Diversification Indexes: Arrangement And Application Possibility For Company Towns

5.1. Concentration Ratio (CRk)

$${\mathit{C}\mathit{R}}_{\mathit{k}}=\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{k}}{\sum }}{\mathit{Y}}_{\mathit{i}}\mathrm{}\mathrm{},$$

where Y _i – share of element i in total summary of elements; k – the number of firms that is chosen arbitrary.

CR is one of the most frequently used measures of a concentration. Bikker& Haaf (2002) estimate CR ₃ , CR ₄ , CR ₅ for 3, 4, and 5 largest banks of mortgage market. Meilak (2008) measures 12 largest merchandise export categories of each country. Hannah& Kay (1977) use the ratio to estimate the share of 100 firms ( CR ₁₀₀ ) of production performance. Considering the type of market competition, some researchers are modifying this index to compare the top corporations and its concentration in the market (Linda, 1976) of the major companies. Linda index combines ranking of the hugest companies with the concentration of 2-3 companies within the districted number of the large companies. Although the CR can be easily used for identifying and analysis in dynamics of the data of town-forming enterprise, this ratio fails to estimate the small and medium business (SMB) at the early stage of their development.

5.2. Herfindal-Hirshman Index (HHI)

$$\mathit{H}\mathit{H}\mathit{I}=\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{n}}{\sum }}{\mathit{Y}}_{\mathit{i}}^{\mathbf{2}},\mathrm{}\mathit{i}=\mathbf{1},\dots ,\mathit{n}$$

where Y _i – the share of element i in the total volume; n – the total number of elements.

HHI is the second of the most widespread indexes of concentration that is used in a variety of fields including banking, production, and social sectors. HHI index represents the key tool of monopoly power estimation due to antitrust laws in the USA. Thus, Y _i can be considered not only in terms of company share in production at the market but even as business and social-economic activities in a town and in a country in general. It may detect the concentration of investments, labour or taxation in a town including town-forming enterprises and medium and small business. Moreover, this index is universal and gives the result even if the list of elements includes 0. That makes this index more preferable than indexes on the basis of logarithm. It is possible to estimate the dynamics of this index, but it is difficult to define the contribution of different elements into the result.

5.3. Rosenbluth Index and Hall-Tideman Index (HT)

$$\mathit{H}\mathit{T}=\frac{\mathbf{1}}{\mathbf{2}\sum {\mathit{R}}_{\mathit{i}}{\mathit{q}}_{\mathit{i}}-\mathbf{1}},$$

where R _i – the rank of company i by market share; q _i – share of company i by sales at homogeneous market. Rosenbluth and Hall-Tideman indexes are generally considered together. The basic difference between them is that Hall & Tideman (1967) offered to rank the market share of a company by sales whereas Rosenbluth Index does not imply such ranking. HT index makes it possible to measure the concentration of company town economic activities by dispatched goods, employment, investments, and taxation showing the opposite rate of diversification.

5.4. The Multigroup Entropy Index or Theil Index (E)

$$\mathit{E}=\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{n}}{\sum }}{\mathit{Y}}_{\mathit{i}}\mathbf{ln}\frac{\mathbf{1}}{{\mathit{Y}}_{\mathit{i}}},$$

where Y _i – share of element i in the total; n – number of elements.

The entropy shows the inverse concentration: the higher its value, the lower the concentration. The former statement makes this index relevant for the rate of company town diversification assessment. The index ranges between [0; $\mathit{l}\mathit{n}$n]. On the one hand, this index is hard to interpret and is of rare use (Reardon & Firebaugh, 2002). Moreover it fails when element Y _i equals 0 . On the other hand, it allows standardization the results with positive and negative smoothed values that can be easily depicted at the matrixes.

5.5. The Hannah and Kay Index (HKI)

$$\mathit{H}\mathit{K}\mathit{I}={\left(\underset{\mathit{a}=\mathbf{1}}{\overset{\mathit{n}}{\sum }}{\mathit{s}}_{\mathit{i}}^{\mathit{\alpha }\mathrm{}}\right)}^{\frac{\mathbf{1}}{\mathbf{1}-\mathit{\alpha }\mathrm{}}}$$

where s - share of element i in total summary of elements; α – the elasticity parameter for entry or exit of the bank from the general group, α >0 and α $\ne \mathit{1}$. Although Hannah & Kay (1977) offer to use this index for the banking sector, it may be widened for other sectors of economy, including economic activities of company towns. Bikker & Haaf  (2002) underline that α is optional and allows alternative views on this measure that makes it ambiguous.

