Lesson Mortgage problems

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Mortgage problems


Problem 1

You bought a home for  $150,000,  paying  10%  as a down payment,
and financing the rest at  8%  interest for  30 years.
    (a)   How much money did you pay as your down payment?
    (b)   How much money was your existing mortgage (loan) for?
    (c)   What is your current monthly payment on your existing mortgage?

Solution 

(a)  Down payment was 10$ of $150,000, or $15,000.     ANSWER to question (a)


(b)  The loaned amount was  $150,000 - $15,000 = $135,000.


(c)  To find monthly payment P, use the standard formula

         P = %28r%2AL%29%2F%281-%281%2Br%29%5E%28-n%29%29

     where L is the loan amount; 
     r is the nominal interest rate per month as a decimal; 
     n is the number of months (the number of equal monthly payments).


     In this problem

         L = $135,000;  

         r = 0.08/12;

         n = 30*12 = 360 months.


     The calculations are here

         P = %28%280.08%2F12%29%2A135000%29%2F%281-%281%2F%281%2B0.08%2F12%29%5E360%29%29 = 990.58 dollars.   ANSWER to question (b)


     You may check these calculations using online calculators

         https://www.calculator.net/loan-calculator.html

     or

         https://www.calculatorsoup.com/calculators/financial/loan-calculator.php

Problem 2

Jane takes out a ​ $320,000 ​ loan at a fixed interest rate of ​ 6% ​ to buy a home.
The loan is to be repaid in a time period of ​ 30 ​ years.
​ ​ ​ ​ (a) ​ ​ Calculate Jane's monthly mortgage payment.
​ ​ ​ ​ (b) ​ ​ Calculate the total amount of money that she will have to repay.
​ ​ ​ ​ (c) ​ ​ Calculate the total interest that she will pay for this loan.

Solution 

Use the formula for the monthly payment for a loan

    M = P%2A%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29


where P is the loan amount; r = 0.06%2F12 is the nominal interest rate per month;
n is the number of payments (same as the number of months); M is the monthly payment.


In this problem  P = $320000;  r = 0.06%2F12;  n = 30*12 = 360.


Substitute these values into the formula and get for monthly payment

    M = 320000%2A%28%28%280.06%2F12%29%29%2F%281-%281%2B0.06%2F12%29%5E%28-360%29%29%29 = $1918.56.


Thus, the monthly payment is $1918.56.


In total, Jane will have to repay  30*12*1918.56 = 690,681.60 dollars in 30 years.


The difference $690,681.60 - $ = $320,000 = $370,681.60  is the interest she will pay for this loan..

Problem 3

You want to buy a  $238,000 home.  You plan to pay  5%  as a down payment,  and take out
a  30  year loan at  5.5%  annual interest for the rest,  paying equal portions at the end of each month.
How much money in interest will you save if you finance for  15  years instead of  30  years?


Solution 

Use the formula for the monthly payment for a loan 

    M = P%2A%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29,


where P is the loan amount; r = 0.055%2F12 is the nominal interest rate per month;
n is the number of payments (same as the number of months 30*12 = 360); M is the monthly payment.


In this problem  P = 238000 - 0.05*238000 = 226100 dollars.


Substitute these values into the formula and get for monthly payment 

    M = 226100%2A%28%28%280.055%2F12%29%29%2F%281-%281%2B0.055%2F12%29%5E%28-360%29%29%29 = $1283.78.


So, the monthly payment is 1283.78 dollars.  

Hence, the total payment for this 30-years $226100 loan is 360*1283.78 = 462160.80 dollars.



Now calculate the 15-years payment. In all, there are 15*12 = 180 payments. The monthly payment is 

    M = 226100%2A%28%28%280.055%2F12%29%29%2F%281-%281%2B0.055%2F12%29%5E%28-180%29%29%29 = $1847.43.


Hence, the total payment for this 15-years $226100 loan is 180*1847.43 = 332537.40 dollars.


You will save  462160.80 - 332537.40 = 129623.40 dollars in interest.    ANSWER

Problem 4

Suppose you can afford to pay at most  $1650  per month for a mortgage payment.
If the maximum amortization period you can get is  15  years,
and you must pay  5%  interest per year compounded monthly,
    (a)   what is the most expensive house you can buy?
    (b)   How much interest will you have paid to the lender at the end of the loan?

Solution 

Use the formula for the monthly payment for a mortgage

    M = P%2A%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29


where P is the loan amount; r = 0.05%2F12 is the effective interest rate per month;
n is the number of payments (same as the number of months); M is the monthly payment.


From this formula, the expression for the maximum loan is

    P = M%2F%28%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29%29.


In this problem  M = $1650;  r = 0.05%2F12,  n = 15*12 = 180 monthly payments.


Substitute these values into the formula and get for the maximum loan amount

    P = 1650%2F%28%28%28%280.05%2F12%29%29%2F%281-%281%2B0.05%2F12%29%5E%28-180%29%29%29%29 = $208,651.15.


Thus, the maximum mortgage amount is $208,651, which means that the most expensive house is about $208,651.    ANSWER to question (a)

The total interest you have paid to the lender at the end of the mortgage is 

    15*12*1650 - 208651 = 297000 - 208651 = 88,349 dollars.   ANSWER to question (b)


My other lessons on Finance problems in this site are
    - Problems on simple interest accounts
    - Problems on discretely compounded accounts
    - Problems on continuously compounded accounts
    - Find future value of an Ordinary Annuity
    - Find regular deposits for an Ordinary Annuity
    - How long will it take for an ordinary annuity to get an assigned value?
    - Find future value for an Annuity Due saving plan
    - Regular withdrawals from an annuity account
    - Ordinary annuity account with non-zero initial deposit as a combined total of two accounts
    - Annual depositing and semi-annual compounding in ordinary annuity saving plan
    - Variable withdrawals from a compounded account (sinking fund)
    - Present value of an ordinary annuity cumulative saving plan
    - Problems on sinking funds
    - Find the compounding rate of an ordinary annuity
    - Accumulate money using ordinary annuity; then spend money via sinking fund
    - Calculating a retirement plan
    - Accumulating money via ordinary annuity and spending simultaneously via sinking fund
    - Loan problems
    - Amortizing a debt on a credit card
    - One level more complicated non-standard problems on ordinary annuity plans
    - One level more complicated problems on sinking funds
    - One level more complicated non-standard problems on loans
    - Using Excel to find the principal part of a certain loan payment
    - Using Excel to find the interest part of a certain loan payment
    - Tricky problems on present values of annuities
    - OVERVIEW of my lessons on Finance section in this site

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.


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