Type: Package
Title: Fully Flexible Probabilities for Stress Testing and Portfolio Construction
Version: 0.2.2
Description: Implements numerical entropy-pooling for portfolio construction and scenario analysis as described in Meucci, Attilio (2008) and Meucci, Attilio (2010) <doi:10.2139/ssrn.1696802>.
License: MIT + file LICENSE
URL: https://github.com/Reckziegel/FFP
BugReports: https://github.com/Reckziegel/FFP/issues
Depends: R (≥ 2.10)
Imports: assertthat (≥ 0.2.1), dplyr (≥ 1.0.10), forcats (≥ 0.5.2), ggdist (≥ 3.2.0), ggplot2 (≥ 3.3.6), lubridate (≥ 1.8.0), magrittr (≥ 2.0.3), methods, mvtnorm (≥ 1.1-3), purrr (≥ 0.3.4), rlang (≥ 1.0.6), scales (≥ 1.2.1), stringr (≥ 1.4.1), stats, tibble (≥ 3.1.8), tidyr (≥ 1.2.1), vctrs (≥ 0.4.1), nloptr (≥ 2.0.3), crayon, NlcOptim (≥ 0.6)
Suggests: copula, covr, ghyp, knitr (≥ 1.40), markdown, rmarkdown, roxygen2, spelling, testthat (≥ 3.1.4), xts (≥ 0.12.1)
Config/testthat/edition: 3
Encoding: UTF-8
Language: en-US
LazyData: true
RoxygenNote: 7.2.1
VignetteBuilder: knitr
NeedsCompilation: no
Packaged: 2022-09-29 13:29:47 UTC; USUARIO
Author: Bernardo Reckziegel [aut, cre]
Maintainer: Bernardo Reckziegel <bernardo_cse@hotmail.com>
Repository: CRAN
Date/Publication: 2022-09-29 15:10:06 UTC

Double-Decay Covariance Matrix

Description

This function computes the covariance matrix using two different decay factors.

Usage

DoubleDecay(x, decay_low, decay_high)

Arguments

x

A set of relevant risk drivers.

decay_low

A numeric value with the low decay (long half-life).

decay_high

A numeric value with the high decay (short half-life).

Details

A common practice is to estimate the covariance of the risk drivers using a high decay (short half-life) for the volatilities and a low decay (long half-life) for the correlations.

Value

A list with the posterior mean ans sigma.

Inspection of a ffp object with ggplot2

Description

Extends the autoplot method for the ffp class.

Usage

## S3 method for class 'ffp'
autoplot(object, color = TRUE, ...)

## S3 method for class 'ffp'
plot(object, ...)

Arguments

object

An objected of the ffp class.

color

A logical flag indicating whether (or not) the color argument should be added to the ggplot2 aesthetics.

...

Additional arguments to be passed to autoplot.

Value

A ggplot2 object.

Examples

library(ggplot2)

x <- exp_decay(EuStockMarkets, 0.001)
y <- exp_decay(EuStockMarkets, 0.01)

autoplot(x) +
  scale_color_viridis_c()
autoplot(y) +
  scale_color_viridis_c()

Stack Flexible Probabilities

Description

This function mimics dplyr bind. It's useful if you have different ffp objects and want to stack them in the tidy (long) format.

Usage

bind_probs(...)

Arguments

...

ffp objects to combine.

Value

A tidy tibble.

The output adds two new columns:

See Also

crisp exp_decay kernel_normal kernel_entropy double_decay

Examples

library(ggplot2)
library(dplyr, warn.conflicts = FALSE)

x <- exp_decay(EuStockMarkets, lambda = 0.001)
y <- exp_decay(EuStockMarkets, lambda = 0.002)

bind_probs(x, y)

bind_probs(x, y) %>%
  ggplot(aes(x = rowid, y = probs, color = fn)) +
  geom_line() +
  scale_color_viridis_d() +
  theme(legend.position="bottom")

Stack Different Views

Description

Bind views for entropy programming.

Usage

bind_views(...)

Arguments

...

Objects of the class ffp_views to combine.

Value

A list of the view class.

