4.4: Methods of Savings (Annuities, Stock, and Bonds)
- Page ID
- 181934
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Distinguish various basic forms of savings plans.
- Compute return on investment for basic forms of savings plans.
- Compute the payment to reach a financial goal.
- Distinguish between basic forms of investments, including stocks, bonds, and mutual funds.
- Read and derive information from a stock table
- Compute return on investment for basic forms of investments.
- Identify and distinguish between retirement savings accounts.
The stock market crash of \(1929\) led to the Great Depression, a decade-long global downturn in productivity and employment. A state of shock swept through the United States; the damage to people’s lives was immeasurable. Americans no longer trusted established financial institutions. By October \(1931,\) the banking industry’s biggest challenge was restoring confidence to the American public. In the next 10 years, the federal government would impose strict regulations and guidelines on the financial industry. The Emergency Banking Act of \(1933\) created the Federal Deposit Insurance Corporation (FDIC), which insures bank deposits. The new federal guidelines helped ease suspicions among the general public about the banking industry. Gradually, things returned to normal, and today we have more investment instruments, many insured through the FDIC, than ever before.
In this section, we will first look at the different types of savings accounts and proceed to discuss the various types of investments. There is some overlap, but we will try to differentiate among these financial instruments. Saving money should be a goal of every adult, but it can also be a difficult goal to attain.
There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money market accounts are the primary options for saving.
Savings Account
There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money market accounts are the primary options for saving.
A savings account is probably the most well-known type of investment, and for many people, it is their first experience with a bank. A savings account is a deposit account held at a bank or other financial institution that bears interest on the deposited money. Savings accounts are intended as a place to save money for emergencies or to achieve short-term goals. They typically pay a low interest rate, but there is virtually no risk involved, and they are insured by the FDIC for up to \($250,000 \ \)
Savings accounts have some strengths. They are highly flexible. Generally, there are no limitations on the number of withdrawals allowed and no limit on how much you can deposit. It is not unusual, however, for a savings account to have a minimum balance in order for the bank to cover maintenance costs. If your account balance drops below the minimum, fees may be applied.
Many banks are covered by FDIC insurance. The FDIC, or Federal Deposit Insurance Corporation, is an independent agency created by the U.S. Congress. One of its purposes is to provide insurance for deposits in banks, including savings accounts and other deposit types. Be aware, not all banks are FDIC insured. The FDIC insures up to \($250,000\) for a savings account, so you do not want your balance to exceed that federally insured limit.
Having your savings account at the same bank as your checking account does offer a real advantage. For example, if your checking account is approaching its lower limit, you can transfer funds from your savings account and avoid any bank fees. Similarly, if you have an excess of funds in your checking account, you can transfer funds to your savings account and earn some interest. Checking accounts rarely pay interest.
The 50-30-20 rule is a simple personal budgeting method designed to help individuals manage their finances by dividing after-tax income into three main spending categories:
50% — Needs (Essential expenses)
- Utilities (electricity, water, gas)
- Groceries
- Transportation (car payment, gas, public transit)
- Insurance (health, auto)
- Minimum loan payments
30% — Wants (Non-essential expenses)
- Entertainment (movies, streaming, events)
- Shopping (clothes, gadgets)
- Vacations
- Hobbies and subscriptions
20% — Savings & Debt Repayment
- Emergency fund
- Retirement accounts (IRA, 401(k))
- Investments
- Extra payments toward credit cards or loans
- Saving for a home, education, etc
Look at an example
David gathers his pay stubs and bills from the past \(6\) months. His income, after taxes, is \($3,450\) per month. His rent, utilities included, is \($925.\) His car payments are \($178.54\) per month, his car insurance is \($129.49\) per month, his credit cards cost him \($117.00\) per month, he spends \($195\) per month on gas, and his food costs are \($290\) per month. He also spends \($21.99\) on Amazon Prime, \($49.99\) on his internet bill, and \($400\) per month going out. Create David’s monthly budget, including totals, based on that information.
