Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 8.1 Percent, Sales Tax, and Income Tax 1. 2. 3. 4. 5. 6. 7. Objectives Express a fraction as a percent. Express a decimal as a percent. Express a percent as a decimal. Solve applied problems involving sales tax and discounts. Compute income tax. Determine percent increase or decrease. Investigate some of the ways percent can be abused. 1/21/2011 Section 8.1 1 Basics of Percent • • Percents are the result of expressing numbers as a part of 100. The word percent means per hundred. Expressing a fraction as a percent: 1. Divide the numerator by the denominator. 2. Multiply the quotient by 100. This is done by moving the decimal point in the quotient two places to the right. 3. Add a percent sign. 1/21/2011 Section 8.1 2 Expressing a Fraction as a Percent 5 Example: Express as a percent. 8 Solution: Step 1. Divide the numerator by the denominator. 5 ÷ 8 = 0.625 Step 2. Multiply the quotient by 100. 0.625 x 100 = 62.5 Step 3. Add a percent sign. 62.5% 5 Thus, = 62.5%. 8 1/21/2011 Section 8.1 3 Expressing a Decimal as a Percent To express a decimal as a percent: • Move the decimal point two places to the right. • Attach a percent sign. Example: Express 0.47 as a percent. Solution: Thus, 0.47 = 47%. 1/21/2011 Section 8.1 4 Expressing a Percent as a Decimal Number To express a percent as a decimal number: 1. Move the decimal point two places to the left. 2. Remove the percent sign. Example: Express each percent as a decimal: a. 19% b. ¼% 1/21/2011 Section 8.1 5 Expressing a Percent as a Decimal Number Example Continued Solution: Use the two steps. a. Thus, 19% = 0.19. b. We change the fraction into a decimal, then use the steps Thus, ¼% = 0.0025. 1/21/2011 Section 8.1 6 Percent, Sales Tax, & Discounts Many applications involving percent are based on the following formula: Note that “of” implies multiplication. • We use this formula to determine sales tax collected by states, counties, cities on sales items to customers. Sales tax amount = tax rate x item’s cost 1/21/2011 Section 8.1 7 Percent and Sales Tax Example: Suppose that the local sales tax rate is 7.5% and you purchase a bicycle for $394. a. How much tax is paid? b. What is the bicycle’s total cost? Solution: a. Sales tax amount = tax rate x item’s cost The tax paid is $29.55. 1/21/2011 Section 8.1 8 Percent and Sales Tax Example Continued b. The bicycle’s total cost is the purchase price, $394, plus the sales tax, $29.55. Total Cost = $394.00 + $29.55 = $423.55 The bicycle’s total cost is $423.55. 1/21/2011 Section 8.1 9 Percent and Sales Price • • Businesses reduce prices, or discount, to attract customers and to reduce inventory. The discount rate is a percent of the original price. Discount price = discount rate x original price. Example: A computer with an original price of $1460 is on sale at 15% off. a. What is the discount amount? b. What is the computer’s sale price? 1/21/2011 Section 8.1 10 Percent and Sales Price Example Continued Solution: a. Discount amount = discount rate x original price The discount amount is $219. 1/21/2011 Section 8.1 11 Percent and Sales Price Example Continued b. A computer’s sale price is the original price, $1460, minus the discount amount, $219. Sale price = $1460 - $219 = $1241 The computer’s sale price is $1241. 1/21/2011 Section 8.1 12 Percent and Income Tax Calculating Income Tax: 1. Determine your adjusted gross income: 2. Determine your taxable income: 1/21/2011 Section 8.1 13 Percent and Income Tax 3. Determine the income tax: 1/21/2011 Section 8.1 14 Percent and Income Tax • The table shows tax rates, standard deductions, and exceptions for the four filing status categories. Tax Rate Single Married Filing Separately Married Filing Jointly Head of Household 10% up to $7300 up to $7300 up to $14,600 up to $10,450 15% $7301 to $29,700 $7301 to $29,700 $14,601 to $59,400 $10,451 to $39,800 25% $29,701 to $71,950 $29,701 to $59,975 $59,401 to $119,950 $39,801 to $102,800 28% $71,951 to $150,150 $59,976 to $91,400 $119,951 to $182,800 $102,801 to $166,450 33% $150,151 to $326,450 $91,401 to $163,225 $182,801 to $326,450 $166,451 to $326,450 35% more than $326,450 more than $163,225 more than $326,450 more than $326,450 Std. Ded. $5000 $5000 $10,000 $7300 Exmt/pers $3200 $3200 $3200 $3200 1/21/2011 Section 8.1 15 Percent and Income Tax Example: Calculate the income tax owed by a single woman with no dependents whose gross income, adjustments, deductions, and credits are given. Use the marginal tax rates from the table. Single Women with no dependents: Gross income: $52,000 Adjustments: $4,000 Tax credit: $500 1/21/2011 Deductions: • $7500: mortgage interest • $2200: property taxes • $2400: charitable contributions • $1500: medical expenses not covered the insurance Section 8.1 16 Percent and Income Tax Example Continued Solution: Step 1. Determine the adjusted gross income. Adjusted gross income = Gross income – Adjustments = $52,000 - $4000 = $48,000 1/21/2011 Section 8.1 17 Percent and Income Tax Example Continued Step 2. Determine taxable income. We substitute $12,100 for deductions in the formula for taxable income. 1/21/2011 Section 8.1 18 Percent and Income Tax Example Continued Taxable income = Adjusted gross income – (Exemptions + Deductions) = $48,000 – ($3200 + $12,100) = $48,000 – $15,300 = $32,700 Step 3. Determine the income tax. Income tax = Tax computation – Tax credits = Tax Computation - $500 Since the single woman taxable income is $32,700 she falls in the tax bracket of 25%. 