Examples

library(ggplot2)

# Invariant
ret <- diff(log(EuStockMarkets))
n   <- nrow(ret)

# Prior probabilities (usually equal weight scheme)
prior <- rep(1 / n, n)

# Prior belief for expected returns (here is 0% for each asset)
view_mean <- view_on_mean(x = ret, mean = rep(0, 4))

#' view on volatility
vol <- apply(ret, 2, stats::sd) * 1.1 # volatility 10% higher than average
view_volatility <- view_on_volatility(x = ret, vol = vol)

views_comb <- bind_views(view_mean, view_volatility)
views_comb

ep <- entropy_pooling(p      = prior,
                      Aeq    = views_comb$Aeq,
                      beq    = views_comb$beq,
                      A      = views_comb$A,
                      b      = views_comb$b,
                      solver = "nlminb")
autoplot(ep)

Flexible Probabilities Driven Bootstrap

Description

Resamples historical scenarios with flexible probabilities.

Usage

bootstrap_scenarios(x, p, n)

## S3 method for class 'numeric'
bootstrap_scenarios(x, p, n)

## S3 method for class 'matrix'
bootstrap_scenarios(x, p, n)

## S3 method for class 'ts'
bootstrap_scenarios(x, p, n)

## S3 method for class 'xts'
bootstrap_scenarios(x, p, n)

## S3 method for class 'tbl'
bootstrap_scenarios(x, p, n)

## S3 method for class 'data.frame'
bootstrap_scenarios(x, p, n)

Arguments

x

A time series defining the scenario-probability distribution.

p

An object of the ffp class.

n

An integer scalar with the number of scenarios to be generated.

Details

The argument x is supposed to have the same size of p.

Value

A tibble with the number of rows equal to n.

Examples

set.seed(123)
ret <- diff(log(EuStockMarkets))
ew  <- rep(1 / nrow(ret), nrow(ret))

bootstrap_scenarios(x = ret, p = as_ffp(ew), n = 10)

Internal function used to check the validity of inputs.

Description

Internal function used to check the validity of inputs.

Usage

check_input(x)

## Default S3 method:
check_input(x)

## S3 method for class 'numeric'
check_input(x)

## S3 method for class 'double'
check_input(x)

## S3 method for class 'matrix'
check_input(x)

## S3 method for class 'xts'
check_input(x)

## S3 method for class 'tbl_df'
check_input(x)

Arguments

x

Any object passed to other functions in the package.

Value

A matrix

Full Information by Market Conditioning

Description

Give full weight to occurrences that satisfies a logical condition.

Usage

crisp(x, lgl)

## Default S3 method:
crisp(x, lgl)

## S3 method for class 'numeric'
crisp(x, lgl)

## S3 method for class 'matrix'
crisp(x, lgl)

## S3 method for class 'ts'
crisp(x, lgl)

## S3 method for class 'xts'
crisp(x, lgl)

## S3 method for class 'data.frame'
crisp(x, lgl)

## S3 method for class 'tbl_df'
crisp(x, lgl)

Arguments

x

An univariate or a multivariate distribution.

lgl

A logical vector with TRUE's and FALSE's indicating which scenarios should considered.

Value

A numerical vector of class ffp with the new probabilities distribution.

See Also

exp_decay kernel_normal

Examples

library(ggplot2)
# invariance (stationarity)
ret <- diff(log(EuStockMarkets))

# full weight on scenarios where CAC returns were above 2%
market_condition <- crisp(x = ret, ret[ , 3] > 0.02)
market_condition

autoplot(market_condition) +
  scale_color_viridis_c()

Dataset used in Historical Scenarios with Fully Flexible Probabilities (matrix format).

Description

Dataset used in Historical Scenarios with Fully Flexible Probabilities (matrix format).

Usage

db

Format

An object of class matrix (inherits from array) with 1083 rows and 9 columns.

See Also

db_tbl

Dataset used in Historical Scenarios with Fully Flexible Probabilities (tibble format).

Description

Dataset used in Historical Scenarios with Fully Flexible Probabilities (tibble format).

Usage

db_tbl

Format

An object of class tbl_df (inherits from tbl, data.frame) with 1083 rows and 9 columns.

See Also

db

Flexible Probabilities using Partial Information

Description

Match different decay-factors on the covariance matrix.