5.6. The Gini Coefficient (G) and the Lorentz Coefficient (L)

$$\mathit{G}=\mathbf{1}-\mathbf{2}\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{m}}{\sum }}{\mathit{x}}_{\mathit{i}}{\mathit{c}\mathit{u}\mathit{m}\mathit{y}}_{\mathit{i}}+\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{m}}{\sum }}{\mathit{x}}_{\mathit{i}}{\mathit{y}}_{\mathit{i}}$$

where x _i – the share of group i in the total; y _i – the share of group i ; cumy _i – the cumulative income share; m – the number of groups.

$$\mathit{L}=\frac{\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{m}}{\sum }}\left|\mathit{d}-\mathit{p}\right|}{\mathbf{2}}\mathrm{}$$

where d – share of i group in total; p – group frequency in the total amount of characteristic in a multiple relationship; m – the number of groups. The index ranges between [0; 1] and basically shows the relative proportion in population income. These indexes are often considered together and are two of the most difficult to calculate due to the necessity of an adequate background for grouping. That makes the index to be hardly used for measurement of economic activity diversification in company towns.

5.7. Coefficient of Variation (CV)

$$\mathit{C}\mathit{V}=\frac{\mathit{\sigma }}{\mathit{a}}\mathrm{*}\mathbf{100}\mathrm{\%}\mathrm{},$$

where $\mathit{\sigma }$- standard deviation, a – arithmetical mean.

CV is a general statistic indicator that shows the standard deviation of the mean value. The deviation can be significant ( CV >33%), average (10%< CV <20%) or insignificant ( CV <10%). Borenstein & Rose (1991) and other researchers use this index for prize discrimination measure. Nevertheless, CV makes it possible to estimate the deviation of economic activities in the company towns as the measure of concentration or diversification. If CV is considered insignificant, that will imply the low diversity of economic activities in the town, whereas company towns at the prime stage of the diversification process will have significant deviation.

5.8. Dominant Firm or Competitive Fringe Model (DF/CF)

$\mathit{K}=\frac{\mathit{D}\mathit{F}}{\mathit{C}\mathit{F}}$,

where DF – dominant firm market share; CF – competitive fringe market share.

This clear model cannot by easily used for concentration and diversification estimation for company towns because the model considers only a homogeneous product for both dominant company and competitive fringe that was stated by Kahai, Kaserman & Mayo (1996) . The fulfilment of this condition means measuring only the basic company town industry since both companies will represent it with a high probability.

5.9. Bank Income Diversification

$$\mathit{D}\mathit{I}\mathit{V}=\mathbf{1}-\left({\mathit{S}\mathit{H}\mathit{N}\mathit{E}\mathit{T}}^{\mathbf{2}}+{\mathit{S}\mathit{H}\mathit{N}\mathit{O}\mathit{N}}^{\mathbf{2}}\right),$$

$$\mathit{S}\mathit{H}\mathit{N}\mathit{E}\mathit{T}=\frac{\mathit{N}\mathit{E}\mathit{T}}{\mathit{N}\mathit{E}\mathit{T}+\mathit{N}\mathit{O}\mathit{N}},$$

$$\mathit{S}\mathit{H}\mathit{N}\mathit{O}\mathit{N}=\frac{\mathit{N}\mathit{O}\mathit{N}}{\mathit{N}\mathit{E}\mathit{T}+\mathit{N}\mathbf{O}\mathit{N}},$$

where NET – net interest income; NON – non-interest income; SHNET is the ratio of net interest income to net operating revenue, and SHNON is the ratio of non-interest income to net operating revenue. DIV measures the degree of diversification.

This index is offered by Lin & Huang (2014) and shows that the higher the value, the more diversified the income. On the contrary, DIV = 0 means that all revenues come from a single source (complete concentration) and DIV =0.5 represents an even split between net interest income and non-interest income (complete diversification). This index can be used for company towns only if a town-forming enterprise is considered as NET and others – as NON . Moreover, the DIV itself is quite interesting because it represents how to convert the concentration, measured as HHI ${(\mathit{S}\mathit{H}\mathit{N}\mathit{E}\mathit{T}}^{\mathbf{2}}\mathbf{+}{\mathit{S}\mathit{H}\mathit{N}\mathit{O}\mathit{N}}^{\mathbf{2}}$), into the diversificasion. Thus, the authors may figure out the proper diversification index that represented in section “Results”.

