| Title: | Portfolio Allocation and Risk Management Applications |
|---|---|
| Description: | Provision of a set of models and methods for use in the allocation and management of capital in financial portfolios. |
| Authors: | Alexios Galanos [aut, cre], Bernhard Pfaff [ctb], Miguel Sousa Lobo [ctb] (SOCP), Lieven Vandenberghe [ctb] (SOCP), Stephen Boyd [ctb] (SOCP), Herve Lebret [ctb] (SOCP) |
| Maintainer: | Alexios Galanos <[email protected]> |
| License: | GPL-3 |
| Version: | 1.7 |
| Built: | 2024-10-23 06:30:54 UTC |
| Source: | CRAN |
Portfolio Allocation and Risk Management. Models and Methods for scenario and moment based optimization of portfolios.
| Package: | parma |
| Type: | Package |
| Version: | 1.5-2 |
| Date: | 2014-07-09 |
| License: | GPL |
| LazyLoad: | yes |
| Depends: | methods |
| Imports: | nloptr, Rglpk, quadprog |
| Suggests: | Rsymphony, truncnorm, timeSeries |
The portfolio allocation and risk managament applications (parma) package
contains a unique set of methods and models for the optimal allocation of
capital in financial portfolios. It uniquely represents certain discontinuous
problems using their smooth approximation counterparts and implements fractional
based programming for the direct optimization of risk-to-reward ratios. In
combination with the rmgarch package, it enables the confident solution to
scenario based optimization problems using such risk and deviation measures as
Mean Absolute Deviation (MAD), Variance (EV), Minimax, Conditional Value at
Risk (CVaR), Conditional Drawdown at Risk (CDaR) and Lower Partial Moments (LPM).
In addition, it implements moment based optimization for use with the quadratic
EV problem, and a higher moment CARA utility expansion using the coskewness and
cokurtosis matrices generated from the GO-GARCH with affine GH or NIG
distributions. Benchmark relative optimization (tracking error) is also
implemented as are basic mixed integer cardinality constraints. Finally, for
non-convex problem formulations such as the upper to lower partial moments
function, global optimization methods using a penalty based method are
available. The key functions in the package are parmaspec
which defines the optimization setup, and parmasolve which solves
the problem given a chosen representation and solver. A portfolio frontier
function is implemented in parmafrontier, utility optimization in
parmautility and a custom translation of the cmaes global
optimization solver of Hansen (2006) with full features is implemented in
cmaes.
Whenever using this package, please cite as
@Manual{Galanos_2014,
author = {Alexios Galanos and Bernhard Pfaff},
title = {{parma}: Portfolio Allocation and Risk Management Applications.},
year = {2014},
note = {R package version 1.5-1.},}
The releases of this package is licensed under GPL version 3.
Alexios Galanos and Bernhard Pfaff
Charnes, A. and Cooper, W. 1962, Programming with linear fractional functionals,
Naval Research Logistics Quarterly, 9, 181–186.
Dinkelbach, W. 1967, On nonlinear fractional programming, Management Science,
13(7), 492–498.
Fishburn, P.C. 1977, Mean-risk analysis with risk associated with below-target
returns, The American Economic Review, 67(2), 116-126.
Galanos, A. 2012, Higher Moment Models for Risk and Portfolio Management,
Thesis (submitted) Cass Business School.
Hansen, N. 2006, The CMA Evolution Strategy: A Comparing Review, Towards
a New Evolutionary Computation (Studies in Fuzziness and Soft Computing),
192, 75–102.
Holthausen, D. 1981, A risk-return model with risk and return measured as
deviations from a target return, The American Economic Review, 71,
182–188.
Konno, H. and Yamazaki, H. 1991, Mean-absolute deviation portfolio optimization
model and its applications to Tokyo stock market, Management Science,
37(5), 519–531.
Markowitz, H. 1952, Portfolio selection, The Journal of Finance, 7(1),
77–91.
Rockafellar, R.T. and Uryasev, S. and Zabarankin, M., 2006, Generalized deviations
in risk analysis, Finance and Stochastics, 10(1), 51–74.
Stoyanov, S.V. and Rachev, S.T. and Fabozzi, F.J. 2007, Optimal financial portfolios,
Applied Mathematical Finance, 14(5), 401–436.
