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calculation.pdf

calculation.tex

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\lhead{Computations with DTI / LTV}
\rhead{Determining Max Loan Amount}

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\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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