equilibrium_mean {BLModel} R Documentation

## 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) ≥ 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)

```

[Package BLModel version 1.0.2 Index]