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Portfolio Selection: Recent ApproachesOptimization and Design
with R
Bernhard [email protected]
Invesco Asset Management Deutschland GmbH, Frankfurt am Main
7th R/Rmetrics Meielisalp WorkshopJune 30 – July 4, 2013,
Meielisalp, Lake Thune Switzerland
Pfaff (Invesco) Portfolio R/Rmetrics 1 / 53
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Contents
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
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Overview
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
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Overview
Overview
Seminal work by Markowitz (1952), i.e., ‘Modern Portfolio
Theory’.
Since then, advances in terms of1 estimators for population
parameters.2 optimization methods.
In general:return-risk space 6= mean-variance space.Purpose of
this talk: Selective survey of more recent portfoliooptimization
techniques and how these can be utilized in R.
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R Resources
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
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R Resources
R Resources IKnowing your friends
Solver-related R packages:DEoptim (Mullen et al., 2011), glpkAPI
(Gelius-Dietrich, 2012),limSolve (Soetaert et al., 2009), linprog
(Henningsen, 2010), lpSolve(Berkelaar, 2011), lpSolveAPI (Konis,
2011), quadprog (Turlach andWeingessel, 2011), RcppDE
(Eddelbuettel, 2012), Rglpk (Theussl andHornik, 2012), rneos
(Pfaff, 2011), Rsocp (Chalabi and Würtz, 2010),Rsolnp (Ghalanos
and Theussl, 2012; Ye, 1987), Rsymphony (Harteret al., 2012)
Portfolio-related R packages:fPortfolio (Würtz et al., 2010a),
fPortfolioBacktest (Würtz et al.,2010b), FRAPO (Pfaff, 2012),
parma (Ghalanos, 2013),PerformanceAnalytics (Carl et al., 2012),
PortfolioAnalytics (Boudtet al., 2011b), rportfolios (Novomestky,
2012), tawny (Rowe, 2012)
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R Resources
R Resources IIKnowing your friends
This should be viewed as a ‘selective’ summary of R packages,
thereare more! Hence, check CRAN Task Views on ‘Finance’
and‘Optimization’ and R-Forge for what is available else and for
recentadditions.
In a nutshell: All kind of portfolio optimization tasks can be
accomplishedfrom/within R.
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Risk-Parity/Budget
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
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Risk-Parity/Budget
Risk-Parity/BudgetMotivation
Characteristic: Diversification directly applied to the
portfolio riskitself.
Motivation: Empirical observation that the risk contributions
are agood predictor for actual portfolio losses. Hence, portfolio
losses canpotentially be limited compared to an allocation which
witnesses ahigh risk concentration on one or a few portfolio
constituents.
Risk concepts:1 Volatility-based, i.e. standard deviation (see
Qian, 2005, 2006, 2011;
Maillard et al., 2009, 2010)2 Downside-based, i.e., CVaR/ES (see
Boudt et al., 2007, 2008; Peterson
and Boudt, 2008; Boudt et al., 2010, 2011a; Ardia et al.,
2010),budgeting (BCC) or min-max (MRC).
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Risk-Parity/Budget
Risk-Parity/BudgetProblem Delineation
Starting point general definition of risk contribution:
CiMω∈Ω = ωi∂Mω∈Ω∂ωi
(1)
whereby Mω∈Ω signify a linear homogeneous risk measure and ωi
isthe weight of the i-th asset.
For volatility-based risk measure:
∂σ(ω)
∂ωi=ωiσ
2i +
∑Ni 6=j ωjσij
σ(ω)(2)
For downside-based risk measure:
Ci CVaRω∈Ω,α = ωi
[µi +
(Σω)i√ω′Σω′
φ(zα)
α
](3)
whereby α signify the confidence level pertinent to the downside
risk.
