56 Optimizing Optimization
positions in futures, indices, currencies, or stocks. For proprietary trading desks
and “ market neutral ” hedge funds, a common constraint is that the long and
short sides are equal in value, or that the overall (absolute) risk is constrained.
If the objective is to minimize the absolute risk, then the problem is known as
optimized hedging.
3.2.5 Active quant management
The active quant is measured against an index or benchmark, but has nonzero
alphas for all the stocks of interest, and attempts to maximize the relative
return (outperformance) while staying within a given risk budget (constraint
on tracking error).
3.2.6 Asset allocation
Asset allocation is characterized by a relatively small number of variables (typi-
cally less than 50) and long-only allocations of weights that must sum to unity.
Asset allocation is an essential part of the portfolio construction for all pension
funds, and for private client investment management.
3.2.7 Index tracking
Index tracking is a form of passive investing in which the benchmark is defined
by a particular index (e.g., the S & P 500) and the problem is to minimize the
active risk or “ tracking error ” of a fund. Typical constraints are to be long-
only in the stocks of the benchmark, hold little or no cash, and to find the min-
imum number of names in the tracking fund to stay within a certain tracking
error. The alpha (expected return) term does not enter into the optimization.
3.3 Mean –variance optimization
3.3.1 A technical overview
The expected return of a portfolio is a linear combination of the “ alphas ” of
each asset within the portfolio, while the risk (absolute risk/volatility or active
risk/tracking error) is most usually expressed as a quadratic expression based
on the covariance matrix of asset returns. Transaction costs for a rebalance
depend on the size and direction of the trades, and may include both linear and
nonlinear terms, to model both commission and market impact of trading.
When creating or rebalancing a portfolio, the optimizer must trade off both
risk and cost against expected return — this is achieved by minimizing a “ utility
function ” that includes positive risk and cost terms and negative return terms.
The general mathematical form of this utility function, when there is a known
benchmark and the risk is defined as active risk, can be expressed as: