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

In quantitative economic modeling and asset downside vulnerability research, nonnegative distributions representing wealth, operational losses, or survival durations frequently exhibit monotonic decreasing densities with varying tail fatness. The Generalized Half-Normal (GHN) distribution, introduced by Nicholas Cooray and Malwane Ananda, extends the standard half-normal distribution by adding a shape parameter that controls hazard rate shapes and tail decay rates. By synthesizing the GHN distribution with Anthony Shorrocks’ inequality decomposition methodology and Stephen Jenkins and Frank Cowell’s asymmetric sensitivity indexes, analysts can construct highly responsive downside tail risk models. 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 GHN Composite Model

The Generalized Half-Normal distribution is characterized by scale parameter (theta) and shape parameter (alpha). Its probability density function takes the form:

f(x) = [sqrt(2 / pi) * (alpha / x) * (x / theta)^alpha] * exp(-(1 / 2) * (x / theta)^(2 * alpha))

for x > 0. When alpha = 1, the model reduces to the standard half-normal distribution. 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_GHN_SHORROCKS:

=LAMBDA(dataset, param_theta, param_alpha,
    LET(
        clean_data, FILTER(dataset, ISNUMBER(dataset) * (dataset > 0)),
        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),
                z_ratio, cohort_mean / param_theta,
                ghn_kernel, (z_ratio^(param_alpha - 1)) * EXP(-0.5 * (z_ratio^(2 * param_alpha))),
                downside_dev, SQRT(AVERAGE((subset - cohort_mean)^2)),
                decile_weight, (11 - r) / 10,
                CHOOSE(c, r, cohort_mean, ghn_kernel, downside_dev, decile_weight * cohort_downside)
            )
        )),
        header_row, HSTACK("Decile", "Mean", "GHN 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 scale parameter theta in cell C2 (e.g. 28000).
3. Enter the shape parameter alpha in cell C3 (e.g. 1.45).
4. In cell E2, execute the formula:

   =CALC_GHN_SHORROCKS(A2:A5000, C2, C3)

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