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Risk-Parity/Budget
Risk-Parity/BudgetExample Risk-Parity vs GMV: R Code
> library(FRAPO)
> library(Rsolnp)
> ## Loading data and computing returns
> data(MultiAsset)
> R ## GMV
> wGmvAll ## ERC for all assets
> SigmaAll wErcAll ## Two-step, by asset class
> SigmaEq wErcEq rEq SigmaBd wErcBd rBd rAsset SigmaCl wErcCl
wErcTwoStage ## comparing the two approaches
> W ## concentration measure
> Concentration
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Risk-Parity/Budget
Risk-Parity/BudgetExample Risk-Parity vs GMV: Results
Assets GmvAll ErcAll ErcTwoStage
EquityS&P 500 4.55 3.72 3.06Russell 3000 0.00 3.59 2.92DAX
4.69 3.47 2.57FTSE 100 0.00 4.12 3.45Nikkei 225 1.35 3.38 2.68MSCI
EM 0.00 2.14 1.92∑
ωEquityi
10.59 20.43 16.60
BondUS Treasury 0.00 16.42 18.57German REX 88.72 42.44 31.97UK
Gilts 0.40 15.93 21.37∑
ωBondi 89.12 74.79 71.91
CommodityGold 0.29 4.78 11.49
Concentration∑ω2i 0.79 0.24 0.20
Table: ERC vs GMV Allocation
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Risk-Parity/Budget
Risk-Parity/BudgetExample BCC and MRC vs GMV: R Code
library(PortfolioAnalytics)
## Defining constraints and objective for CVaR budget
C1
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Risk-Parity/Budget
Risk-Parity/BudgetExample BCC and MRC vs GMV: R Code
Assets Weights Risk-ContributionsGMV ERC BCC MCC GMV ERC BCC
MCC
S&P 500 4.55 3.72 5.84 2.02 9.83 16.63 12.73 6.53Russell
3000 0.00 3.59 2.42 1.01 0.00 16.80 5.57 3.30DAX 4.69 3.47 10.74
1.01 12.19 14.34 18.98 3.20FTSE 100 0.00 4.12 15.85 4.04 0.00 11.20
19.94 10.10Nikkei 225 1.35 3.38 2.90 1.01 3.16 22.36 8.99 4.12MSCI
EM 0.00 2.14 5.72 1.01 0.00 14.22 18.65 5.36US Treasury 0.00 16.42
15.45 18.18 0.00 5.40 2.31 18.88German REX 88.72 42.44 18.11 66.67
74.75 −17.60 −3.42 39.61UK Gilts 0.40 15.93 13.95 1.01 0.50 5.00
0.49 1.18Gold 0.29 4.78 9.03 4.04 −0.43 11.63 15.78 7.70
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Optimal Draw Down
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
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Optimal Draw Down
Optimal Draw DownDefinition
The draw-down of a portfolio at time t is defined as the
difference betweenthe maximum uncompounded portfolio value prior to
t and its value attime period t. More formally, let W (ω, t) = y′tω
signify theuncompounded portfolio value at time t and ω are the
portfolio weightsfor the N assets included in it and yt the
cumulated returns, then thedraw-down, D(ω, t), is defined as:
D(ω, t) = max0≤τ≤t
{W (ω, τ)} −W (ω, t) (4)
The draw down is as such a functional risk measure.
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Optimal Draw Down
Optimal Draw DownProblem Formulations
With respect to portfolio optimization, the following
problemformulations have been introduced by Chekhlov et al. (2000,
2003,2005):
1 Maximum draw down (MaxDD)2 Average draw down (AvDD)3
Conditional draw down at risk (CDaR)
The three portfolio optimization approaches can be formulated as
alinear program (maximizing average annualized returns and
drawdowns are included as constraints).
Implemented in package FRAPO as functions PMaxDD(), PAveDD()and
PCDaR(), respectively.
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Optimal Draw Down
Optimal Draw DownLP: Maximum Draw Down
The linear program for the MaxDD is given as:
PMaxDD = arg maxω∈Ω,u∈R
R(ω) =1
dCy′Tω
uk − y′kω ≤ ν1Cuk ≥ y′kωuk ≥ uk−1u0 = 0
(5)
whereby the maximum allowed draw down in nominal terms is
defined as afraction of the available capital/wealth (ν1C ) and u
signify a (T + 1× 1)vector of slack variables in the program
formulation, i.e., the maximumportfolio values up to time period k
with 1 ≤ k ≤ T .
