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11.1 What You Will Learn • Percent, Fractions, and Decimal Numbers (skip, assume that you known) • Percent Change • Percent Markup and Markdown 11.1-1 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Percent Change Formula: Percent change = • • 11.1-2 amount in amount in latest period − previous period amount in previous period × 100 If percent change is positive, one gets % increase; If percent change is negative, one gets % decrease. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 6: Most Improved Baseball Team In 2009, the Padres won 75 games. In 2010, the Padres won 90 games. Determine the percent increase in the number of games won by the Padres from 2009 to 2010. 20% increase 11.1-3 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Percent Markup percent markup or markdown on cost. • • A positive answer indicates a markup. A negative answer indicates a markdown. Percent markup or = selling price − dealer's cost × 100 dealer's cost markdown on cost 11.1-4 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Determining Percent Markup Holdren Hardware stores pay $48.76 for glass fireplace screens. They regularly sell them for $79.88. At a sale they sell them for $69.99. Determine a) the percent markup on the regular price. 63.8% increase b) the percent markup on the sale price. 43.5% increase 11.1-5 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 10: Down Payment on a Condominium Home Melissa Bell wishes to buy a condominium home for $189,000. To obtain the mortgage loan, she must pay 20% of the selling price as a down payment. Determine the amount of Melissa’s down payment. Down payment: $37,800 11.1-6 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 11.2 What You Will Learn Ordinary Interest The United States Rule (skipped) Banker’s Rule (skipped) 11.2-7 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Simple Interest Formula Interest = Principal × rate × time i=P⋅r⋅t 11.2-8 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Air Conditioning Loan Sherry Tornwall needs to borrow $6200 to replace the air conditioner in her home. From her credit union, Sherry obtains a 30-month loan with an annual simple interest rate of 5.75%. a)Calculate the loan. the simple interest she is charged on The simple interest is $891.25 b)Determine the amount, principal plus interest, Sherry will pay the credit union at the end of the 30 months to pay off her loan. Sherry will pay the credit union $7091.25 at the end of 30 months 11.2-9 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Discount Notes In another type of loan, the discount note, the interest is paid at the time the borrower receives the loan. The interest charged in advance is called the bank discount. 11.210 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: True Interest Rate of a Discount Note Siegrid Cook took out a $500 loan using a 10% discount note for a period of 3 months. Determine a)the interest she must pay to the bank on the date she receives the loan. $12.50 b) the net amount of money she receives from the bank. $487.50 c) the actual rate of interest for the loan. r ≈ 0.1026 11.211 actual rate of interest is about 10.3% rather than the quoted 10% Copyright 2013, 2010, 2007, Pearson, Education, Inc. 11.3 What You Will Learn Compound Interest Present Value 11.312 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Investments An investment is the use of money or capital for income or profit. In a fixed investment, the amount invested as principal is guaranteed and the interest is computed at a fixed rate. In a variable investment, neither the principal nor the interest is guaranteed. 11.313 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Compound Interest Interest that is computed on the principal and any accumulated interest is called compound interest. 11.314 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Compound Interest Formula r A = p 1 + n nt A is the amount that accumulates in the account p is the principal r is the annual interest rate as a decimal t is the time in years n is the number of compound periods per year 11.315 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Using the Compound Interest Formula Kathy Mowers invested $3000 in a savings account with an interest rate of 1.8% compounded monthly. If Kathy makes no other deposits into this account, determine the amount in the savings account after 2 years. The amount in the account after 2 years would be about $3109.88. 11.316 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Present Value Formula p= A r 1 + n nt p is the present value, or principal to invest now A is the amount to be accumulated in the account r is the annual interest rate as a decimal n is the number of compound periods per year t is the time in years 11.317 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Savings for College Will Hunting would like his daughter to attend college in 6 years when she finishes high school. Will would like to invest enough money in a certificate of deposit (CD) now to pay for his daughter’s college expenses. If Will estimates that he will need $30,000 in 6 years, how much should he invest now in a CD that has a rate of 2.5% compounded quarterly? 