You have the option to buy "points" on the loan. This means that you can pay an extra 1% of the initial principal along with your down payment, and doing so will lower the annual interest rate by 0.25% (thereby lowering the monthly payment). Importantly, the points you pay -- the extra 1% -- do NOT count towards the principal of the loan. They are just a fee paid to the bank. In exchange, you get a lower interest rate. Lenders will often advertise the lower rate, to get which you need to pay the points. But is it worth it? You will figure this out in this question.\ \ If you own the home for the entire 30-year loan period, buying points usually lowers the total cost of the loan. But most people dont do that they often move before then. When you sell the house, you use (some of) the proceeds from the sale to repay the remaining mortgage balance, after which you no longer need to make any payments.\ \ Suppose you decide to sell the house after living in it for \ months (and making \ monthly payments). The total cost of the mortgage is the cost of points paid today, plus \ months of monthly payments \ , plus the final balance at the end of the \ 'th month \ .\ \ The total cost of the mortgage is then:\ \ \ Of course, as an experienced finance student, you know about time value of money, which means that you shouldn't make decisions based on discounted total costs of many payments over time. So given a monthly discount rate \ , you compute the present value of mortgage payments as:\ \ \ \ \ What discount rate should you use? This is the age-old question in finance. One way to approach it is to discount cash flows at your opportunity cost, i.e. at the rate you can earn on an alternative investment of similar risk. Intuitively, the reason why money 10 years from now is worth less to you than money today is because, if you got money today, you could invest it and have more in 10 years. Let's assume an annual discount rate of 4%.\ \ In a new part of your notebook, rearrange ("refactor") your code from Part A into a function that takes points, interest rate, and the months until you sell the house as inputs and returns the present value of the mortgage as an output.

Suppose you want to get a 30 -year mortgage on a $400,000 home. You can get a 7.1% annual mortgage rate that is compounded monthly when you make a 20% down payment (so the initial loan principal will be $400,000 minus 20% of that value). The fixed monthly payment PMT on a mortgage is given by the annuity formula: PMT=Pr(1+r)n1(1+r)n where P is the initial principal, r is the monthly interest rate, and n is the number of months in the mortgage term. Each payment consists of a principal and an interest component. The interest is akin to a coupon payment on a bond, compensating the lender for risks and time value of money. The principal payment gradually reduces the outstanding balance of the loan. The gradual repayment of principal is known as amortization. It occurs for mortgages but not for bonds, where the entire principal payment, or face value, is paid at the end of the loan as a balloon payment. The interest component of the payment is calculated as the outstanding balance of the loan at the start of the month times the monthly interest rate. The principal component is the difference between the total payment and the interest component. In other words, INTt=Bt1rPRNt=PMTtINTtBt=Bt1PRNt where INT is the interest portion, PRN is the principal portion, and Bt is the outstanding balance of the loan at the end of month t.B0, the initial balance, is the initial loan amount P