Package 'BLModel'

Computes the Black-Litterman posterior distribution.

Description

BL_post_distr computes posterior distribution in the Black-Litterman model starting from arbitrary prior distribution given as a discrete time series dat and using views_distr – submitted by the user distribution of views.

Usage

BL_post_distr (dat, returns_freq, prior_type = c("elliptic", NULL), market_portfolio,
SR, P, q, tau, risk = c("CVAR", "DCVAR", "LSAD", "MAD"),  alpha = NULL,
views_distr, views_cov_matrix_type = c("diag", "full"), cov_matrix = NULL)

Arguments

dat	Time series of returns data; dat = cbind(rr, pk), where $rr$ is an array (time series) of market asset returns, for $n$ returns and $k$ assets it is an array with $\dim(rr) = (n, k)$, $pk$ is a vector of length $n$ containing probabilities of returns.
returns_freq	Frequency of data in time series dat; given as a number of data rows corresponding to the period of 1 year, i.e. 52 for weekly data or 12 for monthly data.
prior_type	Type of distribution in time series dat; can be "elliptic" – $rr$ is distributed according to (any) elliptical distribution, NULL – $rr$ is distributed according to any non-elliptical distribution.
market_portfolio	Market portfolio – benchmark (equilibrium) portfolio (for details see Palczewski&Palczewski).
SR	Benchmark Sharpe ratio.
P	"Pick" matrix in the Black-Litterman model (see Palczewski&Palczewski).
q	Vector of investor's views on future returns in the Black-Litterman model (see Palczewski&Palczewski).
tau	Confidence parameter in the Black-Litterman model.
risk	Risk measure chosen for optimization; one of "CVAR", "DCVAR", "LSAD", "MAD", where "CVAR" – denotes Conditional Value-at-Risk (CVaR), "DCVAR" – denotes deviation CVaR, "LSAD" – denotes Lower Semi Absolute Deviation, "MAD" – denotes Mean Absolute Deviation.
alpha	Value of alpha quantile in the definition of risk measures CVAR and DCVAR. Can be any number when risk measure is parameter free.
views_distr	Distribution of views. An external function submitted by the user which computes densities of the distribution of views in given data points. It is assumed implicitly that this distribution is an elliptical distribution (but any other distribution type can be used provided calling to this function will preserve described below structure). Call to that function has to be of the following form FUN(x,q,covmat,COF = NULL), where x is a data points matrix which collects in rows the coordinates of the points in which density is computed, q is a vector of investor's views, covmat is covariance matrix of the distribution and COF is a vector of additional parameters characterizing the distribution (if needed).
views_cov_matrix_type	Type of the covariance matrix of the distribution of views; can be: "diag" – diagonal part of the covariance matrix is used; "full" – the complete covariance matrix is used; (for details see Palczewski&Palczewski).
cov_matrix	Covariance matrix used for computation of market expected return (RM) from the formula RM = SR * sqrt( t(w_m) * cov_matrix * w_m) where w_m is market portfolio and SR – benchmark Sharpe ratio. When cov_matrix = NULL covariance matrix is computed from matrix $rr$ in data set dat.

Value

post_distr	a time series of data for posterior distribution; for a time series of length $n$ and $k$ assets
	it is a matrix $(n, k+1)$, where columns (1:k) contain return vectors and the last column
	probabilities of returns.

References

Palczewski, J., Palczewski, A., Black-Litterman Model for Continuous Distributions (2016). Available at SSRN: https://ssrn.com/abstract=2744621.

Examples

library(mvtnorm)
k = 3 
num =100
dat <-  cbind(rmvnorm (n=num, mean = rep(0,k), sigma=diag(k)), matrix(1/num,num,1)) 
# a data sample with num rows and (k+1) columns for k assets;
returns_freq = 52 # we assume that data frequency is 1 week
w_m <- rep(1/k,k) # benchmark portfolio, a vector of length k,
SR = 0.5 # Sharpe ratio
Pe <- diag(k) # we assume that views are "absolute views"
qe <- rep(0.05, k) # user's opinions on future returns (views)
tau = 0.02
BL_post_distr(dat, returns_freq, NULL, w_m, SR, Pe, qe, tau, risk = "MAD", alpha = 0,
views_distr = observ_normal, "diag", cov_matrix = NULL)

---

Solves the inverse optimization to mean-risk standard optimization problem to find equilibrium returns. The function is invoked by BL_post_distr and arguments are supplemented by BL_post_distr.

Description

The function computes the vector of equilibrium returns implied by a market portfolio. The vector of means for the mean-risk optimization problem is found by inverse optimization.
The optimization problem is:
$\min F(w_m^{T} r)$
subject to
$w_m^{T} E(r) \ge RM$,
where
$F$ is a risk measure – one from the list c("CVAR", "DCVAR", "LSAD", "MAD"),
$r$ is a time series of market returns,
$w_m$ is market portfolio,
$RM$ is market expected return.

