Quadratic Risk Penalty

Appearing on the Constraints Worksheet, the risk penalty is a quadratic function that raises the effective risk of an asset by adding an independent risk factor to that asset. (To access the Quadratic Risk Penalty column, activate the quadratic risk option in the Constraints Menu.)  he new effective standard deviation in each simulation will be the square root of the exact quadratic risk penalty, squared, plus the simulated resampled standard deviation, squared.  

The quadratic risk penalty is generally used to increase the risk for assets where historical risk is underestimated and heuristic estimates are required, such as, for example, private equities, hedge funds, and real estate. In these and other cases, historical risk estimates may be poorly measured, or limited by lack of transparency, appraisal data, and lockup periods. A risk penalty may also be sensible for emerging market and other indices where long-term return data may be limited or inadequate as an estimate of future risk.  

Normally, the Optimizer calculates portfolio risk with respect to the standard deviations and correlations specified in the inputs.  However, if quadratic risk penalties are present, it may be advantageous to calculate total risk for portfolios and assets, which includes quadratic risk penalties in the risk calculation. This alternative calculation of risk will affect portfolios with nonzero weights of assets with nonzero quadratic risk penalties, as well as the assets themselves. To visualize the impact of quadratic risk penalties, enable the Add Quadratic Risk Penalty to Risk option in the Charts sub-menu within the Display Menu on the ribbon. The results can be seen on the Efficient Frontier Chart on either the Results or Charts Worksheets.  

Considerations

• Be aware that a quadratic risk penalty is a function of the total portfolio weight, unlike the quadratic return penalty, which is a function of deviations from the initial portfolio.
• Quadratic risk penalties have a greater impact on the lower risk end of the frontier.
• Adding a quadratic risk penalty to an asset will never result in an increased allocation to that asset.  However, you can still have a high allocation to that asset on the higher volatility end of the frontier.
• Note that quadratic risk penalties are equivalent to adding numerical value to the diagonal of the input covariance matrix. Therefore, quadratic risk penalties do not impact the feasibility of optimizations in the same way that asset bound or customized constraints do. Any feasible optimization will also be feasible with different quadratic risk penalties.

Enter any risk penalties in the Quadratic Risk Penalty column.  

Quadratic risk penalties can be saved and loaded as part of a constraint set.

A Note about the Mathematics of Quadratic Risk Penalties:

Quadratic Risk Penalties add additional variance to the covariance matrix. The diagonal elements of the covariance matrix corresponding to any nonzero Quadratic Risk Penalties are increased by the square of the specified Quadratic Risk Penalty. To the Optimizer, this has the effect of reducing the importance of the correlations, since the recalculated correlations with respect to the total risk including Quadratic Risk Penalties will be dividing each off-diagonal element of the covariance by a greater number, thereby reducing the value of that correlation with respect to total risk.

A consequence of this additional component of the total risk is that portfolio risk will be increased.

Portfolio Risk without Risk Penalties is calculated as , where

o     is the portfolio standard deviation,

o     P is the vector of portfolio weights,

o      is the covariance matrix of assets,

o     N is the number of assets in the portfolio,

o      and  are the portfolio weights of assets i and j,

o       is the correlation between assets i and j, and

o      and  are the standard deviations of assets i and j.

With the Quadratic Risk Penalties, the Total Risk is calculated as , where

o     Q is the N by N matrix with the squares of the Quadratic Risk Penalties on the diagonal, and

o      is the Quadratic Risk Penalty corresponding to asset i.

Note that the second calculation, of total risk, is mathematically always greater than the first calculation.