portfolio selection with multiple criteria

Portfolio Selection with Multiple Criteria

Bernhard [email protected]

Invesco Asset Management GmbHFrankfurt am Main

R in Finance, Chicago IL, 20 and 21 May 2016

Pfaff (Invesco) MCO RFinance 2016 1 / 24

Contents

1 Overview

2 Multi Criteria Optimization

3 Example: Multi-asset class portfolio

4 Summary

5 Bibliography


Overview

Overview

Definition of multiple (conflicting) criteria objectives with respect toportfolio optimization problems.

Concept of non-dominated solutions (Pareto efficiency).

Solutions/points can determined by:1 GA/EMO: elitist non-dominated sorting algorithms (e.g. NSGA-II)2 MCDM: (classical) optimization (e.g. weighted-sum method)

Applied to multi-asset portfolio with objectives for:1 Mean return,2 Total Covariance Risk,3 Diversification with respect to assets’ risk contributions.


Multi Criteria Optimization

Multi Criteria OptimizationProblem formulation

minimize fm(ω), m = 1, 2, . . . ,M;

subject to gj(ω) ≥ 0, j = 1, 2, . . . , J;

hk(ω) = 0, k = 1, 2, . . . ,K ;

ω(L)i ≤ ωi ≤ ω

(U)i , i = 1, 2, . . . , n.

Problem: M (conflicting) objective functions and n (constrained)variables.

A solution ω ∈ Ω is efficient (Pareto optimal or non-dominated) ifthere is no ω ∈ Ω such that fk(ω) ≤ fk(ω) for k = 1, . . . , p andfi (ω) < fi (ω) for some i ∈ 1, . . . , k.



Multi Criteria OptimizationGA/EMO: Pareto efficient solutions

Goal: find solutions which lie on Pareto-efficient front and encompassits entire range.

Solutions can be determined by genetic NSGA-II algorithm.

However, no guarantee in finding optimial points on (close to)frontier.

NSGA-II consists of the following steps (a) create random population,(b) selection (non-dominant/constraint-dominant), (c) applyingvariation (crossover, mutation), (d) elitism (crowding distances).

An implementation is provided in the R package mco (see Mersmann,2014).



Multi Criteria OptimizationMCDM: Weighted-sum method

minimizeM∑

m=1

λmfm(ω), with λm ≥ 0;

subject to gj(ω) ≥ 0, j = 1, 2, . . . , J;

hk(ω) = 0, k = 1, 2, . . . ,K ;

ω(L)i ≤ ωi ≤ ω

(U)i , i = 1, 2, . . . , n.

Scaling/normalization of objective functions is important.

Goals and/or satisfycing levels can be included.

Possibility of a hybrid approach, where some objectives are specifiedas ε-constraints.



Multi Criteria OptimizationSynopsis of portfolio applications

Computation of non-dominated surface in tri-criterion problems:Hirschberger et al. (2013).

Bi-criterion EMO problem formulation with discontinuities in searchspace (risk & return with (i) either zero or within bounds allocationsand (ii) cardinality constraint on count of included assets): Deb et al.(2011).

Tri-criterion problems (quad-lin-lin):

risk, return and& transaction costs: Steuer et al. (2005), Steuer et al.(2013).risk, return and& ESG index: Utz et al. (2015).


Example: Multi-asset class portfolio

Example: Multi-asset class portfolioSpecification

Data set MultiAsset contained in FRAPO: month-end data from11/2004 until 11/2011.

MCO conisting of three objectives:1 mean return,2 volatility, and3 dispersion of risk contributions.

Targeted return of 6% p.a.

Targeted volatility of 4% p.a.



Example: Multi-asset class portfolioR code: Initializing

> library(FRAPO)

> library(mco)

> ## Loading of data

> data(MultiAsset)

> Prices <- timeSeries(MultiAsset,

+ charvec = rownames(MultiAsset))

> NAssets <- ncol(Prices)

> R <- returns(Prices, method = "discrete", percentage = TRUE)

> ## Defining parameters

> TargetRpa <- 6 ## percentage p.a.

> TargetR <- 100 * ((1 + TargetRpa / 100)^(1 / 12) - 1)

> TargetVol <- 4 ## percentage p.a.