Usage

equilibrium_mean(dat, w_m, RM, risk = c("CVAR", "DCVAR", "LSAD", "MAD"),
  alpha = 0.95)

Arguments

dat	Time series of returns data; dat = cbind(rr, pk), where $rr$ is an array (time series) of market asset returns, for $n$ returns and $k$ assets it is an array with $\dim(rr) = (n, k)$, $pk$ is a vector of length $n$ containing probabilities of returns.
w_m	Market portfolio.
RM	Market_expected_return.
risk	A risk measure, one from the list c("CVAR", "DCVAR", "LSAD", "MAD").
alpha	Value of alpha quantile in the definition of risk measures CVAR and DCVAR. Can be any number when risk measure is parameter free.

Value

market_returns	a vector of market returns obtain by inverse optimization; this is vector $E(r)$
	from the description of this function.

References

Palczewski, J., Palczewski, A., Black-Litterman Model for Continuous Distributions (2016). Available at SSRN: https://ssrn.com/abstract=2744621.

Examples

# In normal usage all data are supplemented by function BL_post_distr.
library(mvtnorm)
k = 3 
num =100
dat <-  cbind(rmvnorm (n=num, mean = rep(0,k), sigma=diag(k)), matrix(1/num,num,1)) 
# a data sample with num rows and (k+1) columns for k assets;
w_m <- rep(1/k,k) # market portfolio.
RM = 0.05 # market expected return.
equilibrium_mean (dat, w_m, RM, risk = "CVAR", alpha = 0.95)

---

Example of distribution of views – normal distribution

Description

Function observ_normal computes density of normal distribution of views using the formula
$f(x) = c_k*\exp(-((x-q)^{T}*covmat^{-1}*(x-q))/2)$,
where $c_k$ is a normalization constant (depends on the dimension of $x$ and $q$).

Usage

observ_normal(x, q, covmat)

Arguments

x	Data points matrix which collects in rows coordinates of points in which distribution density is computed.
q	Vector of investor's views.
covmat	Covariance matrix of the distribution.

Value

function returns a vector of distribution densities in data points x.

References

Palczewski, J., Palczewski, A., Black-Litterman Model for Continuous Distributions (2016). Available at SSRN: https://ssrn.com/abstract=2744621.

Examples

k =3
observ_normal (x = matrix(c(rep(0.5,k),rep(0.2,k)),k,2), q = matrix(0,k,1), 
               covmat = diag(k))

---

Example of distribution of views – power exponential distribution

Description

Function observ_powerexp computes density of power exponential distribution of views using the formula
$f(x) = c_k*\exp(- ((x-q)^{T}*\Sigma^{-1}*(x-q))^{\beta}/2)$,
where $c_k$ is a normalization constant (depends on the dimension of $x$ and $q$) and $\Sigma$ is the dispersion matrix.

Usage

observ_powerexp(x, q, covmat, beta = 0.6)

Arguments

x	Data points matrix which collects in rows coordinates of points in which distribution density is computed.
q	Vector of investor's views.
covmat	Covariance matrix of the distribution; dispersion matrix $\Sigma$ is computed from covmat.
beta	Shape parameter of the power exponential distribution.

Value

function returns a vector of distribution densities in data points x.

References

Gomez, E., Gomez-Villegas, M., Marin, J., A multivariate generalization of the power exponential family of distributions. Commun. Statist. Theory Methods, 27 (1998), 589–600. DOI: 10.1080/03610929808832115

Examples

k =3
observ_powerexp (x = matrix(c(rep(0.5,k),rep(0.2,k)),k,2), q = matrix(0,k,1),
               covmat = diag(k), beta = 0.6)

---

Example of distribution of views – Student t-distribution

Description

Function observ_ts computes density of Student t-distribution of views using the formula
$f(x) = c_k*(1 +(x-q)^{T}*\Sigma^{-1}*(x-q)/df)^{(-(df+k)/2)}$,
where $c_k$ is a normalization constant (depends on the dimension of $x$ and $q$) and $\Sigma$ is the dispersion matrix.

Usage

observ_ts(x, q, covmat, df = 5)

Arguments

x	Data points matrix which collects in rows coordinates of points in which distribution density is computed.
q	Vector of investor's views.
covmat	Covariance matrix of the distribution; dispersion matrix $\Sigma$ is computed from covmat.
df	Number of degrees of freedom of Students t-distribution.

Value

function returns a vector of observation distribution densities in data points x.

References

Kotz, S., Nadarajah, S., Multivariate t Distributions and Their Applications. Cambridge University Press, 2004.

Examples

k =3
observ_ts (x = matrix(c(rep(0.5,k),rep(0.2,k)),k,2), q = matrix(0,k,1), covmat = diag(k), 
         df=5)

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