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Compute ad-hoc concentration indices for a portfolio

`ci = concentrationIndices(PortfolioData)`

```
[ci,Lorenz]
= concentrationIndices(___,Name,Value)
```

computes
multiple ad-hoc concentration indices for a given portfolio. The `ci`

= concentrationIndices(`PortfolioData`

)`concentrationIndices`

function
supports the following indices:

CR — Concentration ratio

Deciles — Deciles of the portfolio weights distribution

Gini — Gini coefficient

HH — Herfindahl-Hirschman index

HK — Hannah-Kay index

HT — Hall-Tideman index

TE — Theil entropy index

`[`

adds
optional name-value pair arguments. `ci`

,`Lorenz`

]
= concentrationIndices(___,`Name,Value`

)

Compute the concentration indices for a credit portfolio using a portfolio that is described by its exposures. The exposures at default are stored in the `EAD`

array.

Load the `CreditPortfolioData.mat`

file that contains `EAD`

used for the `PortfolioData`

input argument.

```
load CreditPortfolioData.mat
ci = concentrationIndices(EAD)
```

ci = 1x8 table ID CR Deciles Gini HH HK HT TE ___________ ________ _____________ _______ ________ ________ ________ _______ "Portfolio" 0.058745 [1x11 double] 0.55751 0.023919 0.013363 0.022599 0.53485

Use the `CRIndex`

optional input to obtain the concentration ratios for the tenth and twentieth largest exposures. In the output, the `CR`

column becomes a vector, with one value for each requested index.

Load the `CreditPortfolioData.mat`

file that contains the `EAD`

used for the `PortfolioData`

input argument.

load CreditPortfolioData.mat ci = concentrationIndices(EAD,'CRIndex',[10 20])

ci = 1x8 table ID CR Deciles Gini HH HK HT TE ___________ __________________ _____________ _______ ________ ________ ________ _______ "Portfolio" 0.38942 0.58836 [1x11 double] 0.55751 0.023919 0.013363 0.022599 0.53485

Use the `HKAlpha`

optional input to set the alpha parameter for the Hannah-Kay (HK) index. Use a vector of alpha values to compute the `HK`

index for multiple parameter values. In the output, the `HK`

column becomes a vector, with one value for each requested alpha value.

Load the `CreditPortfolioData.mat`

file that contains `EAD`

used for the `PortfolioData`

input argument.

load CreditPortfolioData.mat ci = concentrationIndices(EAD,'HKAlpha',[0.5 3])

ci = 1x8 table ID CR Deciles Gini HH HK HT TE ___________ ________ _____________ _______ ________ ____________________ ________ _______ "Portfolio" 0.058745 [1x11 double] 0.55751 0.023919 0.013363 0.029344 0.022599 0.53485

Compare the concentration measures using an `ID`

optional argument for a fully diversified portfolio and a fully concentrated portfolio.

ciD = concentrationIndices([1 1 1 1 1],'ID','Fully diversified'); ciC = concentrationIndices([0 0 0 0 5],'ID','Fully concentrated'); disp([ciD;ciC])

ID CR Deciles Gini HH HK HT TE ____________________ ___ _____________ ____ ___ ___ ___ ___________ "Fully diversified" 0.2 [1x11 double] 0 0.2 0.2 0.2 -2.2204e-16 "Fully concentrated" 1 [1x11 double] 0.8 1 1 1 1.6094

Use the `ScaleIndices`

optional input to scale the index values of `Gini`

, `HH`

, `HK`

, `HT`

, and `TE`

. The range of `ScaleIndices`

is from `0`

through `1`

, independent of the number of loans.

ciDU = concentrationIndices([1 1 1 1 1],'ID','Diversified, unscaled'); ciDS = concentrationIndices([1 1 1 1 1],'ID','Diversified, scaled','ScaleIndices',true); ciCU = concentrationIndices([0 0 0 0 5],'ID','Concentrated, unscaled'); ciCS = concentrationIndices([0 0 0 0 5],'ID','Concentrated, scaled','ScaleIndices',true); disp([ciDU;ciDS;ciCU;ciCS])

