The Installment Loan Formula: Determining Remaining Balance


– WELCOME TO A LESSON ON HOW TO DETERMINE THE
REMAINING LOAN BALANCE AFTER A CERTAIN NUMBER OF YEARS
USING THE LOAN FORMULA. WITH LOANS IT IS OFTEN DESIRABLE
TO DETERMINE WHAT THE REMAINING LOAN BALANCE
WILL BE AFTER SOME NUMBER OF YEARS. FOR EXAMPLE,
IF YOU PURCHASE A HOME AND PLAN TO SELL IT
IN FIVE YEARS, YOU MIGHT WANT TO KNOW HOW MUCH
OF THE LOAN BALANCE YOU WILL HAVE PAID OFF AND HOW MUCH YOU HAVE TO PAY
FROM THE SALE. TO DETERMINE THE REMAINING LOAN
BALANCE AFTER A NUMBER OF YEARS WE FIRST NEED TO KNOW
THE LOAN PAYMENT IF WE DON’T ALREADY KNOW. REMEMBER THAT ONLY A PORTION
OF A LOAN PAYMENT GOES TOWARDS THE LOAN BALANCE. A PORTION ALSO GOES TOWARD
THE LOAN INTEREST. FOR EXAMPLE, IF YOU’RE PAYMENTS
WERE $1,000 A MONTH, AFTER A YEAR YOU WILL NOT HAVE
PAID OFF $12,000 OF THE LOAN BALANCE. TO DETERMINE THE REAMING LOAN
BALANCE WE CAN THINK HOW MUCH LOAN
WILL THESE LOAN PAYMENTS BE ABLE TO PAY OFF IN THE
REMAINING TIME OF THE LOAN? SO HERE’S OUR LOAN FORMULA, WHICH WE SHOULD ALREADY BE
FAMILIAR WITH, P SUB ZERO IS THE LOAN AMOUNT, D IS THE LOAN PAYMENT
PER UNIT OF TIME, R IS THE ANNUAL INTEREST RATE
AS A DECIMAL. K IS THE NUMBER OF COMPOUNDS
PER YEAR, AND N IS THE LENGTH OF THE LOAN
IN YEARS. LET’S CONSIDER A CAR LOAN
EXAMPLE. IF YOU PURCHASE A NEW CAR
FOR $20,000 USING A LOAN WITH MONTHLY PAYMENTS
FOR 5 YEARS AT 4% INTEREST. WHAT WOULD THE REMAINING BALANCE
BE AFTER THREE YEARS? SO OFTEN WHEN ANSWERING
THIS TYPE OF PROBLEM THERE ARE TWO STEPS. NUMBER ONE, WE CALCULATE THE
MONTHLY PAYMENTS ON THE LOAN, AND THEN, TWO, CALCULATES
THE REMAINING OF THE BALANCE BASED ON THE REMAINING TIME
OF LOAN. SO IT DOES SEEM A BIT STRANGE
THAT IF THIS WAS YOUR NEW CAR YOU WOULDN’T KNOW
THE CAR PAYMENTS. BUT IN EITHER CASE, WE’LL BEGIN BY DETERMINING WHAT
THE MONTHLY PAYMENT WOULD BE. LET’S BEGIN BY IDENTIFYING
THE GIVEN INFORMATION. THE ORIGINAL BALANCE IS $20,000,
SO P SUB 0=20,000. THE PAYMENTS ARE MONTHLY, SO WE’LL ASSUME MONTHLY
COMPOUNDED INTEREST, AND THEREFORE K=12. THE LOAN IS FOR FIVE YEARS
SO N=5. THE INTEREST IS 4%
SO R AS A DECIMAL WOULD BE 0.04. AND WE’RE TRYING TO FIND MONTHLY
PAYMENT WHICH WOULD BE D. NOW WE’LL SUB THESE VALUES
INTO THE FORMULA. SO, AGAIN, WE’VE DONE THIS
BEFORE, BUT P SUB 0=20,000, R=0.04, N=5, AND K=12. K APPEARS THREE TIMES. NOW, WE WANT TO SOLVE THIS
FOR D, SO WE’LL SIMPLIFY INSIDE THE
PARENTHESIS IN THE NUMERATOR, THEN THE DENOMINATOR,
AND THEN SOLVE FOR D. SO IN THE NUMERATOR WE HAVE
1 – THE QUANTITY 1 + 0.04 DIVIDED BY 12. I’M GOING TO RAISE TO THIS
