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Tikz Overlay 与 shift={(label)} 在下一页上产生覆盖

tag-icon • tag-icon tikz-pgf vertical-alignment overlays
Tikz Overlay 与 shift={(label)} 在下一页上产生覆盖

我提前为 MWE 过大而道歉，但这是重现该问题的唯一方法。在当前形式中，注释alignat*后应覆盖在环境上的覆盖层\newpage出现在文档的第二页上。

删除行前的空格\vspace{42pt}Thus,会导致该行上移，从而产生不必要的间距。这也不能解决覆盖定位问题。

为什么覆盖图会出现在下一页上？它似乎占据了正确的shift位置，只是在错误的页面上。

我不知道这是否值得注意，但我的实际 tex 文档的序言要大得多。我不确定是否值得重现，但如果有必要或要求，我可以重现。

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb,mathtools}
\usepackage{actuarialangle,tikz,tikz-cd}

\usepackage[top=40pt,bottom=50pt,left=35pt,right=35pt]{geometry}
\setlength{\abovecaptionskip}{-8pt}
\setlength{\belowcaptionskip}{+10pt}
\newcommand{\minus}{\scalebox{0.55}[1.0]{$-$}}
\usetikzlibrary{decorations.pathreplacing}
\pagestyle{empty}
\begin{document}

\newpage
\subsection{\underline{Exercise 2.}}
\textit{A widow, as beneficiary of a $\$100\,000$ insurance policy, will receive $\$20\,000$ immediately and $\$1800$ every three months thereafter. The company pays interest at $j_4=6\%$; after 3 years, the rate is increased to $j_4=7\%$. $a)$ How many full payments of $\$1800$ will she receive? $b)$ What additional sum paid with the last full payment will exhaust her benefits? $c)$ What payment 3 months after the last full payment will exhaust her benefits?}
$$
\begin{tikzpicture}
\draw (0,0) -- (14,0);
\draw (0,0.2) -- (0,-0.2);
\node [above] at (0.5,0.2) {\$100,000};
\node [below] at (0,-0.2) {0};
\node [align=left,execute at begin node=\setlength{\baselineskip}{10pt}] at (-1.5,0) {Cash\\scenario:};
\end{tikzpicture}
$$
$$
\begin{tikzpicture}
\draw (0,0) -- (6,0);
\draw (6.1,0.2) -- (5.9,-0.2);
\draw (6.2,0.2) -- (6,-0.2);
\draw (6.1,0) -- (12,0);
\draw (12.1,0.2) -- (11.9,-0.2);
\draw (12.2,0.2) -- (12,-0.2);
\draw (12.1,0) -- (14,0);
\draw (0,0.2) -- (0,-0.2);
\draw (2,0.2) -- (2,-0.2);
\draw (4,0.2) -- (4,-0.2);
\draw (8,0.2) -- (8,-0.2);
\draw (10,0.2) -- (10,-0.2);
\draw (14,0.2) -- (14,-0.2);
\draw (0,-1) -- (0,-1.4);
\draw (8,-1) -- (8,-1.4);
\node [above] at (2,0.2) {\$1800};
\node [above] at (4,0.2) {\$1800};
\node [above] at (8,0.2) {\$1800};
\node [above] at (10,0.2) {\$1800};
\node [below] at (0,-0.2) {0};
\node [below] at (2,-0.2) {1};
\node [below] at (4,-0.2) {2};
\node [below] at (8,-0.2) {12};
