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\documentclass[letterpaper,10pt]{article} | |
\usepackage[margin=1in]{geometry} | |
\usepackage{amsthm,amssymb,amsmath} | |
\usepackage{embedfile} | |
\embedfile{\jobname.tex} | |
\usepackage{hyperref} | |
\hypersetup{colorlinks=true,urlcolor=blue} | |
\usepackage{fancyhdr,fancyvrb} | |
\pagestyle{fancy} | |
\lhead{Computations with DTI / LTV} | |
\rhead{Determining Max Loan Amount} | |
\renewcommand{\headrulewidth}{0pt} | |
\renewcommand{\qed}{\(\blacksquare\)} | |
\begin{document} | |
\section{Independent Variables} | |
We have a set of inputs that will be handed down to us | |
\begin{enumerate} | |
\item \(S\) the credit score | |
\item \(A\) the total assets that can be used for down payment and closing | |
\item \(I\) the monthly income | |
\item \(D\) the monthly debts outside the loan | |
\item \(c\) the closing costs as a proportion of the purchase price, for | |
example at 2.5\% (a typical amount) \(c = \frac{2.5}{100}\) | |
\item \(\kappa\) the ``coupon multiplier" that determines the monthly | |
loan payment from the loan amount. This is really a combination of the | |
term \(T\) (in months) and the monthly interest rate \(r\), but we'll never | |
need either of those individually. Starting with those values: | |
\begin{equation} | |
\kappa = \frac{r}{12} \frac{s}{s - 1} \quad \text{where} \quad | |
s = \left(1 + \frac{r}{12}\right)^T. | |
\end{equation} | |
\end{enumerate} | |
\section{Free Variables} | |
\begin{enumerate} | |
\item \(P\) the ``purchasing power", i.e. the maximum purchase price | |
\item \(d\) the DTI (this must be at most 45\%) | |
\item \(\ell\) the LTV (this must be at most 95\%) | |
\end{enumerate} | |
\section{Dependent Variables} | |
Some values depend on others to be computed. We can think of the independent | |
variables as unchanged parameters while the free variables actually impact | |
the "shape" of a given value: | |
\begin{enumerate} | |
\item \(m\left(\ell\right)\) the monthly costs as a proportion of the purchase | |
price, for example if taxes are 0.77\% of the purchase price on a yearly | |
basis and insurance is 0.4\% of it, then on monthly basis | |
\(m = \frac{1}{12}\left(\frac{0.77}{100} + \frac{0.4}{100}\right)\). This | |
depends on \(\ell\) because in the case the LTV exceeds 80\%, mortgage | |
insurance is required. In that case, estimating mortgage insurance at 0.5\% | |
of the purchase price would give | |
\(m = \frac{1}{12}\left(\frac{0.77}{100} + \frac{0.4}{100} + | |
\frac{0.5}{100}\right)\). | |
\item \(R\left(\ell, d, S\right)\) the reserve requirement in months. For | |
example if \(0.36 < d \leq 0.45\), \(\ell < 0.75\) and | |
\(660 \leq S \leq 679\) then \(R = 6\) | |
\item \(M\left(P, \ell, \kappa\right)\) the monthly loan payment | |
\(M = \kappa \ell P\). | |
\end{enumerate} | |
\section{Constraints on Purchase Price} | |
The purchase price will be constrained directly by the assets \(A\) and monthly | |
income \(I\) (as well as related factors of cost) and indirectly by the bounds | |
on \(\ell\) and \(d\). | |
The assets must be split across down payment, closing costs and | |
reserve requirements: | |
\begin{equation} | |
A = P\left(1 - \ell\right) + c P + R\left(\ell, d, S\right) | |
\kappa \ell P. | |
\end{equation} | |
The income similar will be related to total (monthly) debts by the DTI, where | |
total debts come from mortage payments, monthly costs related to the loan | |
and debts outside the loan: | |
\begin{equation} | |
I d = \kappa \ell P + m\left(\ell\right) P + D. | |
\end{equation} | |
These equations can be solved for \(P\) | |
\begin{align} | |
P &= \frac{A}{\left(1 - \ell\right) + c + R\left(\ell, d, S\right) | |
\kappa \ell} = \frac{A}{1 + c - \left[1 - R\left(\ell, d, S\right) | |
\kappa\right] \ell} \label{eq:from-assets} \\ | |
P &= \frac{I d - D}{\kappa \ell + m\left(\ell\right)} \label{eq:from-income}. | |
\end{align} | |
In~\eqref{eq:from-assets}, we expect \(\kappa\) to be small (i.e. never larger | |
than \(0.1\)) and \(R\left(\ell, d, S\right)\) not to exceed \(6\) which | |
means that \(P\) will be increasing in \(\ell\). Additionally, \(P\) is | |
decreasing in \(R\). | |
In~\eqref{eq:from-income}, \(P\) is increasing in \(d\) and decreasing in | |
\(\ell\). | |
Also note that it may be impossible to satisfy~\eqref{eq:from-assets} | |
and~\eqref{eq:from-income} at the same time; this is because for a given | |
choice of \(d\) and \(\ell\), the purchase price may be limited by the | |
assets \(A\) so that not all of the monthly income \(I\) need be used. | |
Similarly if \(I\) is the limiting factor, then not all of \(A\) need be | |
used at purchase time. | |
\end{document} |
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calculation.pdf: calculation.tex | |
pdflatex calculation.tex |
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