1/21/2011 Section 8.1 19 Percent and Income Tax Example Continued • This means that she owes: 10% on the first $7300 of her taxable income, 15% on her taxable income between $7301 to $29,700, inclusive, and 25% on her taxable income above $29,701. 1/21/2011 Section 8.1 20 Percent and Income Tax Example Continued = 0.10 x $7300 + 0.15 x $22,400 + 0.25 x $3000 = $730 + $3360 + $750 = $4840 We substitute $4840 for the tax computation in the formula for income tax. Income tax = Tax computation – Tax credits = $4840 - $500 = $4340 Thus, the income tax owed is $4340. 1/21/2011 Section 8.1 21 Percent and Change If a quantity changes, its percent increase or its percent decrease can be found as follows: 1. Find the fraction for the percent increase or decrease: amount of increase amount of decrease or . original amount original amount 2. Find the percent increase or decrease by expressing the fraction in step 1 as a percent. 1/21/2011 Section 8.1 22 Percent and Change Example: In 2000, world population was approximately 6 billion. The data are from United Nations Family Planning Program and are based on optimistic or pessimistic expectations for successful control of human population growth. a. Find the percent increase in world population from 2000 to 2150 using the high projection data. b. Find the percent decrease in world population form 2000 to 2150 using the low projection data. 1/21/2011 Section 8.1 23 Percent and Change Example Continued Solution: • Use the data shown on the blue, high-projection, graph. amount of increase Percent increase = original amount 30 − 6 24 = = = 4 = 400% 6 4 The projected percent increase in world population is 400%. 1/21/2011 Section 8.1 24 Percent and Change Example Continued • Use the data shown on the green, low-projection, graph. amount of decrease Percent decrease = original amount 6−4 2 1 1 = = = = 33 % 6 6 3 3 • The projected percent decrease in world population is 33⅓%. 1/21/2011 Section 8.1 25 Abuses of Percents Example: John Tesh, while he was still co-anchoring Entertainment Tonight, reported that the PBS series The Civil War had an audience of 13% versus the usual 4% PBS audience, “an increase of more than 300%.” Did Test report the percent increase correctly? Solution: We begin by finding the actual percent increase. amount of increase original amount 13 − 4 9 = = = 2.25 = 225% 4 4 Percent increase = The percent increase for PBS was 225%. This is not more than 300%, so Tesh did not report the percent increase correctly. 1/21/2011 Section 8.1 26 Section 8.2 Simple Interest Objectives 1. Calculate simple interest. 2. Use the future value formula. 3. Use the simple interest formula on discounted loans. 1/21/2011 Section 8.2 1 Simple Interest • Interest is the dollar amount that we get paid for lending money or pay for borrowing money. • The amount of money that we deposit or borrow is called the principal. • The amount of interest depends on the principal, the interest rate, which is given as a percent and varies from bank to bank, and the length of time which the money is deposited. • Simple interest involves interest calculated only on the principal. 1/21/2011 Section 8.2 2 Simple Interest To calculate the simple interest: Interest = principal x rate x time I = Prt The rate r, is expressed as a decimal when calculating simple interest. 1/21/2011 Section 8.2 3 Simple Interest Example: You deposit $2000 in a savings account at Hometown Bank, which has a rate of 6%. Find the interest at the end of the first year. Solution: To find the interest at the end of the first year, we use the simple interest formula. At the end of the first year, the interest is $120. 1/21/2011 Section 8.2 4 Simple Interest Example: A student took out a simple interest loan for $1800 for two years at a rate of 8% to purchase a new car. Find the interest of the loan. Solution: To find the interest of the loan, we use the simple interest formula. The interest on the loan is $288. 1/21/2011 Section 8.2 5 Future Value: Principal Plus Interest The future value, A, of P dollars at simple interest rate r (as a decimal) for t years is given by A = P(1 + rt). • P is also known as the loan’s present value. Example: A loan of $1060 has been made at 6.5% for three months. Find the loan’s future value. 1/21/2011 Section 8.2 6 Future Value: Principal Plus Interest Example Continued Solution: The variables are given as follows: P = $1060 r = 6.5% or 0.065 t = ¼ = 0.25 since 3 months is 3 of 12 a year. Now, we use the future value formula: A = P(1 + rt) A = 1060(1 + 0.065 · 0.25) A ≈ $1077.23 Rounded to the nearest cent, the loan’s future value is $1077.23. 1/21/2011 Section 8.2 7 Future Value: Principal Plus Interest Example: You borrow $2500 from a friend and promise to pay back $2655 in six months. What simple interest will you pay? Solution: We substitute $2655 for A, $2500 for P, and ½ or 0.5 for t since 6 months is ½ of a year. Then solve for the simple interest r. 1/21/2011 Section 8.2 8 Future Value: Principal Plus Interest Example Continued A = P( 1 + rt) 2655 = 2500 (1 + r ⋅ 0 .5 ) 2655 155 155 1250 r = 2500 + 1250 r = 1250 r 1250 r = 1250 = 0 .124 = 12 .4 % You will pay a simple interest rate of 12.4%. 