> l <- rep(1, 3) ## goal weighting

> WeightedSum <- FALSE

> mu <- colMeans(R)

> S <- cov(R)



Example: Multi-asset class portfolioR code: Multiple criteria objective and budget constraint

> f <- function(x)

+ y <- numeric(3)

+ y[1] <- -1.0 * l[1] * drop(crossprod(x, mu)) / TargetR

+ y[2] <- l[2] * drop(sqrt(t(x) %*% S %*% x)) *

+ sqrt(12) / TargetVol

+ y[3] <- l[3] * sum((mrc(x, S) / 100)^2)

+ if(WeightedSum)

+ return(sum(y))

+ else

+ return(y)

+

+

> g <- function(x)

+ c(1.01 - sum(x), sum(x) - 0.99)

+



Example: Multi-asset class portfolioR code: Determining Pareto efficient solutions

> ans <- nsga2(f, NAssets, 3,

+ lower.bounds = rep(0, NAssets),

+ upper.bounds = rep(1, NAssets),

+ constraints = g, cdim = 2, popsize = 500)

> names(ans)

[1] "par" "value" "pareto.optimal"

> ## Preparing objective values for graphics

> mco <- data.frame(ans$value[ans$pareto.optimal, ])

> mco[, 1] <- ((1 + (-1.0 * mco[, 1] * TargetR) / 100)^12

+ - 1.0) * 100

> mco[, 2] <- mco[, 2] * TargetVol

> colnames(mco) <- c("Return", "Risk", "Diversification")



Example: Multi-asset class portfolioR code: 3D scatterplot

> library(scatterplot3d)

> scatterplot3d(mco,

+ main = "Pareto Efficient Solutions",

+ sub = "Pareto Frontier (Surface)",

+ xlab = "Return Objective",

+ ylab = "Risk Objective",

+ zlab = "Dispersion of MRC",

+ angle = 15,

+ highlight.3d = FALSE,

+ box = TRUE,

+ color = "steelblue",

+ pch = 19, type = "p",

+ cex.symbols = 0.6)

Pareto Efficient Solutions

Pareto Frontier (Surface)

4 6 8 10 12 14 16 18 20

0.0

0.2

0.4

0.6

0.8

1.0

2 4

6 8

1012

1416

Return Objective

Ris

k O

bjec

tive

Dis

pers

ion

of M

RC



Example: Multi-asset class portfolioR code: image plot

> library(akima)

> library(fields)

> s <- interp(mco[, 2], mco[, 1], mco[, 3],

+ xo = seq(min(mco[, 2]), max(mco[, 2]), length = 100),

+ yo = seq(min(mco[, 1]), max(mco[, 1]), length = 100),

+ duplicate = "mean"

+ )

> par(mar = c(5, 6, 5, 6))

> image.plot(s, nlevel = 50,

+ main = "Image plot of efficient set",

+ legend.lab = "Dispersion of MRC",

+ xlab = "Risk Objective",

+ ylab = "Return Objective",

+ legend.mar = 4,

+ horizontal = TRUE,

+ legend.shrink = 0.7,

+ col = topo.colors(50))

> contour(s, add = TRUE, nlevels = 20, labcex = 0.8)

> points(mco[, 2], mco[, 1], pch = 18, cex = 0.4, col = "orange")

4 6 8 10 12 14

68

1012

1416

18

Image plot of efficient set

Risk ObjectiveR

etur

n O

bjec

tive

0.2 0.4 0.6 0.8Dispersion of MRC

0.15

0.2 0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.8



Example: Multi-asset class portfolioR code: weighting of objectives

grid <- expand.grid(x = seq(0.05, 0.95, by = 0.05),

y = seq(0.05, 0.95, by = 0.05))

grid <- grid[which(rowSums(grid) <= 1.0), ]

wobj <- as.matrix(cbind(grid, 1 - rowSums(grid)),

nrow = nrow(grid), ncol = 3)

W <- matrix(NA, nrow = nrow(wobj), ncol = NAssets)

WeightedSum <- TRUE

IneqA <- matrix(1, nrow = 1, ncol = NAssets)

ew <- rep(1 / NAssets, NAssets)

library(fPortfolio) ## for donlp2NLP

for(i in 1:nrow(wobj))

l <- c(wobj[i, ])

W[i, ] <- donlp2NLP(start = ew, objective = f,

par.lower = rep(0, NAssets),

ineqA = IneqA, ineqA.lower = 1.0,

ineqA.upper = 1.0)$solution



Example: Multi-asset class portfolioR code: weighting of objectives & ternary plot

> library(PerformanceAnalytics)

