6.7 Investments

A group of stock traders watch a group of individuals ring the closing bell at the New York Stock Exchange. — **Figure 6.13** Stocks are bought and sold to improve investment values.Stocks are bought and sold to improve investment values. (credit: modification of work "FT ringing the Closing Bell at the NYSE" by Financial Times/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

Distinguish between basic forms of investments including stocks, bonds, and mutual funds.
Understand what bonds are and how bond investments work.
Understand how stocks are purchased and gain or lose value.
Read and derive information from a stock table.
Define a mutual fund and how to invest.
Compute return on investment for basic forms of investments.
Compute future value of investments.
Compute payment to reach a financial goal.
Identify and distinguish between retirement savings accounts.

You can save your money in a safe or a vault (or worse, under the mattress!), but that money does not grow. It would be hard to save enough for retirement that way. What can be done to increase the value of the money you already have?

The answer is to invest it. Use the money that you have to earn more money back. For instance, as we saw in Methods of Savings, you can save it in a bank. Or, to reach loftier goals, invest in something more likely to grow, such as stocks.

A great example of this is Apple stock. Anyone who bought stock in Apple Inc. (formerly Apple Computer, Inc.) in 1997 and held onto the shares earned a lot of money. To be more specific, $100 worth of Apple shares bought in 1980, when it was first sold to the public, was valued at $67,564 in 2019, or 676 times more! Perhaps you have heard a story like that, of an investment opportunity taken that paid off, or the story of an investment opportunity missed. But such stories are the exceptions.

In this section, we’ll investigate bonds, stocks, and mutual funds and their comparative strengths and weaknesses. We close the section with a discussion of retirement savings accounts.

Distinguish Between Basic Forms of Investments

Bonds, stocks, and mutual funds tend to offer higher returns, but to varying degrees, come with higher risks. Stocks and mutual funds also vary in how much they earn. Their predicted rates of return on investment are not guaranteed, but educated guesses based on market trends and historical performance.

We will use the methods and formulas we learned earlier to evaluate these forms of investment.

Bonds

Bonds are issued from big companies and from governments. Selling bonds is an alternative to an institution taking a loan from a bank. The funds from the selling of bonds are often used for large projects, like funding the building of a new highway or hospital.

Bonds are considered a conservative investment. They are bought for what is known as the issue price. The interest is fixed (does not change) at the time of purchase and is based on the issue price of the bond. The interest rate is often referred to as the coupon rate; the interest paid is often called the coupon yield. The interest paid is often higher than savings accounts and the risk is exceptionally low. The bond is for a fixed length of time. The end of this time is the maturity date of the bond.

There are several types of bonds:

Treasury bonds are issued by the federal government.
Municipal bonds are issued by state and local governments.
Corporate bonds are issued by major corporations.

There are other types of bonds available, but they are beyond the scope of this section.

Stocks

Stocks are part ownership in a company. They come in units called shares. The performance and earnings of stocks is not guaranteed, which makes them riskier than any other investment discussed earlier. However, they can offer higher return on investment than the other investments. Their value grows in two ways. They offer dividends, which is a portion of the profit made by the company. And the price per share can increase based on how others see that value of the company changing. If the value of the company drops, or the company folds, the money invested in the stock also drops.

Most stock transactions are executed through a broker. Brokers’ commissions can be a percentage of value of the trades made or a flat fee. There are full-service brokers who charge higher commission rates, but they also offer financial advice and perform the research that you may not have the time or the expertise to do on your own. A discount broker only executes the stock transactions, buying or selling, so they charge lower rates than full-service brokers. There are also brokers that offer commission-free trading.

An important thing to remember is that stocks might provide a very large return on investment, but the trade-off is the risk associated with owning stocks.

Example 2 Buying Stock in Company ABC

Haniah buys stock in the ABC company, investing a total of $13,000. She expects the stock to grow, through stock price increase and reinvestment of dividends, by 12.3% per year and compounded annually. If she leaves that money invested, how much will the stocks be worth in 20 years?

Calculating this is a compound interest calculation, if Haniah’s assumption about the stock’s performance is correct. If so, then the principal is $13,000, the rate is 0.123, the number of compounding periods per year is 1, and the time is 20 years. Substituting into the compound interest formula from Methods of Savings, and computing, we have $A = P {(1 + \frac{r}{n})}^{n t} = $ 13,000 {(1 + \frac{0.123}{1})}^{1 \times 20} = $ 13,000 \times 10.1764223996 = $ 132,293.49$ . After 20 years, her stock is now worth $132,293.49.

People in Mathematics

Warren Buffett

Warren Buffett is an investment legend. He began his career as an investment salesman in the 1950s. He formed Buffett associates in 1956. In 1965, he was in control of Berkshire Hathaway, which began as a merger between two textile companies. In his role there, he began to invest in a variety of companies. It is now a conglomerate holding company, and fully owns GEICO, Duracell, Diary Queen, and other large companies.

His investment philosophy involves finding stocks and bonds from companies that have high intrinsic worth compared to their stock or bond prices. This means he focuses not on the supply and demand side of stock investing, but instead on the company’s worth in total. Using this philosophy, he has become one of the world’s most successful investors.

Reading Stock Tables

Information about particular stocks is contained in stock tables. This information includes how much the stock is selling for, and its high and low values form the past year (52 weeks). In a newspaper, the stock table may look like this:

**Table 6.2**
52-Week High Low		Stock	SYM	Div	Yld %	P/E	Vol 100s	High	Low	Close	Net Chg
41.66	18.90	McDonald’s	MCD	.72	2.9	12	7588	25.73	23.87	25.42	+0.31
22.60	13.20	Monsanto	MON	.52	2.4	55	15474	21.86	21.48	21.64	-0.29
17.05	8.30	Motorola	MOT	.16	1.7	dd	16149	10.57	8.88	10.43	+0.14
31.75	22.99	Mueller	MLI	-	-	16	1564	29.32	27.03	27.11	-0.02

The symbols and abbreviations are defined here:

52-week High	52-week Low	The highest and lowest price of the stock over the past 52 weeks
Stock	SYM	The name of the company and the symbol used for trading
	Annual DIV	The current annual dividend per share
	Yld %	Percent yield is $= \frac{annual dividend}{share price} \times 100$
	P/E	Price to earnings ratio, share price divided by earnings per share over past year (dd indicates loss)
	Vol 100s	The number of shares traded yesterday in 100s
High	Low	The highest and lowest prices at which stocks traded yesterday
	Close	The price at which the stock traded at the close of the market yesterday
	Net Chg	Net change; change in price from market close 2 days ago to yesterday’s close

The formulas for yield and price to earnings is a good way to measure how much the stock returns per share. Their values are calculated in the stock table, but deserve attention here.

It should be noted that the price of a stock increases and decreases every moment, and so these value change as the share price changes.

The stock table information is now, and has been, available online, from websites such as cnn.com/markets, markets.businessinsider.com/stocks, and marketwatch.com. The same information is available from these sites as from the newspaper listings, but are often accessed one stock at a time. Figure 6.14 shows the stock table for Lowe’s on September 7, 2022.

A census graph. The x-axis ranges from 10 am to 3 pm in increments of 1 and the y-axis ranges from $192.5 to $202.5 in increments of 2.5. An increasing curve is graphed. — **Figure 6.14** Key data for Lowe's stock 9/7/2022 (data source: marketwatch.com)

Other key data is further down on the website, and is shown in Figure 6.15, below.

The key data factors. The factors are: Open: $ 193.71, 52 week range: 170.12 to 263.31, Shares outstanding: 620.7 M, beta: 1.14, P/E ratio: 15.79, yield: 2.10 percent, Ex-dividend date: October 18, 2022, percentage of float shorted: 1.79 percent, Day range: 193.58 to 201.44, Market cap: $119.77B, Public Float: 620.17 M, Rev. per employee: $ 280.57k, E P S: $12.69, Dividend: $1.05, Short interest: 11.09M, Average volume: 3.62M. The performance for 5 days, 1 month, 3 months, Y T D and 1 year are 3.74, minus 0.01, 4.65, minus 22.08, and minus 1.32 percent. — **Figure 6.15** Key data for Lowe's stock 9/7/2022 (data source: marketwatch.com)

Notice that the 52-week high and low are now shown as the 52-week range. However, you get additional information, including the stock performance over the past 5 days, past month, past 3 months, the year to date (YTD), and over the past year. You can also read the number of shares outstanding, the expected date for the dividend (EX-DIVIDEND DATE), and importantly for the P/E ratio, the earning per share (EPS).

As mentioned, stocks earn money in two ways, through dividends and increase in share price.

Mutual Funds

A mutual fund is a collection of investments that are all bundled together. When you buy shares of a mutual fund, your money is pooled with the assets of other investors. This pooled money is invested in stocks, bonds, money market instruments, and other assets. Mutual funds are typically operated by professional money managers who allocate the fund's assets and attempt to produce capital gains or income for the fund's investors.

A key benefit of mutual funds is that they allow small or individual investors to invest in professionally managed portfolios of equities, bonds, and other securities. This means each shareholder participates proportionally in the gains or losses of the fund. The performance of a mutual fund is usually stated as how much the mutual fund’s total value has increased or decreased. Since there are many different investments inside the mutual fund, the risk is reduced significantly, compared to direct ownership of stocks. Even so, mutual funds historically perform well and can earn more than 10% annually.

The investments that make up a mutual fund are structured and maintained to match stated investment objectives, which are specified in its prospectus. A prospectus is a pamphlet or brochure that provides information about the mutual fund. Before buying shares of a mutual fund, consult its prospectus, consider its goals and strategies to see if they match your goals and values and also research any associated fees.

Example 7 Investing in a Mutual Fund

Kaitlyn has analyzed her $12,862.50 quarterly budget using the 50-30-20 budget philosophy, and sees she should be saving or paying down debt with $2,572.50 per quarter. She decides to invest $1,300 quarterly a mutual fund that reports an average return of 11.62% over the 18-year life of the mutual fund. Assuming that this interest rate continues, and is compounded quarterly, how much will her mutual fund account be worth after 5 years?

Kaitlyn’s plan is an ordinary annuity, and so the future value of her account can be found using the formula $F V = p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n}$ , with a payment of $1,300, a rate of 0.1162, number of compounding periods 4, after 5 years. Substituting these values into the formula and calculating, we find

$\begin{matrix} F V & = & p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n} \\ = & $ 1,300 \times \frac{{(1.02905)}^{20} - 1}{0.02905} \\ = & $ 1,300 \times \frac{0.773084935178}{0.032905} \\ = & $ 1,300 \times 26.6122189784 \\ = & $ 34,595.88 \end{matrix}$

Kaitlyn’s mutual fund will be worth $34,595.88 after 5 years.

Example 8 Investing in a Mutual Fund to Reach a Goal

Kaitlyn wants to retire with $1,500,000 in her mutual fund account. She will invest for 35 years. The mutual fund reports an average return of 11.62% over the 18-year-long life of the mutual fund. Assuming that this interest rate continues, and is compounded quarterly, how much will she need to pay annually into her mutual fund to reach her goal?

Kaitlyn’s plan is an ordinary annuity, and so the payment to reach her goal can be found using the formula $p m t = \frac{F V \times (r / n)}{{(1 + r / n)}^{n \times t} - 1}$ , with a $F V$ , or goal, of $1,500,000, a rate of 0.1162, for 35 years. Substituting these values into the formula and calculating, we find

$\begin{matrix} p m t & = & \frac{F V \times (r / n)}{{(1 + r / n)}^{n \times t} - 1} \\ = & \frac{$ 1,500,000 \times (0.1162 / 1)}{{(1 + 0.1162 / 1)}^{1 \times 35} - 1} \\ = & \frac{$ 174,300}{26.0558103113} \\ = & $ 6,689.49 \end{matrix}$

Kaitlyn needs to invest $6,689.49 per year (or $557.46 per month) into the mutual fund to reach $1,500,000 in 35 years.

