Chapter 2.4, Problem 55AYU

Mortgage Fees Fannie Mae charges a loan-level price adjustment (LLPA) 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-value (LTV) ratio. The LTV ratio is the ratio of amount borrowed to appraised value of the home. For example, a homebuyer who wishes to borrow $\$250,000$ with a credit score of 730 and an LTV ratio of $80\%$ will pay $0.75\%\left(0.0075\right)$ of $\$250,000,$ or $\$1875.$ The table shows the LLPA for various credit scores and an LTV ratio of $80\%$.

$\begin{array}{cc}\text{Credit}\text{Score}& \begin{array}{l}\text{Loan-Level}\\ \text{Price}\text{Adjustment}\text{Rent}\end{array}\\ \le 659& 3.00\%\\ 660-679& 2.75\%\\ 680-699& 1.75\%\\ 700-719& 1.25\%\\ 720-739& 0.75\%\\ \ge 740& 0.50\%\end{array}$

Construct a function $C=C(s),$ where $C$ is the loan-level price adjustment (LLPA) and $s$ is the credit score of an individual who wishes to borrow $\$300,000$ with an $80\%$ LTV ratio.

What is the LLPA on a $\$300,000$ loan with an 80% LTV ratio for a borrower whose credit score is $725?$

What is the LLPA on a $\$300,000$ loan with an 80% LTV ratio for a borrower whose credit score is $670?$

To determine

A function $C(s)$ that models the loan-level price adjustment $C$ for $s$ credit score of an individual who wishes to borrow $\$300,000$ with $80\%$ LTV ratio. Where, Fannie may charge a loan-level price adjustment (LLPA) on all mortgages, which represents a must pay fee for homebuyers seeking a loan. The rate paid depends on the credit score of the borrower, the amount borrowed and the loan-to-value (LTV) ratio. The LTV is the ratio of amount borrowed to appraised value of the home.

The table shows the LLPA for various credit scores and an LTV ratio of $80\%$ is as below,

$\begin{array}{cc}\text{Credit score}& \begin{array}{l}\text{Loan-Level Price}\\ \text{Adjustment Rate}\end{array}\\ \le 659& 3.00\%\\ 660-679& 2.75\%\\ 680-699& 1.75\%\\ 700-719& 1.25\%\\ 720-739& 0.75\%\\ \ge 740& 0.50\%\end{array}$

Answer to Problem 55AYU

Solution:

The function $C(s)$ that models the loan-level price adjustment $C$ for $s$ credit score of an individual who wishes to borrow $\$300,000$ with $80\%$ LTV ratio is:

$C(s)=\{\begin{array}{l}9000s\le 659\\ 8250660\le s\le 679\\ 5250680\le s\le 699\\ 3750700\le s\le 719\\ 2250720\le s\le 739\\ 1500s\ge 740\end{array}$

Explanation of Solution

Given Information:

Fannie may charge a loan-level price adjustment (LLPA) on all mortgages, which represents a must pay fee for homebuyers seeking a loan. The rate paid depends on the credit score of the borrower, the amount borrowed and the loan-to-value (LTV) ratio. The LTV is the ratio of amount borrowed to appraised value of the home.

The table shows the LLPA for various credit scores and an LTV ratio of $80\%$ is as below,

$\begin{array}{cc}\text{Credit score}& \begin{array}{l}\text{Loan-Level Price}\\ \text{Adjustment Rate}\end{array}\\ \le 659& 3.00\%\\ 660-679& 2.75\%\\ 680-699& 1.75\%\\ 700-719& 1.25\%\\ 720-739& 0.75\%\\ \ge 740& 0.50\%\end{array}$

Explanation:

Let $s$ denote the credit score of an individual who wishes to borrow $\$300,000$ with $80\%$ LTV and $C$ is the loan-level price adjustment in dollars.

Given that,

If the individual with a credit score $s\le 659$ will pay $3.00\%$ of $\$300,000$

That is, the loan-level price adjustment for $s\le 659$ is $C(s)=(0.03)300000=9000$…(1)

Now, if the individual with a credit score $660\le s\le 679$ will pay $2.750\%$ of $\$300,000$

That is, the loan-level price adjustment for $660\le s\le 679$ is $C(s)=(0.0275)300000=8250$…(2)

Now, if the individual with a credit score $680\le s\le 699$ will pay $1.75\%$ of $\$300,000$

That is, the loan-level price adjustment for $680\le s\le 699$ is $C(s)=(0.0175)300000=5250$…(3)

Now, if the individual with a credit score $700\le s\le 719$ will pay $1.25\%$ of $\$300,000$

That is, the loan-level price adjustment for $700\le s\le 719$ is $C(s)=(0.0125)300000=3750$…(4)

