6.6 Methods of Savings

Four bundles of currency notes are stacked one above the other. — **Figure 6.8** Money wisely invested grows over time.Money wisely invested grows over time. (credit: “Stack of Cash” by Janak Raja/Flickr, Public Domain Mark 1.0)

Learning Objectives

After completing this section, you should be able to:

Distinguish various basic forms of savings plans.
Compute return on investment for basic forms of savings plans.
Compute payment to reach a financial goal.

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.

Distinguish Various Basic Forms of Savings Plans

There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money market accounts are three main savings account vehicles.

Savings Account

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, which bears some 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, that a savings account will have a minimum balance in order for the bank to pay maintenance costs. If your account should dip below the minimum, there are usually fees attached.

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.

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 called the future value of the account.

Example 1 Single Deposit in a Savings Account

Violet deposits $4,520.00 in a savings account bearing 1.45% interest compounded annually. If she does not add to or withdraw any of that money, how much will be in the account after 3 years?

To find the compound interest, use the formula from Compound Interest, $A = P {(1 + \frac{r}{n})}^{n t}$ , where $A$ represents the amount in the account after $t$ years, with initial deposit (or principal) of $P$ , at an annual interest rate, in decimal form, of $r$ , compounded $n$ times per year. Violet has a principal of $4,520.00, which will earn an interest of $r$ = 0.0145, compounded yearly (so $n$ = 1), for $t$ = 3 years. Substituting and calculating, we find that Violet’s account will be worth

$\begin{matrix} A & = & P {(1 + \frac{r}{n})}^{n t} \\ = & $ 4,520.00 {(1 + \frac{0.0145}{1})}^{1 \times 3} \\ = & $ 4,520.00 {(1.0145)}^{3} \\ = & $ 4,719.48 \end{matrix}$

Or, Violet will have $4,719.48 after 3 years.

Certificates of Deposit, or CDs

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 investment period is up. Also, money cannot be added to an existing CD.

Certificates of deposit have features similar to 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.

Money Market Account

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 have a minimum balance that must be maintained. If the balance in the account drops below the minimum, there is likely to be a penalty. Money market accounts offer the flexibility of checks and ATM cards. Finally, the interest rate on a money market account is typically higher than the interest rate on a savings account.

Return on Investment

If we want to compare the profitability of different investments, like savings accounts versus other investment tools, we need a measure that evens the playing field. Such a measure is return on investment.

Example 4 Calculating Return on Investment

Determine the return on investment for the 5-year CD from Example 2. Round the percentage to two decimal places.
Determine the return on investment for the money market account from Example 3. Round the percentage to two decimal places.

The initial deposit in the CD was $10,000, so $P$ = $10,000. The value at the end of 5 years was $11,239.53. so $F V$ = $11,239.53. Substituting and computing we find the return on investment.

$\begin{matrix} ROI & = & \frac{F V - P}{P} \\ = & \frac{$ 11,239.53 - $ 10,000}{$ 10,000} \\ = & \frac{$ 1,239.53}{$ 10,000} \\ = & 0.123953 \end{matrix}$

The ROI is 12.40%.

The initial deposit in the money market was $2,500, so $P$ = $2,500. The value at the end of 4 years was $2,682.20. so $F V$ = $2,682.20. Substituting and computing we find the return on investment.

$\begin{matrix} ROI & = & \frac{F V - P}{P} \\ = & \frac{$ 2,682.20 - $ 2,500}{$ 2,500} \\ = & \frac{$ 182.20}{$ 2,500} \\ = & 0.07288 \end{matrix}$

The ROI is 7.29%.

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. This agrees with the 50-30-20 budget philosophy, where some income is saved every month. When a deposit is made at the end of each compounding period, such a savings account is called an ordinary annuity.

The formula for the future value of an ordinary annuity is $F V = pmt \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n}$ , where $F V$ is the future value of the annuity, $pmt$ is the payment, $r$ is the annual interest rate (in decimal form), $n$ is the number of compounding periods per year, and $t$ is the number of years.

Example 5 Future Value of an Ordinary Annuity

Jill has an account that bears 3.75% interest compounded monthly. She decides to deposit $250.00 each month, at the end of the compounding period, into this account. What is the future value of this account, after 8 years?

These are regular payments into an account bearing compound interest. She is depositing them at the end of each compounding period. This makes this an ordinary annuity. Substituting the values $pmt$ = 250, $r$ = 0.0375, $n$ = 12, and $t$ = 8 into the formula, we find the future value of the account.

$\begin{matrix} F V & = & pmt \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n} \\ = & 250 \times \frac{{(1 + 0.0375 / 12)}^{12 \times 8} - 1}{0.0375 / 12} \\ = & 250 \times \frac{{(1.003125)}^{96} - 1}{0.003125} \\ = & 250 \times \frac{1.34922752406 - 1}{0.003125} \\ = & 250 \times \frac{0.34922752406}{0.003125} \\ = & 250 \times 111.752807699 \\ = & 27,938.202 \end{matrix}$

The account, after 8 years, will contain $27,938.20.

Example 6 Saving for College

When Yusef was born, Rita and George began to save for Yusef’s college years by investing $2,500 each year in a savings account bearing 3.4% interest compounded annually. How much will they have saved after 18 years?

