The Hidden Cost of Volatility Drag

That said, when it comes to optimizing portfolio returns, the devil is in the details. In our previous analysis, we explored various aspects of kriging, a widely used method for predicting variable values at unmeasured locations. Today, we'll dive deeper into one of its key components: the semivariogram.

The Semivariogram: A Key to Efficient Prediction

The semivariogram is a crucial tool in geostatistical analysis, as it helps determine the system matrix in the prediction step. This equation (6) shows how to minimize the prediction variance σK(x0) under the condition of unbiasedness E( ˆZ(x0)) = Z(x0)), which evaluates to Σi∈Iλi = 1 for ordinary kriging and F⊤λ = f 0 for universal kriging using the design matrix F = (f(xi))⊤ i∈I and f 0 = f(x0). Minimizing σK(x0) finally leads to the kriging equation  C F F⊤ 0   λ θ  =  c0 f 0  (6)

What Does This Mean for Portfolios?

When it comes to portfolios, the semivariogram plays a vital role in determining the system matrix. As discussed earlier, the estimator used by kriging for prediction at a location x0 ∈D has the linear form ˆZ(x0) = λ⊤Z (5) with the kriging weights λ = (λi)i∈I and the data vector Z = (Z(xi))i∈I. Depending on the modeling assumptions regarding m(x), two variants of kriging can be differentiated: Ordinary kriging which uses a constant trend model m(x) = const and universal kriging which models m(x) with a parameter-linear setup m(x) = θ⊤f(x) with a parameter vector θ ∈Rp and a set of regression functions fj(x), j = 1, . . . , p.

Kriging Prediction for the Location x0 Now Consists of Three Steps

Kriging prediction for the location x0 now consists of three steps: 1. Estimation of the semivariogram γ by (4), 2. semivariogram model fitting, and 3. solving equation (6) to determine the kriging weights λ and calcu