The direct translation of the Hansen's cmaes matlab code v3.60.
cmaes(pars, fun, lower = rep(0, length(pars)), upper = rep(1, length(pars)), insigma = 1, ctrl = cmaes.control(), ...) cmaes.control( options = list(StopFitness = -Inf, MaxFunEvals = Inf, MaxIter = '1e3*(N+5)^2/sqrt(popsize)', StopFunEvals = Inf, StopIter = Inf, TolX = '1e-11*max(insigma)', TolUpX = '1e3*max(insigma)', TolFun = 1e-12, TolHistFun = 1e-13, StopOnStagnation = TRUE, StopOnWarnings = TRUE, StopOnEqualFunctionValues = '2 + N/3', DiffMaxChange = Inf, DiffMinChange = 0, WarnOnEqualFunctionValues = FALSE, EvalParallel = FALSE, EvalInitialX = TRUE, Restarts = 0, IncPopSize = 2, PopSize = '4 + floor(3*log(N))', ParentNumber = 'floor(popsize/2)', RecombinationWeights = c("superlinear", "linear", "constant"), DiagonalOnly = '0*(1+100*N/sqrt(popsize))+(N>=1000)', CMA = TRUE, Seed = 'as.integer(Sys.time())', DispFinal = TRUE, DispModulo = 100, Warnings = FALSE), CMA = list(cs = '(mueff+2)/(N+mueff+3)', damps = '1 + 2*max(0,sqrt((mueff-1)/(N+1))-1) + cs', ccum = '(4 + mueff/N) / (N+4 + 2*mueff/N)', ccov1 = '2 / ((N+1.3)^2+mueff)', ccovmu = '2 * (mueff-2+1/mueff) / ((N+2)^2+mueff)', active = 0))cmaes(pars, fun, lower = rep(0, length(pars)), upper = rep(1, length(pars)), insigma = 1, ctrl = cmaes.control(), ...) cmaes.control( options = list(StopFitness = -Inf, MaxFunEvals = Inf, MaxIter = '1e3*(N+5)^2/sqrt(popsize)', StopFunEvals = Inf, StopIter = Inf, TolX = '1e-11*max(insigma)', TolUpX = '1e3*max(insigma)', TolFun = 1e-12, TolHistFun = 1e-13, StopOnStagnation = TRUE, StopOnWarnings = TRUE, StopOnEqualFunctionValues = '2 + N/3', DiffMaxChange = Inf, DiffMinChange = 0, WarnOnEqualFunctionValues = FALSE, EvalParallel = FALSE, EvalInitialX = TRUE, Restarts = 0, IncPopSize = 2, PopSize = '4 + floor(3*log(N))', ParentNumber = 'floor(popsize/2)', RecombinationWeights = c("superlinear", "linear", "constant"), DiagonalOnly = '0*(1+100*N/sqrt(popsize))+(N>=1000)', CMA = TRUE, Seed = 'as.integer(Sys.time())', DispFinal = TRUE, DispModulo = 100, Warnings = FALSE), CMA = list(cs = '(mueff+2)/(N+mueff+3)', damps = '1 + 2*max(0,sqrt((mueff-1)/(N+1))-1) + cs', ccum = '(4 + mueff/N) / (N+4 + 2*mueff/N)', ccov1 = '2 / ((N+1.3)^2+mueff)', ccovmu = '2 * (mueff-2+1/mueff) / ((N+2)^2+mueff)', active = 0))
pars |
A numeric vector of starting parameters. |
fun |
The user function to be minimized. |
lower |
A vector the lower parameter bounds. |
upper |
A vector with the upper parameter bounds. |
insigma |
The initial coordinate wise standard deviations for the search. |
ctrl |
A list with control parameters as returned from calling the ‘cmaes.control’ function. |
... |
Additional arguments passed to the user function. |
options |
The main options in the cmaes.control which may be optionally strings which are evaluated on initialization of the solver. |
CMA |
The options for the active CMA. |
This solver has been translated from the matlab version created by Nikolaus Hansen and available on his website http://www.cmap.polytechnique.fr/~nikolaus.hansen/cmaes_inmatlab.html. There is also a cmaes on CRAN but this does not offer the same level of options and flexibility that the matlab version offers. For more details on what the options mean and generally how the cmaes solver works, consult the relevant website and literature.
Alexios Galanos
Hansen, N. 2006, The CMA Evolution Strategy: A Comparing Review, Towards
a New Evolutionary Computation (Studies in Fuzziness and Soft Computing),
192, 75–102.
