Building an Efficient Frontier from Forecast Returns and Covariances

# Calculating the efficient frontier from expected returns and SD


# Calculating the efficient frontier from expected returns and SD












I'm trying to calculate the efficient frontier (and the optimal portfolio at the Sharpe ratio) given two vectors for a portfolio: (1) expected returns and (2) historical standard deviations. I would like to be able to calculate this in R. Using the `portfolioFrontier()` function of the `fPortfolio` package in R, I have successfully calculated the efficient frontier and optimal portfolio allocation at the Sharpe ratio using a time series of historical returns. However, the `fPortfolio` package only appears to allow back-testing on a time series. I would like to calculate the efficient frontier and optimal portfolio at the Sharpe ratio for future (i.e., expected) returns. How can I do this?

Ideally this would be implemented with a function in R. The closest resource I could find was from this website using Octave code. I successfully translated the code to R, but the efficient frontier doesn't appear to match (or be as accurate) as the one from the R package.

Here's my attempt in R (translating Octave code from the above website):

```
expectedReturns <- c(4, 2, 13, 10)
covarianceMatrix <- matrix(c(185, 86.5, 80, 20, 86.5, 196, 76, 13.5, 80, 76, 411, -19, 20, 13.5, -19, 25), nrow=4, ncol=4, byrow=TRUE)

# Calculate Efficient Frontier
unity <- rep(1, length(expectedReturns))
A <- unity %*% solve(covarianceMatrix) %*% unity
B <- unity %*% solve(covarianceMatrix) %*% expectedReturns
C <- expectedReturns %*% solve(covarianceMatrix) %*% expectedReturns
D <- A*C-B^2

mu = seq(0, 30, by=.1)
minVar = ((A*mu^2)-2*B*mu+C)/D
minSD = sqrt(minVar)

plot(minSD, mu)
```

## Answer by Enrico Schumann (score 3, accepted)

https://quant.stackexchange.com/a/41182

As indicated in my comment, the function `mvFrontier` in the development version of the NMOF package may help you. (Disclosure: I am the package maintainer.) You may get the latest version from GitHub.

Some remarks, first on correlation: an efficient frontier shows portfolio risk, typically volatility, compared with portfolio return. Portfolio volatility is a function of both the volatilities of the assets and the return-correlation between these assets, so you cannot get rid of correlation.

The variance-covariance matrix may be decomposed into the (matrix) product `S` times `C` times `S`, in which `S` is a diagonal matrix with the standard deviations on its main diagonal and zeros elsewhere, and in which `C` is the correlation matrix.

Assume you have the following forecasts:

```
na <- 4                            ## number of assets 
vols <- c(0.10, 0.15, 0.20, 0.22)  ## forecast vols
m <- c(0.06, 0.12, 0.09, 0.07)     ## forecast returns
```

Then a covariance matrix for a constant correlation of 0.5 may be computed in this way:

```
const_cor <- function(rho, na) {
    C <- array(rho, dim = c(na, na))
    diag(C) <- 1
    C
}
var <- diag(vols) %*% const_cor(0.5, na) %*% diag(vols)
```

So you may want to experiment with different assumptions about the correlations: it is difficult to come up with valid arbitrary correlation matrices, but what works is constant positive correlation (i.e. all pairwise correlations are 0.1, or 0.2, ...). Depending on the actual data, the correlation may make little difference to the results (see this note, for instance).

Another remark, on matrix derivations (as in the link you provided): I would prefer to tackle the problem as a optimisation problem instead of following some analytical approach. The advantage is that you may want to introduce restrictions (not allow short shales, say), which becomes more difficult with the analytical approach.

So here would be an example for calling `mvFrontier`:

```
library("NMOF")
wmax <- 1     ## maximum holding size
wmin <- 0.0   ## minimum holding size
rf <- 0.02

## without cash
p1 <- mvFrontier(m, var, wmin = wmin, wmax = wmax, n = 50)

## with cash
p2 <- mvFrontier(m, var, wmin = wmin, wmax = wmax, n = 50, rf = rf)

plot(p1$volatility, p1$return, pch = 19, cex = 0.5, type = "o",
     xlab = "Expected volatility",
     ylab = "Expected return")
lines(p2$volatility, p2$return, col = grey(0.5))
abline(v = 0, h = rf)
```

## Answer by Ajay Verma (score 0)

https://quant.stackexchange.com/a/42412

The same can be achieved for fPortfolio package using SetEstimator. Example below:

```
expectedReturns <- c(4, 2, 13, 10)
covarianceMatrix <- matrix(c(185, 86.5, 80, 20, 86.5, 196, 76, 13.5, 80, 76, 411, -19, 20, 13.5, -19, 25), nrow=4, ncol=4, byrow=TRUE)

covtEstimator <- function (x,data,spec) {
x.mat = as.matrix(x)
list(mu=expectedReturns,Sigma=covarianceMatrix)
}

# Calculate Efficient Frontier
defaultSpec <- portfolioSpec()
setEstimator(defaultSpec) <- 'covtEstimator'
efficientPortfolio(yourreturndata, defaultSpec, constraints = "LongOnly")
```

Additional reference : Page 293 in this pdf