> library(ggtern) ## Wahrschau! version < 2.0.1

> Es95Mod <- apply(W, 1, function(x)

+ r <- timeSeries(R %*% x / 100, time(R))

+ -100 * ES(r)

+ )

> terndat <- data.frame(cbind(wobj, Es95Mod))

> colnames(terndat) <- c("x", "y", "z", "value")

> ## Theme for ternary plot

> terntheme <- function()

+ list(theme_rgbg(),

+ theme(legend.position = c(0, 1),

+ legend.justification = c(0, 1),

+ plot.margin=unit(c(0, 2,0, 2), "cm"))

+ )

+



Example: Multi-asset class portfolioR code: ternary plot, cont’d

> ## ternary plot

> ggtern(terndat, aes(x = x, y = y, z = z, value = value)) +

+ geom_interpolate_tern(aes(value = value, color = ..level..),

+ binwidth = 1.0) +

+ terntheme() +

+ theme_hidegrid_minor() +

+ theme_showgrid_major() +

+ Lline(0.2, color = "blue", linetype = 2) + ## x

+ Tline(0.3, colour = "red2", linetype = 2) + ## y

+ Rline(0.5, color = "brown", linetype = 2) + ## z

+ scale_color_gradient(low = "green", high = "red") +

+ labs(x = "Return", y = "Risk", z = "MRC",

+ title = "Ternary Plot with ES Contour Lines",

+ color = "Level")

100

80

60

40

20

100

80 60 40 20

100

80

60

40

20

Risk

Return MRC

Ris

k

Return

MR

C

0

4

8

12

16Level

Ternary Plot with ES Contour Lines



Example: Multi-asset class portfolioR code: backtest, part I

> library(cccp) ## for ERC portolio

> ## backtest, extending window

> ep <- time(R)[-c(1:59)]

> bs <- length(ep)

> sp <- rep(start(R), bs)

> ## initialising object

> Wmco <- matrix(NA, nrow = bs, ncol = NAssets)

> Wmsr <- Wmdp <- Wgmv <- Werc <- Wmco

> l <- c(0.2, 0.1, 0.7) ## goal weighting



Example: Multi-asset class portfolioR code: backtest, part II

> for(i in 1:bs)

+ rdat <- window(R, start = sp[i], end = ep[i])

+ mu <- colMeans(rdat)

+ S <- cov(rdat)

+ Wmco[i, ] <- donlp2NLP(start = ew, objective = f,

+ par.lower = rep(0, NAssets),

+ ineqA = IneqA, ineqA.lower = 1.0,

+ ineqA.upper = 1.0)$solution

+ ans <- tangencyPortfolio(rdat)

+ Wmsr[i, ] <- getWeights(ans)

+ ans <- PMD(rdat)

+ Wmdp[i, ] <- FRAPO::Weights(ans) / 100

+ ans <- PGMV(rdat)

+ Wgmv[i, ] <- FRAPO::Weights(ans) / 100

+ ans <- rp(ew, S, ew, optctrl = ctrl(trace = FALSE))

+ Werc[i, ] <- c(getx(ans))

+

Iteration: 0

pobj: 0.126801

dobj: -0.96263

pinf: 2.22045e-016

dinf: 3.83253

dgap: 1.08943

Iteration: 1

pobj: 0.124852

dobj: 0.00264633

pinf: 2.22045e-016

dinf: 0.427881

dgap: 0.122206

Iteration: 2

pobj: 0.118802

dobj: 0.0835599

pinf: 4.44089e-016

dinf: 0.111199

dgap: 0.0352419

Iteration: 3

pobj: 0.113548

dobj: 0.0993458

pinf: 7.77156e-016

dinf: 0.0372481

dgap: 0.014202

Iteration: 4

pobj: 0.107482

dobj: 0.104089

pinf: 1.55431e-015

dinf: 1.66533e-016

dgap: 0.00339286

Iteration: 5

pobj: 0.105591

dobj: 0.104892

pinf: 2.9976e-015

dinf: 2.23773e-016

dgap: 0.000698946

Iteration: 6

pobj: 0.105115

dobj: 0.105007

pinf: 5.77316e-015

dinf: 1.8619e-016

dgap: 0.000108626

Iteration: 7

pobj: 0.105047

dobj: 0.105027

pinf: 1.13243e-014

dinf: 1.00074e-016

dgap: 1.98598e-005

Iteration: 8

pobj: 0.105036

dobj: 0.105032

pinf: 2.23155e-014

dinf: 1.90283e-016

dgap: 4.3372e-006

Iteration: 9

pobj: 0.105034

dobj: 0.105033

pinf: 4.44089e-014

dinf: 1.24127e-016

dgap: 8.77542e-007

Iteration: 10

pobj: 0.105033

dobj: 0.105033

pinf: 8.82627e-014

dinf: 1.64204e-016

dgap: 1.12839e-007

Iteration: 11

pobj: 0.105033

dobj: 0.105033

pinf: 1.72196e-013

dinf: 1.94289e-016

dgap: 8.26087e-009

Optimal solution found.