1/21/2011 Section 8.2 9 Discounted Loans • • Some lenders collect the interest from the amount of the loan at the time that the loan is made. This is called a discounted loan. The interest that is the deducted from the loan is the discount. Example: You borrow $10,000 on a 10% discounted loan for a period of 8 months. a. What is the loan’s discount? b. Determine the net amount of money you receive. c. What is the loan’s actual interest rate? 1/21/2011 Section 8.2 10 Discounted Loans Example Continued Solution: a. Because the loan’s discount is the deducted interest, we use the simple interest formula. The loan’s discount is $666.67. 1/21/2011 Section 8.2 11 Discounted Loans Example Continued b. The net amount that you receive is the amount of the loan, $10,000, minus the discount, $666.67: 10,000 – 666.67 = 9333.33. Thus, you receive $9333.33. c. We can recalculate the loan’s actual interest rate, rather than stated 10%, by using the simple interest formula. I = Prt ⎛2⎞ 666.67 = (9333.33)r ⎜ ⎟ ⎝3⎠ 666.67 = 6222.22r 666.67 ≈ 0.107 = 10.7% r= 6222.22 1/21/2011 Section 8.2 The actual rate of interest on the 10% discounted loan is approximately 10.7%. 12 Section 8.3 Compound Interest Objectives 1. Use the compound interest formulas. 2. Calculate present value. 3. Understand and compute effective annual yield. 1/17/11 Section 8.3 1 Compound Interest • Compound interest is interest computed on the original principal as well as on any accumulated interest. To calculate the compound interest paid once a year we use A = P(1 + r)t, where A is called the account’s future value, the principal P is called its present value, r is the rate, and t is the years. 1/17/11 Section 8.3 2 Compound Interest Example: You deposit $2000 in a savings account at Hometown bank, which has a rate of 6%. a. Find the amount, A, of money in the account after 3 years subject to compound interest. b. Find the interest. Solution: a. Principal P is $2000, r is 6% or 0.06, and t is 3. Substituting this into the compound interest formula, we get A = P(1 + r)t = 2000(1 + 0.06)3 = 2000(1.06)3 ≈ 2382.03 1/17/11 Section 8.3 3 Compound Interest Example Continued Rounded to the nearest cent, the amount in the savings account after 3 years is $2382.03. b. The amount in the account after 3 years is $2382.03. So, we take the difference of this amount and the principal to obtain the interest amount. $2382.03 – $2000 = $382.03 Thus, the interest you make after 3 years is $382.03. 1/17/11 Section 8.3 4 Compound Interest To calculate the compound interest paid more than once a year we use where A is called the account’s future value, the principal P is called its present value, r is the rate, n is the number of times the interest is compounded per year, and t is the years. 1/17/11 Section 8.3 5 Compound Interest Example: You deposit $7500 in a savings account that has a rate of 6%. The interest is compounded monthly. a. How much money will you have after five years? b. Find the interest after five years. Solution: a. Principal P is $7500, r is 6% or 0.06, t is 5, and n is 12 since interest is being compounded monthly. Substituting this into the compound interest formula, we get 1/17/11 Section 8.3 6 Compound Interest Example Continued Rounded to the nearest cent, you will have $10,116.38 after five years. b. The amount in the account after 5 years is $10,116.38. So, we take the difference of this amount and the principal to obtain the interest amount. $ 10,116.38 – $7500 = $2616.38 Thus, the interest you make after 5 years is $ 2616.38. 1/17/11 Section 8.3 7 Compound Interest Continuous Compounding • Some banks use continuous compounding, where the compounding periods increase infinitely. After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1. For n compounding per year: 2. For continuous compounding: A = Pert. 1/17/11 Section 8.3 8 Compound Interest Continuous Compounding Example: You decide to invest $8000 for 6 years and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? Solution: The better investment is the one with the greater balance at the end of 6 years. 7% account: The balance in this account after 6 years is $12,160.84. 1/17/11 Section 8.3 9 Compound Interest Continuous Compounding Example Continued 6.85% account: A = Pert = 8000e0.0685(6) ≈ 12,066.60 The balance in this account after 6 years is $12,066.60. Thus, the better investment is the 7% monthly compounding option. 1/17/11 Section 8.3 10 Planning for the Future with Compound Interest Calculating Present Value If A dollars are to be accumulated in t years in an account that pays rate r compounded n times per year, then the present value P that needs to be invested now is given by 1/17/11 Section 8.3 11 Planning for the Future with Compound Interest Example: How much money should be deposited in an account today that earns 8% compounded monthly so that it will accumulate to $20,000 in five years? Solution: We use the present value formula, where A is $20,000, r is 8% or 0.08, n is 12, and t is 5 years. Approximately $13,424.21 should be invested today in order to accumulate to $20,000 in five years. 1/17/11 Section 8.3 12 Effective Annual Yield • The effective annual yield, or the effective rate, is the simple interest rate that produces the same amount of money in an account at the end of one year as when the account is subjected to compound interest at a stated rate. Example: You deposit $4000 in an account that pays 8% interest compounded monthly. a. Find the future value after one year. b. Use the future value formula for simple interest to determine the effective annual yield. 