:

Min CVaR Min CVaR + CVaR Alloc Limit Min CVaR + Return Target

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3

Min CVaR Conc Equal Weight Min CVaR Conc + Return Target

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3

US bond S&P 500 EAFE GSCI CVaR allocation:

Min CVaR Min CVaR + CVaR Alloc Limit Min CVaR + Return Target

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3

Min CVaR Conc Equal Weight Min CVaR Conc + Return Target

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3 1984Q1 1988Q3 1993Q1 1997Q3 2002Q1 2006Q3

US bond S&P 500 EAFE GSCI Out of sample performance:

Equal Min CVaR Min CVaR Conc Weight Position CVaR Alloc Return Return Limit Limit Target Target Mean (in %) 0.40 0.44 0.43 0.43 0.47 0.43 0.43 StdDev (in %) 2.85 1.38 2.41 1.64 1.74 1.85 2.02 Hist 95% CVaR (in %) 6.95 2.72 5.78 3.49 3.59 4.11 4.52 Portfolio turnover (in %) 1.27 2.30 3.10 2.70 4.50 1.82 4.45

T∗− 1 1 Portf. turnover = X w(i)t+1 − w(i)t+ . NT∗ t=1 Out of sample cum performance, relative to min CVaR:

Equal Weight Min CVaR + Position Limit Min CVaR + CVaR Alloc Limit

1.2 1.2 1.2

1.0 1.0 1.0

0.8 0.8 0.8

0.6 0.6 0.6

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

Min CVaR + Return Target Min CVaR Conc Min CVaR Conc + Return Target

1.2 1.2 1.2

1.0 1.0 1.0

0.8 0.8 0.8

0.6 0.6 0.6

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 Drawdowns higher than 10% on portfolio strategies over the period January 1984-December 2009:

Equal Min CVaR Min CVaR Conc Weight Position CVaR Alloc Return Return Limit Limit Target Target ∗ Credit crisis 0.47 0.10 0.37 0.18 0.14 0.24 0.24 ∗∗ Dot-com bubble burst 0.28 0.19 0.11 0.11 ∗∗∗ Asian-Russian crisis 0.13 0.12 0.11 ∗∗∗∗ Black Monday 0.11 0.13 0.12 0.12

∗ Dec 2007-Oct 2008 for the Min CVaR strategy, Nov 2007-Feb 2009 for all other styles.

∗∗ Start: Sept 2000 for all styles. End: Jan 2002 for Min CVaR Conc styles, July 2002 for the Min CVaR with position limit style, September 2002 for all other styles.

∗∗∗ Aug 1997-Aug 1998 for equal-weight strategy, Oct 1997-Aug 1998 for the Min CVaR + Return target and Nov 1997-Aug 1998 for the Min CVaR with position limit style.

∗∗∗∗ Black Monday: Sept-November 1987. Outline

A primer on risk budgets

CVaR budgets as objective or constraint in portfolio allocation

Dynamic portfolio allocation Conclusion Conclusion

Appendix

38 / 42 Conclusion

Outline CVaR budgets are useful for:

A primer on risk budgets CVaR budgets as Ex post analysis of the portfolio risk allocation; objective or constraint in portfolio allocation Dynamic portfolio And input in the portfolio allocation strategy through allocation

Conclusion X minimum CVaR Concentration objective Appendix X and/or risk allocation constraints.

39 / 42 References

Outline Software: R packages DEoptim, PerformanceAnalytics and A primer on risk budgets

CVaR budgets as PortfolioAnalytics objective or constraint in portfolio allocation Related research papers: Dynamic portfolio allocation 1. With B. Peterson and C. Croux: Estimation and decomposition

Conclusion of downside risk for portfolios with non-normal returns. Journal Appendix of Risk, Winter 2008.

2. With B. Peterson: Component VaR for a non-normal world. RISK, November 2008.

3. With D. Ardia, P.Carl, K. Mullen and B. Peterson. DEoptim for non-convex portfolio optimization. SSRN.

4. With P. Carl and B. Peterson. Portfolio optimization with CVaR

budgets. 40 / 42 Outline

A primer on risk budgets

CVaR budgets as objective or constraint in portfolio allocation

Dynamic portfolio allocation Appendix Conclusion

Appendix

41 / 42 %CV aR(1) = f(w(1)) for bivariate normal portfolio:

µ µ ρ − σ σ µ µ ρ − σ σ µ µ ρ σ σ 1= 2=0, = 0.5, 1= 2=1 1=1, 2=0, = 0.5, 1= 2=1 1= 2=0, =−0.5, 1=1, 2=2

1.0 1.0 1.0 0.8 0.8

0.6 0.5 0.6 0.4 0.4

0.2 0.0 0.2 0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

µ µ ρ σ σ µ µ ρ σ σ µ µ ρ σ σ 1= 2=0, =0, 1= 2=1 1=1, 2=0, =0, 1= 2=1 1= 2=0, =0, 1=1, 2=2 1.0 1.0 1.0

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

µ µ ρ σ σ µ µ ρ σ σ µ µ ρ σ σ 1= 2=0, =0.5, 1= 2=1 1=1, 2=0, =0.5, 1= 2=1 1= 2=0, =0.5, 1=1, 2=2 1.0 1.0 1.0

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0.0 0.0 0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0