Usage

double_decay(x, slow, fast)

## Default S3 method:
double_decay(x, slow, fast)

## S3 method for class 'numeric'
double_decay(x, slow, fast)

## S3 method for class 'matrix'
double_decay(x, slow, fast)

## S3 method for class 'ts'
double_decay(x, slow, fast)

## S3 method for class 'xts'
double_decay(x, slow, fast)

## S3 method for class 'tbl'
double_decay(x, slow, fast)

## S3 method for class 'data.frame'
double_decay(x, slow, fast)

Arguments

x

An univariate or a multivariate distribution.

slow

A double with the long half-life (slow decay) for the correlation matrix.

fast

A double with the short-life (high decay) for the volatility.

Value

A numerical vector of class ffp with the new probabilities distribution.

References

De Santis, G., R. Litterman, A. Vesval, and K. Winkelmann, 2003, Covariance matrix estimation, Modern investment management: an equilibrium approach, Wiley.

See Also

kernel_entropy half_life

Examples


  library(ggplot2)

  slow <- 0.0055
  fast <- 0.0166
  ret <- diff(log(EuStockMarkets))

  dd <- double_decay(ret, slow, fast)
  dd

  autoplot(dd) +
    scale_color_viridis_c()

Summary Statistics for Empirical Distributions

Description

Computes the mean, standard deviation, skewness, kurtosis, Value-at-Risk (VaR) and Conditional Value-at-Risk CVaR) under flexible probabilities.

Usage

empirical_stats(x, p, level = 0.01)

## Default S3 method:
empirical_stats(x, p, level = 0.01)

## S3 method for class 'numeric'
empirical_stats(x, p, level = 0.01)

## S3 method for class 'matrix'
empirical_stats(x, p, level = 0.01)

## S3 method for class 'xts'
empirical_stats(x, p, level = 0.01)

## S3 method for class 'ts'
empirical_stats(x, p, level = 0.01)

## S3 method for class 'data.frame'
empirical_stats(x, p, level = 0.01)

## S3 method for class 'tbl_df'
empirical_stats(x, p, level = 0.01)

Arguments

x

A time series defining the scenario-probability distribution.

p

An object of the ffp class.

level

A number with the desired probability level. The default is level = 0.01.

Details

The data in x and p are expected to have the same number of rows (size).

Value

A tidy tibble with 3 columns:

Examples

library(dplyr, warn.conflicts = FALSE)
library(ggplot2)

ret <- diff(log(EuStockMarkets))

# with equal weights (standard scenario)
ew  <- rep(1 / nrow(ret), nrow(ret))
empirical_stats(x = ret, p = as_ffp(ew)) %>%
  ggplot(aes(x = name, y = value)) +
  geom_col() +
  facet_wrap(~stat, scales = "free") +
  labs(x = NULL, y = NULL)

# with ffp
exp_smooth <- exp_decay(ret, 0.015)
empirical_stats(ret, exp_smooth) %>%
  ggplot(aes(x = name, y = value)) +
  geom_col() +
  facet_wrap(~stat, scales = "free") +
  labs(x = NULL, y = NULL)

Effective Number of Scenarios

Description

Shows how many scenarios are effectively been considered when using flexible probabilities.

Usage

ens(p)

Arguments

p

An object of the ffp class.

Value

A single double.

Examples

set.seed(123)
p <- exp_decay(stats::rnorm(100), 0.01)

# ens is smaller than 100
ens(p)

Numerical Entropy Minimization

Description

This function solves the entropy minimization problem with equality and inequality constraints. The solution is a vector of posterior probabilities that distorts the least the prior (equal-weights probabilities) given the constraints (views on the market).

Usage

entropy_pooling(
  p,
  A = NULL,
  b = NULL,
  Aeq = NULL,
  beq = NULL,
  solver = c("nlminb", "solnl", "nloptr"),
  ...
)

Arguments

p

A vector of prior probabilities.

A

The linear inequality constraint (left-hand side).

b

The linear inequality constraint (right-hand side).

Aeq

The linear equality constraint (left-hand side).

beq

The linear equality constraint (right-hand side).

solver

A character. One of: "nlminb", "solnl" or "nloptr".

