# 40 - Notes.pdf

## Mathematics 152 with John Scott at Purdue University *

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Lesson 40 Most loans are repaid in several equal payments. This is called amortization. An amortized loan is also called a mortgage. The payment R (or installment payment) required to repay a loan A is given by R = A i 1 (1 + i) kt : This formula is called the Amortization Formula. Example 1 A couple who wants to purchase a home with a price of $250,000 has $75,000 for a down payment. If they can get a 30-year mortgage at 6% per year on the unpaid balance, nd each of the following. a) What will be their monthly payments? b) What is the total amount they will pay before they own the house outright? c) How much interest will they pay over the life of the loan? What is the di erence between this formula and the Sinking Fund Payment formula? The sinking fund payment formula tells what periodic payments/deposits to make in order to save up a certain amount of money for the future. The Amortization Formula is used to repay a loan. Recap of the formulas: Future Value of an Investment { Use to determine what a given single deposit will grow to in the future. Present Value of an Investment |Use to determine how large a single deposit should be in order to obtain a desired future value. E ective Rate of Interest { Given an interest rate r with k compounding periods per year, this tells what interest rate compounded once per year is equivalent. Use this to compare di erent interest rates with di erent numbers of compounding periods. Future Value of an Annuity { Use to determine what given periodic payments will grow to in the future. Present Value of an Annuity { Tells what single deposit will have the same future value as an annuity (the equivalent lump sum). Sinking Fund Payment for an Annuity {Tells what periodic amounts to save in order to obtain a desired future value. Amortization Formula { Tells the payment amount for repaying a loan. Decide which formula to use for the following situations: 1. If Chris saves $500 each quarter in an account earning 7%, compound quarterly, what will his savings be in 5 years? 2. Bill won a $2 million lottery. Instead of receiving $100,000 a year for 20 years, he wants the lump sum. What should that amount be assuming 4% interest, compounded annually? 3. At the birth of their child, the Fieldsons deposited $7,000 in an account paying 6% interest, compounded quarterly. How much will be available when the child turns 18? 4. A company’s new corporate headquarters will be completed in 2 years. At that time $750,000 will be needed for o ce equipment. How much should be invested monthly to fund that expense? Assume 9.75% interest, compounded monthly. 5. A man estimates that the computer he plans to buy in 18 months will cost $4,200. To meet this goal, how much should he deposit in an account paying 5.75%, compounded monthly? 6. Bob needs to take out a 3 year, $20,000 loan to pay for house renovations. If the interest rate is 6% compounded monthly, what are his monthly payments?

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