- Using David’s Budget, how much income does he have per month after accounting for his expenses?
- Apply the \(50-30-20\) budget philosophy to David’s budget.
- Evaluate David’s budget with respect to the \(50-30-20\) budget philosophy.
|
Amount ($) |
Expense |
Amount ($) |
|---|---|---|
|
\(3450.00\) |
Rent, with utilities |
\(925.00\) |
|
|
Car payments |
\(178.54\) |
|
|
Car Insurance |
\(129.49\) |
|
|
Credit card debt |
\(117.00\) |
|
|
Gas |
\(195.00\) |
|
|
Food |
\(290.00\) |
|
|
Amazon prime |
\(21.99\) |
|
|
Internet |
\(49.99\) |
|
|
Going out |
\(400.00\) |
|
\(3450.00\) |
Total |
\(2307.01\) |
David saved in one month
\[$3450.00 – $2307.01 = $1142.99\nonumber\]
Necessities \(50\%:\) \[ 0.50\times $3450 = $1725\nonumber\]
Wants \(30\%:\)
\[0.30\times$3450 = $1035\nonumber \]
Savings and Debt Repyment \(20\%:\)
\[0.20\times$3450 = $690\nonumber\]
Necessities: \($925 + $178.54 + $129.49 + $117 + $195 + $290 = $1835.03,\) This is very close to the \(50\%.\)
Wants: \($21.99 + $49.99 + $400 = $471.98,\) This is much lower than the \(30\%.\)
Saving and Debt Repayment: \($1142.99,\) This is much greater than the \(20\%.\)
J.P. Morgan was a wealthy banker around the turn of the \(20\) th century. His business interests included railroads and the steel industry. However, it was in \(1907\) that a financial crisis, caused by poor banking decisions and followed by such great distrust in the banking system that a frenzy of withdrawals from banks occurred, that J.P. Morgan and other wealthy bankers lent from their own funds to help stabilize and save the system.
There are some weaknesses to savings accounts. Primarily, it is because savings accounts earn very low interest rates. This means they are not the best way to grow your money. Experts, though, recommend keeping a savings account balance to cover \(3\) to \(6\) months of living expenses in case you should lose your job, have a sudden medical expense, or other emergency.
Around tax time, you will receive a \(1099\)-INT form stating the amount of interest earned on your savings, which is the amount that must be reported when you file your tax return. A \(1099\) form is a tax form that reports earnings that do not come from your employer, including interest earned on savings accounts. These \(1099\) forms have the suffix INT to indicate that the income is interest income.
Savings accounts earn interest, and those earnings can be found using the interest formulas from previous sections. The final value of these accounts is sometimes referred to as the future value of the account.
Banks have not always offered interest on savings accounts. An \(1836\) publication from Indiana noted that banks in other states allow small interest on deposits. It specifically says that in these other states, these deposits are what business transactions are based upon. And that giving interest would encourage deposits, and thus increase the business that banks can do.
Journal of the House of Representatives of the State of Indiana(opens in new window)
Certificates of Deposit, or CDs, and Money Market Accounts
We discussed certificates of deposit (CDs) in earlier sections. CDs differ from savings accounts in a few ways. First, the investment lasts for a fixed period of time, agreed to when the money is invested in the CD. These time periods often range from \(6\) months to \(5\) years. Money from the CD cannot be withdrawn (without penalty) until the end of the investment period. Also, money cannot be added to an existing CD.
Certificates of deposit share features similar to those of savings accounts. They are insured by the FDIC. They are entirely safe. They do, though, offer a better interest rate. The trade-off is that once the money is invested in a CD, that money is unavailable until the investment period ends.
A money market account is similar to a savings account, except the number of transactions (withdrawals and transfers) is generally limited to six each month. Money market accounts typically require a minimum balance to be maintained. If the balance in the account falls below the minimum, a penalty may be incurred. Money market accounts offer the flexibility of checks and ATM cards. Ultimately, the interest rate on a money market account is generally higher than that of a savings account.
