The Cost of Constraints

The CAPM is an equilibrium model in which investors’ supply and demand decisions defines the market portfolio, in which all investors possess identical information and preferences are mean variance optimizers, and for which there are no restrictions on selling. Restrictions make portfolios inefficient relative to the market portfolio. In reality, there are many practical instances of restrictions; institutional investors, for example, may face short-selling restrictions as risk controls and others (like you and me) may find it impractical to engage in short sales for a variety of reasons (brokerage fees, trade size limitations, and downside risk). These restrictions will necessarily push our portfolios off the efficient frontier.

The impact of no short-sales restrictions can be seen in the spreadsheet exercises for this chapter. In this application, we solve for the minimum variance portfolio subject to an additional restriction that all weights must be nonzero. The solution is not analytic as was the case for Port 1 and Port 2 given earlier. Instead, it is a search process requiring iteration toward a combination of asset weights that generate the minimum variance for a targeted return. In all cases, the risk on the no-shorts portfolio will be higher. If, for example, we refer to the sheet containing the monthly returns for our 10 firms and set the targeted portfolio return at 2.5 percent, we get the following minimum variance portfolio:

There is but one short position in this portfolio and that is with Phillip Morris (MO). If we restrict short sales, then we must use Excel's Solver function (we used Solver in Chapters 2 and 6). You can view the Solver setup for the short sales by clicking on Solver under the Data tab on the spreadsheet. This restricted portfolio is:

Notice there are no negative weights, but that the risk is slightly higher at 4.3 percent, which is the cost of the restriction. Put differently, the loss of efficiency is borne by the additional risk. Since Port 2 purportedly lies on the efficient frontier, then the No Shorts portfolio must not; in fact, it lies interior to the frontier directly to the right of Port 2, having increased risk for the same return.