\node [below] at (10,-0.2) {13};
\draw [<->] (0.05,-1.2) to (7.95,-1.2);
\draw [<-] (8.05,-1.2) to (15,-1.2);
\node [below] at (14.6,-0.2) {12+$[n]$ (quarters)};
\node [align=left,execute at begin node=\setlength{\baselineskip}{10pt}] at (-2,0) {Annuity\\scenario:};
\node [fill=white,inner sep=1pt] at (4,-1.2) {$j_4=6\%$};
\node at (3.8,-1.7) {$\Rightarrow i=1.5\%$};
\node [fill=white,inner sep=1pt] at (11,-1.2) {$j_4=7\%$};
\node at (11.2,-1.7) {$i=1.75\%$};
\draw [decorate,decoration={brace,amplitude=10pt}] (0,0.8) to (7.975,0.8);
\draw [decorate,decoration={brace,amplitude=10pt}] (8.025,0.8) to (14,0.8);
\draw [fill=white] (0,0.5) ellipse (35pt and 7.5pt);
\node [inner sep=0pt,execute at begin node=\setlength{\baselineskip}{7pt},fill=white,align=center] at (0,0.5) {\$20,000};
\node at (4.8,1.4) {simple ordinary annuity};
\node at (11.4,1.4) {simple ordinary annuity};
\end{tikzpicture}
$$
After 3 years, the widow has received, in addition to the down payment, 12 full payments of \$1800, for a total present value of 
$$\$20,000+\$1800a_{\angl{12}1.5\%}\cong\$39,633.51<\$100,000$$
Thus, the (wo)man will still receive some payments afterwards. But how many?\\
The ``theoretical'' number of payments to be received after 3 years must satisfy the equation \\
\begin{align*}
    \text{PV of cash scenario }&=\text{ PV of annuity scenario}\\
    \$100,000\quad&=\quad\tikz[remember picture,inner sep=0pt]\node[] (x) {};\$39,633.51\quad+\quad\$1800a_{\angl n1.75\%} \,\,\cdot\,\,(1+1.5\%)^{\minus12}
\end{align*}
\begin{tikzpicture}[overlay, remember picture, shift={(x)}]
\draw [decorate,decoration={brace,amplitude=8pt,mirror}] (0,-0.2) to (2,-0.2);
\draw [decorate,decoration={brace,amplitude=8pt,mirror}] (3.2,-0.2) to (5.7,-0.2);
\tikzstyle{every node}=[align=left,execute at begin node=\setlength{\baselineskip}{8pt}]
\node at (1.1,-1.2) {\footnotesize{PV of first}\\\footnotesize{3 years of}\\\footnotesize{annuity +}\\\footnotesize{\$20,000 down}};
\node at (4.6,-1.2) {\footnotesize{PV of remainder}\\\footnotesize{of annuity, i.e.}\\\footnotesize{value of remainder}\\\footnotesize{at time $n=12$}};
\node at (7.8,-1) {\footnotesize{discount factor}\\\footnotesize{from time $n=12$}\\\footnotesize{to $n=0$}};
\end{tikzpicture}
%\newpage %%%%%%%% WEIRD STUFF HAPPENS IF I DO NOT ADD THIS PAGE BREAK