The return on investment, often denoted ROI, is the percent difference between the initial investment, \(P\), and the final value of the investment, \(FV. \)
\begin{align*}
\text{ROI}=\frac{FV-P}{P}\times100\%
\end{align*}
Note that the length of time of the investment is not considered in ROI.
Example \(\PageIndex{1}\): 5-Year CD ROI
Silvio deposits \($10,000\) in a CD that yields \(2.17\%\) compounded semiannually for \(5\) years. How much is the CD worth after \(5\) years? How much did she earn in interest?
- Answer
-
This also uses the compound interest formula from the Compound Interest section.
Substituting the values \(P=$10,000,\) \(r=0.0217,\) \(n=2\) (semiannually means twice per year), and \(t=5.\) We find the account will be worth\begin{aligned}
A&=P\left(1+\frac{r}{n}\right)^{nt}\\
&=10,000\left(1+\frac{0.0217}{2}\right)^{2\times 5}\\
&=10,000\left(1.01085\right)^{10}\\
&=10,000\times 1.113953736\\
&=11,139.53736\\
\end{aligned} The CD will be worth \($11,139.54\) after \(5\) years.Interest earned during that period is:
\[I = A-P=\$11,139.54 -$10,000=\$1,139.54\nonumber\]
The initial deposit in the CD was \($10,000,\) so = \(P=$10,000.\) The value at the end of \(5 \) years was \($11,139.54.\) and \(
F V \begin{aligned}
\text{ROI}&=\frac{FV-P}{P}\times100\%\\
&=\frac{$11,139.54-$10,000}{$10,000}\times 100\%\\
&=\frac{$1,139.54}{$10,000}\times 100\%\\
&=11.3954\%\\
\end{aligned}The ROI is \(11.40\%.\)
Note: As we did in the compound interest section, you can find the future value using the TVM solver. Put the following values in the given variables
\begin{align*}
\text{N}&=n\times t=2\times5=10\\\text{I% }&=2.17\; \text{ (Do not put percent)}\\
\text{PV} &= 10,000\; \text{(Deposited amount or present value)}\\
\text{FV} &=\text{We need to find it.}\\
\text{PMT} & =0\\
\text{P/Y} &= 2 \\
\text {C/Y} &=2\\
\end{align*}And click solve for FV.
Annuities as Savings
In Compound Interest, we talked about the future value of a single deposit. In reality, people often open accounts that allow them to add deposits or payments to the account at regular intervals. When a deposit is made at the end of each compounding period, such a savings account is called an ordinary annuity. The \(50 - 30 - 20\) is a simple budgeting guideline that helps individuals manage their money effectively. It divides your after-tax income into three broad categories:
\begin{align*}
FV=PMT\left[\frac{(1+\frac{r}{n})^{nt} -1}{\left(\frac{r}{n}\right)}\right]
\end{align*}
where
\(FV\): The future value of the annuity.
\(PMT\): The payment or periodic deposit.
\(r\): The annual interest rate (in decimal form).
\(n\): The number of compounding periods per year.
\(t\): The number of years.
\begin{align*}
\text{Total deposit} &= PMT\times n\times t\\
\text{Interest earned from annuity} &= FV-\text {Total deposit}
\end{align*}
Another form of annuity is the annuity due, which has deposits at the start of each compounding period. This other annuity type has different formulas and is not addressed in this text.
Example \(\PageIndex{2}\): Future Value of an Ordinary Annuity
Jill has an account that bears \(5\%\) interest compounded quarterly. She decides to deposit \($500\) quarterly into this account.
- What is the future value of this account, after \(35\) years?
- How much money will Jill have put into the account in \(35\) years?
- How much interest will Jill have earned?
- Answer 1
-
Here
\(FV\) is the future value of the annuity (We do not know)
\(PMT=$500\) is the payment (Periodic deposit)
\(r=0.05\) is the annual interest rate (in decimal form)
\(n=4\) is the number of compounding periods per year
\(t=35\) is the number of years
\begin{aligned}
FV&=PMT\left[\frac{(1+\frac{r}{n})^{nt} -1}{\left(\frac{r}{n}\right)}\right]\\
&=500\left[\frac{(1+\frac{0.05}{4})^{4\times35} -1}{\left(\frac{0.05}{4}\right)}\right]\\
&=500\left[\frac{(1.0125)^{140} -1}{0.0125}\right]\\
&=500\left[\frac{4.692518676}{0.0125}\right]\\
&=500\times375.4014941\\
&=187,700.747\\
\end{aligned}So the future value of this account, after \(35\) years, will be \($187,700.75.\)
- Answer 2
-
\begin{align*}
\text{Total deposit} &= PMT\times n\times t\\
&= $500\times4\times35\\
&=$70,000
\end{align*} - Answer 3
-
The interest Jill will have earned will be \begin{align*}
\text{Interest earned} &= FV-\text {Total deposit}\\
&=$187,700.75- $70,000\\
&=$117,700.75\\
\end{align*}Note: As we did before, you can check your answer using the TVM solver. Put the following values in the given variables to find the future value of the annuity.
\begin{align*}
\text{N}&=n\times t=4\times35=140\\\text{I% }&=5\; \text{ (Do not put percent)}\\
\text{PV} &= 0\; \text{ (We don't have it, so put it as zero)}\\
\text{FV} &=\text{We need to find it.}\\
\text{PMT} & =500\\
\text{P/Y} &= 4 \\
\text {C/Y} &=4\\
\end{align*}And solve for FV.
There are a number of factors that contribute to the interest rates a bank gives for savings accounts. The interest rate reflects how much the bank values deposits. It also reflects the money that the bank will earn when it lends out money. Finally, interest rates are impacted by the Federal Reserve Bank. When the Fed raises interest rates, so do banks.
The Federal Reserve Board monitors the risks in the financial system to help ensure a healthy economy for individuals, companies, and communities. The Board oversees the \(12\) regional reserve banks. The Chairperson of the Federal Reserve Board testifies to Congress twice per year, meets with the Secretary of the Treasury, chairs the Federal Open Market Committee, and is the face of federal monetary policy. Mr. Jerome Powell is the current Chairman of the Federal Reserve Board. Jerome Powell(opens in new window) was sworn in as chairman on February \(5,\) \(2018.\) He had been first nominated to the position by President Donald Trump(opens in new window) on November \( 2\), \(2017,\) and confirmed by the Senate. He was nominated to a second term by President Joe Biden(opens in new window), confirmed by the Senate, and sworn in on May \(23,\) \(2022.\)
The formula used to calculate the future value of an ordinary annuity is useful for determining the final amount in the account. However, that isn’t how planning works. To plan, we need to determine how much to invest in the ordinary annuity each compounding period in order to reach our goal. Fortunately, that formula exists.
The formula for the amount that needs to be deposited per period.
\begin{align*}
PMT=FV\left[\frac{\left(\frac{r}{n}\right)}{(1+\frac{r}{n})^{nt} -1}\right]
\end{align*}where
\(FV\): The future value of the annuity,
\(PMT\): The periodic payment or deposits
\(r\): The annual interest rate (in decimal form)
\(n\): The number of compounding periods per year
\(t\): The number of years
Example \(\PageIndex{3}\): Saving for a Car
Yaroslava wants to save in order to buy a car in \(3\) years, without taking out a loan. She determines that she’ll need \($35,500\;\) for the purchase. If she deposits money into an ordinary annuity that yields \(4.25\%\) interest compounded monthly.
- How much will she need to deposit each month?
- How much money will she have put into the account?
- How much interest will she have earned?
- Answer 1
-
Yaroslava has a goal and needs to know the payments to make to reach the goal. Her goal is \(FV=$35,500,\) with an interest rate \(r=0.0425,\) compounded per month so \(n=12,\) and for \(3\) years, making \(t = 3.\) Substituting into the formula, Yaroslava finds the necessary payment.
\begin{aligned}
PMT&=FV\left[\frac{\left(\frac{r}{n}\right)}{(1+\frac{r}{n})^{nt} -1}\right]\\
&=35,500\left[\frac{\left(\frac{0.0425}{12}\right)}{(1+\frac{0.0425}{12})^{12\times3} -1}\right]\\
&=35,500\left[\frac{0.003541667}{(1.003541667)^{36} -1}\right]\\
&=\frac{125.72916}{0.13572907}\\
&=926.3248\\
\end{aligned}That means if Yaroslava deposits \($926.33\) every months, after \(3\) years, her account will grow to \($35,500\) with \(4.25\%\) interest rate.
- Answer 2
-
Yaroslava puts $\($926.33\) monthly. So here the total deposit will be
\begin{align*}
\text{Total deposit} &= PMT\times n\times t\\
&= $926.33\times12\times3\\
&=$33,347.88
\end{align*} - Answer 3
-
The interest Yaroslava will have earned will be
\begin{align*}
\text{Interest earned} &= FV-\text {Total deposit}\\
&=$35,500- $33,347.88\\
&=$2,152.12\\
\end{align*}You can check your work by using the TVM solver. Put the following
\begin{align*}
\text{N}&=n\times t=12\times3=36\\\text{I% }&=4.25\; \text{ (Do not put percent)}\\
\text{PV} &=0\; \text{(We don't have it, so put it as a zero.)}\\
\text{FV} &=35500\\
\text{PMT} & =\text{We need to find it.}\\
\text{P/Y} &= 12 \\
\text {C/Y} &=12\\
\end{align*}And click solve for PMT.
You will find PMT = \($926.33\)
To reach her goal, Yaroslava would need to deposit $\(926.33\) (ROUND UP) in her account each month.
Bonds
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 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.
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 the amount of earnings they generate. Their predicted rates of return on investment are not guaranteed, but educated guesses based on market trends and historical performance.
Bonds are issued by big companies and by governments. Selling bonds is an alternative to an institution borrowing from a bank. The funds from the sale of bonds are often used for large projects, such as funding the construction 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 that of savings accounts, and the risk is exceptionally low. The bond is for a fixed term. The end of this time is the maturity date of the bond.
A bond is like a loan you give to a company or government. The face value (or par value) is the amount of money the bond will pay you back when it matures, usually $1,000. The coupon rate is the interest rate the bond pays based on its face value. For example, if a bond has a face value of $1,000 and a 5% coupon rate, it will pay $50 in interest each year. The coupon rate tells you how much income the bond will generate, while the face value tells you how much you will receive at the end of the bond’s term.
There are several types of bonds and their return level
- Treasury bonds are issued by the federal government: safer, lower returns
- Municipal bonds are issued by state and local governments: tax-free, moderate returns
- Corporate bonds are issued by major corporations: riskier, higher returns
The maturity date of a bond is the specific date in the future when the bond issuer must repay the full face value (principal) to the investor
Short-term: \(1–3\) years
Medium-term: \(4–10\) years
Long-term: \(10–30\) years or more
If you buy a \(10\)-year bond on January \(1, 2025\). The maturity date is January \(1, 2035\). You will receive coupon payments each year until \(2035\). On Jan \(1, 2035\), you get the full \($1,000\) face value back
Bonds are often part of larger investment portfolios. These bonds may be traded. However, the interest paid is based on the price when the bond was bought (the issue price). These bonds can be bought and sold for more or less money than the issue price. If the bond is bought for more than the issue price, the interest is still paid on the issue price, not on the purchase price at the time of the trade. This means the actual return on the bond decreases. If the bond is purchased for less than the issue price, the return on the bond increases.