1/17/11 Section 8.3 13 Effective Annual Yield Example Continued Solution: a. We use the compound interest formula to find the account’s future value after one year. Rounded to the nearest cent, the future value after one year is $4332.00. 1/17/11 Section 8.3 14 Effective Annual Yield Example Continued b. The effective annual yield is the simple interest rate. So, we use the future value formula for simple interest to determine rate r. Thus, the effective annual yield is 8.3%. This means that an account that earns 8% interest compounded monthly has an equivalent interest rate of 8.3% compounded annually. 1/17/11 Section 8.3 15 Section 8.4 Annuities, Stocks, and Bonds Objectives 1. Determine the value of an annuity. 2. Determine regular annuity payments needed to achieve a financial goal. 3. Understand stocks and bonds as investments. 4. Read stock tables. 1/17/11 Section 8.4 1 Annuities • • An annuity is a sequence of equal payments made at equal time periods. The value of an annuity is the sum of all deposits plus all interest paid. Example: You deposit $1000 into a savings plan at the end of each year for three years. The interest rate is 8% per year compounded annually. a. Find the value of the annuity after three years. b. Find the interest. 1/17/11 Section 8.4 2 Annuities Example Continued Solution: a. The value of the annuity after three years is the sum of all deposits made plus all interest paid over three years. The value of annuity at the end of three years is $3246.40. 1/17/11 Section 8.4 3 Annuities Example Continued b. You made three payments of $1000 each, depositing a total of 3 x $1000, or $3000. Because the value of annuity is $3246.40, the interest is $3246.40 - $3000 = $246.40. 1/17/11 Section 8.4 4 Annuity Interest Compounded Once a Year If P is the deposit made at the end of each year for an annuity that pays an annual interest rate r (in decimal form) compounded once a year, the value, A, of the annuity after t years is 1/17/11 Section 8.4 5 Annuity Interest Compounded Once a Year Example: To save for retirement, you decide to deposit $1000 into an IRA at the end of each year for the next 30 years. If you can count on an interest rate 10% per year compounded annually, a. How much will you have from the IRA after 30 years? b. Find the interest. 1/17/11 Section 8.4 6 Annuity Interest Compounded Once a Year Example Continued Solution: a. The amount that you will have in the IRA is its value after 30 years. After 30 years, you will have approximately $164,494 from the IRA. 1/17/11 Section 8.4 7 Annuity Interest Compounded Once a Year Example Continued b. You made 30 payments of $1000 each, a total of $30,000. Because the annuity is $164,494, the interest is $164,494 - $30,000 = $134,494. The interest is nearly 4½ times the amount pf your payments. 1/17/11 Section 8.4 8 Annuity Interest Compounded N Times Per Year If P is the deposit made at the end of each compounding period for an annuity that pays an annual interest rate r (in decimal form) compounded n times per year, the value, A, of the annuity after t years is 1/17/11 Section 8.4 9 Annuity Interest Compounded Once a Year Example: At age 25, to save for retirement, you decide to deposit $200 into an IRA at the end of each month at an interest rate of 7.5% per year compounded monthly, a. How much will you have from the IRA when you retire at age 65? b. Find the interest. 1/17/11 Section 8.4 10 Annuity Interest Compounded Once a Year Example Continued Solution: a. The amount that you will have in the IRA is its value after 40 years. At age 65, you will have approximately $604,765 from the IRA. 1/17/11 Section 8.4 11 Annuity Interest Compounded Once a Year Example Continued b. The interest is more than 5 times the amount of your contributions to the IRA. 1/17/11 Section 8.4 12 Planning for the Future with an Annuity P, the deposit that must be made at the end of each compounding period into an annuity that pays an annual interest rate r (in decimal form) compounded n times per year in order to achieve a value of A dollars after t years is 1/17/11 Section 8.4 13 Planning for the Future with an Annuity Example: You would like to have $20,000 to use as a down payment for a home in five years by making regular, end-of-the-month deposits in an annuity that pays 8% compounded monthly. a. How much should you deposit each month? b. How much of the $20,000 down payment comes from deposits and how much comes from interest? 1/17/11 Section 8.4 14 Planning for the Future with an Annuity Example Continued Solution: a. You should deposit $273 each month to be certain of having $20,000 for a down payment on a home. 1/17/11 Section 8.4 15 Planning for the Future with an Annuity Example Continued b. We see that $16,380 of the $20,000 comes from your deposits and the remainder, $3620, comes from interest. 1/17/11 Section 8.4 16 Investments • When depositing money into a bank account, you are making a cash investment. • The account’s interest rate guarantees a certain percent increase in your investment, called its return. • Other kind of investments that are riskier are called stocks and bonds. 1/17/11 Section 8.4 17 Stocks • Investors purchase stock, shares of ownership in a company. For example, if a company has issued a total of 1 million shares and an investor owns 20,000 of these shares, that investor owns of the company. • Any investor who owns some percentage of the company is called a shareholder. 