My Math Forum Calculating Total Interest Paid on a Loan

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 February 19th, 2012, 02:37 PM #1 Newbie   Joined: Feb 2012 Posts: 7 Thanks: 0 Calculating Total Interest Paid on a Loan Having difficulty calculating total interest paid on a loan. Loan: $80,000 Interest Rate: 5% Term: 15 years Once you have the above information I thought it was simply Loan amount * Interest Rate * Term, or$80,000 * .05 * 15 years = $600,000 which is not correct. What am I doing wrong? When I plug these numbers into a total interest paid on a loan calculator I get$33,874.72 for total interest paid. Using http://www.finaid.org/calculators/loanpayments.phtml
 February 19th, 2012, 03:25 PM #2 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Calculating Total Interest Paid on a Loan That total is with 181 payments. What is frequency of the payments? Monthly? Annually? All at once?
February 19th, 2012, 03:36 PM   #3
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Re: Calculating Total Interest Paid on a Loan

Quote:
 Originally Posted by daigo That total is with 181 payments. What is frequency of the payments? Monthly? Annually? All at once?
Is frequency of the payments required to solve my original question? The numbers I made were all hypothetical, I did not have a set frequency for the payments. I though there was enough information there to solve the problem already.

Total interest paid for
$80,000 Loan at 5% interest 15 years February 19th, 2012, 03:40 PM #4 Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407 Re: Calculating Total Interest Paid on a Loan Hello, moreyS! Quote:  Having difficulty calculating total interest paid on a loan. Loan:$80,000 Interest Rate: 5% Term: 15 years Once you have the above information I thought it was simply: Loan amount * Interest Rate * Term [color=beige]. . [/color]or: $80,000 * 0.05 * 15 years =$600,000 which is not correct. [color=beige] . [/color][color=blue]Obviously![/color]

Think about it . . .

You buy a Mercedes for $80,000 and finance it over 15 years. And they charge you over a half million dollars in interest?  February 19th, 2012, 03:45 PM #5 Newbie Joined: Feb 2012 Posts: 7 Thanks: 0 Re: Calculating Total Interest Paid on a Loan @soroban Yes obviously, I pointed that out as joke mostly . But I still want to know how to go about calculating it the correct way.  February 19th, 2012, 04:01 PM #6 Senior Member Joined: Jan 2010 Posts: 205 Thanks: 0 Re: Calculating Total Interest Paid on a Loan Well I am not a finance major so I don't know if there is a default payment frequency, but think about it: The interest is added after every payment you make, the interest of whatever total amount is left over after you make your payment. The longer you take to pay it off, the more interest amount you pay (considering the interest itself does not change). You could set up a hypothetical algebraic problem to prove this, but I just know from personal experience from paying off school loans in real life that this is the case February 19th, 2012, 04:17 PM #7 Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407 Re: Calculating Total Interest Paid on a Loan Hello again, moreyS! You could have given us the full story . . . Quote:  Loan:$80,000 Interest Rate: 5% Term: 15 years The loan is to be paid off in equal monthly payments. The interest is computed monthly. (a) Find the monthly payments. (b) Find the total interest paid.

The problem is an Amortization and requires a special formula,
[color=beige]. . [/color]while you seem to be familiar with Simple Interest only.