...

Further arguments passed to one of the solvers.

Details

When imposing views constraints there is no need to specify the non-negativity constraint for probabilities, which is done automatically by entropy_pooling.

For the arguments accepted in ..., please see the documentation of nlminb, solnl, nloptr and the examples bellow.

Value

A vector of posterior probabilities.

Examples

# setup
ret <- diff(log(EuStockMarkets))
n   <- nrow(ret)

# View on expected returns (here is 2% for each asset)
mean <- rep(0.02, 4)

# Prior probabilities (usually equal weight scheme)
prior <- rep(1 / n, n)

# View
views <- view_on_mean(x = ret, mean = mean)

# Optimization
ep <- entropy_pooling(
 p      = prior,
 Aeq    = views$Aeq,
 beq    = views$beq,
 solver = "nlminb"
)
ep

### Using the ... argument to control the optimization parameters

# nlminb
ep <- entropy_pooling(
 p      = prior,
 Aeq    = views$Aeq,
 beq    = views$beq,
 solver = "nlminb",
 control = list(
     eval.max = 1000,
     iter.max = 1000,
     trace    = TRUE
   )
)
ep

# nloptr
ep <- entropy_pooling(
 p      = prior,
 Aeq    = views$Aeq,
 beq    = views$beq,
 solver = "nloptr",
 control = list(
     xtol_rel = 1e-10,
     maxeval  = 1000,
     check_derivatives = TRUE
   )
)
ep

Full Information by Exponential Decay

Description

Exponential smoothing twists probabilities by giving relatively more weight to recent observations at an exponential rate.

Usage

exp_decay(x, lambda)

## Default S3 method:
exp_decay(x, lambda)

## S3 method for class 'numeric'
exp_decay(x, lambda)

## S3 method for class 'matrix'
exp_decay(x, lambda)

## S3 method for class 'ts'
exp_decay(x, lambda)

## S3 method for class 'xts'
exp_decay(x, lambda)

## S3 method for class 'data.frame'
exp_decay(x, lambda)

## S3 method for class 'tbl'
exp_decay(x, lambda)

Arguments

x

An univariate or a multivariate distribution.

lambda

A double for the decay parameter.

Details

The half-life is linked with the lambda parameter as follows:

For example: log(2) / 0.0166 is approximately 42. So, a parameter lambda of 0.0166 can be associated with a half-life of two-months (21 * 2).

Value

A numerical vector of class ffp with the new probabilities distribution.

See Also

crisp kernel_normal half_life

Examples

library(ggplot2)

# long half_life
long_hl <- exp_decay(EuStockMarkets, 0.001)
long_hl
autoplot(long_hl) +
  scale_color_viridis_c()

# short half_life
short_hl <- exp_decay(EuStockMarkets, 0.015)
short_hl
autoplot(short_hl) +
  scale_color_viridis_c()

Manipulate the ffp Class

Description

Helpers and Constructors from ffp.

Usage

ffp(x = double(), ...)

is_ffp(x)

as_ffp(x)

## Default S3 method:
as_ffp(x)

## S3 method for class 'integer'
as_ffp(x)

Arguments

x

  • For ffp(): A numeric vector.

  • For is_ffp(): An object to be tested.

  • For as_ffp(): An object to convert to ffp.

...

Additional attributes to be passed to ffp.

Details

The ffp class is designed to interact with doubles, but the output of c(ffp, double) or c(double, ffp) will always return a double (not an ffp object), since there is no way to guarantee the interaction between a numeric vector and a probability will also be a probability.

Value

Examples

set.seed(123)
p <- runif(5)
p <- p / sum(p)

is_ffp(p)
as_ffp(p)

Internal vctrs methods

Description

Internal vctrs methods

Usage

new_ffp(x = double(), ...)

## S3 method for class 'ffp'
vec_ptype_abbr(x, ...)

## S3 method for class 'ffp.ffp'
vec_ptype2(x, y, ...)

## S3 method for class 'ffp.double'
vec_ptype2(x, y, ...)

## S3 method for class 'double.ffp'
vec_ptype2(x, y, ...)

## S3 method for class 'ffp.ffp'
vec_cast(x, to, ...)

## S3 method for class 'ffp.double'
vec_cast(x, to, ...)

## S3 method for class 'double.ffp'
vec_cast(x, to, ...)

## S3 method for class 'ffp'
obj_print_data(x, ...)

## S3 method for class 'ffp'
vec_math(.fn, .x, ...)

## S3 method for class 'ffp'
vec_arith(op, x, y, ...)

Arguments

x

A numeric vector.

