How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Le Cam Distance Models in Microsoft Excel

In quantitative statistical decision theory and mathematical econometrics, the Le Cam distance (also known as the deficiency distance between statistical experiments) evaluates the informational loss when deciding between two probability measures. Based on the concept of statistical sufficiency and randomized decision rules, the Le Cam distance between discrete transition distributions provides a bounded, scale-invariant measure of structural discrepancy. When analyzing $10 imes 10$ decile transition probability matrices against a uniform benchmark, decomposing the Le Cam distance directionally reveals whether downside transition risk carries higher statistical deficiency than upward promotion channels.

Using Microsoft Excel’s dynamic array functions—specifically VSTACK, HSTACK, LAMBDA, MAKEARRAY, and LET—analysts can formulate a fully reactive decile downside Le Cam distance engine that evaluates portfolio risk dynamics without legacy VBA macros.

Mathematical Theory of the Le Cam Distance

Let $P$ represent a $10 imes 10$ empirical transition probability matrix and $Q$ represent a uniform benchmark transition matrix ($q_{i,j} = 1/n$). The classical Le Cam distance $Delta(P, Q)$ between two discrete probability vectors is given by:

Le Cam Distance Formula:
Delta(P, Q) = 0.5 * ∑_{i=1}^{n} ∑_{j=1}^{n} ( (p_{i,j} - q_{i,j})^2 / (p_{i,j} + q_{i,j}) )

We partition the Le Cam distance into upside transitions ($j > i$) and downside transitions ($j < i$):

Upside Le Cam (LC_up):
LC_up = 0.5 * (1 / n) * ∑_{i=1}^{n} ∑_{j=i+1}^{n} ( (p_{i,j} - q_{i,j})^2 / (p_{i,j} + q_{i,j}) )

Downside Le Cam (LC_down):
LC_down = 0.5 * (1 / n) * ∑_{i=1}^{n} ∑_{j=1}^{i-1} ( (p_{i,j} - q_{i,j})^2 / (p_{i,j} + q_{i,j}) )

Net Le Cam Asymmetry Ratio (LAR):
LAR = (LC_down - LC_up) / (LC_down + LC_up)

A positive $LAR > 0$ indicates that the statistical deficiency and informational divergence from uniform churn is concentrated in downward transition pathways, signaling structured demotion risk across deciles.

Organising the Transition Matrix in Excel

Structure your $10 imes 10$ decile transition probability grid across cells B4:K13, representing origin deciles D1 through D10 on the rows and destination deciles D1 through D10 on the columns. Verify that each row sums to $1.00$ ($100%$). Set your reference matrix probability $q = 0.10$ ($1/10$).

Authoring the Dynamic LAMBDA Formula

In Excel’s Name Manager (Formulas > Name Manager), create a new formula named DOWNSIDE_LE_CAM_MODEL:

=LAMBDA(matrix,
    LET(
        n, ROWS(matrix),
        q_ref, 1 / n,
        lc_matrix, MAKEARRAY(n, n, LAMBDA(r, c,
            LET(
                p, INDEX(matrix, r, c),
                q, q_ref,
                denom, p + q,
                IF(denom > 0, 0.5 * ((p - q)^2 / denom), 0)
            )
        )),
        lc_up, SUM(MAKEARRAY(n, n, LAMBDA(r, c, IF(c > r, INDEX(lc_matrix, r, c), 0)))) / n,
        lc_down, SUM(MAKEARRAY(n, n, LAMBDA(r, c, IF(c < r, INDEX(lc_matrix, r, c), 0)))) / n,
        lc_diag, SUM(MAKEARRAY(n, n, LAMBDA(r, c, IF(r = c, INDEX(lc_matrix, r, c), 0)))) / n,
        total_lc, lc_up + lc_down + lc_diag,
        lar_ratio, IF((lc_down + lc_up) = 0, 0, (lc_down - lc_up) / (lc_down + lc_up)),
        header, HSTACK("Le Cam Metric", "Distance Value"),
        r1, HSTACK("Upside Le Cam (LC_up)", lc_up),
        r2, HSTACK("Downside Le Cam (LC_down)", lc_down),
        r3, HSTACK("Diagonal Persistence Le Cam", lc_diag),
        r4, HSTACK("Total Matrix Le Cam Distance", total_lc),
        r5, HSTACK("Net Le Cam Asymmetry Ratio (LAR)", lar_ratio),
        VSTACK(header, r1, r2, r3, r4, r5)
    )
)

Interpreting Le Cam Distance in Risk Governance

Entering =DOWNSIDE_LE_CAM_MODEL(B4:K13) into cell B16 outputs a structured analytical report:

• Upside Le Cam: Measures statistical deficiency across upward migration channels.
• Downside Le Cam: Quantifies statistical divergence along downward degradation pathways.
• Net Le Cam Asymmetry Ratio: Bounded strictly between $-1.00$ and $+1.00$. Positive values alert risk committees that downside transitions exhibit structured, non-uniform risk concentration.
• Total Matrix Le Cam Distance: Summarizes the overall departure of the transition matrix from pure unconstrained churn as a bounded statistical metric.

By leveraging Excel’s functional dynamic array formulas, this risk engine provides instantaneous recalculation across stress test scenarios without requiring external code or macros.