\vspace{42pt}Thus,
\begin{spreadlines}{\dimexpr\jot+0.75em\relax}
\begin{alignat*}{2}
&&\quad \$100,000-\$39,633.51&=\frac{\$1800a_{\angl n1.75\%}}{(1.015)^{12}}\\
\iff&& \frac{\$60,366.49(1.015)^{12}}{\$1800}&=\frac{1-(1.0175)^{\minus n}}{0.0175}\\
\iff&&\frac{60,366.49(1.015)^{12}(0.0175)}{1800}&=1-\tikz[remember picture,inner sep=0pt]\node[] (n101) {};(1.0175)^{\minus n}\\
\iff &&(1.0175)^{\minus n}&=1-\frac{60,366.49(1.015)^{12}(0.0175)}{1800}\\
\iff && -n\ln 1.0175&=\ln\left[1-\frac{60,366.49(1.015)^{12}(0.0175)}{1800} \right]\\
\iff && n&=\frac{\ln\left[1-\frac{60,366.49(1.015)^{12}(0.0175)}{1800}\right]}{-\ln1.0175}\\
&& &\cong69.73
\end{alignat*}
\end{spreadlines}
\begin{tikzpicture}[overlay, remember picture, shift={(n101)}]
\draw [->] (0,-0.2) to[out=210,in=330,looseness=0.5] (-2,-0.2);
\draw [->] (-3,-0.4) to[out=330,in=210,looseness=0.5] (-0.25,-0.5);
\node [align=left,execute at begin node=\setlength{\baselineskip}{9pt}] at (-6.4,-3.3) {\small{taking}\\\small{$\ln$ of}\\\small{both sides}};
\end{tikzpicture}
So the widow will receive 69 full payments after the first 3 years of payments, i.e. (s)he will receive a total of $69+12=81$ full payments of $\$1800$.
\vspace{12pt}(b)\vspace{-12pt}
$$
\begin{tikzpicture}
\draw (0,0) -- (6,0);
\draw (6.1,0.2) -- (5.9,-0.2);
\draw (6.2,0.2) -- (6,-0.2);
\draw (6.1,0) -- (12,0);
\draw (12.1,0.2) -- (11.9,-0.2);
\draw (12.2,0.2) -- (12,-0.2);
\draw (12.1,0) -- (14,0);
\draw (0,0.2) -- (0,-0.2);
\draw (2,0.2) -- (2,-0.2);
\draw (4,0.2) -- (4,-0.2);
\draw (8,0.2) -- (8,-0.2);
\draw (10,0.2) -- (10,-0.2);
\draw (14,0.2) -- (14,-0.2);
\draw (0,-1) -- (0,-1.4);
\draw (8,-1) -- (8,-1.4);
\draw (14,-1) -- (14,-1.4);
\node [above] at (2,0.2) {\$1800};
\node [above] at (4,0.2) {\$1800};
\node [above] at (8,0.2) {\$1800};
\node [above] at (10,0.2) {\$1800};
\node [above] at (14.45,0.2) {$\$1800+X$};
\node [below] at (0,-0.2) {0};
\node [below] at (2,-0.2) {1};
\node [below] at (4,-0.2) {2};
\node [below] at (8,-0.2) {12};
\node [below] at (10,-0.2) {13};
\draw [<->] (0.05,-1.2) to (7.95,-1.2);
\draw [<->] (8.05,-1.2) to (13.95,-1.2);
\node [below] at (14.95,-0.2) {81 (quarters)};
\node [fill=white,inner sep=2pt] at (4,-1.2) {$i=1.5\%$};
\node [fill=white,inner sep=2pt] at (11,-1.2) {$i=1.75\%$};
\draw [decorate,decoration={brace,amplitude=10pt}] (0,0.8) to (7.975,0.8);
\draw [decorate,decoration={brace,amplitude=10pt}] (8.025,0.8) to (14,0.8);
\node at (0,1.3) {\$20,000};
\node at (4.8,1.4) {simple ordinary annuity 1};
\node at (11.4,1.4) {simple ordinary annuity 2};
\draw [->] (15.25,1.4) to (15.25,0.8);
\node [align=left,execute at begin node=\setlength{\baselineskip}{9pt}] at (15.25,1.8) {\footnotesize{balloon}\\\footnotesize{payment}};
\end{tikzpicture}
$$
The balloon payment $X$ must be such that
\begin{spreadlines}{\dimexpr\jot+0.75em\relax}
\begin{alignat*}{2}
&& \text{PV of cash scenario}&=\text{PV of annuity scenario}\\
&&\$100,000&=\tikz[remember picture,inner sep=0pt]\node[] (n100) {};\$20,000+\$1800a_{\angl{12}1.5\%}+\$1800a_{\angl{69}1.75\%}(1+1.5\%)^{\minus12}\\
&& &\quad\quad+X(1+1.75\%)^{\minus69}(1+1.5\%)^{\minus18}\\
&& &=\$20,000+\$19,633.51+\$60,040.76+0.2526577697X\\