5.10. The Diversification Coefficient

$${\mathbf{\rho }}_{\mathbf{\omega },{\mathbf{\omega }}^{\mathbf{M}\mathbf{D}\mathbf{P}}}\ge \frac{{\mathbf{D}\mathbf{R}}_{\left(\mathbf{\omega }\right)}}{{\mathbf{D}\mathbf{R}}_{\left({\mathbf{\omega }}^{\mathbf{M}\mathbf{D}\mathbf{P}}\right)}}$$

where $\mathit{D}\mathit{R}\left(\mathit{w}\right)={\left[\mathit{p}\left(\mathit{w}\right)\left(\mathit{1}-\mathit{C}\mathit{R}\left(\mathit{w}\right)\right)+\mathit{C}\mathit{R}\left(\mathit{w}\right)\right]}^{-\raisebox{1ex}{$\mathit{1}$}\!\left/ \!\raisebox{-1ex}{$\mathit{2}$}\right.}$; $\mathit{p}\left(\mathit{w}\right)$ is the volatility-weighted average correlation of the assets in the portfolio; $\mathit{}\mathit{C}\mathit{R}\left(\mathit{w}\right)$ is the volatility-weighted concentration ratio (CR) of the portfolio; w – the weights of a portfolio; MDP – most diversified portfolio.

Choueifaty, Froidure, & Reynier (2013) offer to measure “Maximum Diversification” portfolio invariance properties using the index. They presented the new mathematical properties of the diversification ratio and the most diversified portfolio ( MDP ), and investigated the optimality of the MDP in a mean-variance framework. This index can be hardly applicable for company town diversification measurement.

5.11. Localization Coefficient

$${\mathit{L}\mathit{Q}}_{\mathit{i}}=\frac{{\mathit{S}}_{\mathit{i}}^{\mathit{r}}}{{\mathit{S}}_{\mathit{i}}^{\mathit{N}}}$$

where ${\mathit{S}}_{\mathit{i}}^{\mathit{r}}$ - the share of industry in regional production structure; $\mathit{}{\mathit{S}}_{\mathit{i}}^{\mathit{N}}-$ the share of industry in country production structure.

Hackman & Oldham (1975) calculated this index as the share of sector i in region r compared with the corresponding national economy indicator. The localization coefficient is widely used at the regional level of research as the characteristic of a regional specialization. LQ >1 stands for a specialized sector.

5.12. Hackman Coefficient

$${\mathit{I}}_{\mathit{H}\mathit{A}\mathit{C}}=\frac{\mathbf{1}}{\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{M}}{\sum }}\left({\mathit{L}\mathit{Q}}_{\mathit{i}}\mathrm{*}{\mathit{S}}_{\mathit{i}}^{\mathit{r}}\right)}$$

where ${\mathit{S}}_{\mathit{i}}^{\mathit{r}}$ - the share of industry in regional production structure; $\mathit{}{\mathit{S}}_{\mathit{i}}^{\mathit{N}}-$ the share of industry in country production structure.

The index I _HAC is developed by Hackman (1975). The amount of the regional localization coefficients is weighed by the share of the relevant sector in a common regional indicator. It shows how closed the regional indicators are to the structure of the corresponding indicator for national economy. The coefficient value ranges between [0;1], I _HAC =1 shows the complete match between regional and national structures. Both Localization and Hackman coefficients can be used for industry competitiveness in the company town for considering the proportions of local economy in the regional one.

5.13. Territory Diversification Coefficient

$$\mathit{T}\mathit{K}=\underset{\mathit{i}=\mathbf{1}}{\overset{\mathit{n}}{\sum }}(\left|\left.\frac{\mathbf{100}}{\mathit{N}}\right|-{\mathit{D}}_{\mathit{i}})\right.$$

where N – the number of elements in total; D _i – the weight of element i in per cent.

Turgel (2014) suggests using this coefficient for both production and employment structure diversification. The coefficient shows how homogeneous the industries are in company town economy. This diversification coefficient can be calculated both for assessment of an industry performance and an employment diversity by economic activities in company towns.

5.14. Coefficient of Production per Capita

$${\mathit{K}}_{\mathit{p}}=\left(\frac{{\mathit{I}}_{\mathit{r}}}{{\mathit{I}}_{\mathit{c}}}\mathrm{*}\mathbf{100}\right):\left(\mathrm{}\frac{{\mathit{P}}_{\mathit{r}}}{{\mathit{P}}_{\mathit{c}}}\mathrm{*}\mathbf{100}\right)$$

where I _r – regional industry output; I _c – country industry output; P _р - regional population; P _c – country population.

The Coefficient of proportion per capita ( Uscova , 2008) considers production K _p as the relation between the weight of the regional industry in relevant country industry and the weight of a regional population in the country population. The weight of regional industry can be changed by core industry of company town considering in dynamics.

Trying to evaluate the dynamics and the structural shifts on the territory different authors develop shift-share indexes. The shift-share analysis is conducted by the following indexes that are given in table  2 . This method is used to the regional growth measurement and describes the dynamics of structural changes that makes the method be applicable for structural movements of company town economy for diversification process can be seen as the structural shift of company town economy.