Value

No return value, called for side effects.

Moments with Flexible Probabilities

Description

Computes the location and dispersion statistics under flexible probabilities.

Usage

ffp_moments(x, p = NULL)

## Default S3 method:
ffp_moments(x, p = NULL)

## S3 method for class 'numeric'
ffp_moments(x, p = NULL)

## S3 method for class 'matrix'
ffp_moments(x, p = NULL)

## S3 method for class 'xts'
ffp_moments(x, p = NULL)

## S3 method for class 'data.frame'
ffp_moments(x, p = NULL)

## S3 method for class 'tbl_df'
ffp_moments(x, p = NULL)

Arguments

x

A tabular (non-tidy) data structure.

p

An object of the ffp class.

Value

A list with 2 elements: mu and sigma.

Examples

x <- matrix(diff(log(EuStockMarkets)), ncol = 4)
colnames(x) <- colnames(EuStockMarkets)
p <- stats::runif(nrow(x))
p <- p / sum(p)

ffp_moments(x = x, p = p)

# compare with the standard approach
colMeans(x)
cov(x)

Double-Decay Covariance Matrix by Entropy-Pooling

Description

Internal function that computes the flexible probabilities given by double-decay covariance matrix.

Usage

fit_to_moments(X, m, S)

Arguments

X

A matrix of size TxN with relevant the risk drivers.

m

A matrix of size Nx1 given by Double_Decay with risk drivers location.

S

A matrix of size NxN given by Double_Decay with the dispersion matrix of the risk drivers.

Value

A matrix vector with the posterior probabilities.

Half-Life Calculation

Description

Compute the implied half-life of a decay parameter.

Usage

half_life(lambda)

Arguments

lambda

A number.

Value

A single number with the half-life in days.

See Also

exp_decay double_decay

Examples

half_life(0.0166)
half_life(0.01)

Partial Information Kernel-Damping

Description

Find the probability distribution that can constrain the first two moments while imposing the minimal structure in the data.

Usage

kernel_entropy(x, mean, sigma = NULL)

## Default S3 method:
kernel_entropy(x, mean, sigma = NULL)

## S3 method for class 'numeric'
kernel_entropy(x, mean, sigma = NULL)

## S3 method for class 'matrix'
kernel_entropy(x, mean, sigma = NULL)

## S3 method for class 'ts'
kernel_entropy(x, mean, sigma = NULL)

## S3 method for class 'xts'
kernel_entropy(x, mean, sigma = NULL)

## S3 method for class 'tbl_df'
kernel_entropy(x, mean, sigma = NULL)

## S3 method for class 'data.frame'
kernel_entropy(x, mean, sigma = NULL)

Arguments

x

An univariate or a multivariate distribution.

mean

A numeric vector in which the kernel should be centered.

sigma

The uncertainty (volatility) around the mean. When NULL, only the mean is constrained.

Value

A numerical vector of class ffp with the new probabilities distribution.

See Also

double_decay

Examples

library(ggplot2)

ret <- diff(log(EuStockMarkets[ , 1]))
mean <- -0.01 # scenarios around -1%
sigma <- var(diff(ret))

ke <- kernel_entropy(ret, mean, sigma)
ke

autoplot(ke) +
  scale_color_viridis_c()

Full Information by Kernel-Damping

Description

Historical realizations receive a weight proportional to their distance from a target mean.

Usage

kernel_normal(x, mean, sigma)

## Default S3 method:
kernel_normal(x, mean, sigma)

## S3 method for class 'numeric'
kernel_normal(x, mean, sigma)

## S3 method for class 'matrix'
kernel_normal(x, mean, sigma)

## S3 method for class 'ts'
kernel_normal(x, mean, sigma)

## S3 method for class 'xts'
kernel_normal(x, mean, sigma)

## S3 method for class 'tbl_df'
kernel_normal(x, mean, sigma)

## S3 method for class 'data.frame'
kernel_normal(x, mean, sigma)

Arguments

x

An univariate or a multivariate distribution.

mean

A numeric vector in which the kernel should be centered.

sigma

The uncertainty (volatility) around the mean.

