hi this is Timothy Priscilla and I'm doing a lot of business math with my math 1324 class looking at the present value of an annuity here's an example suppose we're getting a mortgage a 30-year mortgage for house we're going to borrow one hundred thirty three thousand one hundred dollars and that the interest rate is 4.8% fixed for those thirty years we want to find our monthly payment realize that the account balance the lump sum occurs at the present at the beginning of the thirty year time period the account balance the mortgage is one hundred thirty three thousand one hundred dollars at the end of that 30 year time period the account balance would have fallen to zero dollars so that's what tells us that we're going to have to use the present value of an annuity formula the lapsim is in the present and we recall the present value of the annuity formula realize P the present value that's that $133,000 133,000 $100 I should say in how many payments are they going to have over the 30-year life of this mortgage they're making monthly payments for 30 years 12 times 30 is 360 and I the rate per period you take the annual rate that 4.8% written as a decimal point 0 for 8 and divide it by 12 we're splitting that up over the four-year period and what's that going to give us a point zero zero four that terminates pretty nicely now plugging into the oh and by the way that I will unknown is the amount of the payment that's what we're looking for so plug it into the present value of an annuity formula we have one hundred thirty three thousand one hundred dollars is equal to R times one minus one plus point zero zero four is just a one point zero zero four to the negative 360 all divided by point zero zero four and now we'll dig out our calculator I don't do anything you're fancy to uh punch this stuff in what I do is I simplify the numerator I simplify the numerator on the bracket amount I then divide it by the denominator once I know that value to get over by itself it's R times that number to undo the multiplication we'll have to divide but let's see so I need a one minus one point zero zero four raised to the negative 360 and that's just the numerator let's see if we can see that get that so it shows there and then we have to divide it by point zero zero four and so that bracket amount I'm going to write this down just because I want to make a point about that that bracket of male reduces to one ninety point five nine seven six eight one three and I'm writing that down because that's significant in the real world has a very significant meaning to it they're making 360 payments to pay back the Sanu ax T in reality it only took about 190 and a half of those payments to pay the entire loan balance back all the remaining payments which would have amounted to what about a hundred seventy payments were nothing more than just interest that's what the significance about that bracket amount is now it's R times one ninety point stuff so we go one hundred thirty one thousand no one hundred thirty three thousand one hundred dollars divided by our previous answer let's let me see if I can get this clear for you but equals it looks to me that they're a 30-year mortgage payment will be six hundred ninety eight dollars and 33 cents this is called the principal and interest payment because if you get a mortgage you're going to also have to have money going into escrow homeowners insurance taxes what-have-you but this is just the money that's being applied to the principal and your interest and at the very beginning of the mortgage the vast majority of that payment is going to be going just as the interest payment a very small portion of that six hundred ninety eight dollars and 33 cents will actually be going towards a paying down your principal I guess that's not too bad 698 33 but that's a pretty good right that 4.8% anyway this is Timothy Brucella just doing a low business math with this Matt 1324 class ba ba

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