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Welcome to a lesson on the loan formula this lesson is about conventional loans also called amortized loans or installment loans examples include auto loans and home mortgages the techniques in this lesson do not apply to payday loans add-on loans or other loan types where the interest is calculated upfront though I do have lessons on these topics one great thing about loans is that we can use the same formula as a payout annuity to see why imagine that you had $10,000 invested at a bank you start taking out withdrawals while earning interest on the remaining balance as part of a payout annuity after five years your balance is zero flip that around and imagine that you borrow $10,000 from a bank you make payments back to the bank with interest for the money you borrow after five years the loan is paid off the roles are reversed here but the formula to describe the situation is the same so here is the loan formula which again is the same as the payout annuity formula where P Sub Zero is the loan amount or beginning balance or principal D is the loan payment or the payment per unit of time r is the annual interest rate expressed as a decimal K is a number of compounding periods in one year notice K appears three times in the formula and n is the length of the loan in years now the compounding frequency is not always explicitly given but can be determined by how often payments are made before we take a look at two examples though it is important to be very careful about rounding when calculations involve exponents in general keep as many decimals during calculations as you can be sure to keep at least three significant digits meaning three numbers after any leading zeroes for example to round this decimal using three significant digits we would have zero point zero zero zero one two three using three significant digits will usually give you a close enough answer but keeping more digits as always better let's take a look at our first example if you can afford a $150 per month car payment for five years what car price should you shop for the loan interest is 6% let's start by finding all the given information if the monthly payment is $150 then we know that D equals 150 and because the payments are monthly we can assume the number of compounds would be 12 per year or monthly and therefore K equals 12 the loan is for five years so n is five and finally the interest rate is 6% so R equals 6% but this must be expressed as a decimal which would be 0.15 P Sub Zero so D equals 150 just here K equals 12 which is here here and here N equals 5 which is here and finally R is equal to 0.06 which is here and here since we're solving for P Sub Zero we need to evaluate the right side here we'll begin by simplifying inside the parenthesis in the numerator and then the denominator so looking at the numerator in parentheses we'd have 1 minus the quantity 1 plus 0.06 divided by 12 we want to raise this to the power of this would be negative 60 we can hit the exponent key or the caret here and in parentheses we can just type in negative 5 times 12 close parentheses and enter notice how I decided to use all the decimal places here now the denominator is going to be point zero 6 divided by 12 which is 0.0 0.2 the numerator and divided by the denominator to determine what the loan amount would be so we'll put the numerator in parenthesis so we'll have 150 times this decimal here 0.25 8 6 2 7 8 0 3 8 and then we'll divide this by point 0 0 5 rent the nearest cent we have 7750 $8.83 so this tells us that under these conditions if you can afford a $150 payment per month you should shop for a car around this price but it's important to not forget about insurance for the car as well which would be an extra cost now let's take a look at a second example in this example you want to purchase a car for $15,000 and you have been approved for a loan at 4 percent interest for five years what will the monthly payment be again let's start by determining the given information the loan amount would be $15,000 and therefore P sub 0 equals 15,000 the loan is at 4% so R would be 4% expressed as a decimal that would be 0.055 ulness for five years so n is equal to five and the payments are going to be monthly so K would be 12 so our goal here is to find the monthly payment amount which would be D so now we'll substitute these values into our formula and this time we'll be solving for D so P sub 0 equals 15,000 R equals 0.04 and equals five K equals 12 K is here here and here now we want to solve for D so we'll begin by simplifying inside the parenthesis in the numerator and then in the denominator looking at the numerator inside the parenthesis we'd have one minus the quantity one plus 0.04 divided by 12 close parenthesis we're going to raise this to the power of negative 60 we'll raise it to the power of negative five times 12 which gives us this decimal here notice how this is still multiplied by D though in our denominator we have point zero 4 divided by 12 which gives us this decimal here now for the next step we'll find this quotient here and then multiply by D so we'd have point one eight zero nine nine six eight nine six three divided

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