Section 8.1. I. Percent per hundred


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1 1 Section 8.1 I. Percent per hundred a. Fractions to Percents: 1. Write the fraction as an improper fraction 2. Divide the numerator by the denominator 3. Multiply by 100 (Move the decimal two times Right) 4. Attach a % sign. b. Decimals to Percents: 1. Move decimal two times Right 2. Attach % sign c. Percents to Decimals: 1. Move decimal two times Left 2. Remove % sign. II. A is P percent of B A = PB Sales tax = (item s cost)(tax rate) Total price = (item s cost) + (sales tax) Discount = (original price)(discount rate) Total price = (original price) (discount) Example: A CD player costs $380. a. The CD is marked 35% off. What is the discount amount? b. Sales tax in the area is 6%. What s the sale price with 6% tax after the discount?
2 2 III. Percent of Increase/Decrease Ex. For the $380 CD player marked 35% off, what is the percent of decrease? Ex. In 1965, one share was worth $18. In 2008, one share of the same stock was worth $96, 600. Find the percent of increase. Interpret what this means.
3 3 IV. Terms 1. Gross Income Income before all adjustments for the year, including wages, tips, investments, earnings, and unemployment compensation. 2. Adjustments Deductions, payments to taxdeferred savings plans. 3. Exemption A fixed amount of money for yourself, and an amount for all dependents, that is deducted from gross income. 4. Deductions An amount of money to be removed from the adjusted gross income. i. Standard deduction ii. Itemized Deduction Interest on mortgages, state income tax, property tax, charitable contributions, medical expenses greater than 7.5% of the adjusted gross income). Medical Expense >.075AGI 5. Filing Status Single, married filing separately, married filing jointly, head of household.
4 4 V. Income Tax Formulas 1. Adjusted Gross Income = Gross Income Adjustments 2. Taxable Income = AGI (Exemptions + Deductions) Look in the marginal tax table for Exemptions For deductions: Compare the standard deduction from the marginal tax table to the itemized deductions and choose the larger value. Remember to consider medical expenses when calculating the itemized deduction. Medical Expense >.075AGI 3. Income Tax = Tax Computation Tax Credits Find the Tax Computation using the marginal tax table. = Sum[(tax rates)(marginal income)]
5 5 VI. General Form of a Tax Table
6 6 VII. Example Calculate the income tax owed by a single man with no dependents with the given information. Gross Income: $40, 000 Adjustments: $ 1000 Deductions: $3, 000 Charitable Contributions $1, 500 Theft loss $300 cost of tax preparation $1,500 medical expenses Tax Credits: none
7 7 Section 8.2 Definitions 1. Interest Amount of money paid for lending and investing money. The amount owed for borrowing money. 2. Principal The amount borrowed or deposited. 3. Interest Rate Usually given as a percent value. 4. Simple Interest I=Prt I = interest earned P = principal invested r = interest rate (in decimal form!!) t = time (in years) month o if months, use t = 12 day o if days, use t = 360
8 8 Example Find the simple interest owed for borrowing principal P at interest rate r for time period t. a) P = $5,000 r = 5% t = 3 years b) P = $11,000 r = 6.5% t = 10 months c) P = $4,550 r = 11.5% t = 90 days
9 9 Definitions 1. Present Value The amount borrowed or deposited. The principal amount 2. Future value (A) The total amount owed or earned at the end of the loan, including interest. Future Value for Depositing Money A P( 1 rt) A = future value P = principal r = simple interest rate t = time in years How the formula was derived:
10 10 Example The principal of $2000 is deposited at simple interest rate 7.5% for time t. Find the future value when: a) t = 10 years b) t = 7 months c) t = 120 days
11 11 Example A bank offers a CD paying a simple interest at a rate of 5.25%. How much must you put in the CD now to have $5,000 for college tuition in five years? Comment: Generally, t A P Pr
12 12 Definition 1. Discount Interest deducted from a loan. 2. Discounted Loan Lenders collect interest owed from the loan at the time that the loan is made. Words and their implications 1. Deducted Interest or Loan s discount Use I = Prt 2. Net amount of money received Use P  I = P Prt = P(1rt) 3. Actual interest rate Use I = Prt Plug in the deducted interest for I Plug in the net amount received for P I deducted_ int erest Solve for r to get r Pt ( net _ amt _ received)( t)
13 13 Example You borrow $5, 000 on a 9.5% discounted loan for a) 4 years b) 10 months. Determine the loan s discount, net amount of money you receive, and the loan s actual interest rate for each time period.
14 14 Section 8.3 Suppose that you have $15, 000 to invest in either: Money market at 3% interest compounded annually Savings account at 1.5% compounded annually Definitions 1. Compound Interest Interest computed on the original principal and on any accumulated interest. 2. Compound Period Period of time between two interest payments. Compounded annually n = 1 Compounded semiannually n = 2 Compounded quarterly n = 4 Compounded monthly n = 12 Compounded daily n = 365 Compounded continuously: think e General Equation to Calculate Simple Compound Interest A P 1 r t Value of Account = Original_ Investment 1 Interest_ Rate Time _ Period
15 15 Example Consider investing your money in two different accounts: i) a money market account earning 3% compounded annually ii) a savings account earning 1.55% compounded annually. a) Model the potential growth for $1, 500 invested in each account. b) Compare the investments after 10 years.
