Commit: a828e0fa5fbcfe62bae2e13d31927946f3f3eede - myprojects-hg-reborn (hg)

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• Revisión: a828e0fa5fbcfe62bae2e13d31927946f3f3eede (tree)
• Tiempo: 2018-12-04 23:45:57
• Autor: Lorenzo Isella <lorenzo.isella@gmai...>
• Commiter: Lorenzo Isella

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I added an interesting article document in which I modify the margins and the font size.

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• add: latex-documents/weights.tex

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diff -r 642f9158d003 -r a828e0fa5fbc latex-documents/weights.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/latex-documents/weights.tex	Tue Dec 04 15:45:57 2018 +0100
@@ -0,0 +1,314 @@
+% \documentclass[12pt,a4paper]{article}
+\documentclass[14pt, a4paper]{extarticle}
+\usepackage[utf8x]{inputenc}
+\usepackage[english]{babel}
+\usepackage{url}
+\usepackage{graphicx}
+\usepackage{amsmath}  
+\usepackage{xcolor}
+\usepackage{caption}
+\usepackage{hyperref}
+% for placeholder text
+\usepackage{lipsum}
+\usepackage[margin=0.5in]{geometry}
+
+\title{Annex to the Note to File: Calculation of the Growth Rate for
+  FTA and non-FTA Trade}
+% \author{Lorenzo Isella}
+\date{}
+
+\begin{document}
+\maketitle
+  
+\abstract{
+We give the definitions of growth rate and we introduce the main
+formulas for the calculations of the composite growth rate along a
+multi-period time span. After illustrating the shortcomings inherent to
+a straightforward calculation of the
+growth rate of trade flows, we suggest a methodology to bypass these
+issues which borrows from the theory of measurement of investment returns. 
+}
+\section{Basic Definitions and Properties of the Growth Rate}
+
+ % In the simplest case, we aim at calculating the growth rate of all the
+%   trade in the world, regardless of whether it is under FTAs or
+%   not. In the following, I mainly fix the notation. There are plenty
+%   of indexes, but we talk about plain vanilla growth rates in the end
+%   of the day.
+
+Let $l_{ij}^{t}$ be the value of the trade flow between countries $i$
+and $j$ at year $t$. By definition, the growth rate at time $t$ for
+the aforementioned flow is
+given by
+
+\begin{equation}\label{grate}
+g_{ij}^{t}=\frac{l_{ij}^{t}-l_{ij}^{t-1}}{l_{ij}^{t-1}}
+\end{equation}
+
+so that
+
+\begin{equation}
+l_{ij}^{t}=l_{ij}^{t-1}(1+g_{ij}^{t}).
+\end{equation}  
+
+The value of the world total trade at time $t$ is
+
+\begin{equation}
+l_{world}^{t}=\sum_{ij} l_{ij}^{t}
+\end{equation}
+
+then based on Eq. \eqref{grate}, the growth rate of the world total
+trade is
+
+\begin{equation}\label{totalgrowth}
+g_{world}^{t}=\frac{\sum_{ij} l_{ij}^{t} - 
+  l_{ij}^{t-1}}{\sum_{ij} l_{ij}^{t-1}}=\sum_{ij}{w_{ij}^{t} g_{ij}^{t}} 
+\end{equation}  
+
+i.e. a weighted sum of the growth rates for the individual flows with
+weight given by
+
+\begin{equation}
+w_{ij}^{t}=\frac{l_{ij}^{t-1}}{\sum_{ij}l_{ij}^{t-1}}.
+  \end{equation}
+
+The meaning of Eq. \eqref{totalgrowth} is transparent:    if e.g. a trade flow accounts for 10{\% } of the world trade at
+time $t-1$, it will have a weight of 10{\% } in the
+calculation
+of the
+world trade growth at time $t$. % A perfectly analogous formula holds
+% for the growth rate of the total trade under (outside) FTAs, where the
+% sum in Eq. \eqref{totalgrowth} is restricted to the trade flows under
+% (outside) FTAs.