11.318 $25,833.30 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 11.4 What You Will Learn Fixed Installment Loans Open-End Installment Loan 11.419 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Installments A fixed installment loan is one on which you pay a fixed amount of money for a set number of payments. Examples: college tuition loans, loans for cars, boats, appliances, furniture, etc. They are usually repaid in 24, 36, 48 or 60 months. 11.420 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Installments An open-ended installment loan is a loan on which you can make variable payments each month. Example: credit cards 11.421 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Truth in Lending Act in 1968 This law requires that the lending institution tell the borrower two things: The annual percentage rate (APR) is the true rate of interest charged for the loan. The total finance charge is the total amount of money the borrower must pay for borrowing the money: interest plus any additional fees charged. 11.422 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Total Installment Price The total installment price is the sum of all the monthly payments and the down payment, if any. 11.423 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Table 11.2 11.424 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Installment Payment Formula m= r p n r 1 − 1 + n − n⋅t m is the installment payment p is the amount financed r is the APR as a decimal n is the number of payments per year t is the time in years 11.425 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Using the Installment Payment Formula Kristin Aiken wishes to purchase new window blinds for her house at a cost of $1500. The home improvement store has an advertised finance option of no down payment and 6% APR for 24 months. Determine Kristin’s monthly payment. Kristin’s monthly payment would be $66.48. 11.426 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Repaying an Installment Loan Early By paying off a loan early, one is not obligated to pay the entire finance charge. The amount of the reduction of the finance charge from paying off a loan early is called the unearned interest. 11.427 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Repaying an Installment Loan Early Two methods are used to determine the finance charge when you repay an installment loan early. • • 11.428 The actuarial method uses the APR tables. The rule of 78s does not use the APR tables, is less frequently used, and is outlawed in much of the country. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Actuarial Method for Unearned Interest n ⋅ P ⋅V u= 100 + V u is unearned interest n is # of remaining monthly payments P is the monthly payment V is the value from the APR table for the # of remaining payments 11.429 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Using the Actuarial Method Tino Garcia borrowed $9800 to purchase a classic 1966 Ford Mustang. The APR is 7.5% and there are 48 payments of $237. Instead of making his 30th payment of his 48-payment loan, Tino wishes to pay his remaining balance and terminate the loan. 11.430 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Using the Actuarial Method a) Use the actuarial method to determine how much interest Tino will save (the unearned interest, u) by repaying the loan early. n = 18, P = $237, V = $6.04, u ≈ $242.99 Tino will save $242.99 in interest by repaying the loan early, by actuarial method 11.431 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Using the Actuarial Method b) What is the total amount due to pay off the loan early on the day he makes his final payment? Solution Remaining balance (exclude unearned interest): 18(237) – 242.99 = $4,023.01 Total amounts in final payment (remaining balance plus 30th payment): 4,023.01 + 237 = $4,260.01 11.432 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Open-End Installment Loans A credit card is a popular way of making purchases or borrowing money. Typically, credit card accounts report: *These rates vary with different credit card accounts and localities. 11.433 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Open-End Installment Loans Typically, credit card monthly statements contain the following information: balance at the beginning of the period balance at the end of the period (or new balance) the transactions for the period statement closing date (or billing date) payment due date the minimum payment due 11.434 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Open-End Installment Loans For purchases, there is no finance or interest charge if there is no previous balance due and you pay the entire new balance by the payment due date. The period between when a purchase is made and when the credit card company begins charging interest is called the grace period and is usually 20 to 25 days. 11.435 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Open-End Installment Loans However, if you use a credit card to borrow money, called a cash advance, there generally is no grace period and a finance charge is applied from the date you borrowed the money until the date you repay the money. 11.436 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Average Daily Balance Many lending institutions use the average daily balance method of calculating the finance charge because they believe that it is fairer to the customer. 