TO THE POWER OF -60. WE DO HAVE TO BE CAREFUL
WHEN ROUNDING. IF WE DO WANT TO ROUND, WE SHOULD INCLUDE AT LEAST
THREE SIGNIFICANT DIGITS. AND NOTICE IN THIS CASE I’M GOING TO INCLUDE ALL DECIMAL
PLACES, WHICH WE SEE HERE. AND OUR DENOMINATOR
IS 0.04 DIVIDED BY 12, WHICH GIVES US
THIS DECIMAL HERE. AGAIN, IF WE DO ROUND, IT IS GOING TO CHANGE THE ANSWER
SLIGHTLY. NOW, BECAUSE WE STILL HAVE THIS
D HERE WE’LL FIND THIS QUOTIENT, THEN MULTIPLY BY D, THEREFORE, THIS QUOTIENT
WILL GIVE US THIS COEFFICIENT, WHICH WE SEE HERE. SO IF WE TAKE 0.1809968963
AND DIVIDE BY 0.0033333333 IT GIVES US THIS VALUE HERE, WHICH, AGAIN,
IS THE COEFFICIENT OF D. AND NOW TO SOLVE FOR D, WE
DIVIDE BOTH SIDES BY THIS VALUE. SO ON THE RIGHT SIDE
WE JUST HAVE D, AND THIS QUOTIENT WILL GIVE US
THE MONTHLY PAYMENT. SO WE HAVE 20,000
DIVIDED BY 54.29906943. ROUNDED TO THE NEAREST CENT, NOTICE HOW THE MONTHLY PAYMENT
IS $368.33. AND NOW WE CAN ANSWER
THE QUESTION WHAT WOULD THE REMAINING BALANCE
BE AFTER THREE YEARS. WELL, AFTER THREE YEARS NOTICE
HOW BECAUSE THIS IS A FIVE YEAR THERE’S ONLY TWO YEARS
REMAINING. WE’LL BE MAKING THESE PAYMENTS
HERE FOR TWO MORE YEARS. WHICH MEANS, WE NOW WANT TO FIND
P SUB ZERO, THE LOAN AMOUNT, AGAIN, WITH TWO YEARS REMAINING. SO R IS STILL 0.04,
N IS NOW 2 YEAR, K IS STILL 12, AND NOW WE KNOW THE MONTHLY
PAYMENT IS $368.33. SO, AGAIN, PERFORMING
SUBSTITUTION INTO OUR LOAN FORMULA, NOTICE R IS 0.04, N IS 2,
K IS 12, AND D IS 368.33. NOTICE HERE WE’RE SOLVING FOR
P SUB ZERO, WHICH MEANS WE JUST NEED TO
SIMPLIFY THE RIGHT SIDE HERE. AND, AGAIN, WE’LL DO THIS
IN STEPS. SO LOOKING AT THE NUMERATOR
IN THE PARENTHESIS WE’D HAVE 1 – THE QUANTITY
1 + 0.04 DIVIDED BY 12. AND NOW WE’RE GOING TO RAISE
THIS TO THE POWER OF -24, WHICH GIVES US
THIS DECIMAL HERE. AND OUR DENOMINATOR IS THE SAME
AS LAST TIME, WHICH WE SEE HERE, SO NOW WE’LL FIND THE PRODUCT
IN THE NUMERATOR AND THEN DIVIDE
BY THE DENOMINATOR. SO LETS PUT THE NUMERATOR
IN PARENTHESIS. WE HAVE (368.33 X 0.0767608361), CLOSE PARENTHESIS AND THEN WE’LL
DIVIDE BY 0.0033333333, ENTER. NOW, NOTICE HERE
IF WE ROUND TO THE NEAREST CENT WE WOULD ACTUALLY ROUND UP, AND THEREFORE 99 CENTS
ROUNDS TO $1, AND THEREFORE THE REMAINING LOAN
BALANCE AFTER 3 YEARS WOULD BE $8,482. WHICH MEANS, IF YOU WANTED TO
SELL YOUR CAR AFTER THREE YEARS YOU’D WANT TO MAKE SURE
YOU SELL THE CAR FOR MORE THAN THIS AMOUNT. OTHERWISE YOU WOULD STILL OWE
ON THE LOAN. I HOPE YOU FOUND THIS HELPFUL.  

Paul Whisler

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