Simplifying Function with Recursive CTE and/or Window Function

Question

I'm trying to come up with a Recursive CTE and/or Window Function to create a function.

After days, I've boiled the function down to (pseudocode) where I have N and B, and need to generate E:

E_n = B_n * (1 - SUM(E₁, E₂, ... E_n-1))

Examples:

╔═══╦═════════════╦═════════════╗
║ N ║ B           ║ E           ║
╠═══╬═════════════╬═════════════╣
║ 0 ║ 0.142857143 ║ 0.142857143 ║
║ 1 ║ 0.285714286 ║ 0.244897959 ║
║ 2 ║ 0.285714286 ║ 0.174927114 ║
║ 3 ║ 0.285714286 ║ 0.124947938 ║
║ 4 ║ 0.285714286 ║ 0.089248527 ║
║ 5 ║ 0.4         ║ 0.089248527 ║
║ 6 ║ 0.666666667 ║ 0.089248527 ║
║ 7 ║ 1           ║ 0.044624264 ║
╚═══╩═════════════╩═════════════╝

E₀ = 0.143 * (1 - 0) = 0.143
E₁ = 0.286 * (1 - 0.143) = 0.245
E₂ = 0.286 * (1 - (0.143 + 0.245)) = 0.175
E₃ = 0.286 * (1 - (0.143 + 0.245 + 0.175)) = 0.125
E₄ = 0.286 * (1 - (0.143 + 0.245 + 0.175 + 0.125)) = 0.089
E₅ = 0.400 * (1 - (0.143 + 0.245 + 0.175 + 0.125 + 0.089)) = 0.089
E₆ = 0.667 * (1 - (0.143 + 0.245 + 0.175 + 0.125 + 0.089 + 0.089)) = 0.089
E₇ = 1.000 * (1 - (0.143 + 0.245 + 0.175 + 0.125 + 0.089 + 0.089 + 0.089)) = 0.044

If the table above was in Excel, C2 = B2 * (1 - 0) (base) and C3 = B3 * (1 - SUM(C$2:C2)) (recursive)

What I've tried:

Windowed Functions

Tried SUM(...) OVER(ORDER BY [N] ROWS BETWEEN UNBOUNDED PRECEDING AND 1 PRECEDING), but can't reference the column recursively.

Recursive CTE

Tried several iterations of:

WITH B AS ([Num], [Best], [Effective Rate]) AS (
    SELECT *
        , [Best]
    FROM A
    WHERE [Num] = 0
    UNION ALL
    SELECT A.*
        , (1 - [Effective Rate]) * A.[Best]
    FROM B
    JOIN A ON A.[Num] = B.[Num] + 1 
)

and some with an extra column in the CTE, but it only covers 1 previous row and results after 2nd row are wrong.

Recursive CTE with Windowed Function

From all that I've tried, it seems that the recursive segment of the CTE is calculated independently of the other results, and SUM(...) OVER(...) only works on the current row. (With regard to the above table, all values of E would be 0.142857143).

I assume this is because the UNION ALL happens all at once, and not incrementally.

Alternative Solutions

What I would really like to happen is to simplify the above equation, and/or transform it into an iterative function.

Bonus: If anyone cares to know the source of this information, it's used to calculate MACRS depreciation for tax purposes.

Answer 1

You need an extra column to carry along the running total (fiddle).

In the recursive part of the CTE below R refers to the "previous" row and A the current one so referencing the column from R is your SUM(E1, E2, ... En-1).

WITH R
     AS (SELECT N,
                B,
                E = B,
                RunningTotalE = B
         FROM   A
         WHERE  N = 0
         UNION ALL
         SELECT A.N,
                A.B,
                E = A.B * ( 1 - R.RunningTotalE ),
                RunningTotalE = A.B * ( 1 - R.RunningTotalE ) + R.RunningTotalE
         FROM   R
                JOIN A
                  ON A.N = R.N + 1)
SELECT N,
       B,
       E = CAST(E AS DECIMAL(10,9))
FROM   R