## Not run: ctrl = cmaes.control() ctrl$options$StopOnWarnings = FALSE ctrl$cma$active = 1 ctrl$options$TolFun = 1e-12 ctrl$options$DispModulo=100 ctrl$options$Restarts = 0 ctrl$options$MaxIter = 3000 ctrl$options$TolUpX = 5 ctrl$options$PopSize = 300 test1 = cmaes(rnorm(10), fun = parma:::fsphere, lower = -Inf, upper = Inf, insigma = 1, ctrl = ctrl) test2 = cmaes(rnorm(10), fun = parma:::frosenbrock, lower = -Inf, upper = Inf, insigma = 1, ctrl = ctrl) ctrl = cmaes.control() ctrl$options$StopOnWarnings = FALSE ctrl$cma$active = 1 ctrl$options$TolFun = 1e-12 ctrl$options$DispModulo=100 ctrl$options$Restarts = 0 ctrl$options$MaxIter = 3000 ctrl$options$PopSize = 400 test3 = cmaes(rep(1, 10), fun = parma:::frastrigin10, lower = -50, upper = 50, insigma = 1, ctrl = ctrl) ## End(Not run)## Not run: ctrl = cmaes.control() ctrl$options$StopOnWarnings = FALSE ctrl$cma$active = 1 ctrl$options$TolFun = 1e-12 ctrl$options$DispModulo=100 ctrl$options$Restarts = 0 ctrl$options$MaxIter = 3000 ctrl$options$TolUpX = 5 ctrl$options$PopSize = 300 test1 = cmaes(rnorm(10), fun = parma:::fsphere, lower = -Inf, upper = Inf, insigma = 1, ctrl = ctrl) test2 = cmaes(rnorm(10), fun = parma:::frosenbrock, lower = -Inf, upper = Inf, insigma = 1, ctrl = ctrl) ctrl = cmaes.control() ctrl$options$StopOnWarnings = FALSE ctrl$cma$active = 1 ctrl$options$TolFun = 1e-12 ctrl$options$DispModulo=100 ctrl$options$Restarts = 0 ctrl$options$MaxIter = 3000 ctrl$options$PopSize = 400 test3 = cmaes(rep(1, 10), fun = parma:::frastrigin10, lower = -50, upper = 50, insigma = 1, ctrl = ctrl) ## End(Not run)
Provides a number of custom constraints and their jacobians for use with the NLP representation (both minimum risk and the fractional problem).
ineqfun.turnover.min(w, optvars, uservars) ineqjac.turnover.min(w, optvars, uservars) ineqfun.bsturnover.min(w, optvars, uservars) ineqjac.bsturnover.min(w, optvars, uservars) ineqfun.turnover.opt(w, optvars, uservars) ineqjac.turnover.opt(w, optvars, uservars) ineqfun.bsturnover.opt(w, optvars, uservars) ineqjac.bsturnover.opt(w, optvars, uservars) ineqfun.variance.opt(w, optvars, uservars) ineqjac.variance.opt(w, optvars, uservars) ineqfun.variance.min(w, optvars, uservars) ineqjac.variance.min(w, optvars, uservars)ineqfun.turnover.min(w, optvars, uservars) ineqjac.turnover.min(w, optvars, uservars) ineqfun.bsturnover.min(w, optvars, uservars) ineqjac.bsturnover.min(w, optvars, uservars) ineqfun.turnover.opt(w, optvars, uservars) ineqjac.turnover.opt(w, optvars, uservars) ineqfun.bsturnover.opt(w, optvars, uservars) ineqjac.bsturnover.opt(w, optvars, uservars) ineqfun.variance.opt(w, optvars, uservars) ineqjac.variance.opt(w, optvars, uservars) ineqfun.variance.min(w, optvars, uservars) ineqjac.variance.min(w, optvars, uservars)
w |
The decision weight vector. |
optvars |
Problem specific list (not for use by user). |
uservars |
User specific list with some required parameters to be set (see details). |
Functions preceded be “ineqfun” denote the functions which may be
passed to the ineqfun option in the parmaspec function, while
“eqfun” to the eqfun option. Functions which include “jac” in the
first part of the name denote the jacobians of the equivalent constraints and
should be passed to the ineqgrad or eqgrad options in parmaspec. Functions
ending with “min” denote formulations for use with the minrisk type problems
while functions ending with “opt” denote the fractional risk formulation.
For the simple turnover constraint, the uservars list must contain an entry
called ‘wold’ denoting the previous vector of weights with which the
comparison will be made. Additionally, an entry called ‘turnover’
is required which denotes the (positive) value for the maximum turnover.
For the buy and sell turnover constraint (bsturnover), there should instead
be (positive) ‘buyturnover’ and ‘sellturnover’ entries in the
uservars list. Finally note that when using this type of constraints in a fractional
programming setup, care should be taken that the combination of bounds, turnover
limits and the forecast return vector do not result in a negative expected return
in which case the problem is not solvable.
