BL_post_distr {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)