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## CHAPTER 8

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**CHAPTER 8**Consumer Mathematics and Financial Management**8.4**Annuities, Stocks, and Bonds**Objectives**• Determine the value of an annuity. • Determine regular annuity payments needed to achieve a financial goal. • Understand stocks and bonds as investments. • Read stock tables.**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 1: Determining the Value of an Annuity**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. • Find the value of the annuity after three years. • Find the interest. Solution: • The value of the annuity after three years is the sum of all deposits made plus all interest paid over three years. The value at the end of year 1 = $1000 The value at the end of year 2 = $1000(1 + 0.08) + $1000 = $1080 + $1000 = $2080**Example 1: Determining the Value of an Annuity continued**• The value at the end of year 3 = $2080(1 + 0.08) + $1000 • = $2246.40 + $1000 = $3246.40 • The value of the annuity at the end of three years is $3246.40. • b. You made three payments of $1000 each, depositing a total of $3000. • The value of the annuity is $3246.40, the interest is $3246.40 $3000 = $246.40**Value of an 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**Example 2: Determining the Value of an Annuity**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 of 10% per year compounded annually, • How much will you have from the IRA after 30 years? • Find the interest.**Example 2: Determining the Value of an Annuity continued**Solution: • 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.**Example 2: Determining the Value of an Annuity continued**• 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 of your payments.**AnnuityInterest 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**Example 3: Determining the Value of an Annuity**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, • How much will you have from the IRA when you retire at age 65? • Find the interest.**Example 3: Determining the Value of an Annuity continued**Solution: • 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.**Example 3: Determining the Value of an Annuity continued**b. Interest = Value of the IRA – Total deposits ≈ $604,765 $200 12 40 = $604,765 $96,000 = $508,765 The interest is approximately $508,765, more than five times the amount of your contributions.**Planning for the Future with an Annuity**The deposit, P, 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**Example 4: Achieving a Financial Goal**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 6% compounded monthly. • How much should you deposit each month? • How much of the $20,000 down payment comes from deposits and how much comes from interest?**Example 4: Achieving a Financial Goal continued**Solution: a. You should deposit $287 each month to be sure of having $20,000 for a down payment on a home in 5 years.**Example 4: Achieving a Financial Goal continued**b. Total deposits = $287 12 5 = $17,220 Interest = $20,000 $17,220 = $2780**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.**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 • Any investor who owns some percentage of the company is called a shareholder.**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.**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. • Most financial advisors recommend a portfolio with a mixture of low-risk and high-risk investments, called a diversified portfolio.**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. • Vol 100s stands for sales volume in hundreds.**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 exchange closed yesterday. • Net Chgstands for net change. • PE stands for the price-to-earnings ratio.**Example 5: Reading Stock Tables**Use the stock table for Disney to answer the following questions: • What were the high and low prices for the past 52 weeks? • If you owned 3000 shares of Disney stock last year, what dividend did you receive? • What is the annual return for dividends alone? How does this compare to a bank account offering a 3.5% interest rate? • How many shares of Disney were traded yesterday? • What were the high and low prices for Disney shares yesterday?**Example 5: Reading Stock Tables**• What was the price at which Disney shares traded when the stock exchange closed yesterday? • What does “…” in the net change column mean? • Compute Disney’s annual earnings per share using**Example 5: Reading Stock Tables**Solution: • We look under the heading High and Low. The 52-week high was 42.38 or $42.38. The 52-week low was 22.50 or $22.50. • 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. • 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.**Example 5: Reading Stock Tables**d. We look under Vol100s for the number of shares Disney traded yesterday. So, Disney traded 115900 x 100 = 11,590,000 shares yesterday. • 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. • When the stock exchange closed yesterday Disney’s shares were $32.50 per share. • 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.**Example 5: Reading Stock Tables**• 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 the earnings per share.