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Optimal Draw Down
Optimal Draw DownLP: Average Draw Down
Similarly, the linear program for the AveDD is given as:
PAvDD = arg maxω∈Ω,u∈R
R(ω) =1
dCy′Tω
1
T
T∑k=1
(uk − y′kω
)≤ ν2C
uk ≥ y′kωuk ≥ uk−1u0 = 0
(6)
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Optimal Draw Down
Optimal Draw DownLP: Conditional Draw Down at Risk
PCDaR = arg maxω∈Ω,u∈R,z∈R,ζ∈R
R(ω) =1
dCy′Tω
ζ +1
(1− α)T
T∑k=1
zk ≤ ν3C
zk ≥ uk − y′kω − ζzk ≥ 0uk ≥ y′kωuk ≥ uk−1u0 = 0
(7)
whereby ζ signify the threshold draw-down value dependent on the
priorchoosen confidence level α and the (T × 1) vector z represent
thethreshold exceedances.
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Optimal Draw Down
Optimal Draw Down IExample Stock Portfolio: GMV vs. CDaR
> library(FRAPO)
> library(fPortfolio)
> library(PerformanceAnalytics)
> ## Loading of data set
> data(EuroStoxx50)
> ## Creating timeSeries of prices and returns
> pr NAssets RDP ## Backtest of GMV vs. CDaR
> ## Start and end dates
> to from ## Portfolio specifications
> ## CDaR portfolio
> DDbound DDalpha ## GMV portfolio
> mvspec BoxC ## Initialising weight matrices
> wMV for(i in 1:length(to)){
+ series
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Optimal Draw Down
Optimal Draw Down IIExample Stock Portfolio: GMV vs. CDaR
+ cd MVRetFac MVRetFac[1] MVPort CDRetFac CDRetFac[1] CDPort ##
Portfolio returns
> MVRet CDRet ## Draw down table
> table.Drawdowns(MVRet)
> table.Drawdowns(CDRet)
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Optimal Draw Down
Optimal Draw DownExample Stock Portfolio: GMV vs. CDaR
Portfolio From Trough To Depth → ↘ ↗
GMV1 2007-12-10 2008-03-17 NA 20.11 17 152 2007-06-04 2007-08-13
2007-10-08 9.75 19 11 83 2007-10-15 2007-11-05 2007-11-26 3.34 7 4
34 2007-03-12 2007-03-12 2007-03-19 2.30 2 1 15 2007-04-23
2007-04-23 2007-04-30 0.76 2 1 1
CDaR1 2007-11-12 2008-01-21 NA 11.53 21 112 2007-06-04
2007-09-03 2007-10-08 5.58 19 14 53 2007-05-07 2007-05-07
2007-05-14 0.51 2 1 14 2007-03-12 2007-03-12 2007-03-19 0.49 2 1 15
2007-10-22 2007-10-29 2007-11-05 0.30 3 2 1
Table: Overview of Draw Downs (positive, percentages)
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Probabilistic Utility
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
Pfaff (Invesco) Portfolio R/Rmetrics 24 / 53
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Probabilistic Utility
Probabilistic UtilityMotivation
Portfolio selection problems derived from utility functions.
E.g. mean-variance optimisation:U = λω′µ− (1− λ)ω′Σω.Allocation
sensitive to parameters µ,Σ, λ.
Problem-solving approaches: robust/bayesian estimators
and/orrobust optimization.
Nota bene: µ and Σ are random variables; as such the
allocationvector ω is a random variable itself.
Now: probabilistic interpretation of utility functions.
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Probabilistic Utility
Probabilistic UtilityConcept I
Approach introduced by Rossi et al. (2002) and Marschinski et
al.(2007).
Utility function is interpreted as the logarithm of the
probabilitydensity for a portfolio.
Optimal allocation is defined as the expected value of the
portfolio’sweights with respect to that probability, i.e., the
weights are viewed asparameters of this distribution.
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Probabilistic Utility
Probabilistic UtilityConcept II
Given: u = u(ω,U, θ), whereby ω is weight vector, U the
assumedutility function and θ a catch-all parameter vector (e.g.
expectedreturns, dispersion, risk sensitivity).
Expected utility is proportional to the logarithm of a
probabilitymeasure:ω ∼ P(ω|U, θ) = Z−1(ν,U, θ) exp (νu(ω,U,
θ)).Normalizing constant: Z (ν,U, θ) =
∫D(ω)[dω] exp (νu(ω,U, θ)).