Iteration: 0

pobj: 0.362971

dobj: -0.826572

pinf: 3.36575

dinf: 5.07688

dgap: 17.2127

Iteration: 1

pobj: 0.419846

dobj: -0.610122

pinf: 0.352087

dinf: 0.531087

dgap: 2.61477

Iteration: 2

pobj: 0.422811

dobj: -0.146063

pinf: 0.0836643

dinf: 0.126199

dgap: 0.888119

Iteration: 3

pobj: 0.381137

dobj: 0.149456

pinf: 0.0151466

dinf: 0.0228471

dgap: 0.271697

Iteration: 4

pobj: 0.332313

dobj: 0.250453

pinf: 0.00208538

dinf: 0.00314558

dgap: 0.0846849

Iteration: 5

pobj: 0.313591

dobj: 0.288708

pinf: 0.000394894

dinf: 0.000595657

dgap: 0.0251527

Iteration: 6

pobj: 0.303817

dobj: 0.2957

pinf: 6.39925e-005

dinf: 9.65262e-005

dgap: 0.00815032

Iteration: 7

pobj: 0.299024

dobj: 0.297668

pinf: 3.49767e-006

dinf: 5.27588e-006

dgap: 0.00135742

Iteration: 8

pobj: 0.297976

dobj: 0.297794

pinf: 3.25552e-007

dinf: 4.91062e-007

dgap: 0.000182128

Iteration: 9

pobj: 0.297814

dobj: 0.297802

pinf: 1.91646e-008

dinf: 2.89079e-008

dgap: 1.23223e-005

Iteration: 10

pobj: 0.297803

dobj: 0.297802

pinf: 9.70921e-010

dinf: 1.46454e-009

dgap: 6.32737e-007

Iteration: 11

pobj: 0.297802

dobj: 0.297802

pinf: 4.85793e-011

dinf: 7.32765e-011

dgap: 3.16812e-008


Iteration: 0

pobj: 0.123721

dobj: -0.963402

pinf: 1.11022e-016

dinf: 3.82468

dgap: 1.08712

Iteration: 1

pobj: 0.121862

dobj: 0.00123752

pinf: 2.22045e-016

dinf: 0.422508

dgap: 0.120624

Iteration: 2

pobj: 0.115814

dobj: 0.0810899

pinf: 2.22045e-016

dinf: 0.109938

dgap: 0.0347246

Iteration: 3

pobj: 0.11028

dobj: 0.0972651

pinf: 4.44089e-016

dinf: 0.0338733

dgap: 0.0130149

Iteration: 4

pobj: 0.104565

dobj: 0.101339

pinf: 8.88178e-016

dinf: 0.000902096

dgap: 0.00322598

Iteration: 5

pobj: 0.102671

dobj: 0.102068

pinf: 1.77636e-015

dinf: 5.70661e-005

dgap: 0.000603327

Iteration: 6

pobj: 0.102259

dobj: 0.102169

pinf: 3.44169e-015

dinf: 2.69901e-006

dgap: 9.01822e-005

Iteration: 7

pobj: 0.102204

dobj: 0.102187

pinf: 6.77236e-015

dinf: 1.28266e-007

dgap: 1.73152e-005

Iteration: 8

pobj: 0.102194

dobj: 0.10219

pinf: 1.33227e-014

dinf: 2.63174e-009

dgap: 3.7038e-006

Iteration: 9

pobj: 0.102191

dobj: 0.102191

pinf: 2.65343e-014

dinf: 4.96533e-011

dgap: 6.94795e-007

Iteration: 10

pobj: 0.102191

dobj: 0.102191

pinf: 5.19584e-014

dinf: 1.92505e-012

dgap: 7.89682e-008


Iteration: 0

pobj: 0.35774

dobj: -0.832046

pinf: 3.3757

dinf: 5.0986

dgap: 17.3336

Iteration: 1

pobj: 0.415478

dobj: -0.61496

pinf: 0.362862

dinf: 0.548061

dgap: 2.67242

Iteration: 2

pobj: 0.420214

dobj: -0.158925

pinf: 0.0885245

dinf: 0.133706

dgap: 0.921424

Iteration: 3

pobj: 0.379027

dobj: 0.142668

pinf: 0.0154915

dinf: 0.0233981