+As we will note in Section \ref{fta}, the direct application of
+Eq. \eqref{totalgrowth} to the (non-)FTA trade is problematic
+whenever, due to the enforcement of an FTA, one or more trade flow
+transition from being subject to import duties to being tariff-free.
+
+% The last step is how to combine the growth rates.
+Having at hand a
+ set of growth rates $\{ g^{t}, g^{t+1},\cdots, g^{t+n} \}$ at years
+ $\{ t, t+1, \cdots, t+n \}$, we calculate the total growth rate
+in the period $[t, t+n]$ as
+\begin{equation}\label{composite}
+  g^{t,t+n}=\prod_{k=0}^{k=n}(1+g^{t+k})-1=(1+g^{t})\times(1+g^{t+1})\times\cdots\times(1-g^{t+n})-1
+\end{equation}
+Once a growth rate has been calculated on
+any number of consecutive years, it is straightforward to construct
+an annualized growth rate which is essentially the geometric average
+of the yearly growth rates. For instance, in Eq. \eqref{composite}, the
+growth rate is calculated along $n+1$ consecutive years from $t$ to
+$t+n$ and the annualized rate is given by
+\begin{equation}\label{annualized}
+  g^{t,t+n}_{\mathrm{annualized}}=(1+g^{t,t+n})^{\frac{1}{n+1}}-1=\{ (1+g^{t})\times(1+g^{t+1})\times\cdots\times(1-g^{t+n})\}^{\frac{1}{n+1}}-1
+\end{equation}  
+
+\section{Issues with (non-)FTA Trade Flows}\label{fta}
+   The complication arises due to having two classes of trade
+    flows, namely those under FTAs and those outside FTAs with flows
+    moving continuously from the non-FTA to the FTA category.
+
+
+
+Let us fix the ideas with a small example. We have tree countries
+$X$, $Y$ and $Z$ trading with each other as shown in Figure \ref{trade}.
+
+\begin{figure}[htb]       
+  \begin{center}  
+\scalebox{.5}{\input{correct.pdf_t}} %the difference is just this part
+\caption{Example of trade flows between three countries $X$, $Y$ and $Z$.}
+\label{trade} 
+\end{center}  
+\end{figure}
+
+
+We have three possible trade flows (we drop the time index in
+order not to overburden the notation), namely  $l_{XY}$,
+$l_{YZ}$ and  $l_{XZ}$. Let us assume that we have trade data in the period 2013-2016. $X$ and $Y$
+   negotiated an FTA long ago, so the trade between them is always
+   tariff-free in the period under scrutiny. On the contrary, $Y$ and $Z$ have 
+   never liberalised their trade exchanges. 
+$X$ and $Y$ negotiated an FTA which entered into force on the $1^{st}$ of
+January 2015.
+We report the fictitious trade flows between the three
+countries in  Table \ref{tradeexample}.
+Whenever a numerical value is in a
+box, it means that the trade is tariff-free, i.e. it occurs under an FTA.
