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Determining the Monthly Payment of an Installment Loan - YouTube
Channel: Mathispower4u
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- WELCOME TO A LESSON
ON THE LOAN PAYMENT FORMULA.
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THE GOAL OF THE VIDEO
IS TO DETERMINE THE PAYMENT
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FOR A FIXED INSTALLMENT LOAN.
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INSTALLMENT BUYING IS WHEN YOU
PURCHASE SOMETHING TODAY
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WITH A LOAN THAT YOU PAY BACK
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WITH EQUAL PAYMENTS
OVER A PERIOD OF TIME,
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USUALLY MONTHLY
FOR A PERIOD OF YEARS.
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THE TWO MOST COMMON EXAMPLES
WOULD BE FOR A CAR LOAN
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OR A HOME MORTGAGE LOAN,
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AND WE'LL TAKE A LOOK
AT AN EXAMPLE OF BOTH OF THESE.
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HERE IS THE LOAN PAYMENT FORMULA
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FOR FIXED AMOUNT
OF EQUAL PAYMENTS
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WHERE R IS THE ANNUAL NOMINAL
INTEREST RATE
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EXPRESSED AS A DECIMAL,
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T IS THE TIME IN YEARS, N IS THE
NUMBER OF COMPOUNDS PER YEAR.
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P IS THE AMOUNT OF THE LOAN,
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AND PMT REPRESENTS THE MONTHLY
PAYMENT.
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LET'S TAKE A LOOK AT
WHERE THIS FORMULA COMES FROM.
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IT COMES FROM THE COMPOUNDED
INTEREST FORMULA
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AND THE VALUE
OF ANNUITY FORMULA.
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SO IF YOU'RE THE BANK
OR THE LENDER
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YOU WOULD USE THE COMPOUNDED
INTEREST FORMULA
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TO DETERMINE THE RETURN
ON YOUR INVESTMENT.
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AND IF YOU WERE THE PERSON
TAKING OUT THE LOAN
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YOU COULD USE THE VALUE
ANNUITY FORMULA
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WHERE P IS IN REPLACE WITH PMT
FOR PAYMENT
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TO REPRESENT ALL THE PAYMENTS
THAT YOU WOULD MAKE
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TO COVER "A" YOUR LOAN AMOUNT,
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PLUS ALL THE INTEREST
YOU'D ALSO BE CHARGED.
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SO THE LOAN PAYMENT FORMULA
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COMES FROM COMBINING
THESE TWO FORMULAS
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OR SETTING THEM EQUAL
TO EACH OTHER.
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AND SINCE THESE ARE BOTH
EQUAL TO "A,"
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WE CAN SET THE RIGHT SIDES
OF THESE EQUATIONS
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EQUAL TO EACH OTHER.
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AND THEN IF WE SOLVE FOR PMT
OR PAYMENT,
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WE SHOULD HAVE OUR LOAN PAYMENT
FORMULA.
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SO LET'S GO AHEAD
AND TAKE A MOMENT AND DO THAT.
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SO LET'S GO AHEAD AND MULTIPLY
BOTH SIDES OF THE EQUATION
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BY R/N.
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SO IT WOULD SIMPLIFY NICELY
HERE.
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SO WE'D HAVE P x R/N.
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I'M ACTUALLY GO AHEAD
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AND MOVE THIS QUANTITY
DOWN TO THE DENOMINATOR
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SO IT'LL CHANGE THE EXPONENT
TO -NT.
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LET'S GO AHEAD AND CONTINUE THIS
ON THE NEXT PAGE.
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NOW, TO ISOLATE PMT I'M GOING
TO DIVIDE OR MULTIPLY BY 1/1
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+ R/N TO THE NT POWER - 1.
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SO LOOKING AT THE RIGHT SIDE
THIS WOULD SIMPLIFY OUT
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AND WE'RE LEFT WITH THE PAYMENT.
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SO WE ISOLATED PMT FOR PAYMENT.
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LET'S SEE WHAT WE HAVE
ON THE LEFT SIDE.
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OUR NUMERATOR'S STILL GOING
TO BE P x R DIVIDED BY N.
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HERE WE'RE GOING TO MULTIPLY.
