Savings and investments

Transcription

1 Savings and investments 10 FINANCIAL MATHEMATICS As our economy grows and our standard of living improves, Australians are becoming more informed about financial planning and investing their money wisely. People once kept all of their savings in banks, but today investors will also look elsewhere for higher interest rates and better returns for their money. For example, half of adult Australians now invest in the stock market and own shares, especially since well-known companies like Telstra and Woolworths put out their public float offers. This chapter is about the mathematics of saving and investing money, in financial institutions, in the stock market and in items that appreciate in value. It examines the calculations, terminology, graphs and tables involved in managing investment accounts and share portfolios. The formulas and relationships involved with simple and compound interest are also analysed. In this chapter you will learn how to: calculate simple and compound interest using appropriate formulas construct and examine tables and graphs involving simple and compound interest calculate fees and charges associated with savings and investment accounts understand and use the language of the stock market calculate the costs involved in buying and selling shares calculate dividends and dividend yield from shares read and interpret tables and graphs depicting share prices calculate the prices and values of items following inflation and appreciation. SAVINGS AND INVESTMENTS 369

2 SIMPLE INTEREST Interest is money earned from an investment with a bank, credit union or other financial institution. The original amount of money invested is called the principal, and simple interest occurs when the interest is calculated as a percentage of this principal. Simple interest is found by multiplying the principal by the percentage interest rate, then multiplying this result by the term of the investment (the number of months or years): Simple interest = principal interest rate term So simple interest is directly proportional to the principal, unlike compound interest, which varies as the value of the investment increases. Another name for simple interest is flat rate interest. Simple interest is represented algebraically by the following formula: I = Prn where P is the principal (initial value), r is the interest rate per period expressed as a decimal, and n is the number of periods in the term. Example 1 A sum of $ is invested at 10.25% p.a. for 8 months. Calculate the simple interest earned. Because the interest rate r is expressed per year, the number of periods n must also be expressed in years. 8 P = $14 510, r = 10.25% = , n = years 12 I = Prn 8 = $ = $ $ When calculating earned interest to the nearest cent, round down. Financial institutions cannot round up, otherwise they will be paying each investor an extra part of a cent. Example 2 1 Calculate the simple interest earned on $1840 invested at 1.06% per month for 1 -- years. P = $1840, r = 1.06% = , n = = 18 months I = Prn = $ = $ $ n is expressed in months because the interest rate is expressed per month NEW CENTURY MATHS GENERAL: PRELIMINARY

3 Example 3 1 What was the interest rate per annum if $2800 invested for 4 -- years earned $ ? 2 1 P = $2800, n = 4 -- years, I = $ I = Prn 1 $ = $2800 r = $12 600r $ r = $ = The interest rate was 8.75% p.a. Alternative method 1 Interest for 4 -- years = $ $ Interest for 1 year = = $245 $245 Interest rate p.a. = % $2800 = 8.75% Exercise 10-01: Simple interest 1. Calculate the simple interest earned from the following investments. (a) $7400 at 5% p.a. for 4 years 1 (b) $2136 at 6% p.a. for 2 -- years 2 (c) $ at 7.65% p.a. for 3 years (d) $4500 at 8.2% p.a. for 8 months (e) $ at 0.61% per month for 6 months 1 (f) $5946 at 0.32% per month for 4 -- years 2 (g) $4510 at % per day for 31 days (h) $9500 at % per day for 74 days 1 (i) $ at 7 -- % p.a. for 5 months 2 (j) $8250 at 10.5% p.a. for 240 days 2. Kalena earned $ in simple interest from investing an amount for 3 years at 5.4% p.a. What was the amount? 3. Alan earned $ simple interest from an investment of $9835 over 5 years. What was the interest rate per annum? 4. For how long must a principal of $ be invested at 9.8% p.a. for it to earn $3087 in simple interest? 5. Zoran earned $675 interest from $7500 invested for 9 months. What was the interest rate per annum? 6. What principal would earn $ in interest if invested for 3 years at 16.8% p.a.? SAVINGS AND INVESTMENTS 371

4 7. OzExpress Credit Union has the following term deposit accounts, where the principal must be invested for a fixed period. Term Calculate the simple interest earned on the following investments. (a) $6300 for 3 years (b) $ for 8 months (c) $7800 for 5 years (d) $ for 2 months (e) $5000 for 4 years 8. Kylie earned $80.58 interest from an investment of $2530 over 91 days. What was the interest rate per day (correct to 2 significant figures)? 9. For how long would $4720 need to be invested at 0.67% per month to earn $ in simple interest? 10. An amount of $9020 was invested for 2 years and earned $ in simple interest. Calculate the monthly interest rate. 11. Terry earned $1980 in interest from an investment with a term of 4 years at 3% per halfyear. What was the value of his investment? 12. Maxine invested $5660 in an account and earned $ in interest after 15 months. What was the interest rate per quarter? Investigation: Interest rates Visit your local bank or credit union and collect information about the different types of saving accounts available, including the interest rates and conditions imposed on each. Examples: savings account, investment account, fixed term deposits, cheque account, Christmas club account, young saver account, deluxe account, incentive saver account. Current interest rates can also be found in the financial pages of daily newspapers or on the Internet. 1. In which type of account would you choose to save? 2. Compare and contrast the different types of account and compile your findings in a report. Interest rate 1 6 months 8.75% p.a months 9.25% p.a. 1 3 years 10.5% p.a. 4 5 years 12% p.a. Minimum investment $ NEW CENTURY MATHS GENERAL: PRELIMINARY