+
+
+\begin{table}[ht]
+\centering
+  % \captionof{table}{Fictitious data about the trade flows between
+  %   countries $X$, $Y$ and $Z$. The box around  numerical values  indicate
+  % that the trade takes place under an FTA.} 
+ \caption{Fictitious data about the trade flows between
+     countries $X$, $Y$ and $Z$. The box around  numerical values  indicate
+   that the trade takes place under an FTA.}\label{tradeexample}
+\scalebox{1}{
+\begin{tabular}{|r|l|l|l|r|}
+  \hline
+Year & 2013 & 2014 & 2015 & 2016  \\ 
+  \hline
+$l_{XY}$ & \boxed{5} & \boxed{5.2} & \boxed{5.1} & \boxed{5.6} \\ 
+  $l_{YZ}$ & 10 & 10.4 & 11.2 & 12.1  \\ 
+  $l_{XZ}$ & 8 & 8.3 & \boxed{8.1} & \boxed{8.7}  \\
+  \hline
+  Total FTA Trade & 5 & 5.2 & 13.2 & 14.3 \\
+  Total non-FTA Trade & 18 & 18.7 & 11.2 & 12.1  \\
+  \hline
+\end{tabular}
+}
+\end{table}
+
+
+
+Table \ref{tradetable} exemplifies the issues in the calculation of
+the growth rate. If we simply track the amount of trade under FTAs, we
+see a huge increase in 2015 due to the entry into force of a new FTA
+(inclusion of a new trade flow) and similarly a fall in the trade
+outside FTAs for the exclusion of the same trade flow.
+As a consequence, the calculation of the growth rate for the total
+trade under FTAs or outside FTAs in 2015 would be hugely misleading.
+
+\begin{table}[ht]
+\centering
+  % \captionof{table}{Fictitious data about the trade flows between
+  %   countries $X$, $Y$ and $Z$. The box around  numerical values  indicate
+  % that the trade takes place under an FTA.} 
+ \caption{Naive calculation of the growth rates under and outside FTAs
+ for the data already reported in Table \ref{tradeexample}.}\label{tradetable}
+\scalebox{1}{
+\begin{tabular}{|r|l|l|l|r|}
+  \hline
+Year & 2013 & 2014 & 2015 & 2016  \\ 
+  \hline
+$l_{XY}$ & \boxed{5} & \boxed{5.2} & \boxed{5.1} & \boxed{5.6} \\ 
+  $l_{YZ}$ & 10 & 10.4 & 11.2 & 12.1  \\ 
+  $l_{XZ}$ & 8 & 8.3 & \boxed{8.1} & \boxed{8.7}  \\
+    \hline
+  Total FTA Trade & 5 & 5.2 & 13.2 & 14.3 \\
+  Total non-FTA Trade & 18 & 18.7 & 11.2 & 12.1  \\
+    \hline
+  Naive FTA growth rate & & 4{\% } & 154{\%  } & 8{\% }  \\
+  Naive non-FTA growth rate & & 4{\% } & -40{\%  } & 8{\% }  \\
+  \hline
+\end{tabular}
+}  
+\end{table}
+
+\section{Amended Calculation for (non-)FTA Trade Flows}
+
+One possible solution is simply not to include $l_{XZ}$ in 2015 in the
+calculations, i.e. to remove the trade flow which transitions from
+non-FTA to FTA in the year of implementation of the FTA, whereas
+including it in all the calculations for the previous and the
+following year does not pose any problem.
+
+Another tackle on this issue is to calculate the growth rates in 2015
+as if in 2014 (the year before the FTA kicked in) $l_{XY}$ was already
+covered by the FTA. Indeed the idea is
+to artificially shift the FTAs by one year in such a way to get more
+sensible results.
+
+
+As a consequence we create, only for the
+calculations of the growth rate for 2015, an ``artificial'' year 2014
+where the trade flow $l_{XZ}$ is already counted as being part of the
+FTA trade. 
