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

The Generalized Beta of the Second Kind (GB2) distribution is widely recognized in econometric literature as the most versatile parametric model for income and wealth distributions, encapsulating Pareto, Dagum, Singh-Maddala, and Fisk distributions as special cases. By synthesizing the GB2 family 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 Framework of the GB2 Composite Model

The GB2 distribution is parameterized by scale parameter (b) and shape parameters (a, p, q). Its probability density function is expressed as:

f(x) = [a * x^(a * p – 1)] / [b^(a * p) * B(p, q) * (1 + (x / b)^a)^(p + q)]

where B(p, q) is the Euler beta function. 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_GB2_SHORROCKS:

=LAMBDA(dataset, param_a, param_b, param_p, param_q,
    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),
                y, (cohort_mean / param_b)^param_a,
                gb2_kernel, (y^param_p) / ((1 + y)^(param_p + param_q)),
                downside_dev, SQRT(AVERAGE((subset - cohort_mean)^2)),
                asym_weight, (11 - r) / 10,
                CHOOSE(c, r, cohort_mean, gb2_kernel, downside_dev, asym_weight * downside_dev)
            )
        )),
        header_row, HSTACK("Decile", "Mean", "GB2 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 shape parameter a in cell C2 (e.g. 2.85).
3. Enter the scale parameter b in cell C3 (e.g. 38000).
4. Enter the shape parameter p in cell C4 (e.g. 0.75).
5. Enter the shape parameter q in cell C5 (e.g. 1.15).
6. In cell E2, execute the formula:

   =CALC_GB2_SHORROCKS(A2:A5000, C2, C3, C4, C5)

The custom formula immediately spills a comprehensive 11-row by 5-column analysis table detailing decile means, GB2 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.