1/17/11 Section 8.4 18 Stocks • Buying or selling stocks is referred to as trading. • Stocks are traded on a stock exchange. There are two ways to make money by investing in stock: • You sell shares for more money than you paid for them, in which case you have a capital gain on the sale of stock. • While you own the stock, the company distributes all or part of its profits to shareholders as dividends. 1/17/11 Section 8.4 19 Bonds • In order to raise money and dilute the ownership of current stockholders, companies sell bonds. • People who buy a bond are lending money to the company from which they buy the bond. • Bonds are a commitment from a company to pay the price an investor pays for the bond at the time it was purchased, called the face value, along with interest payments at a given rate. • A listing of all the investments that a person holds is called a financial portfolio. 1/17/11 Section 8.4 20 Bonds • A listing of all the investments that a person holds is called a financial portfolio. • Most financial advisors recommend a portfolio with a mixture of low-risk and high-risk investments, called a diversified portfolio. 1/17/11 Section 8.4 21 Reading Stock Tables Daily newspapers and online services give current stock prices and other information about stocks. • 52-week high refers to the highest price at which a company traded during the past 52 weeks. • 52-week low refers to the lowest price at which a company traded during the past 52 weeks. • Stock refers to the company name. • SYM refers to the symbol the company uses for trading. • Div refers to dividends paid per share to stockholders last year. • Yld% stands for percent yield. • Vol100s stands for sales volume in hundreds. 1/17/11 Section 8.4 22 Reading Stock Tables • Hi stands for the highest price at which the company’s stock traded yesterday. • Low stands for the lowest price at which the company’s stock traded yesterday. • Close stands for the price at which shares traded when the stock exchanged closed yesterday. • Net Chg stands for net change. • PE stands for the price-to-earnings ratio. 1/17/11 Section 8.4 23 Reading Stock Tables Example: Use the stock table for Disney to answer the following questions: a. What were the high and low prices for the past 52 weeks? b. If you owned 3000 shares of Disney stock last year, what dividend did you receive? c. What is the annual return for dividends alone? How does this compare to a bank account offering a 3.5% interest rate? d. How many shares of Disney were traded yesterday? e. What were the high and low prices for Disney shares yesterday? 1/17/11 Section 8.4 24 Reading Stock Tables Example Continued f. What was the price at which Disney shares traded when the stock exchange closed yesterday? g. What does “…” in the net change column mean? h. Compute Disney’s annual earnings per share using 52-week High Low Stock SYM Div 42.38 22.50 Disney DIS .21 1/17/11 Yld PE % .6 43 Section 8.4 Vol 100s Hi Lo Close Net Chg 115900 32.50 31.25 32.50 … 25 Reading Stock Tables Example Continued Solution: a. We look under the heading High and Low. The 52week high was 42.38 or $42.38. The 52-week low was 22.50 or $22.50. b. We look under the heading Div for the dividend paid for a share of Disney stock. So, the annual return of a share last year was $0.21. If you owned 3000 shares, then you received a dividend of 3000 x 0.21 = $630. c. The heading Yld% is the percentage that give investors an annual return of their dividends. For Disney, it is 0.6%. This is much lower than a bank account paying 3.5% interest rate. 1/17/11 Section 8.4 26 Reading Stock Tables Example Continued d. We look under Vol100s for the number of shares Disney traded yesterday. So, Disney traded 115900 x 100 = 11,590,000 shares yesterday. e. The high and low prices for Disney shares yesterday are found by looking under Hi and Lo. Hence, the high price was $32.50 and the low price was $31.25 per share yesterday. f. When the stock exchange closed yesterday Disney’s shares were $32.50 per share. g. The “…” under Net Chg means that there was no change in price in Disney stock from the market close two days ago to yesterday’s market close. 1/17/11 Section 8.4 27 Reading Stock Tables Example Continued h. We use the formula given along with 43 for the PE ratio and $32.50 for the closing price per share. Thus, the annual earnings per share for Disney are $0.76. • The PE ratio tells us that yesterdays closing price is 43 times greater than the earning’s per share. 1/17/11 Section 8.4 28 Section 8.5 Installment Buying Objectives 1. Determine the amount financed, the installment price, and the finance charge for a fixed loan. 2. Determine the APR. 3. Compute unearned interest and the payoff amount for a loan paid off early. 4. Find the interest, the balance due, and the minimum monthly payment for credit card loans. 5. Calculate interest on credit cards using three methods. 1/17/11 Section 8.5 1 Fixed Installment Loans • The amount financed is what the consumer borrows: Amount financed = cash price – down payment. • The total installment price is the sum of all monthly payments plus the down payment: Total Installment Price = Total of all monthly payments + down payment. • The finance charge is the interest on the installment loan: Finance charge = Total installment price - Cash price. 