+
+
+
+\begin{table}[ht]
+\centering
+  % \captionof{table}{Fictitious data about the trade flows between
+  %   countries $X$, $Y$ and $Z$. The box around  numerical values  indicate
+  % that the trade takes place under an FTA.} 
+ \caption{Creation of an artificial 2014 year to fix the calculations of
+   the growth rate. In the artificial 2014, $l_{XZ}$ is already
+   considered part of the trade under FTA.}\label{tradetablefixed}
+\scalebox{1}{
+\begin{tabular}{|r|l|l|l|l|r|}
+  \hline 
+Year & 2013 & 2014 & 2014 artificial & 2015 & 2016  \\ 
+  \hline
+$l_{XY}$ & \boxed{5} & \boxed{5.2} & \boxed{5.2} & \boxed{5.1} & \boxed{5.6} \\ 
+  $l_{YZ}$ & 10 & 10.4 & 10.4 & 11.2 & 12.1  \\ 
+  $l_{XZ}$ & 8 & 8.3 & \boxed{8.3} & \boxed{8.1} & \boxed{8.7}  \\
+  \hline
+  Total FTA Trade & 5 & 5.2 & 13.5 & 13.2 & 14.3 \\
+  Total non-FTA Trade & 18 & 18.7 & 10.4 & 11.2 & 12.1  \\
+  \hline
+  Naive FTA growth rate & &  4{\% } & & 154{\%  } & 8{\% }  \\
+  Naive non-FTA growth rate &  & 4{\% } & & -40{\%  } & 8{\% }  \\
+  \hline
+  New FTA growth rate & &  4{\% } & & -2{\%  } & 8{\% }  \\
+  New non-FTA growth rate &  & 4{\% } & & 8{\%  } & 8{\% }  \\
+  \hline
+\end{tabular}
+}
+\end{table}
+  
+The proposed workaround is illustrated in Table
+\ref{tradetablefixed}. We notice
+that, with respect to the naive calculation of the growth rates under
+FTAs, this proposed remedy differs only for the year 2015, when an FTA
+is enforced, but it provides much more sensible results.
+The same strategy can be generalised in case of multiple FTAs being
+enforced on the same year.
+
+An equivalent way of describing this procedure is the following: whenever at
+time $t$ a certain trade flow is covered by an FTA, we calculate its
+growth rate by assuming that at time $t-1$ it was already covered by
+the FTA.
+
+Once we have got the yearly growth rates, we can calculate the
+annualized growth rate according to
+Eq. \eqref{composite}-\eqref{annualized} both of the trade under FTAs
+
+\begin{equation}
+g_{\mathrm{FTA}}^{2014-2016}=\{(1+4\% )\times (1-2\% )\times (1+8\% )\}^{1/3}-1=3\%
+\end{equation}  
+and outside FTAs
+
+\begin{equation}
+g_{\mathrm{non-FTA}}^{2014-2016}=\{ (1+4\% )\times(1+8\% ) \times(1+8\% )    \}^{1/3}-1=7\%.
+\end{equation}
+% Finally, once we have calculated the correct yearly growth rates, we resort to
+% Eq. \eqref{composite} to get the total growth rate along the a period
+% consisting of multiple years.
+
+Our strategy is formally analogous to the calculation
+of the returns of a portfolio in the presence of cash deposits
+and cash withdrawals (see
+
+\href{https://www.fool.com/about/how-to-calculate-investment-returns/}{https://www.fool.com/about/how-to-calculate-investment-returns/}
+and
+
+\href{https://www.investopedia.com/terms/a/annualized-total-return.asp}{https://www.investopedia.com/terms/a/annualized-total-return.asp}):
+signing an FTA means metaphorically depositing the associated trade
+flows into the portfolio of FTA trade while simultaneously removing them
+from the non-FTA trade portfolio.
+
+ The idea is to obtain the growth rate of
+FTA vs non-FTA trade in a way which is not affected by the shift of
+trade flows due to new FTA agreements. In the portfolio management
+analogy, we want to calculate the real performance of the portfolio
+due to the investment choices and without the effects of cash deposits
+and/or withdrawals.
+% The link above also tells how to construct, from the individual growth
+% rates, a growth rate along the whole period (it goes without saying
+% that the simple addition and/or average of the growth rates along
+% different years is not an appropriate method, see also
+
+% \href{https://investinganswers.com/financial-dictionary/investing/average-annual-growth-rate-aagr-2549}{https://investinganswers.com/financial-dictionary/investing/average-annual-growth-rate-aagr-2549}.  )
+
+
+
+
+
+\end{document}
+              
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End: