# Z Risk Ratio

Z risk ratio defined as:

Z = 0.012A + 0.014B + 0.033C+
0.006D + 0.010E

Let T= Total Asset
A= Net Current Asset / T
B= Retained Earning /T
C= Profit b4 interest before tax / T
D= Market Cap / T
E= Sales / T

Z < 1.8 => company bankrupt
Z > 3 => company healthy

Accuracy:
bankruptcy (3 years 95%) (2 yrs 70%)

# Double-Entry Accounting = Algebraic ‘Module’

Accounting Double-Entry Algebraic Structure: Module

Double-entry Accounting forms a structure Module Rⁿ, with scalars from an Abelian Unity Ring Z: (positive=credit, negative = debit).
Note: Change Vector Space’s scalar over Field to over Ring => Module’s scalar is over Ring.

# Combinatoric in Accounting

Prove:
$\displaystyle\sum_{n=2}^{n}{_n}C_r = 2^n-1-n$
$\displaystyle\sum_{n=2}^{n}{_n}C_r 2^r= 3^n-1-2n$
Note:
${_4}C_2=\frac {4.3}{2!}$
Proof:
1.
$\displaystyle\sum_{n=2}^{n}{_n}C_r$ =(1+1)ⁿ -1- $\displaystyle {_n}C_1 =2^n-1-n$
2.
$\displaystyle\sum_{n=2}^{n}{_n}C_r 2^{r}$=(1+2)ⁿ -1 – $\displaystyle {_n}C_1 .2^{1}= 3^n-1-2n$

An accounting transaction = Debit p accounts + Credit q accounts (p, q ≥ 1)

In a company with total n accounts,
Prove: there are $3^n -2^{n+1} + 2$ transactions.

Proof:
Let the Set of all accounts T = {a1, a2, …, an}
aj = account with value ‘+’ (credit), or ‘-‘ (debit), or 0 (nil)
1. Trivial transaction: To= {0, 0,…..0} = 1 way
2. Choose r accounts from n = $\displaystyle {_n}C_r$
3. Slot ‘+’ or ‘-‘ in these r accounts = $2^r$ways
exclude 2 impossible all ‘+’, ‘-‘ transactions = $2^r- 2$ ways
4.
Let T1 = $\displaystyle\sum_{n=2}^{n}{_n}C_r .(2^r-2)$
Total transaction = To + T1
= $\displaystyle 1+ \sum_{n=2}^{n}{_n}C_r .(2^r-2)$
{Apply previous results}
= $\displaystyle 1+ \sum_{n=2}^{n}{_n}C_r .2^r -2.\sum_{n=2}^{n}{_n}C_r$
= $1+3^n-1-2n -2(2^n-1-n)$
= $3^n -2^{n+1} + 2$  [QED]