Maximize the Sharpe Ratio in Axioma Portfolio

Clients occasionally ask, "Is it possible to maximize the Sharpe ratio in Axioma Portfolio?" The answer is, "Absolutely!"

Let's begin by defining our terms. The Sharpe ratio is a measure of risk-adjusted performance or risk-adjusted expected performance that is calculated as follows:

where α is a vector of expected excess returns, Q is a covariance matrix of forecasted risks, and w is a vector of asset holdings. To maximize the Sharpe ratio, we take advantage of a property of the objective known as scale invariance. That is, the holdings w can be scaled up or down arbitrarily without changing the Sharpe ratio of the resulting portfolio, since the weights relative to each other remain unchanged.

Suppose we want to maximize the Sharpe ratio of our long-only portfolio of $1 million, for which we also limit sector exposures. We can formulate the problem we want to solve as maximizing Equation 1, subject to:

where the first constraint specifies a budget of $1 million, s_i gives the definition of sector i, and l_i and u_i give the lower and upper bounds on the holdings in sector i, respectively. In order to implement this within Axioma Portfolio, we introduce a non-negative variable, τ, to produce a homogeneous formulation. Using this additional variable, the problem can be reformulated in an exactly equivalent way as follows:

The additional variable, τ, can be thought of as one additional asset whose weight will be determined by the optimizer. Note that we have scaled the right-hand-side value of the constraints by τ. We have split the range constraints on sectors into two sets of constraints. Once all the asset weights have been determined (including τ's value), τ will be used to scale back down all the other asset weights, to arrive at the final portfolio. So, the optimal solution is the set of weights . Note that the property of scale invariance allowed us to specify any as an arbitrary numerator value in equation two. Because it will really be adjusted by the optimal τ value, the actual expected return of the portfolio is not being limited at at all.

To visualize how this transformation works in the context of Axioma Portfolio, set up an initial workspace with sector constraints and a conventional objective to maximize alpha minus a scaled variance term. This workspace can be used to generate a frontier of possible optimal portfolios. The portfolio with the maximum Sharpe ratio is the portfolio on the frontier where the slope of the tangent is the steepest. We can recognize roughly where along the frontier that portfolio lies, although we won't yet have found that exact portfolio. With some iteration, we could get very close. Another way to approximately find the portfolio with the best Sharpe ratio on the frontier is to use the Sharpe ratio-versus-turnover plot that is part of the Axioma Portfolio frontier feature.

Next, let's modify the objective and constraints as in Equation 2, and add in the synthetic asset called "tau." After re-running the optimizer, we derive a new portfolio. As a last step, we divide all its asset holdings by the derived value of tau. Now let's examine this final optimal portfolio within Axioma Portfolio. We can confirm that the Sharpe ratio (Expected Return divided by Total Risk, taken from the Summary tab) is indeed maximized, and that we have analytically found the precise portfolio along the frontier we had derived above.

Click here for a printable version of this page.