– 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.

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

OMG, thank you so much!

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?

oh and interest is simple.

Great work!!!!!!!!!1

What type of calcurator should i use?

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Thanks is Improve my math

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