11.437 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Using the Average Daily Balance Method The balance on Min Zeng’s credit card account on July 1, the billing date, was $375.80. The following transactions occurred during the month of July. July July July July 11.438 5 10 18 28 Payment Charge: Toy store Charge: Garage Charge: Restaurant Copyright 2013, 2010, 2007, Pearson, Education, Inc. $150.00 $74.35 $123.50 $42.50 Example 8: Using the Average Daily Balance Method a) Determine the average daily balance for the billing period. (i) find the balance by date July July July July July 11.439 1 5 10 18 28 $375.80 $375.80 $225.80 $300.15 $423.65 – $150.00 + $74.50 + $123.50 + $42.50 Copyright 2013, 2010, 2007, Pearson, Education, Inc. = = = = $225.80 $300.15 $423.65 $466.15 Example 8: Using the Average Daily Balance Method Solution (ii) Find the number of days that the balance did not change between each transaction. Count the first day in the period but not the last day. (iii) Multiply the balance due by the number of days the balance did not change. (iv) Find the sum of the products. 11.440 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Using the Average Daily Balance Method Solution 7/1 7/5 7/10 7/18 7/28 11.441 $375.80 $225.80 $300.15 $423.65 $466.15 4 5 8 10 4 (375.80)(4) = $1503.20 (225.80)(5) = $1129.00 (300.15)(8) = $2401.20 (423.65)(10)=$4236.50 (466.15)(4) = $1864.60 31 Sum = $11,134.50 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Using the Average Daily Balance Method Solution (v) Divide the sum by the number of days $11,134.50 ÷ 31 = $359.18 The average daily balance is $359.18. 11.442 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Using the Average Daily Balance Method b) Determine the finance charge to be paid on August 1, Min’s next billing date. Assume that the interest rate is 1.3% per month. Solution Use the simple interest formula: i = prt = $359.18 × 0.013 × 1 ≈ $4.67 11.443 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: Using the Average Daily Balance Method c) Determine the balance due on August 1. The balance due on August 1 = the balance (at last purchase) + finance charge = $466.15 + $4.67 = $470.82. 11.444 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 11.6 What You Will Learn Ordinary Annuities Sinking Funds 11.645 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Annuity An annuity is an account into which, or out of which, a sequence of scheduled payments is made. 11.646 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ordinary Annuity An annuity into which equal payments are made at regular intervals, with the interest compounded at the end of each interval and with a fixed interest rate for each compounding period, is called an ordinary annuity or fixed annuity. 11.647 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ordinary Annuity The amount of money that is present in an ordinary annuity after t years is known as the accumulated amount or the future value of an annuity. Two methods: •Spread sheet •Formula 11.648 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ordinary Annuity 1st half of table 11.649 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ordinary Annuity 2nd half of table 11.650 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ordinary Annuity Another way: the ordinary annuity formula 11.651 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ordinary Annuity Formula nt r p 1+ − 1 n A= r n A: accumulated amount (A is the unknown) p: dollars in each payment n: times of payments per year t: years r: annual interest rate compounded at the end of each payment period 11.652 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Using the Ordinary Annuity Formula Bill and Megan Lutes are depositing $250 each quarter in an ordinary annuity that pays 4% interest compounded quarterly. Determine the accumulated amount in this annuity after 35 years. There will be about $75,677.48 in Bill and Megan’s annuity after 35 years. 11.653 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Timely Tip An ordinary annuity is used when you wish to determine the accumulated amount. A sinking fund is used when you wish to determine how much money an investor must invest each period to reach an accumulated amount at a specific time. 11.654 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sinking Fund A sinking fund is a type of annuity in which the goal is to save a specific amount of money in a specific amount of time. 11.655 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Sinking Fund Payment Formula p= r A n r 1+ n nt −1 p: dollars in each payment (p is the unknown) A: accumulated amount n: times of payments per year t: years r: annual interest rate 11.656 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Using Sinking Fund Payment Formula Abigail and John Clayton would like to have $55,000 in 10 years to remodel their home. The Claytons decide to invest monthly in a sinking fund that pays 3.3% interest compounded monthly. How much should the Claytons invest in the sinking fund each month to accumulate $55,000 in 10 years? Clayton’s need to invest $387.49 each month to accumulate $55,000 in 10 years. 11.657 Copyright 2013, 2010, 2007, Pearson, Education, Inc.