Convergence to maximum utility (ν →∞) or equal-weight solution(ν
→ 0) is controlled by: ν = pNγ .Portfolio solution is then defined
as:ω̄(U, θ) = Z−1(ν,U, θ)
∫D(ω)[dω]ω exp (νu(ω,U, θ))
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Probabilistic Utility
Probabilistic UtilityExample: quadratic utility, one risky
asset, I
## Utility function
U1
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Probabilistic Utility
Probabilistic UtilityExample: quadratic utility, one risky
asset, II
0.0 0.2 0.4 0.6 0.8 1.0
−4
−2
02
Weight
Util
ity /
Den
sity
UtilityDensityexp Utility
Pfaff (Invesco) Portfolio R/Rmetrics 29 / 53
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Probabilistic Utility
Probabilistic UtilityExample: quadratic utility, one risky
asset, III
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
Asymptotic Property of Probabilistic Utility with ν = N
Weight
Den
sity
ν = 1ν = 20ν = 40ν = 60ν = 80ν = 100ν = 120
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Probabilistic Utility
Probabilistic UtilityMarkov Chain Monte Carlo
Class of algorithms for sampling from a probability
distribution; shapeof density suffices.
Purpose of MCMC is the numeric evaluation of
multi-dimensionalintegrals, by (i) searching and (ii) evaluating
the state space.
The state space is searched by means of a Markov
chain-typeprogression of the parameters.
Evaluating proposed move (accepting/rejecting) ordinarily
byMetropolis-Hastings algorithm.
R resources: numerous R packages are available; see CRAN and
taskview ‘Bayesian’ for an annotated listing.
Book resources: Gilks et al. (1995) and Brooks et al.
(2011).
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Probabilistic Utility
Probabilistic UtilityHybrid Monte Carlo I
Introduced by Duane et al. (1987) (see Neal (2011) for a
moretextbook-like exposition).
Inclusion of an auxilliary momentum vector and taking the
gradient ofthe target distribution into account.
Purpose/aim:1 Moving through state space in larger steps.2
Autocorrelation in Markov Chains less pronounced compared to
other
approaches (thinning in principal not necessary).3 High
acceptance rate, ideally all moves are accepted.4 Faster
convergence to equilibrium distribution.
Pfaff (Invesco) Portfolio R/Rmetrics 32 / 53
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Probabilistic Utility
Probabilistic UtilityHybrid Monte Carlo II
Amending density by conjugate variables p:
G (q,p) ∼ exp(U(q)− p
′p
2
)(8)
Algorithm: Starting from a pair (qn,pn)1 Sample η from standard
normal.2 For a time interval T , integrate Hamiltonion
equations:
dpidt
= − δUδpi
(9a)
dqidt
= pi (9b)
together with the boundary constraints p(0) = η and q(0) = qn.3
Accept qn+1 = q(T ) with probability:
β = min(1, exp (G (q(T ),p(T ))− G (qn,η))), (10)else set qn+1 =
qn.
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Probabilistic Utility
Probabilistic Utility IHybrid Monte Carlo III
See http://www.cs.utoronto.ca/~radford/GRIMS.html (adopted
version)
hybridMC
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Probabilistic Utility
Probabilistic Utility IIHybrid Monte Carlo III
## Evaluate potential and kinetic energies at start and end of
trajectory
clogDens
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Probabilistic Utility
Probabilistic Utility IIIHybrid Monte Carlo III
MCMC[i, ]
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Probabilistic Utility
Probabilistic UtilityComparative Simulation: Design
Michaud-type simulation (see Michaud, 1989, 1998) as in
Marschinskiet al. (2007):
1 Treat estimates of location and dispersion as true
populationparameters for a given sample.
2 Obtain optimal ‘true’ MU allocations and hence utility.3 Draw
K random samples of length L from these ‘population’
parameters and obtain MU and PU solutions.4 Compare distances of
these K solutions with ‘true’ utility.
Settings: Sample sizes (L) of 24, 30, 36, 48, 54, 60, 72, 84,
96, 108and 120 observations; length of MC 250 (150 burn-in-periods)
and Kequals 100.
Applied to end-of-month multi-asset data set contained in R
packageFRAPO (see Pfaff, 2012), sample period 2004:11 –
2011:11.
