Determining the Monthly Payment of an Installment Loan


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

11 Replies to “Determining the Monthly Payment of an Installment Loan

  1. thanks, for the annuity problem, I couldn't figure out if that was a negative exponent or if that was a subtraction symbol….thanks for clearing that up

  2. how would weekly payments be calculated? so I have a loan of $10500 @9.96% and I want to pay loan off in 16 weeks..what would the weekly payments be?

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