5 optimization - What Practitioners Need To Know About...

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What Practitioners Need To Know . . . About Optimization I LU ce. g 10 Mark Kritzman Windham Capital Management Optimization is a process by which we determine the most favorable tradeoff between com- peting interests, given the con- straints we face. Within the con- text of portfolio management, the competing interests are risk re- duction and return enhancement. Asset allocation is one form of optimization. We use an opti- mizer to identify the asset weights that produce the lowest levd of risk for various levels of expeaed return. Optimization is also used to construa portfolios of securi- ties that minimize risk in terms of tracking error relative to a bench- mark portfolio. In these applica- tions, 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 ofeetting long and short positions. Suppose, for ex- ample, that we wish to purchase currencies expeaed to yield high returns and to sell currencies ex- pected to yield low returns, with the net result that we are neither long nor short the local currency. In uiis case, we would impose a constraint that the currency ejqjo- sures sum to zero. This column is intended as a tu- torial on optimization. We will demonstrate, through the use of numerical examples, how to opti- mize a two-asset portfolio with only a pencil and me back of an envelope. If you wish to indude three assets, you may need the front of the envelope as well. Beyond three assets, a computer would come in handy. Hie Ol^ective Fanctton Suppose we wish to identify the ccHTibinations of stocks and bonds that produce the lowest levels of risk for varying amounts of ex- pected return. To begin, we must define a portfolio's expected re- turn and risk. The expected return of a portfo- lio comprised of just stocks and bonds is simply the weighted av- erage of the assets' expected re- turns, as shown below: Eq. 1 Rp = (Rs Ws) + (RB WB) where Rp = the portfolio's expeaed return; Rs = the expected return of stocks; RB = the expeaed return of bonds; Wg = the percentage of the portfolio allocatcd to stocks; and WB = the percentage allocated to bonds. Portfolio risk is a little trickier. It is defined as volatility, and it is measured by the standard devia- tion or variance (the standard de- viation squared) around the port- folio's expected return. To compute a portfolio's variance, we must consider not only the \^riance of the component assets' returns, but also the extent to which the assets' returns co-vary.^ The variance of a portfolio of stocks and bonds is computed as follows: Eq. 2 + 2cor(o-s where V = the portfolio variants; (T^ = the standard deviation of stocks; WB = the standard deviation of bonds; and cor = the correlation be- tween stocks and bonds.
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