To find the future value of the account, we use the ordinary annuity formula $F V = pmt \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n}$ . The payment is $2,500, rate is 0.034, the number of compounding periods is 1, and the number of years is 18. Substituting these values and computing, we have

$\begin{matrix} F V & = & pmt \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n} \\ = & $ 2,500 \times \frac{{(1 + 0.034 / 1)}^{1 \times 18} - 1}{0.034 / 1} \\ = & $ 2,500 \times \frac{{(1.034)}^{18} - 1}{0.034} \\ = & $ 2,500 \times \frac{1.82544897331 - 1}{0.034} \\ = & $ 2,500 \times 24.2779109798 \\ = & $ 60,694.77 \end{matrix}$

After saving for 18 years, Rita and George will have $60,694.77 for Yusef’s college.

Tech Check

Google Sheets offers a function to calculate the future value of an ordinary annuity. To get Google Sheets to calculate the future value, you use the following:

=fv(rate,number_of_periods, payment, present_value, end_or_beginning).

To explain, the rate is the rate per compounding period. From our formula, that is $r / n$ . Also, the number of periods must be entered. From our formula, that is $n \times t$ . The payment is the amount deposited each period. Present value is 0 if we begin with no money and rely only on the payments to be made. However, if some money is available to put in the account before the payments start, that amount, an initial deposit, would be the value of $P V$ . Finally, for an ordinary annuity, enter 0 for end or beginning. Using the values for Jill, the payment amount is $250, $r$ = 0.0375, $n$ = 12, $t$ = 8, and that there is no initial deposit, $P V$ = 0, the Google Sheets formula is

=fv(0.0375/12,12*8,250,0,0).

Figure 6.9 shows the formula in Google Sheets.

Figure 6.9 Google Sheets formula

Hitting the enter key shows the payment value (Figure 6.10).

Figure 6.10 Payment value

Notice that the future value is negative, since it is a payment leaving an account.

Compute Payment to Reach a Financial Goal

The formula used to get the future value of an ordinary annuity is useful, finding out what the final amount in the account will be. However, that isn’t how planning works. To plan, we need to know how much to put into the ordinary annuity each compounding period in order to reach a goal. Fortunately, that formula exists.

With this formula, it is possible to plan the amount to be saved.

Example 7 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?

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.

$\begin{matrix} pmt & = & \frac{F V \times (r / n)}{{(1 + r / n)}^{n \times t} - 1} \\ = & \frac{35,500 \times (0.0425 / 12)}{{(1 + 0.0425 / 12)}^{12 \times 3} - 1} \\ = & \frac{35,500 \times (0.003541 \bar{6})}{{(1.003541 \bar{6})}^{36} - 1} \\ = & \frac{125.7291 \bar{6}}{0.13572901696} \\ = & 926.325 \end{matrix}$

To reach her goal, Yaroslava would need to deposit $926.33 in her account each month.

Tech Check

Google Sheets offers a function to calculate the payment necessary to reach a goal using ordinary annuities. To get Google Sheets to calculate the payment, you use the following:

=pmt(rate,number_of_periods, present_value, future_value, end_or_beginning).

To explain, the rate is per compounding period. From our formula, that is $r / n$ . Also, the number of periods must be entered. From our formula, that is $n \times t$ . The present value is the amount of money that the account begins with. If we begin with no money and rely only on the payments to be made, then this number is 0. However, if some money is available to put in the account before the payments start, that amount, an initial deposit, would be the value of $P V$ . Next, enter the future value, $F V$ . Finally, for an ordinary annuity, enter 0 for end or beginning. Using the values for Yaroslava, $r$ = 0.0425, $n$ = 12, $t$ = 3, and that there is no initial deposit, $P V$ = 0, the Google Sheets formula is

=pmt(0.0425/12,12*3,0,35500,0).

Figure 6.11 shows the formula in Google Sheets.

Figure 6.11 Google Sheets formula

Hitting the enter key shows the payment value (Figure 6.12).

Figure 6.12 Payment value

Notice that the payment is negative, since it is a payment leaving an account. Additionally, the payment is $926.32. We rounded that up, but Google Sheets rounded off.

Key Terms

Savings account
1099 form
Certificate of deposit
Money market account
Return on investment
Ordinary annuity

Key Concepts

There are three main types of savings accounts, saving accounts, certificates of deposit (CD), and money market accounts.
Savings account are very risk free, and so yield low interest rates.
The differences in the three types of savings accounts relate to their convenience.
Savings account typically have a lower interest rate that money market accounts, which typically have lower interest rates than CDs.
Ordinary annuities more accurately reflect how we save, in that money is deposited repeatedly over time.
Spreadsheet software, such as Google Sheets, have built in functions that can be used to quickly calculate both the future value of an ordinary annuity account, but also the payment necessary to reach a goal using an ordinary annuity.

Videos

Formulas

$A = P {(1 + \frac{r}{n})}^{n t}$

$ROI = \frac{F V - P}{P}$

$F V = p m t \times \frac{{(1 + r / n)}^{n \times t} - 1}{r / n}$

$p m t = \frac{F V \times (r / n)}{{(1 + r / n)}^{n \times t} - 1}$

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.