dgap: 0.278041

Iteration: 4

pobj: 0.328734

dobj: 0.245164

pinf: 0.00211642

dinf: 0.00319662

dgap: 0.0865185

Iteration: 5

pobj: 0.308997

dobj: 0.284874

pinf: 0.000357781

dinf: 0.000540387

dgap: 0.0243708

Iteration: 6

pobj: 0.299609

dobj: 0.291457

pinf: 6.58067e-005

dinf: 9.93934e-005

dgap: 0.00818694

Iteration: 7

pobj: 0.294797

dobj: 0.2934

pinf: 4.03528e-006

dinf: 6.09483e-006

dgap: 0.00139899

Iteration: 8

pobj: 0.293728

dobj: 0.29354

pinf: 3.8613e-007

dinf: 5.83205e-007

dgap: 0.00018854

Iteration: 9

pobj: 0.293562

dobj: 0.293549

pinf: 2.28877e-008

dinf: 3.45692e-008

dgap: 1.27549e-005

Iteration: 10

pobj: 0.293551

dobj: 0.29355

pinf: 1.16015e-009

dinf: 1.75227e-009

dgap: 6.54844e-007

Iteration: 11

pobj: 0.29355

dobj: 0.29355

pinf: 5.80487e-011

dinf: 8.76747e-011

dgap: 3.27877e-008


Iteration: 0

pobj: 0.123384

dobj: -0.962439

pinf: 1.11022e-016

dinf: 3.81849

dgap: 1.08582

Iteration: 1

pobj: 0.121607

dobj: 0.00201636

pinf: 1.11022e-016

dinf: 0.418756

dgap: 0.11959

Iteration: 2

pobj: 0.115834

dobj: 0.081482

pinf: 2.22045e-016

dinf: 0.108843

dgap: 0.0343518

Iteration: 3

pobj: 0.110506

dobj: 0.0978079

pinf: 3.33067e-016

dinf: 0.0329326

dgap: 0.0126977

Iteration: 4

pobj: 0.10513

dobj: 0.10191

pinf: 6.66134e-016

dinf: 0.00104159

dgap: 0.0032198

Iteration: 5

pobj: 0.103326

dobj: 0.102697

pinf: 1.33227e-015

dinf: 6.32514e-005

dgap: 0.000629017

Iteration: 6

pobj: 0.10292

dobj: 0.102822

pinf: 2.66454e-015

dinf: 2.75908e-006

dgap: 9.83016e-005

Iteration: 7

pobj: 0.102866

dobj: 0.102848

pinf: 5.21805e-015

dinf: 1.30988e-007

dgap: 1.79689e-005

Iteration: 8

pobj: 0.102859

dobj: 0.102856

pinf: 1.03251e-014

dinf: 2.35916e-009

dgap: 3.40098e-006

Iteration: 9

pobj: 0.102859

dobj: 0.102858

pinf: 2.02061e-014

dinf: 7.24439e-011

dgap: 5.93075e-007

Iteration: 10

pobj: 0.102859

dobj: 0.102859

pinf: 3.9746e-014

dinf: 3.26919e-012

dgap: 5.96908e-008


Iteration: 0

pobj: 0.354538

dobj: -0.835719

pinf: 3.37699

dinf: 5.08478

dgap: 17.2936

Iteration: 1

pobj: 0.410595

dobj: -0.619505

pinf: 0.363421

dinf: 0.547207

dgap: 2.66966

Iteration: 2

pobj: 0.415523

dobj: -0.163452

pinf: 0.0886577

dinf: 0.133493

dgap: 0.920655

Iteration: 3

pobj: 0.37489

dobj: 0.138624

pinf: 0.0155736

dinf: 0.0234493

dgap: 0.278065

Iteration: 4

pobj: 0.324781

dobj: 0.240768

pinf: 0.00213858

dinf: 0.00322009

dgap: 0.0869943

Iteration: 5

pobj: 0.304854

dobj: 0.281612

pinf: 0.000325256

dinf: 0.000489741

dgap: 0.0234594

Iteration: 6

pobj: 0.295994

dobj: 0.28797

pinf: 6.22552e-005

dinf: 9.37384e-005

dgap: 0.00805627

Iteration: 7

pobj: 0.291318

dobj: 0.289908

pinf: 3.5414e-006