Value

A numerical vector of class ffp with the new probabilities distribution.

See Also

crisp exp_decay

Examples

library(ggplot2)

ret <- diff(log(EuStockMarkets[ , 1]))
mean <- -0.01 # scenarios around -1%
sigma <- var(diff(ret))

kn <- kernel_normal(ret, mean, sigma)
kn

autoplot(kn) +
  scale_color_viridis_c()

# A larger sigma spreads out the distribution
sigma <- var(diff(ret)) / 0.05
kn <- kernel_normal(ret, mean, sigma)

autoplot(kn) +
  scale_color_viridis_c()

Least Information Kernel-Smoothing

Description

This is an internal function that uses the kernel-smoothing approach to compute the posterior probabilities that satisfies some macroeconomic statement.

Usage

least_info_kernel(Y, y, h2)

Arguments

Y

A matrix with the macroeconomic indicators.

y

A numeric scalar with the target to which Y is expected to be near by.

h2

A matrix with the covariance of the Y factor.

Value

A vector with the new probability distribution.

Relative Entropy

Description

Computes the relative entropy of two distributions.

Usage

relative_entropy(prior, posterior)

Arguments

prior

A prior probability distribution.

posterior

A posterior probability distribution.

Value

A double with the relative entropy.

Examples

set.seed(222)

prior <- rep(1 / 100, 100)

posterior <- runif(100)
posterior <- posterior / sum(posterior)

relative_entropy(prior, posterior)

Plot Scenarios

Description

This functions are designed to make it easier to visualize the impact of a View in the P&L distribution.

Usage

scenario_density(x, p, n = 10000)

scenario_histogram(x, p, n = 10000)

Arguments

x

An univariate marginal distribution.

p

A probability from the ffp class.

n

An integer scalar with the number of scenarios to be generated.

Details

To generate a scenario-distribution the margins are bootstrapped using bootstrap_scenarios. The number of resamples can be controlled with the n argument (default is n = 10000).

Value

A ggplot2 object.

Examples

x <- diff(log(EuStockMarkets))[, 1]
p <- exp_decay(x, 0.005)

scenario_density(x, p, 500)

scenario_histogram(x, p, 500)

Views on Copulas

Description

Helper to construct constraints on copulas for entropy programming.

Usage

view_on_copula(x, simul, p)

## Default S3 method:
view_on_copula(x, simul, p)

## S3 method for class 'matrix'
view_on_copula(x, simul, p)

## S3 method for class 'xts'
view_on_copula(x, simul, p)

## S3 method for class 'tbl_df'
view_on_copula(x, simul, p)

Arguments

x

A multivariate copula.

simul

A simulated target copula.

p

An object of the ffp class.

Value

A list of the view class.

Examples


  set.seed(1)
  library(ggplot2)

  # Invariants
  ret <- diff(log(EuStockMarkets))
  u <- apply(ret, 2, stats::pnorm) # assuming normal copula
  n <- nrow(u)

  #' Prior probability distribution
  prior <- rep(1 / n, n)

  # Simulated marginals
  simul_marg <- bootstrap_scenarios(ret, as_ffp(prior), as.double(n))

  # Copulas derived from the simulated margins
  simul_cop <- apply(simul_marg, 2, stats::pnorm) # assuming normal copula

  views <- view_on_copula(x = u, simul = simul_cop, p = prior)
  views

  ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nloptr")
  autoplot(ep)

Views on Correlation Structure

Description

Helper to construct views on the correlation matrix.

Usage

view_on_correlation(x, cor)

## Default S3 method:
view_on_correlation(x, cor)

## S3 method for class 'matrix'
view_on_correlation(x, cor)

## S3 method for class 'xts'
view_on_correlation(x, cor)

## S3 method for class 'tbl_df'
view_on_correlation(x, cor)

Arguments

x

An univariate or a multivariate distribution.

cor

A matrix for the target correlation structure of the series in x.

