[Solved] 2360 Project1 Main-Playing with Mortgages

1 IntroductionA pair of close friends is currently in the market to buy a house in Boulder. Both have obtained engineering degrees from CU and have an understanding of differential equations but lack the skills necessary to decide what type of mortgage to take out on their new house. In the following sections, you will find some research they have done on commonly used mortgage structures. However, they have become busy and cannot continue their analysis. Since you are in Differential Equations this semester, they have decided to enlist your help in writing a report to understand their options.You need to read the following sections and complete the tasks found in Section 3. Your friends were nice enough to list various problems they would like solved. However, you should be careful to write your responses in a cohesive report, not just a list of answers to their questions. Round all monetary values to the nearest cent and all time values to the nearest hundredth of a year.2 Background Information 2.1 Compounding InterestOften, the interest on a loan is expressed as the annual rate, that is, the percentage of the outstanding balance that is charged as interest over a year. However, the frequency with which the rate is applied to the current balance may vary. This frequency is how often the loan compounds. If the interest is compounded annually, the formula to calculate the amount of money owed after the first year isA(1) = (1 + r)A(0),where A(t) is the outstanding balance after t years, and r is the annual interest rate. Note that the interest rate is usually discussed as a percent but in these formulas it is expressed as a decimal. For example an interest rate of 5% would correspond to r = 0.05.How would this change if, instead, the loan compounded semiannually (twice a year)? Then, half the interest rate would be applied to the loan value every 6 months.

( ( ) ( ~21 + r ) (1 + r )1 + r 1 + rA(1) = A(0.5) = A(0) = A(0)2 2 2 2This pattern continues for any frequency of compounding. That is, if the interest is compounded n times per year, the value of the loan after one year is( 1 + r )nA(1) = A(0)nand after 2 years( ( )n )2n1 + r )n (]1 + r1 + r ~n [( 1 + rA(2) = A(1) = A(0) = A(0)n n n nMore generally, the amount of a loan at time t years, compounded n times per year, is( 1 + r )ntA(t) = A(0) (1)nThe more frequently that a loan compounds, the higher the value at any time t. In the limit as the number of compounding periods increases without bound, which models a continuously compounding loan, we have~ 1 + r ~ntA(t) = A(0) uim = A(0)ert (2)noo nThis model can also be expressed as an initial value problemA = rA, A(0) = A0If the borrower makes a monthly payment of p dollars, the model becomes, assuming that the payments are distributed continuously throughout the year,A = rA 12p, A(0) = A0