dinf: 5.33232e-006

dgap: 0.00141155

Iteration: 8

pobj: 0.290252

dobj: 0.29005

pinf: 3.48668e-007

dinf: 5.24993e-007

dgap: 0.000202496

Iteration: 9

pobj: 0.290074

dobj: 0.29006

pinf: 2.12652e-008

dinf: 3.20192e-008

dgap: 1.45576e-005

Iteration: 10

pobj: 0.290061

dobj: 0.29006

pinf: 1.08233e-009

dinf: 1.62968e-009

dgap: 7.5469e-007

Iteration: 11

pobj: 0.290061

dobj: 0.29006

pinf: 5.41673e-011

dinf: 8.15601e-011

dgap: 3.78077e-008


Iteration: 0

pobj: 0.123232

dobj: -0.961476

pinf: 2.22045e-016

dinf: 3.81415

dgap: 1.08471

Iteration: 1

pobj: 0.121498

dobj: 0.00278445

pinf: 2.22045e-016

dinf: 0.415683

dgap: 0.118714

Iteration: 2

pobj: 0.11584

dobj: 0.0819563

pinf: 6.66134e-016

dinf: 0.107489

dgap: 0.0338839

Iteration: 3

pobj: 0.110571

dobj: 0.0978265

pinf: 9.99201e-016

dinf: 0.0331768

dgap: 0.0127444

Iteration: 4

pobj: 0.105146

dobj: 0.102043

pinf: 1.9984e-015

dinf: 0.000770381

dgap: 0.00310324

Iteration: 5

pobj: 0.103416

dobj: 0.102824

pinf: 3.88578e-015

dinf: 4.96563e-005

dgap: 0.000591971

Iteration: 6

pobj: 0.103052

dobj: 0.102966

pinf: 7.54952e-015

dinf: 2.50231e-006

dgap: 8.59119e-005

Iteration: 7

pobj: 0.103009

dobj: 0.102999

pinf: 1.46549e-014

dinf: 1.30899e-007

dgap: 1.08648e-005

Iteration: 8

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Example: Multi-asset class portfolioR code: backtest, part III

> W <- list("MCO" = Wmco, "MSR" = Wmsr, "MDP" = Wmdp,

+ "GMV" = Wgmv, "ERC" = Werc)

> E <- lapply(W, function(x)

+ wTs <- timeSeries(x, charvec = ep)

+ wTsL1 <- lag(wTs, 1)

+ RetFac <- 1 + rowSums(R[ep, ] * wTsL1) / 100.0

+ RetFac[1] <- 100

+ timeSeries(cumprod(RetFac), charvec = ep)

+ )

> cols <- topo.colors(6)

> plot(E[[1]], lwd = 2,

+ ylab = "Index", xlab = "", col = cols[1],

+ main = "Comparison of Allocation Strategies")

> lines(E[[2]], col = cols[2])




> legend("topleft",

+ legend = c("MCO", "MSR", "MDP", "GMW", "ERC"),

+ col = cols, lty = 1, lwd = 2)

> abline(h = 100, col = "gray")

Comparison of Allocation Strategies

Inde

x

2010−01−01 2010−07−01 2011−01−01 2011−07−01

100

102

104

106

108

110

112

MCOMSRMDPGMWERC



Example: Multi-asset class portfolioR code: backtest, part IV

> Rstrat <- matrix(unlist(lapply(E, Return.calculate)), ncol = 5)

> RstratTs <- na.omit(xts(Rstrat, order.by = as.Date(ep)))

> Bench <- xts(rep(0, nrow(RstratTs)), order.by = as.Date(ep)[-1])

> S1 <- as.matrix(table.AnnualizedReturns(RstratTs, Rf = Bench,

+ scale = 12))

> S2 <- VaR(RstratTs)

> ans <- rbind(S1, -100 * S2)

> colnames(ans) <- c("MCO", "MSR", "MDP", "GMV", "ERC")

> rownames(ans) <- c("Return (p.a.)", "StdDev. Risk (p.a.)",

+ "Sharpe Ratio", "VaR (p.a.)")

