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

Value

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

Value

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

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

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

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)