# Calculation methods

## Annual average growth rate

The annual average growth rate is, unless otherwise specified, computed as the coefficient *b* in the exponential trend function *y* = *a*e^{bt} where *t* stands for time and *y* is the object of measurement. This method takes all observations in the analysed period into account. Therefore, the growth rate reflects trends that are less influenced by exceptional values.

In the Total and urban population page, annual population growth is expressed by the annual exponential rate of growth, defined as:

## Product concentration index of exports

The product concentration index of exports (map in the Trade indicators page) is calculated as a normalized Herfindahl-Hirschmann index:

where X_{i,j} is the value of exports of product *i* from economy *j* and *n* is the number of product groups according to SITC, Revision 3, at the 3-digit level.

## Terms of trade index

The terms of trade index (Trade indicators page, figure 1, tables 1 and 2) with base year 2000 is calculated as follows:

where *UVI*_{exports,i,t} is the unit value index of exports and *UVI*_{imports,i,t} is the unit value of economy *i* at time *t*.

## Market concentration index of exports

The market concentration index of exports (Trade indicators page, figure 2) is calculated as a normalized Herfindahl-Hirschmann index:

where X_{i,j} is the value of exports of product *i* from economy *j* and *n* is the number of economies.

## Trade openness index

The trade openness index (Trade indicators page, figure 3) is calculated as the ratio of the arithmetic mean of merchandise exports (*x*) and imports (*m*) to GDP (*y*):

where *i* designates the economy and *t* the year.

## Purchasing power index of exports

The purchasing power index of exports (Trade indicators page, table 1) is calculated by dividing the export value index by the corresponding import unit value index and scaling up by 100:

where *VI*_{exports,i,t} is the value index of exports (as defined above), *UVI*_{imports,i,t} is the unit value index of imports, *i* designates the economy and *t* the time period.

## Volume index of exports (imports)

The volume index of exports (imports) (Trade indicators page, tables 1 and 2) is calculated by dividing the export (import) value index by the corresponding unit value index and scaling up by 100:

where *VI*_{i,t} is the value index of exports (imports), given by

x_{i,t} is the value of exports (imports), *UVI*_{i,t} is the unit value index of exports (imports), *i* designates the economy and *t* the time period.

## Lorenz curve

The Lorenz curve in the Gross domestic product page (figure 3), plots cumulative population shares ordered by GDP per capita, on the x-axis, against the cumulative shares of global GDP which they account for, on the y-axis. For the construction of the Lorenz curve, the *n* economies of the world are ordered with reference to their GDP per capita, so that

where y_{i} is GDP and *p*_{i} the population of the economy at position *i* in this ranking, counted from below.

The cumulative population shares, measured on the x-axis, are calculated as

with *p* = *p*_{1} + *p*_{2} + ... + *p*_{n}

The cumulative shares of global GDP, measured on the y-axis, are calculated as follows:

with *y* = *y*_{1} + *y*_{2} + ... + *y*_{n}

## Free market commodity price index

The free market commodity price index, in the Prices page, is a fixed base-weight Laspeyres index with base-year 2015=100. It is calculated as

where *i* is the identifier of the commodity group, *q*_{i,2015} is the quantity for which products of commodity group *i* were exported by developing economies during the three years around the base year (from 2014 to 2016), and *p _{i,t}* is the price of a representative product, within commodity group

*i*, in year

*t*. For more details, see UNCTAD (2018e).

## Nowcasts

The nowcasts presented in the Total merchandise trade page (figure 1), the Total trade in services page (figure 1) and the Gross domestic product page (figure 1) represent real-time evaluations of world merchandise exports, world services exports and world GDP based on a large set of relevant and timely indicators. They are based on a dynamic factor model which captures common latent trends in these data through their cross correlations. In its state-space representation, the model can be written as:

where *G*_{t} is a combination of the reference and indicator series, *h*_{t} is the time-varying factor, *B* is a matrix of factor loadings, *D* defines the time structure of the factor, and the error terms *u*_{t} and *v*_{t} are independently distributed according to distributions N(0,*W*) and N(0,*Q*), respectively.

The nowcast for the target variable at time *t* is obtained by extracting the corresponding element from vector *G*_{t} above, once *B* and the latent factor *h*_{t} have been estimated through maximum likelihood. This model is adapted to accommodate variables of different frequencies and unbalanced datasets. It should be noted that the nowcast figures cannot be considered as official data, as they are the result of an estimation. For more details on the methodology, see (UNCTAD, 2018f).