\iff&& \$100,000-\$20,000-\$19,633.51-\$60,040.76&=0.2526577697X\\
\iff&& X&=\frac{\$100,000-\$20,000-\$19,633.51-\$60,040.76}{0.2526577697X}\\
&& &\cong\$1,289.21
\end{alignat*}
\end{spreadlines}
\begin{tikzpicture}[overlay, remember picture, shift={(n100)}]
\node [align=left,execute at begin node=\setlength{\baselineskip}{7pt}] at (1,-0.35) {\scriptsize{down}\\\scriptsize{payment}};
\node [align=left,execute at begin node=\setlength{\baselineskip}{7pt}] at (3,-0.35) {\scriptsize{PV of annuity 1}};
\node [align=left,execute at begin node=\setlength{\baselineskip}{7pt}] at (5.7,-0.45) {\scriptsize{PV of annuity 2}};
\node [align=left,execute at begin node=\setlength{\baselineskip}{7pt}] at (8,-0.5) {\scriptsize{discount}\\\scriptsize{factor}};
\draw [decorate,decoration={brace,amplitude=4pt,mirror}] (4.7,-0.15) to (7.1,-0.15);
\draw [decorate,decoration={brace,amplitude=4pt,mirror}] (7.2,-0.15) to (9.1,-0.15);
\end{tikzpicture}
Thus, the additional sum paid with the last full payment, called balloon payment, to exhaust the widows's benefits is \$1,289.21.
\vspace{12pt}(c)
$$
\begin{tikzpicture}
\draw (0,0) -- (6,0);
\draw (6.1,0.2) -- (5.9,-0.2);
\draw (6.2,0.2) -- (6,-0.2);
\draw (6.1,0) -- (12,0);
\draw (12.1,0.2) -- (11.9,-0.2);
\draw (12.2,0.2) -- (12,-0.2);
\draw (12.1,0) -- (16,0);
\draw (0,0.2) -- (0,-0.2);
\draw (2,0.2) -- (2,-0.2);
\draw (4,0.2) -- (4,-0.2);
\draw (8,0.2) -- (8,-0.2);
\draw (10,0.2) -- (10,-0.2);
\draw (14,0.2) -- (14,-0.2);
\draw (16,0.2) -- (16,-0.2);
\draw (0,-1) -- (0,-1.4);
\draw (8,-1) -- (8,-1.4);
\draw (16,-1) -- (16,-1.4);
\node [above] at (2,0.2) {\$1800};
\node [above] at (4,0.2) {\$1800};
\node [above] at (8,0.2) {\$1800};
\node [above] at (10,0.2) {\$1800};
\node [above] at (14,0.2) {\$1800};
\node [above] at (16,0.2) {$Y$};
\node [below] at (0,-0.2) {0};
\node [below] at (2,-0.2) {1};
\node [below] at (4,-0.2) {2};
\node [below] at (8,-0.2) {12};
\node [below] at (10,-0.2) {13};
\node [below] at (14,-0.2) {81};
\draw [<->] (0.05,-1.2) to (7.95,-1.2);
\draw [<->] (8.05,-1.2) to (15.95,-1.2);
\node [below] at (16.95,-0.2) {82 (quarters)};
\node [fill=white,inner sep=2pt] at (4,-1.2) {$i=1.5\%$};
\node [fill=white,inner sep=2pt] at (12,-1.2) {$i=1.75\%$};
\draw [decorate,decoration={brace,amplitude=10pt}] (0,0.8) to (7.975,0.8);
\draw [decorate,decoration={brace,amplitude=10pt}] (8.025,0.8) to (14,0.8);
\node at (0,1.3) {\$20,000};
\node at (4.8,1.4) {simple ordinary annuity 1};
\node at (11.4,1.4) {simple ordinary annuity 2};
\end{tikzpicture}
$$
The drop payment $Y$ is such that 
\begin{alignat*}{2}
&& \text{PV of cash scenario}&=\text{PV of annuity scenario}\\
&&\$100,000&=\$20,000+\$1800a_{\angl{12}1.5\%}+\$1800a_{\angl{69}1.75\%}(1_1.5\%)^{\minus12}\\
&& &\quad\quad+Y(1+1.75\%)^{\minus70}(1+1.5\%)^{\minus12}\\
&& &=\$20,000+\$19,633.51+\$60,040.76+0.2483123044Y\\
%&& &\quad\quad+0.
\iff&& Y&=\frac{\$100,000-\$20,000-\$19,633.51-\$60,040.76}{0.2483123044}\\
&& &\cong\$1311.78
\end{alignat*}
Hence, the payment 3 months after the last full payment, also called drop payment, which will exhaust the widow's benefits is \$1311.78.
%\vspace{8em}\hyperref[toc]{\textit{\underline{\footnotesize{Return to the Table of Contents}}}}
\end{document}

