Optimization is a process by which we determine the most favorable trade-off between competing interests, given the constraints we face. Within the context of portfolio management, the competing interests are risk reduction and return enhancement. Asset allocation is one form of optimization. We use an optimizer to identify the asset weights that produce the lowest level of risk for various levels of expected return. Optimization is also used to construct portfolios of securities that minimize risk in terms of tracking error relative to a benchmark portfolio. In these applications, we are usually faced with the constraint that the asset weights must sum to one.

We can also employ optimization techniques to manage strategies that call for offsetting long and short positions. Suppose, for example, that we wish to purchase currencies expected to yield high returns and to sell currencies expected to yield low returns, with the net result that we are neither long nor short the local currency. In this case, we would impose a constraint that the currency exposures sum to zero.

This column is intended as a tutorial on optimization. We will demonstrate, through the use of numerical examples, how to optimize a two-asset portfolio with only a pencil and the back of an envelope. If you wish to include three assets, you may need the front of the envelope as well. Beyond three assets, a computer would come in handy.