> round(ans, 3)

MCO MSR MDP GMV ERC

Return (p.a.) 0.061 0.060 0.058 0.051 0.060

StdDev. Risk (p.a.) 0.038 0.039 0.037 0.034 0.034

Sharpe Ratio 1.605 1.532 1.585 1.513 1.746

VaR (p.a.) 1.301 1.380 1.308 1.211 1.017


Summary

Summary

Aiding decision makers by making portfolio choices for conflictingobjectives (a posteriori analysis).

Allows amendment of classical portfolio optimization formulations(e.g. GMV, ERC, MDP and/or MSR) by additional goals.

MCDM: For tri-criterion formulations, depiction of solutions by afourth portfolio characteristic/measure is feasible by means of ternaryplots.

Caveat/strength of EMO: It is at the user’s discretion to chose his’optimal’ allocation out of the Pareto efficient set, which might be achallenge on its own.


Bibliography

Bibliography I

Akima, H. and A. Gebhardt (2015). akima: Interpolation of Irregularly andRegularly Spaced Data. R package version 0.5-12.

Deb, K. (2001). Multi-Objective Optimization using Evolutionary Algorithms.Chichester, UK: John Wiley & Sons, Ltd.

Deb, K. (2011). Multi-objective Evolutionary Optimisation for Product Designand Manufacturing, Chapter Multi-Objective Optimization Using EvolutionaryAlgorithms: An Introduction, pp. 3–34. New York, NY: Springer.

Deb, K., R. Steuer, R. Tewari, and R. Tewari (2011). On the effectiveness of ansga-ii local search approach customized for portfolio optimization. KanGALReport 2011007, Indian Institute of Technology Kanpur, Kanpur, India.

Ehrgott, M. (2005). Multicriteria Optimization (second ed.). Berlin, Heidelberg,New York: Springer.

Goldberg, D. (1989). Genetic Algorithms in Search, Optimization, and MachineLearning. New York, NY: Addison-Wesley.


Bibliography

Bibliography II

Hamilton, N. (2015). ggtern: An Extension to ’ggplot2’, for the Creation ofTernary Diagrams. R package version 1.0.6.1.

Hirschberger, M., R. Steuer, S. Utz, and M. Wimmer (2013, January–February).Computing the nondominated surface in tri-criterion portfolio selection.Operations Research 61(1), 169–183.

Jones, D. and M. Tamiz (2010). Pratical Goal Programming. International Seriesin Opertations Research and Management Science. New York: Springer.

K., D., A. Pratap, S. Agarwal, and T. Meyarivan (2002). A fast and elitistmultiobjective genetic algorithm: Nsga-ii. IEEE Transaction on EvolutionaryComputation 6(2), 181–197.

Ligges, U. and M. Machler (2003). Scatterplot3d - an r package for visualizingmultivariate data. Journal of Statistical Software 8(11), 1–20.

Mersmann, O. (2014). mco: Multiple Criteria Optimization Algorithms andRelated Functions. R package version 1.0-15.1.

Mitchell, M. (1996). An Introduction to Genetic Algorithms. Cambridge, MA:MIT Press.


Bibliography

Bibliography III

Nychka, D., R. Furrer, J. Paige, and S. Sain (2016). fields: Tools for SpatialData. R package version 8.3-6.

Peterson, B. and P. Carl (2014). PerformanceAnalytics: Econometric tools forperformance and risk analysis. R package version 1.4.3541.

Pfaff, B. (2013). Financial Risk Modelling and Portfolio Optimisation with R.Chichester, UK: John Wiley & Sons, Ltd.

Steuer, R., Y. Qi, and M. Hirschberger (2005, June). Multiple objectives inportfolio selection. Journal of Financial Decision Making 1(1).

Steuer, R., M. Wimmer, and M. Hirschberger (2013, February). Overviewing thetransition of markowitz bi-criterion portfolio selection to tri-criterion portfolioselection. Journal of Business Economics 83(1), 61–85.

Team, R. C., D. Wuertz, T. Setz, and Y. Chalabi (2014). fPortfolio: Rmetrics -Portfolio Selection and Optimization. R package version 3011.81.

Utz, S., M. Wimmer, and R. Steuer (2015, October). Tri-criterion modeling forconstructing more-sustainable mutual funds. European Journal of OperationalResearch 246(1), 331–338.


portfolio selection with multiple criteria

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