5 COMPOUND INTEREST In reality, calculating interest is not so simple and straightforward. Simple interest is used only when the interest earned is collected by the investor and not added to the investment, such as in a term deposit account. With most accounts, however, the balance plus the interest becomes the new balance on which the interest is calculated next time. In other words, the interest will increase because you also earn interest on your interest. This is called compound interest. Compound means to combine. Example 4 If a principal of $3000 is invested at 6% p.a. interest, compounded over 3 years, what is: (a) the value of the investment after 3 years? (b) the compound interest earned? (a) After the 1st year, Interest = $ Amount of investment = old principal + interest = $ ($ ) = $3000( ) = $3000(1.06) After the 2nd year, Amount of investment = $3000(1.06) + interest = $3000(1.06) 1.06 = $3000(1.06) 2 After the 3rd year, Amount of investment = $3000(1.06) 2 + interest = $3000(1.06) = $3000(1.06) 3 = $ $ Rounding down to the nearest cent Adding 6% interest to the principal is equivalent to increasing the principal by 6% Increasing $3000(1.06) by another 6% Increasing $3000(1.06) 2 by another 6% Rounding down to the nearest cent Notice from the pattern that the principal $3000 is being increased by the interest rate 0.06 successively 3 times, where 3 is the number of years. Generally, this can be summarised by the formula A = P(1 + r) n, where P is the principal, r is the decimal interest rate, n is the number of periods, and A is the final amount of the investment. (b) The compound interest earned after 3 years is simply found by subtracting the original principal from the final amount of the investment. Compound interest earned = $ $3000 = $ The compound interest formulas are A = P(1 + r) n and I = A P where A is the final amount, I is the compound interest, P is the principal, r is the interest rate per compounding period expressed as a decimal, and n is the number of compounding periods in the term. SAVINGS AND INVESTMENTS 373

6 Example 5 Calculate the final amount and compound interest earned when $3200 is invested at 8.5% p.a. for 4 years. P = $3200, r = 8.5% = 0.085, n = 4 A = P(1 + r) n = $3200( ) 4 = $3200(1.085) 4 = $ $ The final amount is $ I = A P = $ $3200 = $ The compound interest earned is $ Example 6 If $ is invested at 4.8% p.a. with the interest compounded monthly, calculate the final balance and total interest earned over 2 years. As interest is compounded monthly, r and n must be expressed in months: 4.8 P = $12 500, r = % = 0.4% = 0.004, n = 2 12 = Final balance = $12 500( ) 24 = $12 500(1.004) 24 = $ $ Compound interest = $ $ = $ Example 7 What principal must be invested at 4.5% p.a. for 8 years so that it grows to $10 000? A = $10 000, r = 4.5% = 0.045, n = 8 A = P(1 + r) n $ = P( ) 8 = P(1.045) 8 $ P = ( 1.045) 8 = $ $ A principal of $ is needed. Round principal up to give a required final amount. 374 NEW CENTURY MATHS GENERAL: PRELIMINARY

7 Technology: Interest on graphics calculators and spreadsheets Many graphics calculators, financial calculators and spreadsheet software have financial modes for calculating simple interest, compound interest and final values of investments. Graphics calculators have a TVM mode that stands for time-value-money, while spreadsheets have special financial functions. In the financial world, the compound interest formula A = P(1 + r) n is written FV = PV(1 + r) n where FV stands for future value and PV stands for present value (principal). Investigate the interest calculation functions on your graphics calculator or spreadsheet. Exercise 10-02: Compound interest 1. Calculate the final amount of each of the following investments and hence the compound interest earned. (a) $7400 at 5% p.a. for 4 years (b) $2840 at 6.5% p.a. for 5 years (c) $4500 at 4.9% p.a. for 2 years (d) $ at 0.5% per month for 10 months (e) $9250 at 0.82% per month for 6 months (f) $9000 at 8.4% p.a. for 8 months, compounded monthly (g) $ at 10.8% p.a. for 1 year, compounded monthly (h) $ at 7.5% p.a. for 2 years, compounded half-yearly (i) $ at 7.5% p.a. for 2 years, compounded quarterly (j) $6920 at 9% p.a. for 240 days, compounded daily 1 year = 365 days. 2. Paul wants to invest some money so that it will grow to $ in 5 years time, when he will travel through Europe. If the interest rate is 5.5% p.a., what amount should he invest, to the nearest cent? 3. A sum of $8500 is invested at 7% p.a. for 5 years. (a) Calculate the total interest earned if it is: (i) calculated at a flat rate (simple interest) (ii) compounded yearly (b) Which type of interest is greater: simple or compound? By how much? (c) Why is one type of interest greater than the other? 4. A principal of $ is invested at 5% p.a. Calculate the value of the investment after 2 years if the interest is compounded: (a) yearly (b) half-yearly (c) quarterly (d) monthly (e) daily 5. Judging by your results from question 4, what happens to the amount of interest earned as the frequency of compounding increases? Why? 6. Zara has $4000 in an account earning 4% p.a. interest, compounded yearly. By guessing and checking, determine how long it will take her to double her money. Answer to the nearest year. 7. How long will an investment of $2400 take to grow to $3265 at 8% p.a. interest compounded yearly (to the nearest year)? 8. After 8 years, the value of Corrina s investment grew to $ What was the initial amount of her investment if the interest rate was 8.75% p.a.? SAVINGS AND INVESTMENTS 375

8 9. Determine the sum to be deposited if $ is required in 6 years time and terms of 3.75% p.a. (compounded quarterly) are available. 10. A principal of $ is to be invested for 3 years. Determine which is the best investment option: A. 6% p.a. simple interest B. 5.9% p.a. compounded annually C. 5.85% p.a. compounded half-yearly D. 5% p.a. compounded monthly 11. An investment of $ earns 7.3% p.a. compounded half-yearly for 4 years. (a) Calculate the interest earned and the equivalent flat (simple) interest rate per annum. (b) Does this flat rate change if the principal is different? Spreadsheet activity: Simple vs compound interest 1. Compare the growth of an investment over 20 years under simple and compound interest. Create a spreadsheet that allows you to enter a principal and interest rate (% p.a.) and then calculates the interest and value of the investment at the end of each year, from the 1st to the 20th year. A B C D E 1 Principal $ Interest rate (% pa) SIMPLE INTEREST COMPOUND INTEREST 5 Year Interest Investment Interest Investment 6 1 $40 $1040 $40.00 $ $40 $1080 $41.60 $ $40 $1120 $43.26 $ $40 $1160 $44.99 $ : : : : : : : : : : : : $40 $1760 $81.03 $ $40 $1800 $84.27 $ Total $800 Total $ Draw graphs illustrating the growth of an investment under simple and compound interest, using graph paper, the spreadsheet s graphing function, a graphics calculator or computer graphing software. Modelling activity: Double your money 1. Use the spreadsheet above to determine how long it will take to double an investment of $8000 if invested at 7% p.a. compound interest. Answer to the nearest year. 2. Investigate whether this period changes if the principal changes. 3. Investigate whether this period is halved if the interest rate is doubled, and vice-versa. 4. Financial experts say that a law of seventies exists for finding when an original investment will be doubled. They claim that when the product of the percentage interest rate and the number of periods is around 70, the original investment will be doubled for example, when r = 14% and n = 5. Investigate this claim. 376 NEW CENTURY MATHS GENERAL: PRELIMINARY

9 INTEREST TABLES AND GRAPHS Example 8 Simple interest Rachel invests $2000 in an account that earns 6% p.a. simple interest. Construct a graph that shows the simple interest I earned in dollars over n years, for values of n from 0 to 8. (a) What does the gradient of this graph represent? 1 (b) Use the graph to estimate the simple interest earned after 4 -- years. 2 I = Prn = n = 120n I = 120n is a linear function. Its graph is a straight line of the form y = mx + b with gradient 120 and vertical intercept 0. This line passes through the origin, so simple interest is an example of direct linear variation; that is, the interest I is directly proportional to the number of years n. To help us graph the line, we can complete a small table of values for I = 120n. n (years) I ($) Simple interest earned from an investment of $2000 at 6% p.a (8, 960) Interest, I ($) (2, 240) (b) Term, n (years) (a) The gradient, 120, is the interest earned per year ($120). 1 (b) Reading from the graph, when n = 4 --, I = $ SAVINGS AND INVESTMENTS 377

10 Example 9 Compound interest Ross invests $2000 in an account that earns 8% p.a. compound interest. Construct a graph that shows the amount A in dollars of the investment, over n years for values of n from 0 to 10. A = P(1 + r) n = 2000( ) n = 2000(1.08) n This is not a linear function because n appears in the power or index of the formula. In fact, this is called an exponential function, and the exponential graph is an increasing curve. We can use a table of values where A is expressed to the nearest whole dollar. n (years) A ($) Amount of an investment of $2000 at 8% p.a. compound interest Amount, A ($) Term, n (years) Note that the curve grows steeper than a straight line graph, illustrating that an investment grows faster with compound interest than with simple interest NEW CENTURY MATHS GENERAL: PRELIMINARY

11 Example 10 Financial tables Before calculators became widely available, accountants used financial tables like the one below to calculate compound interest. This table is also useful when your calculator does not have a power x y key. It lists the value of (1 + r) n to 3 decimal places for different values of r and n, so that you can calculate the final amount of an investment (per dollar) undergoing compound interest. No. of periods (n) COMPOUND INTEREST TABLE (1 + r) n FINAL AMOUNT OF INVESTMENT (PER DOLLAR) Interest rate per compounding period (r) % % % % % Use the table to calculate the final amount of: (a) $2000 invested at 8% p.a. for 5 years (b) $ invested at 15% per month for 8 months (c) $4780 invested at 12% p.a. for 10 months, compounded monthly (a) A = $2000(1.08) 5 = $ = $2938 From the table: r = 0.08, n = 5 (b) A = $11 000(1.15) 8 From the table: r = 0.15, n = 8 = $ = $ (c) A = $4780(1.01) % p.a. = = 1% per month 12 = $ From the table: r = 0.01, n = 10 = $ % SAVINGS AND INVESTMENTS 379

12 Technology: Interest tables and graphs Equipment: Computer and spreadsheet software, graphics calculator or graphing software 1. Create a spreadsheet that reproduces the compound interest table on page 379 (Example 10), or modify it so that it calculates values of (1 + r) n for any value of r that you enter. 2. Use a graphics calculator, spreadsheet or graphing software to construct graphs that show simple interest or the growth of an investment under simple and compound interest. 3. Use a graphics calculator, spreadsheet or graphing software to graph on the same axes the value of an investment of $5000 over 10 years at 9% p.a., compounded: (a) yearly (b) half-yearly (c) monthly Exercise 10-03: Interest tables and graphs 1. Use the simple interest graph on page 377 (Example 8) to estimate the interest earned when $2000 are invested at 6% p.a. for: (a) 9 years (b) 2.5 years (c) 6.5 years (d) 4.5 years 2. Graph on the same axes the simple interest I earned when $4000 are invested in an account earning: (a) 4% p.a. (b) 12% p.a. over n years, for values of n from 0 to (a) Use the graph from question 2 to estimate the simple interest earned from an investment of $4000 over 6 years: (i) at 4% p.a. (ii) at 12% p.a. (b) How do the two different interest rates appear on the graphs? 4. Use the compound interest graph on page 378 (Example 9) to estimate the final value of an investment of $2000 earning 8% p.a. interest after: (a) 9 years (b) 2.5 years (c) 6.5 years (d) 4.5 years 5. Use the compound interest graph on page 378 (Example 9) to estimate when the investment will reach the following amounts. Answer to the nearest year. (a) $2500 (b) $ Graph on the same axes the amount A of a $5000 investment earning compound interest at a rate of: (a) 4% p.a. (b) 12% p.a. over n years, for values of n from 0 to (a) Use the graph from question 6 to estimate the value of an investment of $5000 after 7 years: (i) at 4% p.a. (ii) at 12% p.a. (b) How are the two different interest rates indicated on the graphs? 380 NEW CENTURY MATHS GENERAL: PRELIMINARY

13 8. Use the compound interest table on page 379 (Example 10) to calculate the final value of the following investments. Principal Interest rate Period (a) $ % p.a. 4 years (b) $ % p.a. 8 years (c) $1 15% p.a. 10 years (d) $ % p.a. 2 years (e) $ % p.a. 6 years (f) $ % per month 3 months (g) $ % per month 6 months (h) $ % per month 7 months (i) $ % p.a. compounded monthly 8 months (j) $ % p.a. compounded half-yearly 2 years 9. Use the same compound interest table to find how long will it take a principal to double its value if it is invested at 15% p.a. compound interest (to the nearest year)? Spreadsheet activity: Compounding periods 1. Create a spreadsheet that will allow you to enter a principal, an interest rate in percentage per annum, and a term in years, then calculate: (a) the total simple interest earned (b) the total compound interest earned if the interest is compounded: (i) yearly (ii) half-yearly (iii) quarterly (iv) monthly (v) daily taking 1 year as days. A B C D E 1 Principal $ Interest rate (% p.a.) 14 3 Term (years) Interest rate (% per period) Number of periods Final amount Total interest 6 Simple interest 14 5 $ $ Compound interest 8 compounded yearly 14 5 $ $ compounded half-yearly 7 10 $ $ compounded quarterly $ $ compounded monthly $ $ compounded daily $ $ Banks once calculated interest on savings accounts monthly but computer technology now allows them to calculate this interest daily. Is this better for the investor? SAVINGS AND INVESTMENTS 381

14 ACCOUNT FEES AND CHARGES Banks and credit unions impose fees and charges on their savings accounts, for situations such as: excessive withdrawals made in a month accounts with low balances (e.g. less than $500) ATM withdrawals made at another bank s ATM dishonoured cheques: bad cheques deposited or withdrawn when there are insufficient funds in the account overdrawn accounts: accounts where more money has been withdrawn than there were funds in the account. Example 11 OziBank s CashCard savings account has the following associated fees and charges: First 8 withdrawals per month (not including withdrawals at another bank s ATM, direct debit or transfers to another bank account: see below)* Free Each subsequent withdrawal ATM/EFTPOS/cheque withdrawals $0.50 Phone/Internet banking (for paying bills and transfers) $0.20 Withdrawal at bank $2.00 * Withdrawal at another bank s ATM $1.80 * Direct debit Free Monthly account-keeping fee (for balances below $500) $5.00 Overdrawn account $20.00 Cheque/direct-debit dishonour fee $25.00 Card replacement $5.00 New chequebook $10.00 * Transfers to another bank account (made at bank) $5.00 Yearly dormancy fee (if no transactions made in a year) $10.00 There is also a Government Debits Tax (GDT) on cheque withdrawals, at the following rates: Debit amount Less than $1.00 GDT nil $1 to $99.99 $0.30 $100 to $ $0.70 $500 to $ $1.50 $5000 to $ $3.00 $ and over $ NEW CENTURY MATHS GENERAL: PRELIMINARY

15 Manuel has $ in his CashCard account. (a) Does he need to pay the monthly account-keeping fee? (b) During the month, he makes 10 withdrawals from his account, the last two being EFTPOS and a cash withdrawal at the bank. Calculate the charges incurred. (c) Manuel also writes a cheque for $ What amount of GDT must he pay? (d) His deposits in the month total $ and his withdrawals total $ If Manuel also buys a new chequebook, calculate his final balance after all charges have been applied. (a) No, because his account balance is above $500. (b) EFTPOS charge = $0.50, Bank withdrawal = $2.00 Total charges = $ $2.00 = $2.50 (c) For a $ cheque, GDT = $0.70. (d) Old balance = $ Total deposits = $ Total withdrawals = $ Total charges = $ $ $10.00 = $13.20 Final balance = $ $ $ $13.20 = $ Exercise 10-04: Account fees and charges Questions 1 8 refer to OziBank s CashCard savings account on page 382 (Example 11). 1. Jeremy has already made 8 withdrawals from his CashCard account this month. What will it cost him now to withdraw some money from an ATM? 2. (a) After 8 withdrawals have been made from a CashCard account, which method of withdrawing money incurs the lowest charge? Why do you think? (b) Which method of withdrawing money incurs the highest charge? Why do you think? 3. Jess has $ in her account and has lost her card. (a) What is the cost of acquiring a new card? (b) Does she have to pay the monthly account-keeping fee? Give a reason for your answer. (c) During the month, she makes 11 withdrawals from her account, all at her bank s ATM. How much will she be charged for this? (d) She then writes a cheque for $ Calculate the charge and the GDT for this transaction. (e) If Jess s deposits for the month total $ and her withdrawals (including the cheque) total $541.22, calculate her final balance after all fees have been applied. 4. OziBank imposes a fee if no transactions are made on the CashCard account during the year. (a) What is this fee called and how much is it? (b) Why do you think a bank would impose such a fee? 5. Claire wrote a cheque for $340 when she had only $ in her account. (a) What type of fee will she be charged? (b) How much is this? SAVINGS AND INVESTMENTS 383

16 6. Samir had $8061 in his account and made 9 withdrawals, the last one being the payment of his electricity bill by Internet banking. He then went to his bank branch to transfer $2400 to an account at another bank, and then made a withdrawal of $120 from another bank s ATM. All of this happened in the same month. (a) Calculate all of the charges incurred on the account during the month. (b) If Samir made no deposits and his first 9 withdrawals totalled $831.90, calculate his final balance after all withdrawals and charges had been deducted. 7. Why do you think Ozibank imposes: (a) a charge for more than 8 withdrawals made in the month? (b) a monthly account-keeping fee for balances below $500? (c) a cheque/direct-debit dishonour fee? 8. Ozibank published a brochure for its customers listing ways of minimising account charges. Explain how the following suggestions will reduce account charges. (a) If you shop by EFTPOS, ask the retailer if you can also withdraw extra cash. (b) If you currently withdraw small amounts nearly every day, try withdrawing larger amounts once or twice a week instead. (c) Pay by credit or direct debit rather than by ATM, EFTPOS or cheque. (d) Combine all of your separate accounts into one account. (e) Bank via the telephone or the Internet. 9. Why do you think banks prefer you to make a withdrawal at one of their ATMs rather than: (a) inside one of their branches? (b) at another bank s ATM? Investigation: Account-keeping fees 1. Obtain a brochure from one or more of the banks or credit unions in your local area and compare the fees and charges of the different types of savings accounts available. Examine a monthly statement of your own account (or your family s) and see how and when these fees and charges are applied. 2. Prepare a report comparing the main fees and charges of two different savings accounts, either within the same bank or between two banks or credit unions. Study tips MORE EXAM TIPS Read all of the instructions of the exam carefully. Don t rush. Don t panic. Aim to work steadily. Plan your time well and keep an eye on it. When you have finished a question, make sure you have actually answered it. Do you need to write the answer in a sentence? Put a circle or box around the answer to highlight it. Make sure that your answer sounds reasonable, especially if it involves money or measurement. Did you round off correctly and include the correct units? If your working-out is taking too long, stop and think: Am I on the wrong track? Don t get bogged down. You may need to retrace your steps, start again or come back later. Once you have completed the exam, go over it again. Double-check your answers, especially the more difficult ones or those of which you are unsure. 384 NEW CENTURY MATHS GENERAL: PRELIMINARY

17 INVESTING IN SHARES As well as saving at a bank or credit union, an investor may buy stocks in an Australian company such as BHP, Woolworths or Qantas. This means that they own part of the company, with their investment being used with those of other shareholders to run the business. The company s annual profit is divided between all shareholders, and each share of the profit is called a dividend. Dividends are paid once or twice per year. The price of a share changes continually during each weekday, influenced by demand and supply, the Australian economy and market confidence in the company. A shareholder may profit by buying shares at a low price and selling them at a higher price. The original price of a share, usually 50c, $1 or $2, is called its face value while the current price is called its market value or market price. Investing in shares is riskier than investing in a bank, but the dividends earned can be far greater. There are two types of shares: ordinary shares and preference shares. Preference shareholders receive their dividends first, at a fixed predetermined rate (like bank interest), while the dividend from an ordinary share is variable. The dividend yield (or annual yield) is the dividend expressed as a percentage of the share s market value. Dividend yield = dividend per share 100% market price of share Agents who buy and sell shares are called stockbrokers. A stockbroker s commission, usually around 2%, is called brokerage. Most investors own shares in more than one company. A collection of shares from different companies is called a portfolio. Note: Stocks and shares mean the same thing. You can buy and sell stock in a company, and one unit of stock is called a share. Example 12 Joanne owns 1000 $2 ordinary shares and 700 $1 preference shares in Coles Myer. The current prices of the ordinary and preference shares are $7.87 and $5.40 respectively. (a) What is the market value of Joanne s shares? (b) If the dividend on the ordinary shares is 28c per share and the dividend on the preference shares is 6% of face value, calculate Joanne s total dividends. (c) Calculate the dividend yield of Joanne s ordinary shares correct to 1 decimal place. (a) Market value = 1000 $ $5.40 = $ (b) Dividends on ordinary shares = 1000 $0.28 = $280 Dividends on preference shares = 6% 700 $1 = $42 Total dividends = $280 + $42 = $322 SAVINGS AND INVESTMENTS 385

18 dividend per share (c) Dividend yield = % market price of share $0.28 = % $7.87 = % 3.6% Example 13 Anh bought 1000 Telstra shares at $5.80 each, with a dividend yield of 5.6%. Brokerage costs were 2.5% of the purchase price, while stamp duty (state tax) was 15c per $100 or part thereof. (a) Calculate the total cost of purchasing the shares. (b) One year later, Anh sold all of his shares at $6.50 each. Calculate his total earnings for the year, after costs. (a) Cost of shares = 1000 $5.80 = $5800 Brokerage = 2.5% $5800 = $145 Stamp duty = $ c per $100 = $8.70 Total cost = $ $145 + $8.70 = $ (b) Dividend = 5.6% $5800 = $ Selling price = 1000 $6.50 = $6500 Earnings = selling price + dividend total costs = $ $ $ = $ Investigation: Share prices and the Australian Stock Exchange Information about share prices is supplied by the Australian Stock Exchange (ASX) and can be found in the Finance section of daily newspapers or on the Internet. Over 1200 Australian companies are listed on the ASX, in industries such as mining and oil (e.g. Rio Tinto), banking (St George), media (Prime TV), transport (TNT), food (Coca-Cola Amatil) and retail (Harvey Norman). 1. Visit the Australian Stock Exchange s website and play the share market game. It asks you to choose how many shares you wish to purchase in six companies, then calculates the value of your investment after a given number of years, using real prices. 2. Read the Finance section of a newspaper and interpret the information about shares presented in a table. 3. Write the names of eight companies that are listed on the Australian Stock Exchange and list their market prices and dividend yields. 386 NEW CENTURY MATHS GENERAL: PRELIMINARY

19 Exercise 10-05: Investing in shares 1. Adrian wants to purchase 2500 shares in C & W Optus. The market price of the shares is $3.54. (a) Calculate the total cost of the shares. (b) The stockbroker charges a basic order fee of $20 plus a commission of 2% of the cost of the shares. Calculate the total brokerage. (c) Calculate the stamp duty if it is 60c per $100 or part thereof. (d) A dividend of 47c was paid when the market price of the share was $2.62. Calculate the dividend yield correct to 2 decimal places. 2. Terry bought 4000 $2 preference shares for $8920. They have a 12% dividend of face value. Calculate: (a) the market price of 1 share (b) the total dividend earned (c) the dividend yield correct to 2 decimal places 3. A stockbroker charges the following rates of brokerage: 2.5% of share value up to $5000 2% of the next $ % of the next $ % of the remainder Colin buys 800 shares in David Jones at $1.49 each. Calculate: (a) the cost of the shares, including the brokerage (b) the stamp duty if it is 30c per $100 or part thereof (c) his total dividend and dividend yield to 2 decimal places, given that the dividend is 11c per share when the market price is $ Eliza bought 200 $2 shares in Fairfax and paid $780 for them. Calculate: (a) the market price of 1 share (b) the dividend per share (to the nearest cent) if the dividend yield is 6.2% 5. Zeli bought shares in Western Metals at $0.58 each. Her stockbroker charged 2.5% brokerage for the first $ of shares and 0.75% thereafter. Stamp duty was 30c per $100 or part thereof. (a) Calculate Zeli s total cost of purchasing the shares. (b) If Zeli collects a dividend yield of 4.8% and then she sells all her shares at $0.74 each, calculate her total earnings after costs. 6. Vicky owns 4200 $1 ordinary shares and 3000 $2 preference shares in the Ten Network. (a) Calculate the total face value of her shares. (b) The shares cost $2.70 per ordinary share and $4.33 per preference share, and the additional costs were: Broker s service fee: $5.50 Brokerage: 2.5% of share value up to $5000 2% of the next $ % of the next $ % of the remainder. Calculate the total cost of buying the shares. (c) If the company pays a dividend of 28c per ordinary share and 7% of face value per preference share, determine Vicki s total dividend. (d) Calculate the dividend yield of an ordinary share correct to 2 decimal places. SAVINGS AND INVESTMENTS 387

20 7. Village Roadshow shares have a face value of $2, a market value of $3.40 and pay a dividend of 12.5% of face value. (a) Adrian owns 300 shares. What is their total market value? (b) What is the total dividend earned? (c) Calculate the dividend yield correct to 1 decimal place. 8. Karl manages the following share portfolio. No. of shares Company Market value Dividend yield 1000 Pacific Mining $ % 600 Air New Zealand $2.49 6% 800 TAB $ % 1700 Cadbury $ % (a) Calculate the total value of the shares. (b) Calculate the total dividends earned. 9. George owns 2500 AGL shares. The dividend per share is 42c and the market price is $8.37. (a) What dividend will George receive on his shares? (b) Calculate the dividend yield correct to 1 decimal place. (c) How many extra shares could George purchase if he reinvested his dividend? 10. Caltex 50c shares have a market price of $2.60. If the dividend is 35% of face value, calculate the dividend yield correct to 2 decimal places. 11. Paula bought c ordinary shares and 2500 $2 preference shares in Qantas. (a) Calculate the total face value of her shares. (b) If the company pays a dividend of 14c per ordinary share and 7.3% per preference share, determine Paula s total dividend. (c) If the market prices of the ordinary and preference shares are $2.80 and $4.93 respectively, calculate correct to 1 decimal place: (i) the value of Paula s share portfolio (ii) the dividend yield on Paula s ordinary shares 12. Sandy bought 3500 shares in Fosters Brewing at $6.49 with a dividend yield of 7%. Brokerage was 2.5% of the first $5000, 2% of the next $10 000, 1.5% of the next $ and 1% thereafter. Stamp duty was 30c per $100 or part thereof. (a) Calculate the total cost of purchasing the shares. (b) If Sandy sells her shares 1 year later at $7.14 each, find her total profit after costs. Investigation: The language of the share market Like any specialised field, the share market has its own jargon or specific terminology. You may have seen or heard the following words used in the financial section of a newspaper or on TV/radio news programs. Find out what they mean: Bear market bid blue-chip stocks Bull market capital gains tax CHESS dividend reinvestment float par value P/E ratio prospectus speculation 388 NEW CENTURY MATHS GENERAL: PRELIMINARY

