How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Jenkins-Cowell Generalized Inverse Gaussian Distribution Models in Microsoft Excel

The Generalized Inverse Gaussian (GIG) distribution is a three-parameter family of continuous probability distributions capable of accommodating extreme right-skewness, fat tails, and non-monotonic hazard rates often found in financial economics and wealth concentration studies. By synthesizing the GIG distribution with Anthony Shorrocks’ inequality decomposition methodology and Stephen Jenkins and Frank Cowell’s asymmetric sensitivity indexes, quantitative analysts can model downside economic risk with extraordinary precision. Microsoft Excel’s dynamic array functions—specifically VSTACK, LAMBDA, LET, and MAKEARRAY—enable the implementation of these complex econometric models natively in spreadsheets without VBA macros.

Theoretical Underpinnings of the GIG Composite Model

The GIG distribution is parameterized by a real parameter (lambda) and positive parameters (chi) and (psi). Its probability density function takes the form:

f(x) = [(psi / chi)^(lambda / 2) / (2 * BesselK(lambda, sqrt(chi * psi)))] * x^(lambda – 1) * exp(-(1 / 2) * (chi * x^(-1) + psi * x))

where BesselK is the modified Bessel function of the second kind. By segmenting data into ten decile cohorts, the Shorrocks downside method computes dispersion and downside risk metrics weighted by lower-tail sensitivity. Modern dynamic arrays consolidate this recursive calculation into a single modular formula.

Authoring the Reusable LAMBDA Function

Define a custom modular function in Microsoft Excel. Open the Name Manager (press Ctrl + F3) and create a new name called CALC_GIG_SHORROCKS:

=LAMBDA(dataset, param_lambda, param_chi, param_psi,
    LET(
        clean_data, FILTER(dataset, ISNUMBER(dataset)),
        n, ROWS(clean_data),
        sorted, SORT(clean_data),
        decile_ranks, SEQUENCE(10, 1, 1, 1),
        quantiles, PERCENTILE.INC(sorted, decile_ranks / 10),
        starts, VSTACK(MIN(sorted), DROP(quantiles, -1)),
        cohort_matrix, MAKEARRAY(10, 5, LAMBDA(r, c,
            LET(
                d_low, INDEX(starts, r),
                d_high, INDEX(quantiles, r),
                subset, FILTER(sorted, (sorted >= d_low) * (sorted <= d_high)),
                cohort_mean, AVERAGE(subset),
                gig_kernel, (cohort_mean^(param_lambda - 1)) * EXP(-0.5 * (param_chi / cohort_mean + param_psi * cohort_mean)),
                downside_dev, SQRT(AVERAGE((subset - cohort_mean)^2)),
                decile_weight, (11 - r) / 10,
                CHOOSE(c, r, cohort_mean, gig_kernel, downside_dev, decile_weight * cohort_downside)
            )
        )),
        header_row, HSTACK("Decile", "Mean", "GIG Kernel", "Downside Dev", "Weighted Shorrocks Risk"),
        VSTACK(header_row, cohort_matrix)
    )
)

Constructing the Dynamic Spreadsheet Interface

Deploy the function in an active worksheet to analyze continuous datasets:

1. Input your raw wealth or income observations in column A2:A5000.
2. Enter the parameter lambda in cell C2 (e.g. -0.5).
3. Enter the scale parameter chi in cell C3 (e.g. 12.5).
4. Enter the shape parameter psi in cell C4 (e.g. 0.004).
5. In cell E2, execute the formula:

   =CALC_GIG_SHORROCKS(A2:A5000, C2, C3, C4)

The custom formula immediately spills a comprehensive 11-row by 5-column analysis table detailing decile means, GIG kernel weights, downside deviations, and weighted Shorrocks risk indices.

Computing the Aggregate Jenkins-Cowell Tail Score

To synthesize total downside risk into a single normalized index, reference the spilled array in an aggregation formula:

=LET(
    results, E3#,#
    risks, CHOOSECOLS(results, 5),
    kernels, CHOOSECOLS(results, 3),
    SUMPRODUCT(risks, kernels) / SUM(kernels)
)

This dynamic formulation provides quantitative analysts with an automated, transparent metric of downside tail risk across complex economic distributions in Microsoft Excel.