The variance constraint allows the targeting of a maximum acceptable variance.
The extra arguments which must be passed to the uservars list are ‘Cov’
for the asset covariance matrix and ‘varbound’ representing the maximum
acceptable upper variance.
Used internally by the NLP solver. The fun return a scalar, while the jac return a matrix with n.cols equal to the length of the decision vector (which may be greater than the length of the weights as in the fractional problem which contains the fractional multiplier as well as other formulations which have additional decision variables).
Alexios Galanos
The xts dataset consists of the adjusted daily closing prices of 15 iShare ETFs for the period 2003-05-28 to 2012-06-01 (2272 periods) representing a selection of US style and international equity benchmark. The iShare series are IWF (Russell 1000 Growth Index), IWD (Russell 1000 Value Index), IWO (Russell 2000 Growth Index), IWN (Russell 2000 Value Index), EEM (MSCI Emerging Markets Index), TLT (Barclays 20+ Year T-Bond), EWC (MSCI Canada Index), EWA (MSCI Australia Index), EWJ (MSCI Japan Index), EWG (MSCI Germany Index), EWL (MSCI Switzerland Index), EWQ (MSCI France Index), EWU (MSCI UK Index), EPP (MSCI Pacific ex-Japan), EZA (MSCI South Africa Index).
data(etfdata)data(etfdata)
An xts matrix containing 2272x15 observations.
Yahoo Finance
Solves for the portfolios on the efficient frontier given a specification object.
parmafrontier(spec, n.points = 100, miny = NULL, maxy = NULL, type = NULL, solver = NULL, solver.control = list(), parma.control = list(ubounds = 10000, mbounds = 1e+05, penalty = 10000), cluster = NULL)parmafrontier(spec, n.points = 100, miny = NULL, maxy = NULL, type = NULL, solver = NULL, solver.control = list(), parma.control = list(ubounds = 10000, mbounds = 1e+05, penalty = 10000), cluster = NULL)
spec |
A |
n.points |
The number of portfolios to solve for along the frontier. |
miny |
(Optional) Minimum return from which to calculate the frontier. If not provided, will be calculated by a first pass optimization. |
maxy |
(Optional) Maximum return for which to calculate the frontier. If not provided, will be based on the maximum forecast. |
type |
The problem type to use (the show method on |
solver |
For a GNLP type problem the choice of global solver, either
“cmaes” or “crs” from the |
solver.control |
A list with optional control parameters passed to the
|
parma.control |
Internal NLP tuning parameters, where ‘ubounds’ represents the symmetric unconstrained parameter bounds in the fractional problem, ‘mbounds’ the multiplier upper bounds in the fractional formulation, ‘penalty’ the GNLP penalty parameter. |
cluster |
A precreated cluster object from the parallel package for the parallel evaluation of the frontier portfolios (see note). |
If using a cluster object, care should be taken since some of the LP based problems are quite memory intensive. Though some care some gone into using slam matrices and forcing garbage collection in intermediate setup steps of these problems, there is no guarantee that memory will not be quickly depleted for large problems using many cores/sockets.
A matrix object with columns for the weights, risk, reward and
termination status of the solver (not available for the QP solver), and
n.points rows. The user should investigate the non-converged solutions as well
as remove any NA based rows (non-converged with error).
Alexios Galanos
"parmaPort"
The parma optimal portfolio class.
Objects can be created by calls to parmasolve..
solution:Object of class "vector" The list with the
optimal values.
model:Object of class "vector" A list with details of
the risk model.
signature(object = "parmaPort"): Summary.
signature(object = "parmaPort"): Extracts the optimal
weights.
signature(object = "parmaPort"): Extracts the time
elapsed to solve the problem.
signature(object = "parmaPort"): Extracts the
arbitrage check on the scenario.
signature(object = "parmaPort"): Extracts the expected
risk of the optimized portfolio.
signature(object = "parmaPort"): Extracts the expected
reward of the optimized portfolio.
signature(object = "parmaPort"): Solver termination
code for the LP and NLP solvers.