21 Just for the record ALL ORDINARIES INDEX One statistic that summarises the state of the Australian stock market is a figure called the All Ordinaries Index, sometimes abbreviated All Ords. Like share prices, this is a value that changes continually each weekday. It is calculated from the prices of 500 different shares representing 90% of share transactions. It is called an all ordinaries index because it is based on the prices of ordinary shares. The All Ordinaries Index was created in January 1980, beginning with a base value of 500. In 1999, its value was around the mark. Every day, the index rises or falls by a number of points, reflecting the movements of share prices, and the final closing value is reported in newspapers, on the nightly TV news and on the Internet. 1. Find the value of today s All Ordinaries Index. 2. Other countries have different indices for measuring the state of their stock markets. Which stock exchanges are associated with these indices: the Dow Jones, the Nikkei, the FTSE (also called FT100) and the Hang Seng? SHARE TABLES AND GRAPHS Daily information about share prices can be found in the Finance section of a newspaper or on the Internet, listed in a table like this. High 52 week Low Company Last sale Move Buy Sell Div. cents Div. yield % AGL Angus and Coote Just Jeans Qantas Seven Network Sydney Gas TAB Westpac Key 52 week high/low: the maximum and minimum prices of the share in the last 52-week period (in dollars) Last sale: the market price of the share (in dollars) Move: the change in cents of the market price compared to yesterday s price Buy: the highest buying price offered during the day Sell: the lowest selling price offered during the day Div. cents: the dividend per share in cents Div. yield %: the dividend yield SAVINGS AND INVESTMENTS 389

22 Example 14 From the share table on page 389: (a) What is the market price of Qantas shares? (b) What was their price yesterday? (c) What was the lowest price of AGL shares over the past 52 weeks? (d) Which share paid a dividend of 20.5 cents? (e) What was the lowest selling price of Just Jeans shares during the day? (f) What was the dividend yield on TAB shares? (g) If the TAB dividend was paid today, what would it be (to the nearest 0.1c)? (a) $4.93 (b) Qantas rose 19c today, so its share price yesterday was $4.93 $0.19 = $4.74. (c) $8.53 (d) Seven Network (e) $1.28 (f) 3.38% (g) 3.38% $2.66 = $ c Example 15 This graph shows the performance market price of Orica shares over 6 months. (a) What was its highest price during this period? (b) How many times did the price reach $8.40? (c) Describe the performance of the share over the period. (d) Predict what will happen to the share price after 15 October. Orica SHARE PRICE $ Apr 15 Oct Source: Bloomberg, Sydney Morning Herald, 16 October (a) Approximately $9.50. (b) 6 times. (c) The price shot up rapidly around May, then gradually declined while still fluctuating daily, then started gradually increasing around 15 October. (d) It looks like the price may gradually increase again. Technology: Graphs of share performances 1. Use a graphics calculator, spreadsheet or graphing software to input and graph the market price of a particular share over a period. Use published share data to create a table of the company s share price over the period first. 2. Graph the changes in the All Ordinaries Index over the same period and compare its performance with that of your share. 390 NEW CENTURY MATHS GENERAL: PRELIMINARY

23 Exercise 10-06: Share tables and graphs Questions 1 8 refer to the share table on page For Angus and Coote shares: (a) What was the dividend yield? (b) What is the market price? (c) There were no sales yesterday. How is this shown in the table? (d) What was the lowest price in the past 52 weeks? (e) What was the lowest buying price in the past 24 hours? (f) What was the dividend? 2. (a) Which share showed the smallest change in price from yesterday? (b) What was this change? 3. (a) Which share showed the greatest change in price from yesterday? (b) What was this change? 4. (a) Which share has the highest market price? (b) Which share has the lowest market price? 5. Which share(s) paid a dividend of: (a) 45c? (b) 10c? (c) 9c? 6. Which share had the lowest market price in the past 52 weeks? 7. (a) What is Sydney Gas s current share price? (b) What was its price yesterday? (c) What would be the dividend per share if it was paid today? 8. (a) Which share has a market price of $9.548? (b) Which share had a high of $12.11 in the past 52 weeks? (c) Which had the largest drop in price from yesterday? (d) Which had a dividend yield of 7.81%? (e) Which had a highest selling price of $8.70 during the day? 9. For the graph of Orica share prices on page 390 (Example 15), find: (a) the market price on 15 April (b) the lowest price over the period (c) the number of months covered by the period (d) the number of times the price reached $9.00 (e) the price on 15 October (f) an estimate for the price on 15 November National 10. This graph shows the performance of National Mutual Mutual shares over 6 months. (a) What was the highest price over this period? (b) What was the lowest price? (c) How many times did the price reach $2.60? (d) Estimate the price at 22 July. (e) Describe the performance of the share in the first 3 months. (f) Describe the performance of the share in the 22 Apr 22 Oct last 3 months. Source: Bloomberg, Sydney Morning Herald, 23 October SHARE PRICE $ SAVINGS AND INVESTMENTS 391