答案1

以下是使用 的方法tikzmark。使用tikzmark可以指定叠加层前标记是在文本中定义的，这样您就可以简单地移动覆盖图片前必要时它们所属的文本。

我还消除了大部分坏盒子（剩下1个）并进行了一些清理。

  • 不要用于$$在 LaTeX 中显示数学运算。
  • 不要使用\\来结束特殊环境之外的行，例如tabular和array。
  • 不要把tikzpictures 放在数学模式中！（完全不知道为什么有人会这样做，但是你做过，所以大概是有动机的。不管是什么原因，都不是一个好的原因。
  • 字体大小命令（其中包括）例如\Huge、\small等等\footnotesize。不接受争论.\small{abc} def就像\small abc def。在两种情况下，abc def都会以比默认字体更小的字体排版。
  • 请使用 Ti钾Z 的样式并制作自定义样式来简化您的代码。

在下面的代码中，我使用了颜色来使覆盖层更加明显。显然，在生产使用中应该删除它。

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb,mathtools}
\usepackage{actuarialangle,tikz,tikz-cd}

\usepackage[top=40pt,bottom=50pt,left=35pt,right=35pt]{geometry}
\setlength{\abovecaptionskip}{-8pt}
\setlength{\belowcaptionskip}{+10pt}
\newcommand{\minus}{\scalebox{0.55}[1.0]{$-$}}
\usetikzlibrary{decorations.pathreplacing,tikzmark}
\tikzset{%
  node font size/.style n args=2{%
    align=left, font=#1, execute at begin node=\setlength\baselineskip{#2}%
  },
}
\pagestyle{empty}
\begin{document}
\subsection{\underline{Exercise 2.}}
\textit{A widow, as beneficiary of a $\$100\,000$ insurance policy, will receive $\$20\,000$ immediately and $\$1800$ every three months thereafter. The company pays interest at $j_4=6\%$; after 3 years, the rate is increased to $j_4=7\%$. $a)$ How many full payments of $\$1800$ will she receive? $b)$ What additional sum paid with the last full payment will exhaust her benefits? $c)$ What payment 3 months after the last full payment will exhaust her benefits?}
\begin{center}
  \begin{tikzpicture}
    \draw (0,0) -- (14,0);
    \draw (0,0.2) -- (0,-0.2);
    \node [above] at (0.5,0.2) {\$100,000};
    \node [below] at (0,-0.2) {0};
    \node [node font size={\normalsize}{10pt}] at (-1.5,0) {Cash\\scenario:};
  \end{tikzpicture}

  \makebox[0pt]{%
    \begin{tikzpicture}
      \draw (0,0) -- (6,0);
      \draw (6.1,0.2) -- (5.9,-0.2);
      \draw (6.2,0.2) -- (6,-0.2);
      \draw (6.1,0) -- (12,0);
      \draw (12.1,0.2) -- (11.9,-0.2);
      \draw (12.2,0.2) -- (12,-0.2);
      \draw (12.1,0) -- (14,0);
      \draw (0,0.2) -- (0,-0.2);
      \draw (2,0.2) -- (2,-0.2);
      \draw (4,0.2) -- (4,-0.2);
      \draw (8,0.2) -- (8,-0.2);
      \draw (10,0.2) -- (10,-0.2);
      \draw (14,0.2) -- (14,-0.2);
      \draw (0,-1) -- (0,-1.4);
      \draw (8,-1) -- (8,-1.4);
      \node [above] at (2,0.2) {\$1800};
      \node [above] at (4,0.2) {\$1800};
      \node [above] at (8,0.2) {\$1800};
      \node [above] at (10,0.2) {\$1800};
      \node [below] at (0,-0.2) {0};
      \node [below] at (2,-0.2) {1};
      \node [below] at (4,-0.2) {2};
      \node [below] at (8,-0.2) {12};
      \node [below] at (10,-0.2) {13};
      \draw [<->] (0.05,-1.2) to (7.95,-1.2);
      \draw [<-] (8.05,-1.2) to (15,-1.2);
      \node [below] at (14.6,-0.2) {12+$[n]$ (quarters)};
      \node [node font size={\normalsize}{10pt}] at (-2,0) {Annuity\\scenario:};
      \node [fill=white,inner sep=1pt] at (4,-1.2) {$j_4=6\%$};
      \node at (3.8,-1.7) {$\Rightarrow i=1.5\%$};
      \node [fill=white,inner sep=1pt] at (11,-1.2) {$j_4=7\%$};
      \node at (11.2,-1.7) {$i=1.75\%$};
      \draw [decorate,decoration={brace,amplitude=10pt}] (0,0.8) to (7.975,0.8);
      \draw [decorate,decoration={brace,amplitude=10pt}] (8.025,0.8) to (14,0.8);
      \draw [fill=white] (0,0.5) ellipse (35pt and 7.5pt);
      \node [inner sep=0pt,node font size={\normalsize}{7pt},fill=white,align=center] at (0,0.5) {\$20,000};
      \node at (4.8,1.4) {simple ordinary annuity};
      \node at (11.4,1.4) {simple ordinary annuity};
    \end{tikzpicture}%
  }
\end{center}

