![](media/image1.jpeg){width="7.241666666666666in" height="0.825in"}

**Method #2: Allocating Total Reserve Requirement Across Causes**

**Goal**

The goal of this paper is to provide a quantitatively robust method for
allocating a control area's balancing requirement among the relevant
elements causing the requirement. By accounting for the correlations
among factors, one may calculate the component balancing requirement
such that the individual components sum to the diversified total
requirement. Proceeding further, one may take the concept of the
Gaussian Copula to make the calculation relevant for non-normal
distributions.

**A Simple Example**

To begin the discussion, an example in terms of standard normal
distributions is examined. Consider two normal distributions, both with
mean = 0.0 and standard deviation = 1.0. The joint standard deviation
considering correlation is calculated by:

s~p~ = \[**sRs**^T^\]^1/2^

where:

s~p~ is the total standard deviation of the combined distributions,

> **s** is a row vector, {s~1~, s~2~,. . .s~n~}, of size n containing
> standard deviation values of the variables contained in the total
> portfolio (in this example the values are 1.0 and 1.0),
>
> **R** is the correlation matrix of size n x n relating each of the
> portfolio variables to one another,
>
> **s**^T^ is a column vector, {s~1~, s~2~,. . .s~n~}, of size n of
> standard deviation values of the variables contained in the total
> portfolio (again, in this example the values are 1.0 and 1.0).

Here it is observable that the correlation matrix is the key element in
calculating the standard deviation of the total portfolio. Continuing
with the example and given the above equation, the 95^th^ percentile, or
a z-value of 1.64, for the combined distribution under varying
correlation scenarios is as follows:

> **Correl P95 a P95 b P95 a + b**
>
> 1.00 1.64 1.64 3.29
>
> 0.90 1.64 1.64 3.21
>
> 0.80 1.64 1.64 3.12
>
> 0.70 1.64 1.64 3.03
>
> 0.60 1.64 1.64 2.94
>
> 0.50 1.64 1.64 2.85
>
> 0.40 1.64 1.64 2.75
>
> 0.30 1.64 1.64 2.65
>
> 0.20 1.64 1.64 2.55
>
> 0.10 1.64 1.64 2.44
>
> 0.00 1.64 1.64 2.33

Observe that under the correlation value of 1.00 the 95^th^ percentile
values are directly additive, but under all other circumstances the
percentiles are not directly additive.

The question now is how to make the above percentiles additive.
Calculating the incremental standard deviation values allows for the
direct addition of the percentile values:

**s** ~Inc~ = **Rs**^T^ / s~p~

where;

> **s** ~Inc~ is a vector, {s~Inc\ 1~, s~Inc\ 2~,. . .s~Inc\ n~}, of
> size n of incremental standard deviation values.

The incremental standard deviation calculates percent contribution of
any individual component to the total portfolio standard deviation. For
each given correlation value, the incremental standard deviation values
are as follows:

**Correl s~inc~ a s~inc~ b**

1.00 1.00 1.00

0.90 0.97 0.97

0.80 0.95 0.95

0.70 0.92 0.92

0.60 0.89 0.89

0.50 0.87 0.87

0.40 0.84 0.84

0.30 0.81 0.81

0.20 0.77 0.77

0.10 0.74 0.74

0.00 0.71 0.71

Observing the above table of incremental standard deviation values
reveals why under the case of a 1.00 correlation the standard deviation
values, and hence percentile values, are directly additive: each
component is contributing 100% of its variation to the aggregate
standard deviation. Note that he percent contribution of a and b to the
total in the table above is the same under all correlation values
because in this example a and b both have the same standard deviation
values. If the standard deviation values differed, the incremental
standard deviations would not be equal for both a and b. It may be worth
noting that the sum of **s** ~Inc~ does not necessarily equal 1.0.