24 (g) What do you think may happen to the share price after 22 October? (h) Would after 22 October be a good time to sell? Give reasons for your answer. 11. This graph shows the performance of Flight Centre shares over 6 months. (a) How is the performance of Flight Centre shares different from the performances of Orica and National Mutual shares (pages 390 and 391)? (b) Describe the performance of Flight Centre shares over the period. (c) Estimate the share price in: (i) April (ii) July (iii) October (d) When did the price of the share reach the $8 mark? (e) When did the share price rise the quickest? (f) Would October be a good time to buy or sell Flight Centre shares? Why? Modelling activity: Investing in the stock market Flight Centre SHARE PRICE $ Apr 99 Oct Source: Bloomberg, Sydney Morning Herald, 12 October In groups of three to five, pretend that you are investing in the stock market. Given a spending account of $10 000, create a share portfolio and purchase shares in up to eight different companies. For every week over a term of 10 weeks, keep track of the prices of your shares. You can buy and sell your shares at any time. Also take note of the All Ordinaries Index. At the end of the term, sell all of your shares at the buyer s price. 2. Calculate your total dividends and the total profit or loss on your investment. 3. Construct graphs showing the performances of one of your shares and the All Ordinaries Index. Compare the two graphs. INFLATION AND APPRECIATION Inflation The prices of goods and services rise almost every year, and this is usually accompanied by increases in workers wages and salaries. This is called inflation, and the percentage by which prices increase each year is called the annual inflation rate. Recently, the inflation rate in Australia has remained fairly low at 0 to 5%, but it has been as high as 13% in the past. Calculating price rises after inflation is an application of increasing an amount by a percentage; but if the inflation rate remains constant, then we can use the compound interest final amount formula. A = P(1 + r) n where A is the final price, P is the initial price, r is the annual inflation rate expressed as a decimal, and n is the number of years. 392 NEW CENTURY MATHS GENERAL: PRELIMINARY

25 Example 16 A particular model of car costs $ today. Calculate the price of a similar model of car in 5 years time (to the nearest $100) if the inflation rate remains constant at 3% p.a. P = $28 500, r = 0.03, n = 5 Price in 5 years, A = $28 500( ) 5 = $28 500(1.03) 5 = $ $ Example 17 The current price of a magazine is $5.50. Calculate its price 10 years ago (to the nearest 10c) if the inflation rate during this time was 1.4% p.a. A = $5.50, r = 0.014, n = 10 Current price, $5.50 = P( ) 10 = P(1.014) 10 $5.50 Price 10 years ago, P = ( 1.014) 10 = $ $4.80 Appreciation Most items decrease in value over time as they become old and out-of-date. This is called depreciation. However, some items increase in value over time, such as jewellery, gold, antiques, prestige cars, art, stamp collections, land and houses. They become more valuable as time passes because they become more rare or scarce. This is called appreciation. Some people like to invest their money by buying and selling such items. Calculating the value of an item after appreciation is another application of percentage increase and the compound interest formula (where r is the annual rate of appreciation). Example 18 A block of land with a value of $ appreciates at a rate of 6.5% p.a. Calculate its value after 3 years. P = $85 000, r = 0.065, n = 3 Value after 3 years, A = $85 000( ) 3 = $85 000(1.065) 3 = $ $ SAVINGS AND INVESTMENTS 393

26 Just for the record CONSUMER PRICE INDEX (CPI) While the All Ordinaries Index summarises the prices of Australian shares, the Consumer Price Index (CPI) summarises the prices of Australian goods and services. It is a value that is calculated quarterly by the Australian Bureau of Statistics, based on the prices of food, clothes, houses, household equipment, transport, health care, recreational and educational items, tobacco and alcohol. Like the All Ordinaries Index, the CPI means nothing on its own. It is used for comparing prices and the cost of living between two different times, and noting changes in them over the period. The CPI was last set at a base value of 100 in the financial year and during the 1990s it rose to the 120 mark. The annual rate of inflation is calculated from the CPI of consecutive years, using the formula new CPI old CPI Inflation rate = % old CPI 1. Find the current CPI by visiting the Australian Bureau of Statistics Internet website 2. Calculate this year s inflation rate using this year s and last year s CPIs. Exercise 10-07: Inflation and appreciation 1. A loaf of bread currently costs $2.00. Calculate its price in 5 years if the inflation rate is 4.7% p.a. 2. A 15-day holiday package costs $2300. Calculate its cost in 2 years time if the inflation rate is 6.1% p.a. 3. A textbook today costs $ What was its price 8 years ago if the inflation rate over this period was 4% p.a.? 4. If gold appreciates at 9.3% p.a. and its current value is $459 per ounce, calculate its value in 5 years time. 5. A collection of Beatles memorabilia appreciates at 4.8% p.a. and is currently valued at $ What will be its value in 9 years? 6. Name two or three items that appreciate in value over time. 7. A house increased in value this year from $ to $ What was the rate of appreciation? 8. An accounting executive s salary increases with the annual inflation rate of 2.1%. Calculate her salary in 3 years if it is currently $ A pair of jeans costs $45 today. What was its value 10 years ago if the inflation rate was 3.5% p.a.? 10. How much will $50 be worth by today s standards in 5 years time if the inflation rate is 2.4% p.a.? Hint: Final value = $ Calculate the cost of a $649 refrigerator in 6 years if the inflation rate is 4% p.a. 394 NEW CENTURY MATHS GENERAL: PRELIMINARY