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Probabilistic Utility
Probabilistic Utility IComparative Simulation: R Code
## Load packages
library(FRAPO)
library(MASS)
library(numDeriv)
library(parallel)
library(compiler)
enableJIT(3)
## Loading data and computing returns
data(MultiAsset)
Assets
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Probabilistic Utility
Probabilistic Utility IIComparative Simulation: R Code
PUW
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Probabilistic Utility
Probabilistic Utility IIIComparative Simulation: R Code
for(i in 1:LS){
cat(paste("Computing for Sample Size", Samples[i], "\n"))
SampleL
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Probabilistic Utility
Probabilistic UtilityComparative Simulation: R Code, Distances
from true utility
Sample Sizes
Dev
iatio
n fr
om 't
rue'
Util
ity (
mea
n an
d m
ean
+/−
Std
Dev
)
● ●
●
●●
● ●●
● ● ●
24 30 36 48 54 60 72 84 96 108 120
0
20
40
60
80
100
120●
ProbUtilMaxUtil
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Optimal Risk/Reward
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
Pfaff (Invesco) Portfolio R/Rmetrics 42 / 53
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Optimal Risk/Reward
Optimal Risk/RewardDefinition
Fractional (non-)linear programming problem:
PRatio = arg minω∈Ω
fRisk(R, ω, θ)
fReward(R, ω)
ω′i = 1
ω ≥ 0l ≤ Aω ≤ u
(11)
Key developments by Charnes and Cooper (1969) (linear case)
andDinkelbach (1967); Schaible (1967a,b); Stoyanov et al.
(2007)(non-linear case).
Risk measures: Variance, MAD, minimizing maximum loss,
lowerpartial moment, CVaR, CDaR.
Pfaff (Invesco) Portfolio R/Rmetrics 43 / 53
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Optimal Risk/Reward
Optimal Risk/RewardOptimal portfolio with LPM: Closing the
loop
Using the semi-standard deviation as risk-measure has
beenmentioned in Markowitz (1952).
The lower partial moment is defined as:
LPMn,τ =
∫ τ−∞
(τ − x)nf (x)dx , (12)
whereby x is the random variable, f (x) the associated
densityfunction, τ is the target for which the deviations are
measured and nsignify the weighting of the deviations from the
threshold.
The semi-variance results as a special for τ = E(x) and n =
2.
Pfaff (Invesco) Portfolio R/Rmetrics 44 / 53
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Optimal Risk/Reward
Optimal Risk/Reward IOptimal portfolio with LPM: R Code
> library(parma)
> rlpm parmasolve(rlpm, type = "NLP")
+---------------------------------+
| PARMA Portfolio |
+---------------------------------+
No.Assets : 10
Problem : NLP
Risk Measure : LPM
Objective : optimal
Risk : 0.6766982
Reward : 0.5376806
Optimal_Weights
GREXP 0.8176
GLD 0.1019
GDAXI 0.0805
> ## Charming outcome: Allocate to German Bonds & Equity
and Gold :-)
Pfaff (Invesco) Portfolio R/Rmetrics 45 / 53
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Summary
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
Pfaff (Invesco) Portfolio R/Rmetrics 46 / 53
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Summary
Summary
More than sixty years after seminal work of Markowitz, progress
hascentred on how the risk-return space is modeled.
Advances were driven by financial market crisis.
Basically, all of these newly proposed portfolio
optimizationapproaches can addressed within/from R.
In a kaleidoscopic fashion, some of these advances have
beenintroduced in this talk, but . . .
Pfaff (Invesco) Portfolio R/Rmetrics 47 / 53
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Summary
. . . more examples in . . .
Pfaff (Invesco) Portfolio R/Rmetrics 48 / 53
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Bibliography
1 Overview
2 R Resources
3 Risk-Parity/Budget
4 Optimal Draw Down
5 Probabilistic Utility
6 Optimal Risk/Reward
7 Summary
8 Bibliography
Pfaff (Invesco) Portfolio R/Rmetrics 49 / 53
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Bibliography
Bibliography I
Ardia, D., K. Boudt, P. Carl, K. Mullen, and B. Peterson (2010).
Differential Evolution (DEoptim) for non-convex
portfoliooptimization.
Berkelaar, M. (2011). lpSolve: Interface to Lp solve v. 5.5 to
solve linear/integer programs. R package version 5.6.6.