Return on Investment

As in Methods of Savings, the formula for return on investment is $ROI = \frac{F V - P}{P}$ . As indicated before, this formula does not take into account how long the investment took to reach its current value. It depends only on the initial value, $P$ , and the value at the end of the investment, $F V$ .

As mentioned, the ROI does not address the length of time of the investment. A good way to do that is to equate the ROI to an account bearing interest that is compounded annually.

The annual return is the average annual rate, or the annual percentage yield (APY) that would result in the same amount were the interest paid once a year.

We apply this to the previous example.

In Example 10 and Your Turn, the annual return was lower than the interest rate of the investment. This is because the interest from a bond is simple interest, but annual yield equates to compounded annually.

Example 11 Return on Investment for Stock in Company ABC

Haniah buys stock in the ABC company, investing a total of $13,000. After 20 years, the stock is worth $132,293.49, including reinvestment of dividends.

What is Haniah’s return on investment?
What is Haniah’s annual return?

To calculate Hanniah’s return on investment, substitute $13,000 for $P$ and $132,293.49 for $F V$ in the formula $ROI = \frac{F V - P}{P}$ and calculate. Doing so we find Haniah’s return on investment to be $ROI = \frac{F V - P}{P} = \frac{$ 132,293.49 - $ 13,000}{$ 13,000} = 9.176422$ , or 917.64%
To calculate Haniah’s annual return, substitute $13,000 for $P$ and $132,293.49 for $F V$ in the formula $annual return = {(\frac{F V}{P})}^{(\frac{1}{t})} - 1$ and calculate. Doing so we find her annual return to be $annual return = {(\frac{F V}{P})}^{(\frac{1}{t})} - 1 = {(\frac{$ 132,293.49}{$ 13,000})}^{(\frac{1}{20})} - 1 = 0.1230$ , or 12.3%

You should see that the annual return is equal to the annual compounded interest that was assumed for the stocks.

Compute Payment to Reach a Financial Goal

As in Methods of Savings, determining the payment necessary to reach a financial goal uses the payment formula for an ordinary annuity, $p m t = \frac{F V \times (r / n)}{{(1 + r / n)}^{n \times t} - 1}$ . If dealing with mutual funds or stocks, an assumed annual interest rate, compounded, will be used. This value is often determined through research and informed speculation.

Retirement Savings Plans

We close this section by investigating the three main forms of retirement savings accounts: traditional individual retirement accounts (IRAs), Roth IRAs, and 401(k) accounts. Each has distinct characteristics that are suited to different investors’ needs.

Individual Retirement Accounts

A traditional IRA lets you contribute up to an amount set by the government, which may change from year to year. For example, the maximum contribution for 2022 is $6,000; $7,000 over age 50. Anyone is eligible to contribute to a traditional IRA, regardless of your income level. Your money grows tax-deferred, but withdrawals after age 59½ are taxed at current rates. Traditional IRAs also allow you to use the contribution itself as a deduction on a current year tax return.

Roth IRAs allow contributions at the same levels as traditional IRAs, with a maximum $6,000 for 2022; $7,000 over age 50. However, to be eligible to make contributions, your earned income must be below a certain level. A Roth IRA allows after-tax contributions. In other words, the contribution itself is not tax-deductible, as it is with the traditional IRA. However, your money grows tax-free. If you make no withdrawals until you are age 59½, there are no penalties. IRAs pay a modest interest rate.

In either case, IRA deposits have to be from earned income, which in effect means if your earned income is over $6,000 ($7,000) then you can deposit the maximum.

401(k) Accounts

Your employer may offer a retirement account to you. These are often in the form of a 401(k) account. There are traditional and Roth 401(k) accounts, which differ in how they are taxed, much as with other IRAs. In the traditional 401(k) plans, the money is deposited before tax is assessed, which means you do not pay taxes on this money. However, that means when money is withdrawn, it is taxed. These accounts are similar to mutual funds, in that the money is invested in a wide range of assets, spreading the risk.

One of the perks some employers offer is to match some amount of your contributions to the 401(k) plan. For instance, they may match your deposits up to 5% of your income. This is an instant 100% return on the money that was matched.