After 3 years, the widow has received, in addition to the down payment, 12 full payments of \$1800, for a total present value of
\[\$20,000+\$1800a_{\angl{12}1.5\%}\cong\$39,633.51<\$100,000\]
Thus, the (wo)man will still receive some payments afterwards. But how many?\\
The ``theoretical'' number of payments to be received after 3 years must satisfy the equation \\
\begin{tikzpicture}[overlay, remember picture, shift={({pic cs:x})}, red]
  \draw [decorate,decoration={brace,amplitude=8pt,mirror}] (0,-0.2) to (2,-0.2);
  \draw [decorate,decoration={brace,amplitude=8pt,mirror}] (3.2,-0.2) to (5.7,-0.2);
  \tikzset{node font size={\footnotesize}{8pt}}
  \node at (1.1,-1.2) {{PV of first}\\{3 years of}\\{annuity +}\\{\$20,000 down}};
\node at (4.6,-1.2) {{PV of remainder}\\{of annuity, i.e.}\\{value of remainder}\\{at time $n=12$}};
\node  at (7.8,-1) {discount factor\\from time $n=12$\\to $n=0$};
\end{tikzpicture}
\begin{align*}
  \text{PV of cash scenario }&=\text{ PV of annuity scenario}\\
  \$100,000\quad&=\quad\tikzmark{x}\$39,633.51\quad+\quad\$1800a_{\angl n1.75\%} \,\,\cdot\,\,(1+1.5\%)^{\minus12}
\end{align*}

