equilibrium_mean {BLModel}

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)