As you see in the ROI formula above, it does not account for the duration 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 if the interest were paid once a year.
The formula for the annual return of the investment
\begin{align*}
\text{AR} &=\left[\left(\frac{FV}{P}\right)^{\frac{1}{t}}-1\right]\times100\%
\end{align*}
where
\(t\) = the number of years
\(FV\) = new value
\(P\) = starting principal
\(AR\) = Annual Return
Also, if you know the annual return (AR), we can find the value of investment (FV) after \(t\) years by the following formula
\begin{align*}
\text{Future Value: FV} &=P\left(1+AR\right)^{t}
\end{align*}
Example \(\PageIndex{4}\): Bond Investment
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 what is the total amount earned with the bond?
- What is Mauriel’s return on investment?
- What was Muriel’s annual return on investment? Interpret this as compound interest.
- Answer 1
-
The coupon rate is \(5.5\%.\) per year.
Money paid in one year is \[0.055\times$3,000=$165\nonumber\]
Each year Muriel receives \($165.\) Total amount earned in the bond is \($165\times4=$660.\)
- Answer 2
-
Each year, Muriel received \($165.\) She received this money four times, so she earned a total of \($660.\) Muriel's bond face value is
\($3,000+$660=$3,660\) and investment on the bond is \(3,000,\) which is \(P.\)
\begin{align*}
\text{ROI}&=\frac{\text{FV-P}}{\text{P}}\times 100\%\\
\text{ROI}&=\frac{\text{660}}{3000}\times 100\%\\
\text{ROI}&=22\%
\end{align*}So \(22\%\) is Muriel's ROI.
- Answer 3
-
Muriel purchases a \($3,000\;\) bond so her investment \(P = $3,000.\) Muriel earned a total of \($660.\)
\(FV= $3,000 + $660 = $3,660\). Using that, we find that the annual return is
\begin{align*}
\text{Annual return } &=\left[\left(\frac{FV}{P}\right)^{\frac{1}{t}}-1\right]\times100\%\\
&=\left[\left(\frac{3,660}{3000}\right)^{\frac{1}{4}}-1\right]\times100\%\\
&=(1.22^{\frac{1}{4}}-1)\times100\%\\
&==5.0969\%\\
\end{align*}\(5.1\%\) is Muriel's annual return.
The \(5.5\%\) bond earned the equivalent of \(5.10\%\) compounded annually.
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 a higher return on investment than 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 the 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 the 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 may provide a substantial return on investment, but the trade-off is the risk associated with owning them.
In the fall of \(2022,\) the parent company of Regal Theaters, named Cineworld, filed for Chapter \(11\) bankruptcy. According to news articles, the bankruptcy was necessitated due to its heavy debt load. Generally, a company can file for Chapter \(11\) bankruptcy to allow them time to reorganize and restructure debts. When this happens, the company, after the Chapter \(11\) process is over, offers new stock. This makes the previous stock worthless. However, the company may allow an exchange of old stock for a discounted amount of the new stock. This, in effect, reduces (maybe vastly) the wealth held by those who owned the original stock.
The question of risk hovers over every investment. How risky can it get? Volkswagen seems to be a rather safe investment. But in \(2015,\) Volkswagen’s stock tumbled \(30\%\) over a few days when it was revealed that the company had installed software that altered the emission performance of some of its diesel engines. Volkswagen’s hope was that lower emissions would bolster US sales of some of their diesel models. This was a drastic drop, and many investors lost a lot of money. However, the stock has since recovered. This was mild compared to the \(65\%\) drop in the Martha Stewart Living Omnimedia stocks.
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.
A stock trading platform is a digital tool—typically a website or mobile app—that enables investors to buy and sell stocks, mutual funds, bonds, and other securities. These platforms connect users to financial markets and provide the necessary tools to manage their investments. Some popular stock trading platforms include Robinhood, E*TRADE, Charles Schwab, Fidelity, TD Ameritrade, Vanguard, and Webull.