\vspace{42pt}%
Thus,
\begin{tikzpicture}[overlay, remember picture, shift={({pic cs:n101})},blue]
  \draw [->] (0,-0.2) to[out=210,in=330,looseness=0.5] (-2,-0.2);
  \draw [->] (-3,-0.4) to[out=330,in=210,looseness=0.5] (-0.25,-0.5);
  \node [node font size={\small}{9pt}] at (-6.4,-3.3) {taking\\$\ln$ of\\both sides};
\end{tikzpicture}
\begin{spreadlines}{\dimexpr\jot+0.75em\relax}
  \begin{alignat*}{2}
    &&\quad \$100,000-\$39,633.51&=\frac{\$1800a_{\angl n1.75\%}}{(1.015)^{12}}\\
    \iff&& \frac{\$60,366.49(1.015)^{12}}{\$1800}&=\frac{1-(1.0175)^{\minus n}}{0.0175}\\
    \iff&&\frac{60,366.49(1.015)^{12}(0.0175)}{1800}&=1-\tikzmark{n101}(1.0175)^{\minus n}\\
    \iff &&(1.0175)^{\minus n}&=1-\frac{60,366.49(1.015)^{12}(0.0175)}{1800}\\
    \iff && -n\ln 1.0175&=\ln\left[1-\frac{60,366.49(1.015)^{12}(0.0175)}{1800} \right]\\
    \iff && n&=\frac{\ln\left[1-\frac{60,366.49(1.015)^{12}(0.0175)}{1800}\right]}{-\ln1.0175}\\
    && &\cong69.73
  \end{alignat*}
\end{spreadlines}
So the widow will receive 69 full payments after the first 3 years of payments, i.e. (s)he will receive a total of $69+12=81$ full payments of $\$1800$.
\vspace{12pt}(b)\vspace{-12pt}
\begin{center}
  \begin{tikzpicture}
    \draw (0,0) -- (6,0);
    \draw (6.1,0.2) -- (5.9,-0.2);
    \draw (6.2,0.2) -- (6,-0.2);
    \draw (6.1,0) -- (12,0);
    \draw (12.1,0.2) -- (11.9,-0.2);
    \draw (12.2,0.2) -- (12,-0.2);
    \draw (12.1,0) -- (14,0);
    \draw (0,0.2) -- (0,-0.2);
    \draw (2,0.2) -- (2,-0.2);
    \draw (4,0.2) -- (4,-0.2);
    \draw (8,0.2) -- (8,-0.2);
    \draw (10,0.2) -- (10,-0.2);
    \draw (14,0.2) -- (14,-0.2);
    \draw (0,-1) -- (0,-1.4);
    \draw (8,-1) -- (8,-1.4);
    \draw (14,-1) -- (14,-1.4);
    \node [above] at (2,0.2) {\$1800};
    \node [above] at (4,0.2) {\$1800};
    \node [above] at (8,0.2) {\$1800};
    \node [above] at (10,0.2) {\$1800};
    \node [above] at (14.45,0.2) {$\$1800+X$};
    \node [below] at (0,-0.2) {0};
    \node [below] at (2,-0.2) {1};
    \node [below] at (4,-0.2) {2};
    \node [below] at (8,-0.2) {12};
    \node [below] at (10,-0.2) {13};
    \draw [<->] (0.05,-1.2) to (7.95,-1.2);
    \draw [<->] (8.05,-1.2) to (13.95,-1.2);
    \node [below] at (14.95,-0.2) {81 (quarters)};
    \node [fill=white,inner sep=2pt] at (4,-1.2) {$i=1.5\%$};
    \node [fill=white,inner sep=2pt] at (11,-1.2) {$i=1.75\%$};
    \draw [decorate,decoration={brace,amplitude=10pt}] (0,0.8) to (7.975,0.8);
    \draw [decorate,decoration={brace,amplitude=10pt}] (8.025,0.8) to (14,0.8);
    \node at (0,1.3) {\$20,000};
    \node at (4.8,1.4) {simple ordinary annuity 1};
    \node at (11.4,1.4) {simple ordinary annuity 2};
    \draw [->] (15.25,1.4) to (15.25,0.8);
    \node [node font size={\footnotesize}{9pt}] at (15.25,1.8) {balloon\\payment};
  \end{tikzpicture}
\end{center}
The balloon payment $X$ must be such that
\begin{spreadlines}{\dimexpr\jot+0.75em\relax}
  \begin{alignat*}{2}
    && \text{PV of cash scenario}&=\text{PV of annuity scenario}\\
    &&\$100,000&=\tikzmark{n100}\$20,000+\$1800a_{\angl{12}1.5\%}+\$1800a_{\angl{69}1.75\%}(1+1.5\%)^{\minus12}\\
    && &\quad\quad+X(1+1.75\%)^{\minus69}(1+1.5\%)^{\minus18}\\
    && &=\$20,000+\$19,633.51+\$60,040.76+0.2526577697X\\
    \llap{$\iff$}&& \$100,000-\$20,000-\$19,633.51-\$60,040.76&=0.2526577697X\\
    \llap{$\iff$}&& X&=\frac{\$100,000-\$20,000-\$19,633.51-\$60,040.76}{0.2526577697X}\\
    && &\cong\$1,289.21
  \end{alignat*}
\end{spreadlines}
\begin{tikzpicture}[overlay, remember picture, shift={({pic cs:n100})}, node font size={\normalsize}{7pt}, green]
  \node [font=\scriptsize] at (1,-0.35) {down\\payment};
  \node  at (3,-0.35) {{PV of annuity 1}};
  \node  at (5.7,-0.45) {{PV of annuity 2}};
  \node  at (8,-0.5) {{discount}\\{factor}};
\draw [decorate,decoration={brace,amplitude=4pt,mirror}] (4.7,-0.15) to (7.1,-0.15);
\draw [decorate,decoration={brace,amplitude=4pt,mirror}] (7.2,-0.15) to (9.1,-0.15);
\end{tikzpicture}