Value

A list of the view class.

Examples

library(ggplot2)

# Invariant
ret <- diff(log(EuStockMarkets))

# Assume that a panic event throws all correlations to the roof!
co <- matrix(0.95, 4, 4)
diag(co) <- 1
co

# Prior probability (usually the equal-weight setting)
prior <- rep(1 / nrow(ret), nrow(ret))

# View
views <- view_on_correlation(x = ret, cor = co)
views

# Optimization
ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nlminb")
autoplot(ep)

# prior correlation structure
stats::cor(ret)

# posterior correlation structure matches the initial view very closely
stats::cov2cor(ffp_moments(x = ret, p = ep)$sigma)

Views on Covariance Matrix

Description

Helper to construct views on variance-covariance matrix.

Usage

view_on_covariance(x, mean, sigma)

## Default S3 method:
view_on_covariance(x, mean, sigma)

## S3 method for class 'matrix'
view_on_covariance(x, mean, sigma)

## S3 method for class 'xts'
view_on_covariance(x, mean, sigma)

## S3 method for class 'tbl_df'
view_on_covariance(x, mean, sigma)

Arguments

x

An univariate or a multivariate distribution.

mean

A double for the location parameter of the series in x.

sigma

A matrix for the target variance-covariance parameter of the series in x.

Value

A list of the view class.

Examples

library(ggplot2)

# Invariant (stationarity)
ret <- diff(log(EuStockMarkets))

# Expectations for location and dispersion parameters
mean <- colMeans(ret) # No active expectations for returns
cor <- matrix(0, ncol = 4, nrow = 4) # diagonal covariance matrix
diag(cor) <- 1                       # diagonal covariance matrix
sds <- apply(ret, 2, sd)             # diagonal covariance matrix
covs <- diag(sds) %*% cor %*% diag(sds) ## diagonal covariance matrix

# prior probabilities (usually equal weight scheme)
prior <- rep(1 / nrow(ret), nrow(ret))

# Views
views <- view_on_covariance(x = ret, mean = mean, sigma = covs)
views

# Optimization
ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nlminb")
autoplot(ep)

# original covariance matrix
stats::cov(ret)

# Posterior covariance matrix
ffp_moments(x = ret, p = ep)$sigma

Views on Joint Distribution

Description

Helper to construct constraints on the entire distribution.

Usage

view_on_joint_distribution(x, simul, p)

## Default S3 method:
view_on_joint_distribution(x, simul, p)

## S3 method for class 'matrix'
view_on_joint_distribution(x, simul, p)

## S3 method for class 'xts'
view_on_joint_distribution(x, simul, p)

## S3 method for class 'tbl_df'
view_on_joint_distribution(x, simul, p)

Arguments

x

An univariate or a multivariate distribution.

simul

An univariate or multivariate simulated panel.

p

An object of the ffp class.

Details

Value

A list of the view class.

Examples

set.seed(1)
library(ggplot2)

# Invariants
ret <- diff(log(EuStockMarkets))
n <- nrow(ret)

#' Prior probability distribution
prior <- rep(1 / n, n)

# Simulated marginals
simul <- bootstrap_scenarios(ret, as_ffp(prior), as.double(n))

views <- view_on_joint_distribution(x = ret, simul = simul, p = prior)
views

ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nlminb")
autoplot(ep)

# location matches
colMeans(simul)
ffp_moments(x = ret, p = ep)$mu

# dispersion matches
cov(simul)
ffp_moments(x = ret, p = ep)$sigma

Views on Marginal Distribution

Description

Helper to construct constraints on the marginal distribution.

Usage

view_on_marginal_distribution(x, simul, p)

## Default S3 method:
view_on_marginal_distribution(x, simul, p)

## S3 method for class 'matrix'
view_on_marginal_distribution(x, simul, p)

## S3 method for class 'xts'
view_on_marginal_distribution(x, simul, p)

## S3 method for class 'tbl_df'
view_on_marginal_distribution(x, simul, p)

Arguments

x

An univariate or a multivariate distribution.

simul

An univariate or multivariate simulated panel.

p

An object of the ffp class.

Details

Value

A list of the view class.

