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Investments

Distinguish between basic forms of investments including stocks, bonds, and mutual funds.

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.

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 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.

Example

Try it.

Muriel purchases a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. How much is Muriel paid each year, and how much does she receive on the maturity date?

Solution

The coupon rate is 5.5%. 5.5% of her bond value is $0.055\times \text{\$}3,000=\text{\$}165$. After year 1, Muriel receives $165. She receives $165 after years 2 and 3 also. In year 4, when the bond matures, Muriel receives $3,165, or the interest and the initial investment, or principal.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Return on Investment

As in Methods of Savings, the formula for return on investment is $\text{ROI}=\frac{FV-P}{P}$. 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, $FV$.

Return on Investment for a Bond

Try it.

Recall , in which Muriel purchased a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. What was Muriel’s return on investment?

Solution

Each year, Muriel received $165. She received this money four times, so earned a total of $660. This represents $FV$ – $P$, or just the earnings. Using that we find that the ROI is $\text{ROI}=\frac{660}{3000}=0.22$, or 22%.

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.

Annual Return on Investment for a Bond

Try it.

Recall , in which Muriel purchased a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. What was Muriel’s annual return on investment? Interpret this as compound interest.

Solution

Muriel earned a total of $660. This represents $FV$ – $P$, or just the earnings. The starting principal was $3,000. The value at the end of 4 years was $3,000 + $660 = $3,660. The time of the investment was 4 years. Using that we find that the annual return is $\text{annual return}={(\frac{FV}{P})}^{(\frac{1}{t})}-1={(\frac{\text{\$}3,660}{\text{\$}3,000})}^{(\frac{1}{4})}-1={1.22}^{\frac{1}{4}}-1=1.050969=0.050969$, or 5.10%. The 5.5% bond earned the equivalent of 5.10% compounded annually.

In 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.

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

Condensed — the full section is in OpenStax Contemporary Mathematics.

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, $pmt=\frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1}$. 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.

Example

Try it.

Richard is saving for new siding for his home. He and his partner believe they will need $37,500 in 10 years to pay for the siding. How much should they invest yearly in a mutual fund they believe will have an annual interest rate of 12%, compounded annually, in order to reach their goal?

Solution

The necessary annual payment is found using the function $pmt=\frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1}$ with $FV$ = 37,500, $r$ = 0.12, and $n$ = 1. Substituting and calculating, we find the annual payment should be

\[\begin{array}{lll}pmt & = & \frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1} \\ & = & \frac{\text{\$}37,500\times (0.12/1)}{{(1+0.12/1)}^{1\times 10}-1} \\ & = & \frac{\text{\$}4,500}{{(1.12)}^{10}-1} \\ & = & \frac{\text{\$}4,500}{2.10584820834} \\ & = & \text{\$}2,136.91\end{array}\]

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.

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.

Comparing Roth IRAs to Traditional IRAs

Try it.

Which type of IRA, Roth or traditional, has an income limit for its use?

Solution

Roth IRAs require income to be below a certain limit.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

Formulas

$\text{annual return}={(\frac{FV}{P})}^{(\frac{1}{t})}-1$

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

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

Practice (15)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Muriel purchases a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. How much is Muriel paid each year, and how much does she receive on the maturity date?

Tunjukkan jawapan

The coupon rate is 5.5%. 5.5% of her bond value is $0.055\times \text{\$}3,000=\text{\$}165$. After year 1, Muriel receives $165. She receives $165 after years 2 and 3 also. In year 4, when the bond matures, Muriel receives $3,165, or the interest and the initial investment, or principal.
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?

Tunjukkan jawapan

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 $A=P{(1+\frac{r}{n})}^{nt}=\text{\$}13,000{(1+\frac{0.123}{1})}^{1\times 20}=\text{\$}13,000\times 10.1764223996=\text{\$}132,293.49$. After 20 years, her stock is now worth $132,293.49.
1. Find the percent yield for a stock with a price of $30.69 and an annual dividend of $1.48.
2. Find the percent yield for a stock with a price of $62.25 and an annual dividend of $1.76.
Tunjukkan jawapan
1. Substituting the values for price, $30.69, and annual dividend, $1.48, we find the percent yield for the stock to be $\text{Y}\text{l}\text{d}\%=\frac{\text{Annual Dividend}}{\text{Share Price}}\times 100\%=\frac{\text{\$}1.48}{\text{\$}30.69}\times 100\%=4.82\%$
2. Substituting the values for price, $62.25, and annual dividend, $1.76, we find the percent yield for the stock to be $Yld\%=\frac{\text{Annual Dividend}}{\text{Share Price}}\times 100\%=\frac{\text{\$}1.76}{\text{\$}62.25}\times 100\%=2.83\%$
Consider the stock table (), and answer the questions based on the table.
1. What is the current price for McDonald’s Corp on this date?
2. What is the 52-wk high? 52-wk low?
3. When is the dividend expected?
4. What is its yield?
5. What is the earnings per share?
Tunjukkan jawapan
1. Looking at the table, the current price of a share is $258.87.
2. The high was $271.15, and the low was $217.68.
3. August 31, 2022
4. 2.13%
5. The EPS value is $8.12.
Darma owns 150 shares of stock in the GDW company. This quarter, GDW is paying $0.87 per share in dividends. How much will Darma earn in dividends this quarter?

Tunjukkan jawapan

Each share pays $0.87, so Darma earns $150\times \text{\$}0.87=\text{\$}130.50$.
Vincent buys 100 stocks in the REM company for $21.87 per share. One year later, he sells those 100 shares for $29.15 per share.
1. How much money did Vincent make?
2. What was his return on investment for that one year?
Tunjukkan jawapan
1. Vincent spent $21.87 per share to buy the stock. The total he spent on the stock was $\text{\$}21.87\times 100=\text{\$}2,187.00$. When he sold the stock, the price was $29.15, so he received $\text{\$}29.15\times 100=\text{\$}2,915.00$. He made $\text{\$}2,915.00-\text{\$}2,187.00=\text{\$}728.00$.
2. His return on investment was $\frac{\text{Earnings}}{\text{Original Price}}=\frac{\text{\$}728}{\text{\$}2,187}=33.29\%$.
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?

Tunjukkan jawapan

Kaitlyn’s plan is an ordinary annuity, and so the future value of her account can be found using the formula $FV=pmt\times \frac{{(1+r/n)}^{n\times t}-1}{r/n}$, 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
\[\begin{array}{lll}FV & = & pmt\times \frac{{(1+r/n)}^{n\times t}-1}{r/n} \\ & = & \text{\$}1,300\times \frac{{(1.02905)}^{20}-1}{0.02905} \\ & = & \text{\$}1,300\times \frac{0.773084935178}{0.032905} \\ & = & \text{\$}1,300\times 26.6122189784 \\ & = & \text{\$}34,595.88\end{array}\]
Kaitlyn’s mutual fund will be worth $34,595.88 after 5 years.
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?

Tunjukkan jawapan

Kaitlyn’s plan is an ordinary annuity, and so the payment to reach her goal can be found using the formula $pmt=\frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1}$, with a $FV$, or goal, of $1,500,000, a rate of 0.1162, for 35 years. Substituting these values into the formula and calculating, we find
\[\begin{array}{lll}pmt & = & \frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1} \\ & = & \frac{\text{\$}1,500,000\times (0.1162/1)}{{(1+0.1162/1)}^{1\times 35}-1} \\ & = & \frac{\text{\$}174,300}{26.0558103113} \\ & = & \text{\$}6,689.49\end{array}\]
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.
Recall , in which Muriel purchased a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. What was Muriel’s return on investment?

Tunjukkan jawapan

Each year, Muriel received $165. She received this money four times, so earned a total of $660. This represents $FV$ – $P$, or just the earnings. Using that we find that the ROI is $\text{ROI}=\frac{660}{3000}=0.22$, or 22%.
Recall , in which Muriel purchased a $3,000 bond with a maturity of 4 years at a fixed coupon rate of 5.5% paid annually. What was Muriel’s annual return on investment? Interpret this as compound interest.

