Sharpe's Algorithm in Asset Allocation
Team Latte
October 1, 2006

Every time we do training in the area of asset allocation models and portfolio engineering we are invariably asked by some trainee as to whether Sharpe's algorithm (of portfolio optimization) is more effective than "classical" efficient frontier modelling of Markowitz. Invariably, the discussion gravitates, with some degree of aggressiveness on the part of the trainees, towards proving how the classical efficient frontier construction is too theoretical in its approach and that Sharpe's Algorithm is way more practical. Of late we have encountered this question more than three times in a single month which means that we need to bring this issue out in the public.

Sharpe's algorithm, published in 1987, is highly intuitive and lends itself to real world model building by the fund managers and investors. But in our opinion it is mathematically much more tractable. One can do away with cumbersome matrix algebra (not that it is an issue at all given the computer power these days) though of course, there could some conceptual issues with the notion of "risk aversion" coefficient. A big appeal of this algorithm is that it is much more dynamic than the classic efficient frontier optimization models. A manager can continuously, at discrete time intervals, keep adjusting his portfolio to keep it "efficient".

In Sharpe's model, we simply maximize our utility function of the investor under certain simple constraints.

In the above equation utility is a linear function of the expected return of the portfolio, is the variance of the portfolio and is the risk aversion coefficient. The risk aversion coefficient measures by how many units an investor is willing to forgo the return in order to reduce the risk (measured by the square root of variance) by one unit. Let's take a simple example to illustrate the algorithm (in an Excel spreadsheet with VBA macros one can easily handle a 100 asset portfolio).

In a two portfolio world the investors portfolio consists of only two assets 1 and 2 and the respective returns are and and the respective volatilities are and . The correlation between the assets is .

Now we maximize the modified objective (utility) function:

Mathematically, the maximization process results in the following partial derivatives:

Say, a fund manager is invested between stock A and stock B. If the expected return of asset A is 8% and its volatility is 9% and the expected return of stock B is 14% and the volatility is 19% and the correlation between the two stocks is 0.35. Further, say, the fund manager has invested 25% of his funds in stock A and 75% in stock B. Most importantly, let us assume that the risk aversion coefficient, lambda, is 2.

If we plug in the above values in the above partial derivatives we'll get

This implies that the fund manager can increase the expected utility by 0.054 units if he increases his exposure to stock A by 1% and he can increase the expected utility by 0.126 units if he increases his exposure to stock B by 1%. But since the weight constraint implies that the portfolio allocation can only be 100% (and not more) therefore the manager needs to increase the exposure to stock A by 1% and decrease the exposure to stock B by the same amount.

This will make the new asset allocation percentages as 26% for stock A and 74% for stock B. Using these new asset weights the process needs to repeated again and again until both the partial derivatives are equal. At that point the portfolio's expected utility cannot be improved any further and the portfolio is efficient.

Questions :

1. What if the partial derivatives are negative in value?
2. If we need to construct a number of efficient frontiers of stock A and stock B (think of them as indifference curves) what variable needs to be adjusted?

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