ID CR Deciles Gini HH HK HT TE ________________________ ___ _____________ ____ __________ ___________ ___________ ___________ "Diversified, unscaled" 0.2 [1x11 double] 0 0.2 0.2 0.2 -2.2204e-16 "Diversified, scaled" 0.2 [1x11 double] 0 3.4694e-17 -3.4694e-17 -6.9389e-17 -1.3796e-16 "Concentrated, unscaled" 1 [1x11 double] 0.8 1 1 1 1.6094 "Concentrated, scaled" 1 [1x11 double] 1 1 1 1 1

Load the `CreditPortfolioData.mat`

file that contains EAD used for the `PortfolioData`

input argument.

```
load CreditPortfolioData.mat
P = EAD;
ci = concentrationIndices(P);
```

Visualize an approximate Lorenz curve using the deciles information and also the concentration at the decile level.

Proportion = 0:0.1:1; figure; subplot(2,1,1) area(Proportion',[ci.Deciles' Proportion'-ci.Deciles']) axis([0 1 0 1]) title('Lorenz Curve (By Deciles)') xlabel('Proportion of Loans') ylabel('Proportion of Value') subplot(2,1,2) bar(diff(ci.Deciles)) axis([0 11 0 1]) title('Concentration by Decile') xlabel('Decile') ylabel('Weight')

Load the `CreditPortfolioData.mat`

file that contains EAD used for the `PortfolioData`

input argument. The optional output `Lorenz`

contains the data for the exact Lorenz curve.

load CreditPortfolioData.mat P = EAD; [~,Lorenz] = concentrationIndices(P); figure; area(Lorenz.ProportionLoans,[Lorenz.ProportionValue Lorenz.ProportionLoans-Lorenz.ProportionValue]) axis([0 1 0 1]) title('Lorenz Curve') xlabel('Proportion of Loans') ylabel('Proportion of Value')

`PortfolioData`

— Nonnegative portfolio positions in numeric array

Nonnegative portfolio positions in *N* assets,
specified as an `N`

-by-`1`

(or `1`

-by-`N`

)
numeric array.

**Data Types: **`double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside single quotes (`' '`

). You can
specify several name and value pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`[ci,Lorenz] = concentrationIndices(PortfolioData,'CRIndex',100)`

`'CRIndex'`

— Index of interest for concentration ratio`1`

(default) | nonnegative integerIndex of interest for the concentration ratio, specified as
the comma-separated pair consisting of `'CRIndex'`

and
an integer value between `1`

and *N*,
where *N* is the number of assets in the portfolio.
The default value for `CRIndex`

is `1`

(the
default `CR`

is the largest portfolio weight). If `CRIndex`

is
a vector, the concentration ratio is computed for the index value
in the given order.

**Data Types: **`double`

`'HKAlpha'`

— Alpha parameter for Hannah-Kay index`0.5`

(default) | nonnegative numericAlpha parameter for Hannah-Kay index, specified as the comma-separated
pair consisting of `'HKAlpha'`

, and a positive number
that cannot be equal to `1`

. If `HKAlpha`

is
a vector, the Hannah-Kay index is computed for each alpha value in
the given order.

**Data Types: **`double`

`'ID'`

— User-defined ID for portfolio`"Portfolio"`

(default) | character vector | string objectUser-defined ID for the portfolio, specified as the comma-separated
pair consisting of `'ID'`

and a scalar string object
or character vector.

**Data Types: **`char`

| `string`

`'ScaleIndices'`

— Flag to indicate whether to scale concentration indices`false`

(no scaling) (default) | logical Flag to indicate whether to scale concentration indices, specified
as the comma-separated pair consisting of `'ScaleIndices'`

and
a logical scalar. When the `ScaleIndices`

is set
to true, the value of the `Gini`

, `HH`

, `HK`

, `HT`

,
and `TE`

indices are scaled so that all these indices
have a minimum value of `0`

(full diversification)
and a maximum value of `1`

(full concentration).

Scaling is applied only for portfolios with at least two assets. Otherwise, the scaling capability is undefined.