Alexios Galanos
showClass("parmaPort")showClass("parmaPort")
Solves for the optimal weights given parmaSpec defined model.
parmasolve(spec, type = NULL, solver = NULL, solver.control = list(), x0 = NULL, w0 = NULL, parma.control = list(ubounds = 1e4, mbounds = 1e5, penalty = 1e4, eqSlack = 1e-05), ...)parmasolve(spec, type = NULL, solver = NULL, solver.control = list(), x0 = NULL, w0 = NULL, parma.control = list(ubounds = 1e4, mbounds = 1e5, penalty = 1e4, eqSlack = 1e-05), ...)
spec |
A |
type |
The problem type to use (the show method on |
solver |
For a GNLP type problem the choice of global solver, either
“cmaes” or “crs” from the |
solver.control |
A list with optional control parameters passed to the
|
x0 |
Optional starting parameters for the NLP type problems. This may be of size greater than the number of assets depending on whether it takes extra optimization parameters (CVaR has VaR in position 1, whilst the fractional formulation has the multiplier in the last position). |
w0 |
Optional starting parameters for only the asset weights. |
parma.control |
Internal NLP tuning parameters, where ‘ubounds’ represents the symmetric unconstrained parameter bounds in the fractional problem, ‘mbounds’ the multiplier upper bounds in the fractional formulation, ‘penalty’ the GNLP penalty parameter and ‘eqSlack’ the slack value to use for converting equalities to inequalities in the SOCP formulation. |
... |
Currently only the ‘verbose’ argument (logical) which is used and passed to the GLPK solver (similar to ‘trace’). |
In most of the cases, the intersection of objective and constraints will define whether a problem is LP, MILP, QP, MIQP, QCQP, NLP, MINLP or GNLP. However, there are cases when the problem can be solved by more than one type of solver so that the ‘type’ option allows the user to choose between the options. For the GNLP type, the ‘solver’ allows for a choice of “cmaes” or “crs”, while the LP type there is a choice of “glpk” or “symphony”. Future development will likely expand on these choices.
A parmaPort object containing details of the PARMA
optimized portfolio.
Alexios Galanos
"parmaSpec"
Object returned from calling parmaspec.
Objects can be created by calls of the form new("parmaSpec").
model:A list with details of the risk and optimization model.
modeldata:A list with the data.
constraints:A list with details on the optimization constraints.
signature(spec = "parmaSpec"):
Solves for the optimal weights.
signature(object = "parmaSpec"): Summary method.
signature(object = "parmaSpec"): Set a specified
‘arg’ from the list of arguments defined in the parmaspec
to some ‘value’ and return a modified specification.
signature(object = "parmaSpec"): Get a specified
‘arg’ from the list of arguments defined in the parmaspec.
Alexios Galanos
showClass("parmaSpec")showClass("parmaSpec")
Defines the type model and method for optimization using either a scenario or covariance matrix.
parmaspec(scenario = NULL, probability = NULL, S = NULL, Q = NULL, qB = NULL, benchmark = NULL, benchmarkS = NULL, forecast = NULL, target = NULL, targetType = c("inequality", "equality"), risk = c("MAD", "MiniMax", "CVaR", "CDaR", "EV", "LPM", "LPMUPM"), riskType = c("minrisk", "optimal", "maxreward"), riskB = NULL, options = list(alpha = 0.05, threshold = 999, moment = 1, lmoment=1, umoment=1, lthreshold = -0.01, uthreshold = 0.01), LB = NULL, UB = NULL, budget = 1, leverage = NULL, ineqfun = NULL, ineqgrad = NULL, eqfun = NULL, eqgrad = NULL, uservars = list(), ineq.mat = NULL, ineq.LB = NULL, ineq.UB = NULL, eq.mat = NULL, eqB = NULL, max.pos = NULL, asset.names = NULL, ...)parmaspec(scenario = NULL, probability = NULL, S = NULL, Q = NULL, qB = NULL, benchmark = NULL, benchmarkS = NULL, forecast = NULL, target = NULL, targetType = c("inequality", "equality"), risk = c("MAD", "MiniMax", "CVaR", "CDaR", "EV", "LPM", "LPMUPM"), riskType = c("minrisk", "optimal", "maxreward"), riskB = NULL, options = list(alpha = 0.05, threshold = 999, moment = 1, lmoment=1, umoment=1, lthreshold = -0.01, uthreshold = 0.01), LB = NULL, UB = NULL, budget = 1, leverage = NULL, ineqfun = NULL, ineqgrad = NULL, eqfun = NULL, eqgrad = NULL, uservars = list(), ineq.mat = NULL, ineq.LB = NULL, ineq.UB = NULL, eq.mat = NULL, eqB = NULL, max.pos = NULL, asset.names = NULL, ...)