On the other hand, a crypto trading platform is an online exchange—also available as a website or mobile app—where you can buy, sell, trade, and store cryptocurrencies such as Bitcoin, Ethereum, Solana, and many others. Some well-known crypto trading platforms include Coinbase, Binance, Kraken, and Crypto.com.
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 from the past year \(52\) weeks).
| \(52\)-Week High Low | The highest and the lowest share prices during the past \(52\) weeks. |
|---|---|
| SYM | The company name and the ticker symbol are used to identify the company. |
| DIV | The current annual dividend per share. |
| Yld % | \(\frac{\text{annual divided}}{\text{share price}}\times100\%\) |
| Vol | The number of shares that have been traded today. |
| Open/High/Low | Opening, highest, and lowest prices so far today |
| Close | The price at which the stock traded at the close of the in prior trading day. |
| Net Chg | The difference between the prior trading period’s closing price and the current trading period’s closing price |
| Market Cap | Total stock value of the company. |
| Share Outstanding | The total number of shares that exist for the company to trade. |
| P/E | The share price divided by the earnings per share over the past year (dd indicates loss.) |
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.
\begin{aligned}
\text{P/E} &=\frac{\text{Share price}}{\text{Dividend}}\\
\text{Yld%}&=\frac{\text{Annual Dividend}}{\text{Share price}}\times100\%\\
\text{Earning per share}&=\frac{\text{Share price in opening day}}{\text{P/E ratio}}\\
\end{aligned}
It should be noted that the price of a stock increases and decreases every moment, and so these values change as the share price changes.
Example \(\PageIndex{5}\): Computing Percent Yield
Find the Annual percent yield for a stock with a price of \($30.69\;\) and quarterly dividends of \($1.48 \) per share.
- Answer
-
Substituting the values for price, \($30.69,\) and annual dividend, \($4\times 1.48,\), becuase quarterly dividends of \($1.48 \) per share. We find the Annual percent yield for the stock to be
\begin{aligned}
\text{Yld%} &=\frac{4\times$1.48}{$30.69}\times100\%\\
&=19.28\%
\end{aligned}So the percent yield for a stock is \(19.28\%\)
Example \(\PageIndex{6}\): Reading an Online Stock Table
Consider the stock table Figure \(\PageIndex{1}\), and answer the questions based on the table.
- What is the current price of McDonald’s Corp on this date?
- What is the price of the stock (share) at the start of the day?
- What is the \(52\)-week high? \(52\)-week low?
- When is the dividend expected?
- What is its yield?
- Sujan owns \(500\) shares of McDonald’s Corp stock. How much will Sujan earn in dividends in the quarter?
- What are the earnings per share (EPS)?
- What is the total value of the company?
- Answer
-
- Looking at the table, the current price of a share is \($258.87\).
- The high was \($271.15,\) and the low was \($217.68.\)
- \($255.14\)
- August \(31,\) \(2022\)
- \(2.13%\%\)
- \($500\times1.38 =$690\)
- The EPS value is \($8.12.\)
- \($187.16\) billion
Example \(\PageIndex{7}\): Stock Price Increases
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.
- How much money did Vincent make?
- What was his return on investment for that one year?
- Answer
-
- Vincent spent \($21.87\) per share to buy the stock. The total he spent on the stock was \($21.87\times100=$2,187.\) When he sold the stock, the price was \($29.15,\;\) so he receive \($29.15\times100=$2,915.\)
- He made \($2,915-2,187=$728.\) His return on investment was
\text{ROI} &=\frac{\text{earning }}{\text{original price}}\\
&=\frac{$728}{$2,187}\\
&=0.3329\\
&=33.29\%
\end{aligned}
Mutual Funds
A mutual fund is a collection of investments that are combined into a single portfolio. 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 managed by professional money managers who allocate the fund's assets and strive to generate capital gains or income for the fund's investors. Some mutual funds are
Examples of some mutual funds
-
Vanguard 500 Index Fund (VFIAX)
-
Fidelity 500 Index Fund (FXAIX)
-
Schwab S&P 500 Index Fund (SWPPX)
An example of a mutual fund, VFIAX, is provided online below, with some parts labeled.