Examples

set.seed(1)
library(ggplot2)

# Invariants
ret <- diff(log(EuStockMarkets))
n <- nrow(ret)

#' Prior probability distribution
prior <- rep(1 / n, n)

# Simulated marginals
simul <- bootstrap_scenarios(ret, as_ffp(prior), as.double(n))

views <- view_on_marginal_distribution(x = ret, simul = simul, p = prior)
views

ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nlminb")
autoplot(ep)

# location matches
colMeans(simul)
ffp_moments(x = ret, p = ep)$mu

# dispersion matches
cov(simul)
ffp_moments(x = ret, p = ep)$sigma

Views on Expected Returns

Description

Helper to construct views on expected returns.

Usage

view_on_mean(x, mean)

## Default S3 method:
view_on_mean(x, mean)

## S3 method for class 'matrix'
view_on_mean(x, mean)

## S3 method for class 'xts'
view_on_mean(x, mean)

## S3 method for class 'tbl_df'
view_on_mean(x, mean)

Arguments

x

An univariate or a multivariate distribution.

mean

A double for the target location parameter of the series in x.

Value

A list of the view class.

Examples

library(ggplot2)

# Invariant
ret <- diff(log(EuStockMarkets))
n   <- nrow(ret)

# View on expected returns (here is 2% for each asset)
mean <- rep(0.02, 4)

# Prior probabilities (usually equal weight scheme)
prior <- rep(1 / n, n)

# View
views <- view_on_mean(x = ret, mean = mean)
views

# Optimization
ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nlminb")
autoplot(ep)

# Probabilities are twisted in such a way that the posterior
# `mu` match's exactly with previously stated beliefs
ffp_moments(x = ret, p = ep)$mu

Views on Relative Performance

Description

Helper to construct views on relative performance of assets.

Usage

view_on_rank(x, rank)

## Default S3 method:
view_on_rank(x, rank)

## S3 method for class 'matrix'
view_on_rank(x, rank)

## S3 method for class 'xts'
view_on_rank(x, rank)

## S3 method for class 'tbl_df'
view_on_rank(x, rank)

Arguments

x

An univariate or a multivariate distribution.

rank

A integer with the assets rank (from the worst to the best performer).

Details

If rank = c(2, 1) it is implied that asset in the first column will outperform the asset in the second column. For longer vectors the interpretation is the same: assets on the right will outperform assets on the left.

Value

A list of the view class.

Examples

library(ggplot2)

# Invariants
x <- diff(log(EuStockMarkets))
prior <- rep(1 / nrow(x), nrow(x))

# asset in the first col will outperform the asset in the second col (DAX will
# outperform SMI).
views <- view_on_rank(x = x, rank = c(2, 1))
views

ep <- entropy_pooling(p = prior, A = views$A, b = views$b, solver = "nloptr")
autoplot(ep)

# Prior Returns (SMI > DAX)
colMeans(x)[1:2]

# Posterior Returns (DAX > SMI)
ffp_moments(x, ep)$mu[1:2]

Views on Volatility

Description

Helper to construct views on volatility.

Usage

view_on_volatility(x, vol)

## Default S3 method:
view_on_volatility(x, vol)

## S3 method for class 'matrix'
view_on_volatility(x, vol)

## S3 method for class 'xts'
view_on_volatility(x, vol)

## S3 method for class 'tbl_df'
view_on_volatility(x, vol)

Arguments

x

An univariate or a multivariate distribution.

vol

A double for the target volatility structure of the series in x.

Value

A list of the view class.

Examples

library(ggplot2)

# Invariant
ret <- diff(log(EuStockMarkets))
n   <- nrow(ret)

# Expected a volatility 30% higher than historical average
vol <- apply(ret, 2, stats::sd) * 1.3

# Prior Probabilities
prior <- rep(1 / n, n)

# Views
views <- view_on_volatility(x = ret, vol = vol)
views

# Optimization
ep <- entropy_pooling(p = prior, Aeq = views$Aeq, beq = views$beq, solver = "nlminb")
autoplot(ep)

# Desired volatility
vol

# Posterior volatility matches very closely with the desired volatility
sqrt(diag(ffp_moments(x = ret, p = ep)$sigma))