1/17/11 Section 8.5 2 Fixed Installment Loans Example: The cost of a used pick-up truck is $9345. We can finance the truck by paying $300 down and $194.38 per month for 60 months. a. Determine the amount financed. b. Determine the total installment price. c. Determine the finance charge. 1/17/11 Section 8.5 3 Fixed Installment Loans Example Continued Solution: • Amount financed = cash price – down payment = $9345 - $300 = $9045 The amount financed is $9045. b. Total Installment Price = Total of all monthly payments + down payment = $194.37 x 60 + $300 = $11,662.80 + $300 = $11,962.80 The total installment price is $11,962.80. 1/17/11 Section 8.5 4 Fixed Installment Loans Example Continued c. Finance charge = Total installment price - Cash price = $11,962.80 - $9345 = $2617.80 The finance charge for the loan is $2617.80. 1/17/11 Section 8.5 5 Annual Percentage Rate APR Steps in using the APR table below: 1. Compute the finance charge per $100 financed: 2. Look in the row (in table below) corresponding to the number of payments to be made and find the entry closest to the value in step 1. 3. Find the APR at the top of the column in which the entry from step 2 is found. This is the APR rounded to the nearest ½%. 1/17/11 Section 8.5 6 Annual Percentage Rate APR Table 1/17/11 Section 8.5 7 Annual Percentage Rate APR Example: In the previous example, we found that the amount financed for the truck was $9045 and the finance charge was $2617.80. The borrower financed the truck with 60 monthly payments. Use the table to determine the APR. 1/17/11 Section 8.5 8 Annual Percentage Rate Example Continued Solution: Step 1. Find the finance charge per $100 of the amount financed. This means that the borrower pays $28.94 interest for each $100 being financed. 1/17/11 Section 8.5 9 Annual Percentage Rate Example Continued Step 2. We look for 60 in the left column since we make 60 monthly payments. Move to the right until you find the amount closest to the finance charge of $28.94. Step 3. Find the APR. Look at the corresponding APR for $28.96 at the top of the column. Thus, the APR for the fixed installment truck loan is approximately 10.5%. 1/17/11 Section 8.5 10 Methods for Computing Unearned Interest The total amount due on the day that a loan is paid off early is called the payoff amount. 1/17/11 Section 8.5 11 Methods for Computing Unearned Interest Example: You got a big bonus at work—and a raise. Instead of making the thirty-sixth payment on your truck, you decide to pay the remaining balance and terminate the loan. a. Use the actuarial method to determine how much interest will be saved (the unearned interest, u) by repaying the loan early. b. Find the payoff amount. 1/17/11 Section 8.5 12 Methods for Computing Unearned Interest Example Continued Solution: a. We use the formula to find u, the unearned interest. We need to find k, R, and V. k: k=60 – 36 = 24, since the current payment is number 36. R: R = $194.38 V: V = 11.30, from the APR table with 24 monthly payments left. Thus, . 1/17/11 Section 8.5 13 Methods for Computing Unearned Interest Example Continued Approximately $473.64 will be saved in interest by terminating the loan early using the actuarial method. b. The pay off amount, the amount due on the day of the loan’s termination is determined as follows: The payment needed to terminate the loan at the end of 36 months is $4385.86. 1/17/11 Section 8.5 14 Methods for Computing Unearned Interest Example: The loan in the previous example is paid off at the time of the thirty-sixth monthly payment. Use the rule of 78 to find a. the unearned interest. b. the payoff amount. 1/17/11 Section 8.5 15 Methods for Computing Unearned Interest Example Continued Solution: a. We use the formula to find u, the unearned interest. k: k=60 – 36 = 24, since the current payment is number 36. n: n = 60, the original number of payments. F: F = $2617.80, the finance charge (interest). 1/17/11 Section 8.5 16 Methods for Computing Unearned Interest Example Continued Now, Thus, by the rule of 78, the borrower will save $429.15 in interest. b. Next, we determine the payoff amount. Thus, the payoff amount is $4430.35. 1/17/11 Section 8.5 17 Open-end Installment Loans • Using a credit card is an example of an open-end installment loan, commonly called revolving credit. • Customers receive a statement, called an itemized billing. • One method for calculating interest on credit cards is the unpaid balance method. 1/17/11 Section 8.5 18 Open-end Installment Loans Example: A particular VISA card calculates interest using the unpaid balance method. The monthly interest rate is 1.3% on the unpaid balance on the first day of the billing period less payments and credits. Here are some of the details in the May 1–May 31 itemized billing: May 1 Unpaid Balance: $1350 Payment Received May 8: $250 Purchases Charged to the VISA Account: Airline tickets: $375, Books: $57, Meals: $65 Last Day of the Billing Period: May 31 Payment Due Date: June 9 1/17/11 Section 8.5 19 Open-end Installment Loans Example Continued a. Find the interest due on the payment due date. b. Find the total balance owed on the last day of the billing period. c. This credit card requires a $10 minimum monthly payment if the total balance owed on the last day of the billing period is less than $360. Otherwise, the minimum monthly payment is of the balance owed on the last day of the billing period, rounded to the nearest whole dollar. What is the minimum monthly payment due by June 9? 