\noindent Thus, the additional sum paid with the last full payment, called balloon payment, to exhaust the widows's benefits is \$1,289.21.
\vspace{12pt}(c)
\begin{center}
  \makebox[0pt]{%
    \begin{tikzpicture}
      \draw (0,0) -- (6,0);
      \draw (6.1,0.2) -- (5.9,-0.2);
      \draw (6.2,0.2) -- (6,-0.2);
      \draw (6.1,0) -- (12,0);
      \draw (12.1,0.2) -- (11.9,-0.2);
      \draw (12.2,0.2) -- (12,-0.2);
      \draw (12.1,0) -- (16,0);
      \draw (0,0.2) -- (0,-0.2);
      \draw (2,0.2) -- (2,-0.2);
      \draw (4,0.2) -- (4,-0.2);
      \draw (8,0.2) -- (8,-0.2);
      \draw (10,0.2) -- (10,-0.2);
      \draw (14,0.2) -- (14,-0.2);
      \draw (16,0.2) -- (16,-0.2);
      \draw (0,-1) -- (0,-1.4);
      \draw (8,-1) -- (8,-1.4);
      \draw (16,-1) -- (16,-1.4);
      \node [above] at (2,0.2) {\$1800};
      \node [above] at (4,0.2) {\$1800};
      \node [above] at (8,0.2) {\$1800};
      \node [above] at (10,0.2) {\$1800};
      \node [above] at (14,0.2) {\$1800};
      \node [above] at (16,0.2) {$Y$};
      \node [below] at (0,-0.2) {0};
      \node [below] at (2,-0.2) {1};
      \node [below] at (4,-0.2) {2};
      \node [below] at (8,-0.2) {12};
      \node [below] at (10,-0.2) {13};
      \node [below] at (14,-0.2) {81};
      \draw [<->] (0.05,-1.2) to (7.95,-1.2);
      \draw [<->] (8.05,-1.2) to (15.95,-1.2);
      \node [below] at (16.95,-0.2) {82 (quarters)};
      \node [fill=white,inner sep=2pt] at (4,-1.2) {$i=1.5\%$};
      \node [fill=white,inner sep=2pt] at (12,-1.2) {$i=1.75\%$};
      \draw [decorate,decoration={brace,amplitude=10pt}] (0,0.8) to (7.975,0.8);
      \draw [decorate,decoration={brace,amplitude=10pt}] (8.025,0.8) to (14,0.8);
      \node at (0,1.3) {\$20,000};
      \node at (4.8,1.4) {simple ordinary annuity 1};
      \node at (11.4,1.4) {simple ordinary annuity 2};
    \end{tikzpicture}%
  }
\end{center}
The drop payment $Y$ is such that
\begin{alignat*}{2}
  && \text{PV of cash scenario}&=\text{PV of annuity scenario}\\
  &&\$100,000&=\$20,000+\$1800a_{\angl{12}1.5\%}+\$1800a_{\angl{69}1.75\%}(1_1.5\%)^{\minus12}\\
  && &\quad\quad+Y(1+1.75\%)^{\minus70}(1+1.5\%)^{\minus12}\\
  && &=\$20,000+\$19,633.51+\$60,040.76+0.2483123044Y\\
  %&& &\quad\quad+0.
  \iff&& Y&=\frac{\$100,000-\$20,000-\$19,633.51-\$60,040.76}{0.2483123044}\\
  && &\cong\$1311.78
\end{alignat*}
Hence, the payment 3 months after the last full payment, also called drop payment, which will exhaust the widow's benefits is \$1311.78.
\end{document}

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