Figure \(\PageIndex{4}\):Top \(10\) holding
As we mentioned, a mutual fund is a collection of investments. Figure \(\PageIndex{4}\) shows the top \(10\) collection of stocks in the VFIAX mutual fund. Those collection is called holdings in a mutual fund.
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 that each shareholder participates proportionally in the fund's gains or losses. 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 within the mutual fund, the risk is significantly reduced compared to direct ownership of stocks. Even so, mutual funds historically perform well and can earn more than \(10\%\) annually.
The investments that comprise a mutual fund are structured and managed to align with the stated investment objectives, as specified in its prospectus. A prospectus is a document that provides information about a mutual fund. Before buying shares of a mutual fund, consult its prospectus to consider its goals and strategies and determine if they align with your objectives and values. Additionally, research any associated fees.
Example \(\PageIndex{8}\): Reading Mutual Fund Table
Answer the following questions based on an actual Fidelity Select Communication Service mutual fund.
\(a.\) If you invest \($5,000\) in this fund today, how many shares will you be able to buy?
\(b.\) If you had invested \($5,000\) in this fund three years ago, how much would it be worth today?
\(c.\) If you had invested \($5,000\) in this fund ten years ago, how much would it be worth today?
- Answer
-
Figure \(\PageIndex{1}\): Copy and Paste Caption here. (Copyright; author via source)\(a.\) Since each shares cost \($147.90\), the umber of share we can buy with \($5000\) is: \(\frac{5000}{147.90}=33.8066\)
\(b.\) Here we need to find future value of \($5,000\) investment after \(3\) years.
\begin{align*}
\text{Future Value: FV} &=P\left(1+r\right)^{t}\\
&=5,000\left(1+0.3699\right)^{3}\\
&=12853.94986
\end{align*}\(c.\) Here we need to find future value of \($5,000\) investment after \(10\) years.
\begin{align*}
\text{Future Value: FV} &=P\left(1+r\right)^{t}\\
&=5,000\left(1+0.1503\right)^{10}\\
&=20280.6188
\end{align*}
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 the needs of different investors.
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 \(2026\), if you're 49 or younger, you can contribute up to \($7,500\), and if you're \(50\) or older, you can contribute up to \($8,600\).
Anyone is eligible to contribute to a traditional IRA, regardless of their income level. Your money grows tax-deferred( You pay taxes later when you withdraw the money in retirement), 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. For \(2026\), if you're 49 or younger, you can contribute up to \($7,500\), and if you're \(50\) or older, you can contribute up to \($8,600\). 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, unlike 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.
\(401\)(k) plans with matching funds provide great value, as their rates of return are high compared to savings accounts, and are less risky than 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 \(\PageIndex{9}\): Contribution in \(401\)K
Alice signs up for her employer-based \(401\)(k). The employer matches any \(401\)(k) contribution up to \(6\%\) of the employee's salary. Alice’s annual salary is \(\$51,600.\)
- What is the maximum amount that Alice can deposit, which will be fully matched by the company?
- How much total will be deposited into Alice’s account if she deposits the full \(6\%?\)
- Answer
-
- The employer will match up to \(6\%\) of any employee’s salary. The \(6\%\) of Alice’s salary is \[0.06\times$51,600=$3,096\nonumber\]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\[$3,096+$3,096=$6,192\nonumber\] which is the total that is deposited into Alice’s account.
- She earns a \(100\%\) return on the day she deposits her \($3,096.\)
Which of the types of investments listed would be considered to have the highest level of risk?
Bonds are considered the lowest-risk investment, while stocks are considered the highest-risk investment.