scenario |
An n-by-m scenario matrix. |
probability |
An optional n-by-1 vector of scenario probabilities which must sum to 1 (only currently used in LP problems). Default is to assign equal weights to each row of the scenario. |
S |
An m-by-m positive definite covariance matrix. |
Q |
A list of m-by-m positive definite matrices for QCQP type problems. |
qB |
a vector of the same length as Q denoting the upper bound on Q. |
benchmark |
A n-by-1 scenario benchmark matrix, used when scenario is not NULL. |
benchmarkS |
An m+1 vector consisting of the benchmark variance (1) and covariances (m) with the other m assets, used when S is not NULL. |
forecast |
A vector (m) of forecast values for the assets. If a benchmark is included, this should then be the active forecast over the benchmark. |
target |
The target return required when riskType is minrisk. If a benchmark is used, then this is the active target over the benchmark (given the active forecasts above). |
targetType |
Whether the target should be a hard equality or inequality. |
risk |
The risk measure. |
riskType |
The type of optimization to use, with a choice of minimizing the risk given the relevant constraints, on optimizing directly the risk to reward ratio using established fractional programming methods or maximizing the reward subject to a risk upper bound and other constraints (only currently supported by the SOCP solver for covariance matrix type problems). |
riskB |
For the case that riskType is “maxreward”, then riskB is the upper bound for the risk constraint. |
options |
A vector of optional parameters related to the tail risk measures CVaR, CDaR and LPM. |
LB |
The lower bounds for the asset weights. If using a benchmark, this should be the maximum deviation below the benchmark weights. |
UB |
The upper bounds for the asset weights. If using a benchmark, this should be the maximum deviation above the benchmark weights. |
budget |
The investment constraint. If using a benchmark, this is usually set to zero so that weights represent the active bets on the benchmark. |
leverage |
The leverage constraint for Long/Short optimization. |
ineqfun |
A list of user inequality functions for use in an NLP type setup (see details). |
ineqgrad |
A list of user inequality gradients if ineqfun was provided. |
eqfun |
A list of user equality functions for use in an NLP type setup (see details). |
eqgrad |
A list of user equality gradients if eqfun was provided. |
uservars |
A list of any additional user required values to be used with user defined inequality and equality functions (for NLP). |
ineq.mat |
A k-by-m inequality matrix (for LP and QP problems). |
ineq.LB |
A vector (k) of the lower bounds for the ineq.mat. |
ineq.UB |
A vector (k) of the upper bounds for the ineq.mat. |
eq.mat |
A l-by-m equality matrix (for LP and QP problems). |
eqB |
A vector (l) of equalities for the eq.mat. |
max.pos |
Cardinality Constraints. The maximum assets to include in the solution, effectively making this a MILP, MIQP or MINLP problem. |
asset.names |
An optional character vector of asset names. |
... |
Not used. |
The parmaspec method is the entry point for specifying and solving portfolio
problems in the parma package. Currently 7 measures of risk are supported, 3
based on tail measures: Conditional Value at Risk (CVaR), Conditional Drawdown at
Risk (CDaR) and Lower Partial Moments (LPM), and 3 based on the Lp-Norm: Mean
Absolute Deviation (L_1, MAD), Mean Variance (L_2, EV) and MiniMax
(L_inf, Minimax). The LPMUPM measure is the ratio of lower to upper partial moments,
a non convex measure discussed in Holthausen (1981).
Additionally, the problems may be solved based on minimization
of risk subject to a target return, else on the optimal risk-reward ratio using
fractional programming (see references), thus avoiding the estimation of the
entire frontier. Problems are classified and solved according to whether they
can be formulated as Linear (LP), Mixed Integer LP (MILP), Quadratic (QP),
Mixed Integer Quadratic (MIQP), Second Order Cone Programming (SOCP),
Non-Linear (NLP), Mixed Integer NLP (MINLP) and Global NLP (GNLP). This in turn
depends on the intersection of objectives and constraints. It is possible that a
problem may be solved both as an LP and NLP (or QP and NLP), and this can be
defined during the solver stage (parmasolve). Because all NLP models,
make use of analytical derivatives, the results should be the same for any
formulation chosen, and considerations such as memory usage should guide the
choice of formulation (with some LP models being particularly expensive). Not
all problem types are supported, but this might change subject to the availability
of solvers in R which can deal with these specific types e.g. MINLP and MIQP.
The parmaspec also allows the input of a benchmark so that benchmark relative
optimization is carried out.
User defined equality and inequality functions for NLP problems need to be
properly defined to be accepted by the model, and their analytic gradients
also provided, unless the problem is solved as a GNLP in which case a derivative
free penalty function is used. These custom constraint functions should be
provided in a list, and should take as arguments the vector of decision
variables ‘w’, an argument called ‘optvars’ which is used
by the program internally, and an argument called ‘uservars’ which is a
list with optional user defined values for the constraints. The examples in the
inst folder provide some guidance, and the user is left to his own devices to
study the underlying workings of the program to understand how to supply these.
