Portfolio Optimization Algorithms and Corner Portfolios
Team Latte
September 5, 2006

Many a times we land up in discussions with long only fund managers, of which one happened last week, whereby it is argued that the math behind strategic portfolio optimization (Markowitz's efficient frontier techniques using mean-variance analysis) problems don't work well in real life. Some fund managers say: " math is all bunk ", some say "reliance on math models results in erroneous investment decisions ". Last week we had a chance meeting with an exceptionally bright fund manager (not necessarily in terms of making money) who commented that the "every analyst touts the basic problem as the holy grail and moves on but it (basic problem) does not capture the reality of a large mutual or pension fund manager's portfolio..........what matters for us is the corner portfolios........ ".

He was referring to the "basic problem" in portfolio math, first enunciated clearly by William Sharpe, and what we now take it as the classical efficient frontier problem. We were impressed by the fund manager because he went on to talk about the various boundary conditions that are contained in any large fund mandate which really becomes the crux of the portfolio decisions. We thought of writing a few lines on this on our site.

All strategic portfolio optimization algorithms can be broadly classified into either (1) a basic problem or (2) a standard problem. This classification follows William Sharpe's original work. Strategic portfolio optimization entails finding the optimal weights of assets in a portfolio by using an efficient frontier (mean-variance analysis). The assets are held in the portfolio for a certain period of time (investment horizon) and there is no dynamic hedging of the portfolio either through derivatives or portfolio insurance.

Basic Problem

In a basic problem the optimization algorithm is:

The above is a quadratic programming problem and can be solved using the Lagrangian multiplier technique. The lambda in the above equation is a lagrangian multiplier. One needs to find the partial differentials of the above objective function with respect to and all the asset weights and equate them to zero to get the local minima of the function and from those set equations back out the asset weights. In fact the math solution and its application in Excel it is pretty simple and can easily be done using Excel's matrix functions.

Standard Problem

The above problem does not contain any inequalities or boundary conditions. But most mutual fund managers (or long only and pension fund managers) face such constraints in their portfolios. For example, a fund manager may have a mandate to invest exactly 10% in asset 10 or his fund mandate may stipulate that he can invest between 5% and 15% in asset 23 or the funds invested in a certain asset class cannot exceed 7%. These inequalities and boundary conditions lead to what we call a standard problem in portfolio optimization. A standard problem is an enhanced basic problem with boundary conditions and inequalities.

In a standard problem the optimization algorithm is:

The above inequality constraints make the above optimization a mathematical programming problem and the problem is difficult to solve. In the above problem if the fund mandate says that between 5% and 10% should be invested in asset 10 then we'll have . If exactly 20% must be invested in asset 14 then we'll have .

Another complicating constraint becomes that of the lambda, which is also a proxy measure of risk tolerance. What happens in a standard problem - and one that is mathematically difficult to handle on a piece of paper - is that the asset weights, don't vary linearly with throughout the entire range of values of . In some regions it is linear but at other regions we get corners (what is known as "corner portfolios"). Corner portfolios is optimal for a given risk tolerance at which a variable changes status. It is called a corner portfolio because in a graph that plots asset holdings (asset weights) against risk tolerance (lambda in our equation above), two or more variables "turn a corner". As we will see, corner portfolios play a central role in the critical line algorithm. They also are attractive candidates for mutual funds.

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