16 16 Compounding n times a year for t years A P 1 r n nt A = future value P = present value = initial investment = principal r = nominal interest rate as a decimal. (APR) n = number of compounding per year. t = number of years r n n 11 = effective interest rate (APY) This is the amount that you actually pay or earn in an account. Compounding continuously t years A rt Pe A = future value P = present value = initial investment = principal r = nominal interest rate as a decimal. (APR) t = number of years r e  1 = effective interest rate (APY) This is the amount that you actually pay or earn in an account.
17 17 Example You invest $50 into a mutual fund paying a nominal interest rate of 7.2%. Find the future value for each case. a) Compounded annually for 6 months. b) Compounded quarterly for 2 years. Also find the effective interest rate and the nominal interest rate. c) Compounded daily for 2 years. d) Compounded continuously for 8 months. Also find the effective interest rate and the nominal interest rate.
18 18 Definition Effective annual yield / effective rate / annual percentage rate The simple interest rate that produces the same amount I = Prt of money in an account at the end of one year as when the account is subjected to compounding interest at the stated rate. The true interest rate paid/earned. Investing: You want to choose the highest effective annual yield. Borrowing: you want to choose the lowest annual percentage rate (all other factors being equal) Y (1 r n ) 1 r n Y e 1 or Where: r is the interest rate as a decimal n is the number of compounding per year Y is the effective annual yield (APY). Be sure to write your final answer as a %!
19 19 Example An account has a nominal rate of 4.6%. Find the effective annual yield, rounded to the nearest tenth of a percent, with: a) quarterly compounding b) monthly compounding c) daily compounding. How does changing the compounding period affect the annual yield?
20 20
21 21 Section 8.4 I. Pensions Goal: Here s my income, what can I afford to invest today? You have no control over the amount of money available at retirement (income received per month in the future) because it s a present amount. Pensions are usually paid by an employer and are not guaranteed money because payments are based on investments in stocks. Pensions are paid after a certain age, not after a certain time period. Real Life Example of why pensions are NOT guaranteed retirement money: A United Employee retired with a pension. United went bankrupt and used all of its employees pension money to try to save the business. In other words, the retiree s pension was gone! The government stepped in and paid pennies on the dollar, which means that the retirees pension was essentially gone.
22 22 II. Annuities Goal: Here s the income I need in the future, what s my monthly payment? Definition: A sequence of equally payments or receipts made at equal time intervals. In other words, payouts are made in equal amounts after a certain time period (not a certain age). You control and plan for the amount of money available at retirement (income received per month in the future). Annuity formulas amortize your deposit to calculate the future value of the money based on the present value. Annuity money is safe/guaranteed. Example: When you win the lottery, you can get a lump sum payment (present value) or you can get a series of payments (future value/annuity) Value of an Annuity = all deposits + all interest paid Compounded Once per Year: P[(1 A r) r t 1] Compounded n times per Year: A P[(1 r ) n r n nt 1] Think of A as the ending value, the goal that you want to reach. Total deposits = Pt Interest Earned = (Ending value) (Total deposits) = A  Pt
23 23 Example At age 21, you decide to save for retirement by investing $150 at regular intervals into an IRA paying 6.55% compounded at regular intervals for 20 years. a) Find the value at the end of each year when interest is compounded annually. b) Find the value at the end of each month when interest is compounded annually. c) Find the value at the end of 3 months when interest is compounded quarterly. A P[(1 r ) n r n nt 1]
24 24 Ordinary Annuity Deposit made at the end of each compounding period. Annuity Due Deposit make at the beginning of the compounding period. So how do you decide on the amount of money to deposit? P A r n nt 1 r 1 n P = deposit/principal A = value of the annuity after t years. The goal you want to reach. The ending value. t = time in years. n = compounding n times per year r = interest rate as a decimal.
25 25 Example Determine the periodic deposit and how much of the financial goal comes from deposits and how much comes from interest at the end of every six months at a rate of 4.75% compounded semiannually for 35 years with a financial goal of $150, 000. P A r n nt 1 r 1 n
26 26 III. Definitions Cash Investment Deposit money into an account. Return The percent increase on investment. ex) annually/monthly/quarterly return is r%. Investment portfolio A listing of all investments a person holds. You want a diversified portfolio: mixture of low risk and high risk investments. Mutual funds: A group of stocks and bonds managed for you by the fund manager. You re money is combined with other people s money. The fund manager invests all the money to obtain maximum returns. Bond Investors lend money to a company. In return, the company agrees to pay back the bond and interest. Investors do not own a percentage of the company. Low risk. Face value: the amount that the bond is bought for. Share Percent of ownership in a company that an investor can purchase. Shareholder Any investor owning a percentage of the company.