1/17/11 Section 8.5 20 Open-end Installment Loans Example Continued Solution: a. The monthly interest rate is 1.3% on the unpaid balance on the first day of the billing period, $1350, less payments and credits, $250. The interest due is computed using I = Prt. The interest due on the payment due date is $14.30. 1/17/11 Section 8.5 21 Open-end Installment Loans Example Continued b. The total balance owed on the last day of the billing cycle is determine as follows: The total balance owed on the last day of the billing period is $1611.30. 1/17/11 Section 8.5 22 Open-end Installment Loans Example Continued c. Because the balance owed on the last day of the billing period, $1611.30, exceeds $360, the customer must pay a minimum of of the balance owed. Rounded to the nearest whole dollar, the minimum monthly payment due by June 9 is $45. 1/17/11 Section 8.5 23 Methods for Calculating Interest on Credit Cards For all three methods, I = Prt, where r is the monthly rate and t is one month. Unpaid balance method: The principal, P, is the balance on the first day of the billing period less payments and credits. Previous balance method: The principal, P, is the unpaid balance on the first day of the billing period. Average daily balance method: The principle, P, is the average daily balance. This is determined by adding the unpaid balances for each day in the billing period and dividing by the number of days in the billing period. 1/17/11 Section 8.5 24 Methods for Calculating Interest on Credit Cards Example: A credit card has a monthly rate of 1.75%. In the January 1-31 itemized billing, the January 1 unpaid balance is $2500. A payment of $1000 was received on January 8. There are no purchases or cash advances in this billing period. The payment due date is February 9. Find the interest due on this date using each of the three methods for calculating credit card interest. Solution: Because the monthly rate is 1.75%, for all three methods I = Prt = P x 0.0175 x 1 month. The principal, P, is different for each method. 1/17/11 Section 8.5 25 Methods for Calculating Interest on Credit Cards Example Continued a. The Unpaid Balance Method The principal, P, is the unpaid balance on the first day of the billing period, $2500, less payments and credits, $1000. The interest is I = Prt = ($2500 - $1000) x 0.0175 x 1 = $1500 x 0.0175 x 1 = $26.25. The interest due on the payment due date is $26.25. 1/17/11 Section 8.5 26 Methods for Calculating Interest on Credit Cards Example Continued b. The Previous Balance Method The principal, P, is the unpaid balance on the first day of the billing period, $2500. The interest is I = Prt = $2500 x 0.0175 x 1 = $43.75. The interest due on the payment due date is $43.75. 1/17/11 Section 8.5 27 Methods for Calculating Interest on Credit Cards Example Continued c. The Average Daily Balance Method The principal, P, is the average daily balance. Add the unpaid balances for each day in the billing period and divide by the number of days in the billing period, 31. The unpaid balance on the first day of the billing period is $2500, and a $1000 payment is recorded on January 8. The sum of the balances owed for each day of the billing period is $2500 added for each of the first 7 days, expressed as $2500 x 7, plus $1500 added for each of the remaining 24 days, expressed as $1500 x 24. 1/17/11 Section 8.5 28 Methods for Calculating Interest on Credit Cards Example Continued Average daily balance The average daily balance serves as the principal. 1/17/11 Section 8.5 29 Methods for Calculating Interest on Credit Cards Example Continued The interest is I = Prt = $1725.81 x 0.0175 x 1 ≈ $30.20. The interest due on the payment due date is $30.20. 1/17/11 Section 8.5 30 Section 8.6 Amortization and the Cost of Home Ownership Objectives 1. Understand mortgage options. 2. Compute the monthly payment and interest costs for a mortgage. 3. Prepare a partial loan amortization schedule. 4. Compute payments and interest for other kinds of installment loans. 1/17/11 Section 8.6 1 Mortgages • A mortgage is a long-term loan for the purpose of buying a home. • The down payment is the portion of the sale price of the home that the buyer initially pays to the seller. • The amount of the mortgage is the difference between the sale price and the down payment. • Some companies, called mortgage brokers, offer to find you a mortgage lender willing to make you a loan. • Fixed-rate mortgages have the same monthly payment during the entire time of the loan. • Variable-rate mortgages known as adjustable-rate mortgages (ARMs), have payment amounts that change from time to time depending on changes in the interest rate. 1/17/11 Section 8.6 2 Computation Involved with Buying a Home • Most lending institutions require the buyer to pay one or more points at the time of closing—that is, the time at which the mortgage begins. • A point is a onetime charge that equals 1% of the loan amount. For example, two points means that the buyer must pay 2% of the loan amount at closing. • A document, called the Truth-in-Lending Disclosure Statement, shows the buyer the APR for the mortgage. • In addition, lending institutions can require that part of the monthly payment be deposited into an escrow account, an account used by the lender to pay real estate taxes and insurance. 