Boudt, K., P. Carl, and B. Peterson (2010, April). Portfolio
optimization with cvar budgets. Presentation at
r/financeconference, Katholieke Universteit Leuven and Lessius,
Chicago, IL.
Boudt, K., P. Carl, and B. Peterson (2011a, September). Asset
allocation with conditional value-at-risk budgets. Technicalreport,
http://ssrn.com/abstract=1885293.
Boudt, K., P. Carl, and B. Peterson (2011b). PortfolioAnalytics:
Portfolio Analysis, including Numeric Methods for Optimizationof
Portfolios. R package version 0.6.1/r1849.
Boudt, K., B. Peterson, and C. Croux (2007, September).
Estimation and decomposition of downside risk for portfolios
withnon-normal returns. Working Paper KBI 0730, Katholieke
Universteit Leuven, Faculty of Economics and Applied
Economics,Department of Decision Sciences and Information
Management (KBI), Leuven.
Boudt, K., B. Peterson, and C. Croux (2008). Estimation and
decomposition of downside risk for portfolios with
non-normalreturns. The Journal of Risk 11(2), 79–103.
Brooks, S., A. Gelman, G. Jones, and X.-L. Meng (Eds.) (2011).
Handbook of Markow Chain Monte Carlo. Boca Raton, FL:Chapman &
Hall / CRC.
Carl, P., B. Peterson, K. Boudt, and E. Zivot (2012).
PerformanceAnalytics: Econometric tools for performance and
riskanalysis. R package version 1.0.4.4.
Chalabi, Y. and D. Würtz (2010). Rsocp: An R extenstion library
to use SOCP from R. R package version 271.1/r4910.
Charnes, A. and W. Cooper (1969). Programming with linear
fractional functionals. Naval Research logistics quarterly
9(3–4),181–186.
Chekhlov, A., S. Uryasev, and M. Zabarankin (2000). Portfolio
optimization with drawdown constraints. Research report2000-5,
Department of Industrial and Systems Engineering, University of
Florida, Gainesville, FL.
Pfaff (Invesco) Portfolio R/Rmetrics 50 / 53
http://ssrn.com/abstract=1885293
-
Bibliography
Bibliography II
Chekhlov, A., S. Uryasev, and M. Zabarankin (2003, January).
Portfolio optimization with drawdown constraints. Workingpaper,
University of Florida, Gainesville, FL.
Chekhlov, A., S. Uryasev, and M. Zabarankin (2005). Drawdown
measure in portfolio optimization. International Journal
ofTheoretical and Applied Finance 8(1), 13–58.
Dinkelbach, W. (1967). On nonlinear fractional programming.
Management Science 13(7), 492–498.
Duane, S., A. Kennedy, B. Pendleton, and D. Roweth (1987).
Hybrid monte carlo. Physical Letters B195, 216–222.
Eddelbuettel, D. (2012). RcppDE: Global optimization by
differential evolution in C++. R package version 0.1.1.
Gelius-Dietrich, G. (2012). glpkAPI: R Interface to C API of
GLPK. R package version 1.2.1.
Ghalanos, A. (2013). parma: portfolio allocation and risk
management applications. R package version 1.03.
Ghalanos, A. and S. Theussl (2012). Rsolnp: General Non-linear
Optimization Using Augmented Lagrange Multiplier Method. Rpackage
version 1.14.
Gilks, W., S. Richardson, and D. Spiegelhalter (1995). Markov
Chain Monte Carlo in Practice. Interdisciplinary Statistics.
BocaRaton, FL.: Chapman & Hall / CRC.
Harter, R., K. Hornik, and S. Theussl (2012). Rsymphony:
Symphony in R. R package version 0.1-14.
Henningsen, A. (2010). linprog: Linear Programming /
Optimization. R package version 0.9-0.
Konis, K. (2011). lpSolveAPI: R Interface for lp solve version
5.5.2.0. R package version 5.5.2.0-5.
Maillard, S., T. Roncalli, and J. Teiletche (2009, May). On the
properties of equally-weighted risk contributions
portfolios.Working paper, SGAM Alternative Investments and Lombard
Odier and University of Paris Dauphine.
Maillard, S., T. Roncalli, and J. Teiletche (2010). The
properties of equally weighted risk contribution portfolios. The
Journal ofPortfolio Management 36(4), 60–70.