Tunjukkan jawapan

Muriel earned a total of $660. This represents $FV$ – $P$, or just the earnings. The starting principal was $3,000. The value at the end of 4 years was $3,000 + $660 = $3,660. The time of the investment was 4 years. Using that we find that the annual return is $\text{annual return}={(\frac{FV}{P})}^{(\frac{1}{t})}-1={(\frac{\text{\$}3,660}{\text{\$}3,000})}^{(\frac{1}{4})}-1={1.22}^{\frac{1}{4}}-1=1.050969=0.050969$, or 5.10%. The 5.5% bond earned the equivalent of 5.10% compounded annually.
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.
1. What is Haniah’s return on investment?
2. What is Haniah’s annual return?
Tunjukkan jawapan
1. To calculate Hanniah’s return on investment, substitute $13,000 for $P$ and $132,293.49 for $FV$ in the formula $\text{ROI}=\frac{FV-P}{P}$ and calculate. Doing so we find Haniah’s return on investment to be $\text{ROI}=\frac{FV-P}{P}=\frac{\text{\$}132,293.49-\text{\$}13,000}{\text{\$}13,000}=9.176422$, or 917.64%
2. To calculate Haniah’s annual return, substitute $13,000 for $P$ and $132,293.49 for $FV$ in the formula $\text{annual return}={(\frac{FV}{P})}^{(\frac{1}{t})}-1$ and calculate. Doing so we find her annual return to be $\text{annual return}={(\frac{FV}{P})}^{(\frac{1}{t})}-1={(\frac{\text{\$}132,293.49}{\text{\$}13,000})}^{(\frac{1}{20})}-1=0.1230$, or 12.3%
Richard is saving for new siding for his home. He and his partner believe they will need $37,500 in 10 years to pay for the siding. How much should they invest yearly in a mutual fund they believe will have an annual interest rate of 12%, compounded annually, in order to reach their goal?

Tunjukkan jawapan

The necessary annual payment is found using the function $pmt=\frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1}$ with $FV$ = 37,500, $r$ = 0.12, and $n$ = 1. Substituting and calculating, we find the annual payment should be
\[\begin{array}{lll}pmt & = & \frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1} \\ & = & \frac{\text{\$}37,500\times (0.12/1)}{{(1+0.12/1)}^{1\times 10}-1} \\ & = & \frac{\text{\$}4,500}{{(1.12)}^{10}-1} \\ & = & \frac{\text{\$}4,500}{2.10584820834} \\ & = & \text{\$}2,136.91\end{array}\]
Which type of IRA, Roth or traditional, has an income limit for its use?

Tunjukkan jawapan

Roth IRAs require income to be below a certain limit.
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.
1. What is the most money that Alice can deposit that will be fully matched by the company?
2. How much total will be deposited into Alice’s account if she deposits the full 6%?
3. How much return does Alice earn if she deposits exactly 6% in her 401(k)?
Tunjukkan jawapan
1. The employer will match up to 6% of any employee’s salary. 6% of Alice’s salary is $0.06\times \text{\$}51,600=\text{\$}3,096$. So Alice can deposit up to $3,096 and receive that amount in matching funds in her account.
2. Alice’s contribution plus the company’s contribution is $\text{\$}3,096+\text{\$}3,096=\text{\$}6,192$, which is the total that is deposited into Alice’s account.
3. She earns a 100% return on the day she deposits her $3,096.
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.
1. How much will DeJean’s account be worth if he keeps making those payments for 30 years?
2. What will his account be worth without the matching funds?
Tunjukkan jawapan
1. This is a form of an ordinary annuity, so the formula $FV=pmt\times \frac{{(1+r/n)}^{n\times t}-1}{r/n}$ 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 $12\times \text{\$}600=\text{\$}7,200$. 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
  \[\begin{array}{lll}FV & = & pmt\times \frac{{(1+r/n)}^{n\times t}-1}{r/n} \\ & = & \text{\$}7,200\times \frac{{(1+0.1/1)}^{1\times 30}-1}{0.1/1} \\ & = & \text{\$}7,200\times \frac{{(1.1)}^{30}-1}{0.1} \\ & = & \text{\$}7,200\times 164.494022689 \\ & = & \text{\$}1,184,356.96\end{array}\]
2. This is a form of an ordinary annuity, so the formula $FV=pmt\times \frac{{(1+r/n)}^{n\times t}-1}{r/n}$ will be used but the deposit is now only $300 per month without the matching funds. For the year, he will deposit $12\times \text{\$}300=\text{\$}3,600$. 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
  \[\begin{array}{lll}FV & = & pmt\times \frac{{(1+r/n)}^{n\times t}-1}{r/n} \\ & = & \text{\$}3,600\times \frac{{(1+0.1/1)}^{1\times 30}-1}{0.1/1} \\ & = & \text{\$}3,600\times \frac{{(1.1)}^{30}-1}{0.1} \\ & = & \text{\$}3,600\times 164.494022689 \\ & = & \text{\$}592,178.48\end{array}\]

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Investments

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.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Cubalah sendiri

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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