1/17/11 Section 8.6 3 Computation Involved with Buying a Home Loan Payment Formula for Installment Loans The regular payment amount, PMT, required to repay a loan of P dollars paid n times per years over t years at an annual rate r is given by 1/17/11 Section 8.6 4 Computation Involved with Buying a Home Example: The price of a home is $195,000. The bank requires a 10% down payment and two points at the time of closing. The cost of the home is financed with a 30-year fixed rate mortgage at 7.5%. a. b. c. d. Find the required down payment. Find the amount of the mortgage. How much must be paid for the two points at closing? Find the monthly payment (excluding escrowed taxes and insurance). e. Find the total interest paid over 30 years. 1/17/11 Section 8.6 5 Computation Involved with Buying a Home Example Continued Solution: a. The required down payment is 10% of $195,000 or 0.10 x $195,000 = $19,500. b. The amount of the mortgage is the difference between the price of the home and the down payment. 1/17/11 Section 8.6 6 Computation Involved with Buying a Home Example Continued c. To find the cost of two points on a mortgage of $175,500, find 2% of $175,500. 0.02 x $175,000 = $3510 The down payment ($19,500) is paid to the seller and the cost of two points ($3510) is paid to the lending institution. 1/17/11 Section 8.6 7 Computation Involved with Buying a Home Example Continued d. We need to find the monthly mortgage payment for $175,000 at 7.5% for 30 years. We use the loan payment formula for installment loans. The monthly mortgage payment for principal and interest is approximately $1227.00. 1/17/11 Section 8.6 8 Computation Involved with Buying a Home Example Continued e. The total cost of interest over 30 years is equal to the difference between the total of all monthly payments and the amount of the mortgage. The total of all monthly payments is equal to the amount of the monthly payment multiplied by the number of payments. We found the amount of each monthly payment in (d): $1227. The number of payments is equal to the number of months in a year, 12, multiplied by the number of years in the mortgage, 30: 12 x 30 = 360. Thus, the total of all monthly payments = $1227 x 360. 1/17/11 Section 8.6 9 Computation Involved with Buying a Home Example Continued Now we calculate the interest over 30 years. The total interest paid over 30 years is approximately $266,220. 1/17/11 Section 8.6 10 Loan Amortization Schedules • When a loan is paid off through a series of regular payments, it is said to be amortized, which literally means “killed off.” • Although each payment is the same, with each successive payment the interest portion decreases and the principal portion increases. • A document showing important information about the status of the mortgage is called a loan amortization schedule. 1/17/11 Section 8.6 11 Loan Amortization Schedules Example: Prepare a loan amortization schedule for the first two months of the mortgage loan shown in the following table: 1/17/11 Section 8.6 12 Loan Amortization Schedules Example Continued Solution: We begin with payment number 1. Interest for the month = Prt = $130,000 x 0.095 x ≈ $1029.17 Principal payment = Monthly payment - Interest payment = $1357.50 - $1029.17 = $328.33 Balance of loan = Principal balance - Principal payment = $130,000 - $328.33 = $129,671.67 1/17/11 Section 8.6 13 Loan Amortization Schedules Example Continued Now, starting with a loan balance of $129,671.67, we repeat these computations for the second month. Interest for the month = Prt = $129,671.67 x 0.095 x = $1026.57 Principal payment = Monthly payment – Interest payment = $1357.50 – $1026.57 = $330.93 Balance of loan = Principal balance – Principal payment = $129,671.67 – $330.93 = $129,340.74 1/17/11 Section 8.6 14 Loan Amortization Schedules Example Continued 1/17/11 Section 8.6 15 Monthly Payments and Interest Costs for Other Kinds of Installment Loans Example: You decide to take a $20,000 loan for a new car. You can select one of the following loans, each requiring regular monthly payments: Installment Loan A: 3-year loan at 7%. Installment Loan A: 5-year loan at 9%. a. Find the monthly payments and the total interest for Loan A. c. Find the monthly payments and the total interest for Loan B. d. Compare the monthly payments and total interest for the two loans. 1/17/11 Section 8.6 16 Monthly Payments and Interest Costs for Other Kinds of Installment Loans Solution: a. We first determine monthly payments and total interest for Loan A. The monthly payments are approximately $618. 1/17/11 Section 8.6 17 Monthly Payments and Interest Costs for Other Kinds of Installment Loans Now we calculate the interest paid over 3 years or 36 months. The interest paid over 3 years is approximately $2248. 1/17/11 Section 8.6 18 Monthly Payments and Interest Costs for Other Kinds of Installment Loans b. Next, we determine monthly payment and total interest for Loan B. The monthly payments are approximately $415. 1/17/11 Section 8.6 19 Monthly Payments and Interest Costs for Other Kinds of Installment Loans Now we calculate the interest over 5 years, or 60 months. The total interest paid over 5 years is approximately $4900. 1/17/11 Section 8.6 20 Monthly Payments and Interest Costs for Other Kinds of Installment Loans c. The monthly payments and total interest for the two loans is given in the table below. 1/17/11 Section 8.6 21