**Data Types: **`logical`

`ci`

— Concentration indices information for given portfoliotable

Concentration indices information for the given portfolio, returned as a table with the following columns:

`ID`

— Portfolio ID string. Use the`ID`

name-value pair argument to set it.`CR`

— Concentration ratio. By default, the concentration ratio for the first index (largest portfolio weight) is reported. Use the`CRIndex`

name-value pair argument to choose a different index. If`CRIndex`

is a vector of length*m*, then`CR`

is a row vector of size`1`

-by-*m*. For more information, see Definitions.`Deciles`

— Deciles of the portfolio weights distribution is a`1`

-by-`11`

row vector containing the values`0`

, the nine decile cut points, and`1`

. For more information, see Definitions.`Gini`

— Gini coefficient. For more information, see Definitions.`HH`

— Herfindahl-Hirschman index. For more information, see Definitions.`HK`

— Hannah-Kay index (reciprocal). By default, the`'alpha'`

parameter is set to`0.5`

. Use the`HKAlpha`

name-value pair argument to choose a different value. If`HKAlpha`

is a vector of length*m*, then`HK`

is a row vector of size`1`

-by-*m*. For more information, see Definitions.`HT`

— Hall-Tideman index. For more information, see Definitions.`TE`

— Theil entropy index. For more information, see Definitions.

`Lorenz`

— Lorenz curve datatable

Lorenz curve data, returned as a table with the following columns:

`ProportionLoans`

— (`N`

+`1`

)-by-`1`

numeric array containing the values`0`

,`1`

/`N`

,`2`

/`N`

, ...`N`

/`N`

=`1`

. This is the data for the horizontal axis of the Lorenz curve.`ProportionValue`

— (`N`

+`1`

)-by-`1`

numeric array containing the proportion of portfolio value accumulated up to the corresponding proportion of loans in the`ProportionLoans`

column. This is the data for the vertical axis of the Lorenz curve.

All the concentration indices for `concentrationIndices`

assume
a credit portfolio with an exposure to counterparties.

Let *P* be a given credit portfolio with exposure
to *N* counterparties. Let *x _{1}*,...

`0`

for all `1`

,...$$x={\displaystyle \sum _{i=1}^{N}{x}_{i}}$$

Assume that *x* > `0`

,
that is, at least one exposure is nonzero. The portfolio weights are
given by *w _{1}*,...,

$${w}_{i}=\frac{{x}_{i}}{x}$$

The weights are sorted in nondecreasing order. The following standard notation uses brackets around the indices to denote ordered values.

$${w}_{[1]}\le {w}_{[2]}\le \mathrm{...}\le {w}_{[N]}$$

The *concentration ratio* (CR)
answers the question “what proportion of the total exposure
is accumulated in the largest *k* loans?”

The formula for the concentration ratio (CR) is:

$$C{R}_{k}={\displaystyle \sum _{i=1}^{k}{w}_{{}_{[N-i+1]}}}$$

For example, if *k*=`1`

, *CR _{1}* is
a sum of the one term

`0`

through `1`

.The *Lorenz curve* is
a visualization of the cumulative proportion of portfolio value (or
cumulative portfolio weights) against the cumulative proportion of
loans.

The cumulative proportion of loans (*p*) is
defined by:

$${p}_{0}=0,{p}_{1}=\frac{1}{N},{p}_{2}=\frac{2}{N},\mathrm{...},{p}_{N}=\frac{N}{N}=1$$

The cumulative proportion of portfolio value *L* is
defined as:

$${L}_{0}=0,{L}_{k}={\displaystyle {\sum}_{i=1}^{k}{w}_{[i]}}$$

The Lorenz curve is a plot of *L* versus *p*,
or the cumulative proportion of portfolio value versus cumulative
proportion of the number of loans (sorted from smallest to largest).

The diagonal line is indicated in the same plot because it represents the curve for the portfolio with the least possible concentration (all loans with the same weight). The area between the diagonal and the Lorenz curve is a visual representation of the Gini coefficient, which is another concentration measure.

*Deciles* are commonly
used in the context of income inequality.

If you sort individuals by their income level, what proportion
of the total income is earned by the lowest 10% and the lowest 20%
of the population? In a credit portfolio, loans can be sorted by exposure.
The first decile corresponds to the proportion of the portfolio value
that is accumulated by the smallest 10% loans, and so on. Deciles
are proportions, therefore they always take values from `0`

through `1`

.