27 27 Stock Share of ownership in a company that an investor can purchase. Shares are bought and sold at the stock exchange. Prices depend on the supply and demand, the economy, and the success of a business. Trading buying or selling stocks. Capital Gain sell shares for more money than they were purchased for. Capital Loss sell shares for less money than they were purchased for. Dividend A company distributes profit to each shareholder through dividends. The amount paid to a shareholder depends on the number of shares owned. Types of Stock According to Risk and Rewards
28 28 Stock Tables
29 29 Example a) What were the high and low prices for a share for the past 52 weeks? b) If you owned 700 shares of this stock last year, what dividend did you receive? c) What is the annual return on the dividends alone? How does this compare to a bank offering a 3% interest rate?
30 30 d) How many shares of this company s stock were traded yesterday? e) What were the high and low prices for a share yesterday? f) What was the price at which a share traded when the stock exchange closed yesterday? g) What was the change in price for a share of stock from the market close two days ago to yesterday s market close? h) Compute the company s annual earnings per share using yesterday ' s _ closin g _ price _ per _ share annual _ earning _ per _ share PE _ ratio
31 31 Section 8.5 Installment buying repay a loan for the cost of a product on a weekly basis. Installment loans loans paid off with payments over a time period. Exs. Mortgage and credit cards. 1. Mortgage a long term installment loan for buying a house. Property is a security for payment. When there is no payment, the lender reposses the property. Fixed rate same monthly payment Adjustable rate/variable rate payment amount changes depending on changes in the interest rate. Down payment portion of the sale price the buyer immediately pays to the seller. Amount of mortgage = sale price down payment Point 1% of the amount of the mortgage paid to the lending institution. o 1 point =.01(amount of mortgage) o Fewer points = higher interest rate. o More points = lower interest rate. Escrow account holding account used by a seller. Your mortgage payment can include payments into an escrow account to pay real estate tax (property tax) and house insurance.
32 32 Summary for the parts of a mortgage: Selling price $120, % down payment Amount of mortgage 2 points at closing Parts of a monthly $1,000 mortgage payment: It is the bank s responsibility not to spend the escrow account money so that your tax and insurance gets paid. If the bank does not pay the money, then collectors come to you. Escrow Account Example: A lawyer asks for a retainer of $2000. That retainer goes into an escrow account. You re prepaying the lawyer, and all unused funds will be returned to you. If the case is reconciled, if he is disbarred, or if he does no work, then the full retainer is returned to you. Comment: We will not be considering escrow accounts when handling mortgages.
33 33 Regular Payment amount for principle and interest due: PMT PMT 1 1 r p n r n nt P = loan amount n = number of payments per year t = number of years r = annual interest rate. Cost of Interest = (Total of all monthly payments for one year) (mortgage amount) = (PMT)(number of payments per year)(number of years) (mortgage amount down payment)
34 34 Example Selling price $120, % down payment Amount of mortgage Now let s say that we were financed for 15 years at 7.5%. a) Find the monthly payment PMT 1 1 r p n r n nt b) Find the total interest paid over 15 years.
35 35 II. Amortized to pay off a loan through a series of regular payments. Loan amortization schedule a document showing how monthly payments are split between interest and principal. Beginning: most of the payment goes to interest, and very little to principle. o Ex) Monthly payment $ payment #1: $ goes to interest and $ goes to principal. End: most of the payment goes to principal, little to interest. o Ex) payment # 179: $21.26 goes to interest and $ goes to principal. Schedule
36 36 Example Prepare a loan amortization schedule for Selling price $120, % down payment Amount of mortgage = 108, points at closing Financed for 15 years at 7.5% with monthly payment $ payments total.
37 37 III. Revolving Credit no schedule for paying a fixed amount each period; only a minimum monthly payment that depends on the unpaid balance and the interest rate. Ex) Credit Cards You re not charged interest if you buy a product and pay the entire balance during the same billing cycle. You are charged daily for cash advances. When you make the minimum payment, most of the payment goes to interest. Itemized billing a bill showing the details of your account. Looks like:
38 38 Formulas Credit Card Interest: I = Prt r = monthly interest rate P = average daily balance t = time in months (t = 1 for one billing cycle) Example: Credit card rate of 1.6% annual rate of Credit cards have HIGH interest rates. Average Daily Balance = Sum unpaid balance for each day in the billing cycle number of days in the billing period
39 39 Average Daily Balance = Sum unpaid balance #days in the billing period
40 40 Example A credit card calculates interest using the average daily balance method. The monthly interest rate is 1.1% of the average daily balance. The following transactions occurred during the November 1November 30 billing period. a) Find the average daily balance. b) Find the interest paid for the next billing date, December 1. c) Find the balance due on December 1. d) This credit card requires a $10 minimum monthly payment if the balance due at the end of the billing period is less than $360. Otherwise, the minimum monthly payment is 1 of the balance due at 36 the end of the billing period, rounded up to the nearest whole dollar. What is the minimum monthly payment due by December 9?
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