In the portfolio optimization models that we considered in this chapter, risk is represented by variance or standard deviation of portfolio return.

An alternative is using MAD (mean absolute deviation):

E· y ^ RiWi - E ^RkWk

.fc = l where Ri is the random return of asset i and w¿ is its portfolio weight. Suppose that we do not trust any probability distribution for return, but we have a time series of historical data. Let r¿¿ be the observed return of asset i in time bucket t, t = Ι,.,.,Τ.

• Build a MILP model to find the minimum MAD portfolio subject to the following constraints:

— Short selling is not allowed.

— Expected return should not be below a given target.

— To avoid a fragmented portfolio, no more than k < n assets can be included in the portfolio, and if an asset is included, there is a lower bound on its weight.

— Assets are partitioned according to industrial sectors (e.g., banks, energy, chemicals, etc), as well as according to geographic criteria

(Asia, Europe, etc.). For each set of assets, overall lower and upper bounds are to be satisfied.

• What is the danger of this modeling approach, based on observed time series?