equilibrium_mean {BLModel} | R Documentation |
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) ≥ 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.
equilibrium_mean(dat, w_m, RM, risk = c("CVAR", "DCVAR", "LSAD", "MAD"), alpha = 0.95)
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. |
market_returns | a vector of market returns obtain by inverse optimization; this is vector E(r) |
from the description of this function. |
Palczewski, J., Palczewski, A., Black-Litterman Model for Continuous Distributions (2016). Available at SSRN: https://ssrn.com/abstract=2744621.
# 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)