Finally, the NLP functions which are known to be discontinuous because of the
presence of functions such as the min and max, have been re-written to take
advantage of smooth approximations to such functions, details of which may be
founds in the vignette.
The package support for GNLP is based on a choice of the cmaes solver of the
cmaes package (which is not production level) or the crs solver of the
nloptr package which may be defined in solver.control option of the
parmasolve method with named argument ‘solver’. High quality
GNLP solvers are not available in R and as such support for these types of
problems is experimental at best and your mileage will vary. The problems which
must be solved as GNLP include the ‘LPMUPM’ measure, all problems with
risk type ‘optimal’ AND cardinality constraints (‘max.pos’), and
all problems with custom NLP constraints without derivatives, non-convex
inequalities or non-affine equalities.
A parmaSpec object containing details of the PARMA
specification.
Alexios Galanos
Utility based portfolio optimization using either Taylor series expansion of utility function with moments or scenario based.
parmautility(U = c("CARA", "Power"), method = c("moment", "scenario"), scenario = NULL, M1 = NULL, M2 = NULL, M3 = NULL, M4 = NULL, RA = 1, budget = 1, LB = rep(0, length(M1)), UB = rep(1, length(M1)))parmautility(U = c("CARA", "Power"), method = c("moment", "scenario"), scenario = NULL, M1 = NULL, M2 = NULL, M3 = NULL, M4 = NULL, RA = 1, budget = 1, LB = rep(0, length(M1)), UB = rep(1, length(M1)))
U |
The utility function (only CARA curretly implemented). |
method |
Whether to use moment or scenario based optimization (only moment currently implemented). |
scenario |
A n-by-m scenario matrix. |
M1 |
A vector (m) of forecasts. |
M2 |
An m-by-m positive definite covariance matrix. |
M3 |
An m-by-m^2 third co-moment matrix. |
M4 |
An m-by-m^3 fourth co-moment matrix. |
RA |
Risk Aversion Coefficient for CARA. |
budget |
The investment constraint. |
LB |
The lower bounds for the asset weights (positive). |
UB |
The upper bounds for the asset weights. |
The function currently only implements the CARA moment based approach, but will be expanded in the future. The moment approach can take as inputs either M1 and M2 (2-moment approximation), or M1, M2, M3 and M4 (4-moment approximation). Not many models generate M3 and M4, but the “gogarch” model with manig or magh distribution will.
A parmaPort object containing details of the PARMA
optimized portfolio.
Alexios Galanos
Galanos, A. and Rossi, E. and Urga, G. 2012, Independent Factor Autoregressive
Conditional Density Model submitted-TBA
Calculates a given portfolio risk/deviation measure given a set of weights and matrix of returns, possible representing a forecast scenario.
riskfun(weights, Data, risk = c("mad", "ev", "minimax", "cvar", "cdar", "lpm"), benchmark = NULL, alpha = 0.05, moment = 1, threshold = 0, VaR = NULL, DaR = NULL)riskfun(weights, Data, risk = c("mad", "ev", "minimax", "cvar", "cdar", "lpm"), benchmark = NULL, alpha = 0.05, moment = 1, threshold = 0, VaR = NULL, DaR = NULL)
weights |
vector of weights. |
Data |
Matrix of returns. |
risk |
Choice of measure. |
benchmark |
(Optional) vector of benchmark returns with same number of rows as Data. |
alpha |
The lower quantile for the “cvar” and “cdar” measures. |
moment |
The “lpm” measure moment. |
threshold |
The “lpm” measure threshold. A value of 999 will subtract the portfolio mean. |
VaR |
(Optional) The pre-calculated VaR for the “cvar” measure. |
DaR |
(Optional) The pre-calculated DaR for the “cdar” measure. |
A simple utility function for the calculation and understanding of some of the risk and deviation measures implemented in the package.
A numeric value representing the risk/deviation measure.
Alexios Galanos
The function solves second-order cone problem by
primal-dual interior point method. It is a wrapper function to the
C-routines written by Lobo, Vandenberghe and Boyd (see
reference below).