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WE MULTIPLY 1 + R/N
TO THE -NT POWER x 1
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+ R/N TO THE +NT POWER.
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WE ADD OUR EXPONENTS
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AND THAT WOULD GIVE US
AN EXPONENT OF ZERO.
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SO THIS FIRST PRODUCT
IS EQUAL TO 1,
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AND THEN WE HAVE - 1 + R/N
TO THE -NT.
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AND THIS IS THE FORMULA
THAT WE USE
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TO DETERMINE THE PAYMENT AMOUNT
FOR A GIVEN LOAN
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IN THE AMOUNT OF P.
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LET'S GO AND TAKE A LOOK
AT OUR EXAMPLES.
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DETERMINE THE MONTHLY PAYMENT
FOR A 30 YEAR MORTGAGE LOAN
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OF 150,000 WITH A 5% FIXED
INTEREST COMPOUNDED MONTHLY.
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THEN DETERMINE THE TOTAL
INTEREST THAT WILL BE PAID
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OVER THE 30 YEARS.
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SO MONTHLY PAYMENT IS GOING
TO BE EQUAL TO P THE LOAN AMOUNT
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x R DIVIDED BY N,
THAT'LL BE 0.05.
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IT'S COMPOUNDED MONTHLY
SO N IS 12.
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AND DIVIDE ALL OF THIS BY 1 - 1
+ R/N TO THE -N x T POWER.
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WELL, N IS 12 AND T IS TIME
IN YEARS SO IT'LL BE 30.
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SO OUR EXPONENT HERE
IS GOING TO BE -360.
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LET'S GO AHEAD
AND EVALUATE THIS.
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WE'LL PUT OUR NUMERATOR
IN A SET OF PARENTHESIS.
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SO THERE'S OUR NUMERATOR
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AND WE'LL DIVIDE THIS
BY OUR DENOMINATOR.
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AND WE CAN SEE
THAT OUR MONTHLY PAYMENT
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IS GOING TO BE APPROXIMATELY
$805.23.
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NOW, THE SECOND PART ASK US
DETERMINE THE TOTAL INTEREST
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THAT WILL BE PAID
OVER THE 30 YEARS.
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WELL, WE'RE GOING TO MAKE 360
PAYMENTS OF $805.23 FOR A LOAN
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IN THE AMOUNT OF $150,000.
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SO LETS FIRST DETERMINE
HOW MUCH MONEY WE'RE PAYING
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OVER THE 30 YEARS.
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IT'LL BE 805.23 x 12 MONTHS
A YEAR x 30 YEARS.
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SO WE'RE GOING TO PAY
$289,882.80 OVER THE 30 YEARS
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FOR A LOAN AMOUNT OF $150,000.
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SO IF WE SUBTRACT $150,000
FROM THIS AMOUNT
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THE REST WILL BE THE AMOUNT
OF INTEREST PAID.
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SO THE TOTAL INTEREST THAT WE'LL
BE PAYING OVER THE 30 YEARS
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IS ALMOST $140,000
OR $139,882.80.
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SO NOTICE THAT WE'RE PAYING
ALMOST AS MUCH INTEREST
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AS THE TOTAL LOAN AMOUNT.
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NOW, FOR OUR SECOND EXAMPLE WE
WANT TO COMPARE THE SAME LOAN,
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BUT NOW INSTEAD OF A 30 YEAR
MORTGAGE
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WE'LL TAKE A LOOK
AT THE DIFFERENCE IN PAYMENTS
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AND INTEREST AMOUNT IF WE HAVE
A 15 YEAR MORTGAGE INSTEAD.
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SO EVERYTHING IS THE SAME HERE
EXCEPT NOW T,
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THE NUMBER OF YEARS, WILL BE 15
INSTEAD OF 30.
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SO OUR EXPONENT HERE IS GOING TO
BE -12 x 15 NOW INSTEAD OF 30.
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SO LET'S SEE HOW THIS AFFECTS
OUR MONTHLY PAYMENT,
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AS WELL AS THE TOTAL INTEREST
PAID OVER 15 YEARS.
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HERE'S OUR NUMERATOR.
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AND, AGAIN, OUR EXPONENT HERE
IS GOING TO BE -180.