Markowitz, H. (1952, March). Portfolio selection. The Journal of
Finance 7(1), 77–91.
Marschinski, R., P. Rossi, M. Tavoni, and F. Cocco (2007).
Portfolio selection with probabilistic utility. Annals of
OperationsResearch 151, 223–239.
Pfaff (Invesco) Portfolio R/Rmetrics 51 / 53
-
Bibliography
Bibliography III
Michaud, R. (1989). The markowitz optimization enigma: Is
optimized optimal. Financial Analyst Journal 45, 31–42.
Michaud, R. (1998). Efficient Asset Management: A Practical
Guide to Stock Portfolio Optimization and Asset Allocation.
NewYork: Oxford University Press.
Mullen, K., D. Ardia, D. Gil, D. Windover, and J. Cline (2011).
DEoptim: An R package for global optimization by
differentialevolution. Journal of Statistical Software 40(6),
1–26.
Neal, R. (2011). Handbook of Markov Chain Monte Carlo, Chapter
MCMC using Hamiltonian dynamics, pp. 113–162.Handbooks of Modern
Statistical Methods. Boca Raton, FL: Chapman & Hall / CRC.
Novomestky, F. (2012). rportfolios: Random portfolio generation.
R package version 1.0.
Peterson, B. and K. Boudt (2008, November). Component var for a
non-normal world. Risk. Reprint in Asia Risk.
Pfaff, B. (2011). rneos: XML-RPC Interface to NEOS. R package
version 0.2-6.
Pfaff, B. (2012). Financial Risk Modelling and Portfolio
Optimisation with R. London: John Wiley & Sons, Ltd.
Qian, E. (2005). Risk parity portfolios: Efficient portfolios
through true diversification. White paper, PanAgora, Bostan,
MA.
Qian, E. (2006). On the financial interpretation of risk
contribution: Risk budgets do add up. Journal of
InvestmentManagement 4(4), 1–11.
Qian, E. (2011, Spring). Risk parity and diversification. The
Journal of Investing 20(1), 119–127.
Rossi, P., M. Tavoni, F. Cocco, and R. Marschinski (2002,
November). Portfolio selection with probabilistic utility,
bayesianstatistics and markov chain monte carlo. eprint arXiv
arXiv:cond-mat/0211480, 1–27. http://arxiv.org.
Rowe, B. (2012). tawny: Provides various portfolio optimization
strategies including random matrix theory and shrinkageestimators.
R package version 2.0.2.
Schaible, S. (1967a). Fractional programming. i, duality.
Management Science 22(8), 858–867.
Schaible, S. (1967b). Fractional programming. ii, on
dinkelbach’s algorithm. Management Science 22(8), 868–873.
Soetaert, K., K. Van den Meersche, and D. van Oevelen (2009).
limSolve: Solving Linear Inverse Models. R package 1.5.1.
Stoyanov, S., S. Rachev, and F. Fabozzi (2007). Optimal
financial portfolios. Applied Mathematical Finance 14(5),
401–436.
Pfaff (Invesco) Portfolio R/Rmetrics 52 / 53
-
Bibliography
Bibliography IV
Theussl, S. and K. Hornik (2012). Rglpk: R/GNU Linear
Programming Kit Interface. R package version 0.3-8.
Turlach, B. A. and A. Weingessel (2011). quadprog: Functions to
solve Quadratic Programming Problems. R package version1.5-4.
Würtz, D., Y. Chalabi, W. Chen, and A. Ellis (2010a, April).
Portfolio Optimization with R/Rmetrics. Rmetrics Association
&Finance Online, www.rmetrics.org. R package version
2130.80.
Würtz, D., Y. Chalabi, W. Chen, and A. Ellis (2010b, April).
Portfolio Optimization with R/Rmetrics. Rmetrics Association
&Finance Online, www.rmetrics.org. R package version
2110.4.
Ye, Y. (1987). Interior Algorithms for Linear, Quadratic, and
Linearly Constrained Non-Linear Programming. Ph. D.
thesis,Department of ESS, Stanford University.
Pfaff (Invesco) Portfolio R/Rmetrics 53 / 53
OverviewR ResourcesRisk-Parity/BudgetOptimal Draw
DownProbabilistic UtilityOptimal Risk/RewardSummaryBibliography