Socp(f, A, b, C, d, N, x = NULL, z = NULL, w = NULL, control = list())Socp(f, A, b, C, d, N, x = NULL, z = NULL, w = NULL, control = list())
f |
Vector defining linear objective, |
A |
Matrix with the |
b |
Vector with the |
C |
Matrix with the |
d |
Vector with the |
N |
Vector of size |
x |
Primal feasible initial point. Must satisfy: |
z |
Dual feasible initial point. |
w |
Dual feasible initial point. |
control |
A list of control parameters. |
The primal formulation of an SOCP is given as:
subject to
for . Here, is the
vector to be optimised. The dual form of an SOCP is expressed as:
subject to
and
for , given strictly feasible primal and dual
initial points.
The algorithm stops, if one of the following criteria is met:
abs.tol – maximum absolute error in objective
function; guarantees that for any x: .
rel.tol – maximum relative error in objective
function; guarantees that for any x: . Negative
value has special meaning, see target next.
target – if , stops when
.
max.iter – limit on number of algorithm outer iterations.
Most problems can be solved in less than 50 iterations. Called with
max_iter = 0 only checks feasibility of x and z,
(and returns gap and deviation from centrality).
The target value is reached. rel\_tol is negative and
the primal objective is less than the target.
A list-object with the following elements:
x |
Solution to the primal problem. |
z |
Solution to the dual problem. |
iter |
Number of iterations performed. |
hist |
see |
convergence |
A logical code. |
info |
A numerical code. It indicates if the convergence was successful. |
message |
A character string giving any additional information returned by the optimiser. |
This function has been ported from the Rsocp package contained in the Rmetrics-Project on R-Forge. In contrast to the former implementation, allowance is made for specifying more than one cone constraint.
Bernhard Pfaff
Lobo, M. and Vandenberghe, L. and Boyd, S., SOCP: Software for Second-order Cone Programming, User's Guide, Beta Version, April 1997, Stanford University.
SocpPhase1, SocpPhase2, SocpControl
This function returns a list object of control parameters that
are passed down to the C-function SOCP. It's default
values are used in Socp.
SocpControl(abs.tol = 1e-18, rel.tol = 1e-16, target = 0, max.iter = 500, Nu = 10, out.mode = 0, BigM.K = 2, BigM.iter = 5)SocpControl(abs.tol = 1e-18, rel.tol = 1e-16, target = 0, max.iter = 500, Nu = 10, out.mode = 0, BigM.K = 2, BigM.iter = 5)
abs.tol |
Absolute tolerance. |
rel.tol |
Relative tolerance. |
target |
Target value < 0, only used if rel.tol < 0. |
max.iter |
The maximum number of iterations, socp is aborted if more are required for convergence. |
Nu |
The parameter that controls the rate of convergence, Nu > 1, recommended range 5 to 50. |
out.mode |
Specifies what should be output: 0 - nothing, 1 - duality gap for initial point and after each iteration, 2 - duality gap and deviation from centrality, for initial point and after each iteration. |
BigM.K |
Iterataion parameter. The default values is |
BigM.iter |
Iterataion parameter. The default values is
|
For details about these control parameters, the reader is referred to
the reference below, in particular sections 2.7, 2.8 and 4.3 to 4.5. A
pdf-version of the user's guide is shipped in the packages doc
subdirectory.
A list object with the control parameters.
This function has been ported from the Rsocp package contained in the Rmetrics-Project on R-Forge.
Bernhard Pfaff
Lobo, M. and Vandenberghe, L. and Boyd, S., SOCP: Software for Second-order Cone Programming, User's Guide, Beta Version, April 1997, Stanford University.
This function determines values for x, whence they have not
been specified by the user. Here, a feasibility problem is solved
first and its solution is then used as an initial point for the
original problem.
SocpPhase1(f, A, b, N, control)SocpPhase1(f, A, b, N, control)
f |
|
A |
|
b |
|
N |
|
control |
|
The finding of an initial point x is described in the user's
guide, sectionb 2.8.
A vector with the initial point for x.
This function has been ported from the Rsocp package contained in the Rmetrics-Project on R-Forge.
Bernhard Pfaff
Lobo, M. and Vandenberghe, L. and Boyd, S., SOCP: Software for Second-order Cone Programming, User's Guide, Beta Version, April 1997, Stanford University.
This function determines values for z, whence they have not
been specified by the user.
SocpPhase2(f, A, b, N, x, control)SocpPhase2(f, A, b, N, x, control)
f |
|
A |
|
b |
|
N |
|
x |
|
control |
|
A vector with the initial point for z (dual form of
SOCP).
This function has been ported from the Rsocp package contained in the Rmetrics-Project on R-Forge.
Bernhard Pfaff
Lobo, M. and Vandenberghe, L. and Boyd, S., SOCP: Software for Second-order Cone Programming, User's Guide, Beta Version, April 1997, Stanford University.