Q. 55 - Precalculus Enhanced with Graphing Utilities

Question

Mortgage Fees Fannie Mae charges an adverse market delivery charge on all mortgages, which represents a fee homebuyers seeking a loan must pay. The rate paid depends on the credit score of the borrower, the amount borrowed, and the loan-to-valuc (ITV) ratio. The ITV ratio is the ratio of amount borrowed to appraised value of the home. For example, a homebuyer who wishes to borrow $\backslash (250,000$ with a credit score of $730$ and an LTV ratio of $80\%$ will pay $0.5 \%(0.005)$ of $\backslash )250,000$or $\backslash (1250$. The table shows the adverse delivery charge for various credit scores and an LTV ratio of $80 \%$

(a) Construct a function $C=C\left(s\right)$ where $C$ is the adverse market delivery charge and $s$ is the credit score of an individual who wishes to borrow $\backslash )300,000$ with an $80\%$ LTV ratio.

(b) What is the adverse market delivery charge on a $\backslash (300,000$loan with an $80\%$ LTV ratio for a borrower whose credit score is $725$ ?

(c) What is the adverse market delivery charge on a $\backslash )300,000$ loan with an $80\%$ LTV ratio for a borrower whose credit score is $670$ ?

Step-by-Step Solution

Answer

The values are $\$1500,\$8250$ respectively

Part (a) Step 1: Given information

Given the borrowing as $\$2,50,000$ and LTV ratio is $80\%$

Part (a) Step 2: Forming the equation

Assume $x$ denote the bill received by the credit card holders.

If the bill received is less than $\$10$, the due is the entire amount or $x$. It can be represented as $x$ if $0\le x<10$.

If the bill received is greater than or equal to $\$10$, but less than $\$500$, the due is $\$10$. It can be represented as $10$ if $10\le x<500$.

Part (a) Step 3: Creating the function

The function is

 $$\begin{gathered} C\left(s\right)=9750,s\le 659 \\ 8250 ,660\le s\le 679 \\ 4500 ,680\le s\le 699 \\ 3000 ,700\le s\le 719 \\ 1500 ,720\le s\le 739 \\ 750 ,s\ge 740 \end{gathered}$$

Part (b) Step 1: Given information

Given the table $C\left(s\right)$

Part (b) Step 2: Checking from the table

As $725$ is within the interval $[720,739]$ we conclude according to one that this is the second case from the end (adverse market delivery charge rate is $0.5\%$ ), i.e. for $s=725$ is

$C\left(s\right)=\$1500$

Part (c) Step 1: Given information

Given the table $C\left(s\right)$

Part (c) Step 2: Checking from the table

Checking, we get

As $670$ is within the interval $[660,679]$ we conclude according to one that this is the second case (adverse market delivery charge rate is $2.75\%$, i.e. for $s=670$ is

$C\left(s\right)=\$8250$