Example 14 Matching 401(k) Deposit

Alice signs up for her employer-based 401(k). The employer matches any 401(k) contribution up to 6% of the employee salary. Alice’s annual salary is $51,600.

What is the most money that Alice can deposit that will be fully matched by the company?
How much total will be deposited into Alice’s account if she deposits the full 6%?
How much return does Alice earn if she deposits exactly 6% in her 401(k)?

The employer will match up to 6% of any employee’s salary. 6% of Alice’s salary is $0.06 \times $ 51,600 = $ 3,096$ . So Alice can deposit up to $3,096 and receive that amount in matching funds in her account.
Alice’s contribution plus the company’s contribution is $$ 3,096 + $ 3,096 = $ 6,192$ , which is the total that is deposited into Alice’s account.
She earns a 100% return on the day she deposits her $3,096.

401(k) plans with matching funds provide great value, as their rates of return are high compared to savings accounts, and are less risky that stocks since such funds invest across many investment vehicles. The next example demonstrates the power of constant deposits into a 401(k) plan that has some employer match.

Example 15 Constant Deposits into a 401(k) Plan

DeJean begins depositing $300 per month from his paycheck each month in his employer-based 401(k) account. The employer matches this deposit as it falls below their matching threshold. DeJean expects the return to average 10% per year, compounded annually.

How much will DeJean’s account be worth if he keeps making those payments for 30 years?
What will his account be worth without the matching funds?

This is a form of an ordinary annuity, so the formula $F V = p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n}$ will be used. The company matches DeJean’s full deposit, so each month $600 will be deposited. He is assuming the money will compound annually, so the amount deposited each year is needed as the value of pmt. For the year, he will deposit $12 \times $ 600 = $ 7,200$ . The rate is 0.1, the number of compounding periods is 1, and the number of years is 30. Substituting and calculating, the value of DeJean’s account after 30 years will be

$\begin{matrix} F V & = & p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n} \\ = & $ 7,200 \times \frac{{(1 + 0.1 / 1)}^{1 \times 30} - 1}{0.1 / 1} \\ = & $ 7,200 \times \frac{{(1.1)}^{30} - 1}{0.1} \\ = & $ 7,200 \times 164.494022689 \\ = & $ 1,184,356.96 \end{matrix}$
This is a form of an ordinary annuity, so the formula $F V = p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n}$ will be used but the deposit is now only $300 per month without the matching funds. For the year, he will deposit $12 \times $ 300 = $ 3,600$ . The rate is 0.1, the number of compounding periods is 1, and the number of years is 30. Substituting and calculating, the value of DeJean’s account after 30 years will be

$\begin{matrix} F V & = & p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n} \\ = & $ 3,600 \times \frac{{(1 + 0.1 / 1)}^{1 \times 30} - 1}{0.1 / 1} \\ = & $ 3,600 \times \frac{{(1.1)}^{30} - 1}{0.1} \\ = & $ 3,600 \times 164.494022689 \\ = & $ 592,178.48 \end{matrix}$

Key Terms

Bonds
Maturity date
Stocks
Dividend
Mutual fund
Prospectus
Issue price
Shares
Stock table
Individual retirement account
Roth IRA
401(k)

Key Concepts

There are many different investments with different returns and risks.
Bonds are loans form the purchaser to the entity selling the bond.
Bonds have some tax benefits, low to no risk, and a low return.
Stocks represent part ownership in a company. As such, stock holders share in the profits, and losses, of the company.
Information, including price, P/E, yearly highs and lows, and dividend amount can be found in online stock tables available on many websites.
Mutual funds represent collections of professionally administered investment vehicles. Have shares in a mutual fund has lower risk than ownership of stocks.
Retirement accounts employ some of the same strategies as mutual funds, in that they spread the risk and are professionally managed.
IRAs and Roth IRAs differ on when taxes are paid on the money, and who can use them. Roth IRAs have income limits while traditional IRAs do not.

Videos

Formulas

$annual return = {(\frac{F V}{P})}^{(\frac{1}{t})} - 1$

$P / E = \frac{Share Price}{Dividend}$

$Yld % = \frac{Annual Dividend}{Share Price} \times 100 %$

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.