Defining the cumulative proportion of loans (*p*)
and the cumulative proportion of values *L* as in Lorenz Curve, the deciles are a subset of the proportion
of value array. Given indices *d1*, *d2*,…, *d9* such
that the proportion of loans matches exactly these values:

$${p}_{d1}=0.1,{p}_{d2}=0.2,\mathrm{...},{p}_{d9}=0.9$$

The deciles *D*_{0},*D*_{1},....,*D*_{9},*D*_{10} are
defined as the corresponding proportion of values:

$${D}_{0}={L}_{0}=0,{D}_{1}={L}_{d1},{D}_{2}={L}_{d2},\mathrm{...},{D}_{9}={L}_{d9,}{D}_{10}={L}_{N}=1$$

When the total number of loans *N* is not divisible
by 10, no indices match the exact proportion of loans 0.1, 0.2, and
so on. In that case, the decile values are linearly interpolated from
the Lorenz curve data (that is, from the *p* and *L* arrays).
With this definition, there are 11 values in the deciles information
because the end points 0% and 100% are included.

The *Gini index* (or coefficient)
is visualized on a Lorenz curve plot as the area between the diagonal
and the Lorenz curve.

Technically, the Gini index is the ratio of that area to the area of the full triangle under the diagonal on the Lorenz curve plot. The Gini index is also defined equivalently as the average absolute difference between all the weights in the portfolio normalized by the average weight.

Using the proportion of values that array *L* defined
in the Lorenz curve section, the Gini index is given by the formula:

$$Gini=1-\frac{1}{N}{\displaystyle {\sum}_{i=1}^{N}({L}_{i-1}+{L}_{i})}$$

Equivalently, the Gini index can be computed from the sorted weights directly with the formula:

$$Gini=\frac{1}{N}{\displaystyle {\sum}_{i=1}^{N}(2i-1){w}_{[i]}-1}$$

The Gini coefficient values are always between `0`

(full
diversification) and `1`

– `1`

/*N* (full
concentration).

The *Herfindahl-Hirschman index * is
commonly used as a measure of market concentration.

The formula for the Herfindahl-Hirschman index is:

$$HH={\displaystyle {\sum}_{i=1}^{N}{w}_{i}{}^{2}}$$

The Herfindahl-Hirschman index takes values between `1`

/*N* (full
diversification) and `1`

(full concentration).

The *Hannah-Kay index* is
a generalization of the Herfindahl-Hirschman index.

The formula for the Hannah-Kay depends on a parameter α
> `0`

, α ≠ `1`

,
as follows:

$$H{K}_{\alpha}={\left({\displaystyle {\sum}_{i=1}^{N}{w}_{i}{}^{\alpha}}\right)}^{1/(\alpha -1)}$$

This formula is the reciprocal of the original Hannah-Kay index,
which is defined with `1`

/(`1`

−
α) in the exponent. For concentration analysis, the reciprocal
formula is the standard because it increases as the concentration
increases. This is the formula implemented in `concentrationIndices`

.
The Hannah-Kay index takes values between `1`

/*N* (full
diversification) and `1`

(full concentration).

The *Hall-Tideman index * is
a measure commonly used for market concentration.

The formula for the Hall-Tideman index is:

$$HT=\frac{1}{2{\displaystyle {\sum}_{i=1}^{N}(N-i+1){w}_{[i]}-1}}$$

The Hall-Tideman index takes values between `1`

/*N* (full
diversification) and `1`

(full concentration).

The *Theil entropy index*,
based on a traditional entropy measure (for example, Shannon entropy),
is adjusted so that it increases as concentration increases (entropy
moves in the opposite direction), and shifted to make it positive.

The formula for the Theil entropy index is:

$$TE={\displaystyle {\sum}_{i=1}^{N}{w}_{i}\mathrm{log}({w}_{i)}+\mathrm{log}(N)}$$

The Theil entropy index takes values between `0`

(full
diversification) and `log`

(*N*)
(full concentration).

[1] Basel Committee on Banking Supervision. *"Studies
on Credit Risk Concentration".* Working paper no. 15. November,
2006.

[2] Calabrese, R., and F. Porro. "Single-name concentration risk in credit portfolios: a comparison of concentration indices." working paper 201214, Geary Institute, University College, Dublin, May, 2012.

[3] Lütkebohmert, E. *Concentration Risk in Credit
Portfolios.* Springer, 2009.

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