Now, if the individual with a credit score $720\le s\le 739$ will pay $0.75\%$ of $\$300,000$

That is, the loan-level price adjustment for $720\le s\le 739$ is $C(s)=(0.0075)300000=2250$…(5)

Now, if the individual with a credit score $s\ge 740$ will pay $0.50\%$ of $\$300,000$

That is, the loan-level price adjustment for $s\ge 740$ is $C(s)=(0.0050)300000=1500$…(6)

From (1) to (6),

Therefore, the function $C(s)$ that models the loan-level price adjustment $C$ for $s$ credit score of an individual who wishes to borrow $\$300,000$ with $80\%$ LTV ratio is:

$C(s)=\{\begin{array}{l}9000s\le 659\\ 8250660\le s\le 679\\ 5250680\le s\le 699\\ 3750700\le s\le 719\\ 2250720\le s\le 739\\ 1500s\ge 740\end{array}$

To determine

The LLPA for a borrower whose credit score is $725$ on a loan of $\$300,000$ with an $80\%$ LTV ratio.

Answer to Problem 55AYU

Solution:

The LLPA on a loan of $\$300,000$ with an $80\%$ LTV ratio for a borrower with credit score $725$ is $\$2250$.

Explanation of Solution

Given Information:

Fannie may charge a loan-level price adjustment (LLPA) on all mortgages, which represents a must pay fee for homebuyers seeking a loan. The rate paid depends on the credit score of the borrower, the amount borrowed, and the loan-to-value (LTV) ratio. The LTV is the ratio of amount borrowed to the appraised value of the home.

The table shows the LLPA for various credit scores and an LTV ratio of $80\%$ is as below:

$\begin{array}{cc}\text{Credit score}& \begin{array}{l}\text{Loan-Level Price}\\ \text{Adjustment Rate}\end{array}\\ \le 659& 3.00\%\\ 660-679& 2.75\%\\ 680-699& 1.75\%\\ 700-719& 1.25\%\\ 720-739& 0.75\%\\ \ge 740& 0.50\%\end{array}$

Explanation:

From part a)

The function $C(s)$ that models the loan-level price adjustment $C$ for $s$ credit score of an individual who wishes to borrow $\$300,000$ with $80\%$ LTV ratio is:

$C(s)=\{\begin{array}{l}9000s\le 659\\ 8250660\le s\le 679\\ 5250680\le s\le 699\\ 3750700\le s\le 719\\ 2250720\le s\le 739\\ 1500s\ge 740\end{array}$

Since, a borrower credit score is $s=725$ that is $720\le s\le 739$

$\Rightarrow C(s)=2250$.

Therefore, the LLPA on a loan of $\$300,000$ with an $80\%$ LTV ratio for a borrower with credit score $725$ is $\$2250$

To determine

The LLPA for a borrower whose credit score is $670$ on a loan of $\$300,000$ with an $80\%$ LTV ratio.

Answer to Problem 55AYU

Solution:

The LLPA on a loan of $\$300,000$ with an $80\%$ LTV ratio for a borrower with credit score $670$ is $\$8250$.

Explanation of Solution

Given Information:

Fannie may charge a loan-level price adjustment (LLPA) on all mortgages, which represents a must pay fee for homebuyers seeking a loan. The rate paid depends on the credit score of the borrower, the amount borrowed and the loan-to-value (LTV) ratio. The LTV is the ratio of amount borrowed to appraised value of the home.

The table shows the LLPA for various credit scores and an LTV ratio of $80\%$ is as below:

$\begin{array}{cc}\text{Credit score}& \begin{array}{l}\text{Loan-Level Price}\\ \text{Adjustment Rate}\end{array}\\ \le 659& 3.00\%\\ 660-679& 2.75\%\\ 680-699& 1.75\%\\ 700-719& 1.25\%\\ 720-739& 0.75\%\\ \ge 740& 0.50\%\end{array}$

Explanation:

From part a)

The function $C(s)$ that models the loan-level price adjustment $C$ for $s$ credit score of an individual who wishes to borrow $\$300,000$ with $80\%$ LTV ratio is:

$C(s)=\{\begin{array}{l}9000s\le 659\\ 8250660\le s\le 679\\ 5250680\le s\le 699\\ 3750700\le s\le 719\\ 2250720\le s\le 739\\ 1500s\ge 740\end{array}$

Since, a borrower credit score is $s=670$ that is $660\le s\le 679$

$\Rightarrow C(s)=8250$.

Therefore, the LLPA on a loan of $\$300,000$ with an $80\%$ LTV ratio for a borrower with credit score $670$ is $\$8250$