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SO FOR A 15 YEAR MORTGAGE THE
LOAN PAYMENT WOULD BE $1,186.19.
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SO GOING BACK AND COMPARING THIS
TO THE 30 YEAR MORTGAGE,
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LOOKS LIKE OUR LOAN PAYMENT
WENT UP MORE THAN $350
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BUT THE PAYMENTS WOULD ONLY BE
FOR HALF THE TIME.
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LET'S ALSO COMPARE THE INTEREST
PAID OVER 15 YEARS
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COMPARED TO THE 30 YEAR
MORTGAGE.
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SO WE'LL BE MAKING THIS MONTHLY
PAYMENT FOR 15 YEARS
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OR 12 TIMES A YEAR FOR 15 YEARS.
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SO HERE'S THE TOTAL AMOUNT PAID
OVER THE 15 YEARS,
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AND THEN WE'LL SUBTRACT OUT
THE LOAN AMOUNT,
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AND THAT'LL LEAVE US WITH
THE AMOUNT OF INTEREST PAID.
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SO WE'LL BE PAYING $63,514.20
OF INTEREST OVER THE 15 YEARS.
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AGAIN, COMPARING THIS
TO THE 30 YEAR MORTGAGE
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WE'LL BE PAYING OVER $70,000
MORE OF INTEREST
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IF WE SELECT
THE 30 YEAR MORTGAGE.
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LET'S GO AND TAKE A LOOK
AT ONE MORE EXAMPLE
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DEALING WITH A CAR LOAN.
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SO DETERMINE THE MONTHLY PAYMENT
OF A FIVE YEAR CAR LOAN
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OF $20,000 WITH A 5.5% FIXED
INTEREST COMPOUNDED MONTHLY.
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SO IT'S THE SAME FORMULA
THAT WE HAVE.
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$20,000 x 0.055 DIVIDED BY 12
AS OUR NUMERATOR.
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OUR DENOMINATOR'S GOING TO BE 1
- THE QUANTITY 1 + 0.055
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DIVIDED BY 12
RAISED TO THE -NT POWER.
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WELL, N IS 12 BECAUSE IT'S
STILL COMPOUNDED MONTHLY.
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AND IT'S FOR FIVE YEARS
SO T IS 5.
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SO OUR EXPONENT HERE
IS GOING TO BE -60.
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LET'S GO BACK TO OUR CALCULATOR.
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OUR EXPONENT HERE IS GOING TO BE
-12 x 5 THAT'LL BE -60,
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AND THERE'S OUR DENOMINATOR.
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SO OUR MONTHLY PAYMENT WOULD BE
APPROXIMATELY $382.02.
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THEN, AGAIN, THE TOTAL AMOUNT OF
INTEREST PAID OVER FIVE YEARS,
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WE'LL DETERMINE
THE TOTAL AMOUNT PAID
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AND THEN WE'LL SUBTRACT
THE LOAN AMOUNT OF $20,000.
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SO WE'LL HAVE $382.02 x 12
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THAT'LL BE THE AMOUNT PAID
PER YEAR x 5 YEARS,
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SO $22,921.20
IS THE TOTAL AMOUNT PAID.
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MINUS THE LOAN AMOUNT LEAVES US
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WITH THE AMOUNT
OF INTEREST PAID.
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SO ALMOST $3,000 OF INTEREST
OR $2,921.20.
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I LIKE TO MAKE A COUPLE CLOSING
COMMENTS ON MORTGAGE LOANS.
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SOME MORTGAGE LOANS HAVE
ORIGINATION FEES OR POINTS.
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FOR EACH POINT THE BUYER
MUST PAY A COST OF 1%
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OF THE TOTAL LOAN.
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AND SOME MORTGAGES
WILL ALSO REQUIRE
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AN ADDITIONAL MONTHLY PAYMENT
INTO AN ESCROW ACCOUNT
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TO PAY YEARLY PROPERTY TAXES
AND INSURANCE.
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IT'S IMPORTANT TO BE AWARE
OF ALL OF THE COST
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WHEN TAKING A LOAN.
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I HOPE YOU FOUND THIS VIDEO
HELPFUL.
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THANK YOU FOR WATCHING.
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