From the incremental standard deviation values contained in the above
table, the marginal standard deviation values are calculated resulting
in 95^th^ percentile values that also are now directly additive:

**Correl P95 a P95 b P95 a + b**

1.00 1.64 1.64 3.29

0.90 1.60 1.60 3.21

0.80 1.56 1.56 3.12

0.70 1.52 1.52 3.03

0.60 1.47 1.47 2.94

0.50 1.42 1.42 2.85

0.40 1.38 1.38 2.75

0.30 1.33 1.33 2.65

0.20 1.27 1.27 2.55

0.10 1.22 1.22 2.44

0.00 1.16 1.16 2.33

Again, in the above example, the proportion is equal for both
distributions because each distribution has the same standard deviation.
Had differing standard deviation values been chosen, the proportioning
would be different.

**Applying to Non-Normal Distributions: Taking a Notion from the
Gaussian Copula**

The preceding machinations may be applied to non-normal distributions to
make the percentile values directly additive by borrowing a concept from
the Gaussian Copula method of correlating distributions. In general
terms, a Gaussian Copula is a method of correlating n variables in terms
of n standard normal distributions. This results in a matrix of size i x
n of correlated z-values. Since calculating the cumulative distribution
function for all z-values results in n uniform distributions with values
u~i~, 0.0\>= u~i~, \<= 1.0, correlated cumulative probabilities may be
selected from any empirical or other known distribution type. The
process for the Gaussian Copula is given by:

**P** = CDF\[ **CZ** \],

where;

**P** is a i x n matrix of correlated values 0.0\>= u~i~, \<= 1.0 for
each n,

CDF is the cumulative distribution function for a standard normal
distribution,

**C** is a matrix of Cholesky factors given correlation matrix **R**,

where;

**C** = **LD**^1/2^,

where **L** and **D** are the lower diagonal and the diagonal of matrix
**R**.

> **Z** is a i x n matrix of independent, identically distributed,
> standard normal random variables

This notion of processing in terms of standard normal distributions and
transforming from standard normal probability values to those of the
distributions in question is what allows for the component decomposition
of the aggregate balancing requirement; despite the balancing
distribution being non-normal. In the specific application used for the
purposes of allocating a control area's balancing requirement among the
relevant elements, the above notion is performed partly in reverse.

**The Application to Allocating the Balancing Requirement (Error
Signal)**

Recalling that the goal is to provide a quantitatively robust method for
allocating the control area balancing requirement among the relevant
elements causing the requirement, the aforementioned calculations and
notions are now applied to the specific case of allocating the control
area balancing requirement among between two causal elements; Element 1
(E2) and Element 2 (E2).

Given that the desire is to allocate the balancing requirement between
E1 and E2, total calculated balancing is:

**B**~T~ = **B**~E1~ + **B**~E2~

where;

**B**~T~ is a vector of length n of calculated total balancing
requirements,

> **B**~E1~ is a vector of length n of calculated balancing requirements
> for Element 1 only,
>
> **B**~E2~ is a vector of length n of calculated balancing requirements
> for Element 2 only,

The first step toward being able to sum the percentiles is to translate
the P~99.5~\[**B**~T~\] value into a normal distribution z-value:

z**B**~T~ = (P~99.5~\[**B**~T~\] - Mean\[**B**~T~\]) / s\[**B**~T~\]

Having translated the desired P~99.5~\[**B**~T~\] off of the empirical
distribution, **B**~T~, into terms of a normal distribution, calculating
the incremental standard deviation values for E1 and E2 may begin. From
the earlier discussion, the component standard deviation is given by:

**s** ~Inc~ = **Rs**^T^ / s~p~

applying to the specific problem at hand:

**R** is the correlation matrix of **B**~E1~ and **B**~E2~,

> **s**^T^ is a column vector of standard deviation values for **B**~E1~
> and **B**~E2~,

s~p~ is the standard deviation of **B**~T~,

and finally,

**s** ~Inc~ is a vector containing the component contribution of
**B**~E1~ and **B**~E2~ to **B**~T~,.

Calculating the amount that **B**~E1~ and **B**~E2~ contribute to
**B**~T~ at a given percentile, the 99.5^th^ in this case, is given by:

P~99.5~\[**B**~T~\] = z**B**~T~ \* **s**{**B**~E1~}\* **s**
~Inc~{**B**~E1~} + Mean\[**B**~E1~\] +

> z**B**~T~ \* **s**{**B**~E2~}\* **s** ~Inc~{**B**~E2~} +
> Mean\[**B**~E2~\]