 February 19th, 2012, 07:13 PM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 465 Math Focus: Calculus/ODEs Re: Calculating Total Interest Paid on a Loan Let P = monthly payment, A = amount borrowed, i = monthly interest rate, and n = the number of payments. Also, let $D_n$ be the debt amount after payment n. Consider the recursion: (1) $D_{n}=(1+i)D_{n-1}-P$ (2) $D_{n+1}=(1+i)D_{n}-P$ Subtracting (1) from (2) yields the homogeneous recursion: $D_{n+1}=(2+i)D_{n}-(1+i)D_{n-1}$ whose associated auxiliary equation is: $r^2-(2+i)r+(1+i)=0$ $$$r-(1+i)$$$$r-1$$=0$ Thus, the closed-form for our recursion is: $D_n=k_1(1+i)^n+k_2$ Using initial values, we may determine the coefficients $k_i$: $D_0=k_1+k_2=A$ $D_1=k_1(1+i)+k_2=(1+i)A-P$ Solving this system, we find: $k_1=\frac{Ai-P}{i},\,k_2=\frac{P}{i}$ and so we have: $D_n=$$\frac{Ai-P}{i}$$(1+i)^n+$$\frac{P}{i}$$=\frac{(Ai-P)(1+i)^n+P}{i}$ Now, equating this to zero, we can solve for P: $\frac{(Ai-P)(1+i)^n+P}{i}=0$ $(Ai-P)(1+i)^n+P=0$ $(P-Ai)(1+i)^n=P$ $P$$(1+i)^n-1$$=Ai(1+i)^n$ $P=\frac{Ai(1+i)^n}{(1+i)^n-1}$ $P=\frac{Ai}{1-(1+i)^{-n}}$ Using the numbers you gave (assuming monthly payments): $A=80000$ $i=\frac{0.05}{12}=\frac{1}{240}$ $n=15\cdot12=180$ and hence: $P\approx632.63$ And so the total amount paid back is: $nP\approx113874.28$ Subtracting the amount borrowed gives the total interest I: $I=nP-A\approx33874.28$
 February 19th, 2012, 10:53 PM #9 Senior Member   Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Calculating Total Interest Paid on a Loan If you really want to mess with this junk, we've got the following equations for compounding lump sums and compounding annuities. (Explanation to follow) $\text{FV=Pmt\,\left(\frac{(1+i)^n-1}i\right)}$ is for future value of an annuity $\text{PV=Pmt\,\left(\frac{1-(1+i)^{-n}}i\right)}$ is for present value of an annuity $FV=PV(1+i)^n$ Future value of a lump sum Where i = interest per compounding period, n = total number of compounding periods, and us finance people tend to use FV for future value and PV for present value. Yes, how it compounds absolutely does matter. If you took out a loan, and were presumably paying on it monthly, then you would only be charged 1/12 of the interest for just that month. Then after making a payment, a portion would count towards the principal of the loan. Once the principal goes down, so does the interest. But again, you would only be paying 1/12 of it. Interest rates are always quoted annually unless stated otherwise, so if the payment is not annual, then you must adjust. An annuity is when an equal series of payments is involved - has to be the same amount for each payment, done at equal time periods. (Yes, it's also a type of investment, but the name comes from the finance use of the term.) So a loan is an annuity. Putting the same amount into your IRA each year would be an annuity. Etc. Versus, you just stick a lump of money into something and let it sit there earning interest, without adding more payments, that's not an annuity and that's the lump sum equation, but the interest can still can compound. To figure out if it's a present value or future value, it's relative to the payments. That is, a present value annuity starts with an amount and payments come off it, and it goes downwards to zero. (Paying off loans and taking money from a retirement account are examples.) A future value starts at nothing, and you add payments to it, and it grows into some future value. (Saving for your child's college or saving for retirement are examples.) If you don't say otherwise, it's assumed you're talking annually, so if it's anything else, it has to state so. Most loans are paid monthly, but not always. If we assume monthly, then i = .05/12 and n = 12 times per year * 15 years. If you can do a bit of algebra, the top two can be used to solve for payment. (Notice Mark's equation is just mine flipped around a bit to solve for the payment.) And the bottom one can be used to solve for either FV or PV, again with a bit of algebra. For the top two, the interest is the difference between the present or future value, and the total of all payments made. For the bottom one, the interest is just the difference between FV and PV. These equations also assume the payments are at the end of the period. That is, you borrow money and your first payment is at the end of the first month. And yes you can find a gazillion calculators online, if you're entering the numbers correctly.
 February 20th, 2012, 02:53 AM #10 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,840 Thanks: 764 Re: Calculating Total Interest Paid on a Loan A = Amount borrowed (80000) i = interest per month (.05/12) n = number of months (180) Total Interest = A * {i * n / [1 - (1 + i)^(-n)] - 1} = 33874.28225....

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# mathematics equation total interest payment over life of loan

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