Objectives: Savings Plans 4.7
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.
Section 4.7 Savings Plans
For most of us it is not practical to deposit a large sum of money in the bank. Instead, we save by depositing smaller amounts of money regularly. We might save in an IRA or 401-K for retirement. We might also save for a down payment on a car or house, or in a college savings plan for our children.
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.
Subsection Distinguish Various Basic Forms of Savings Plans
There are at least three types of savings accounts:
- Traditional Savings Account
- Certificate of Deposit
- Money Market
Subsubsection 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.
Who Knew.
Many banks are covered by FDIC insurance. The FDIC is the Federal Deposit Insurance Corporation and 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. 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.
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.
Subsection Savings Plan Formulas
This formula is used when we want to calculate the future value of a single payment or deposit. Here is the future value formula again:
The mathematical formula is shown below.
Savings Plan Formulas.
\begin{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t}
\end{equation*}
- A(t)
- is the balance in the account after t years (future value)
- r
- is the annual interest rate in decimal form
- n
- is the number of compounding periods in one year
- t
- is the number of years
If the compounding is done annually (once a year), \(n=1\text{.}\)
If the compounding is done quarterly, \(n=4\text{.}\)
If the compounding is done monthly, \(n=12\text{,}\)
If the compounding is done weekly, \(n=52\text{,}\)
If the compounding is done daily, \(n=365\)
Example 4.7.1.
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?
Solution.
To find the compound interest, use the formula from Compound Interest,
\begin{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t},
\end{equation*}
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{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t}
\end{equation*}
\begin{equation*}
A(t) = \displaystyle $4,520 \left(1+\frac{0.0145}{1}\right)^{1*3}
\end{equation*}
\begin{equation*}
A(t) = $4,520(1.0145)^{3}
\end{equation*}
\begin{equation*}
A(t) = $4,719.48
\end{equation*}
Violet will have $4,719.48 after 3 years.
Checkpoint 4.7.2.
Brian deposits $5,600 in a savings account that yields 1.23% interest compounded semi-annually. If he leaves that deposit in the account and adds nothing new to the account, what will the account be worth in 5 years?
Solution.
$5,954.09
Who Knew.
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.
Subsection Certificate of Deposits, or CDs
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.
Example 4.7.3.
Certificate of Deposit
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?
Solution.
This also uses the compound interest formula from Compound Interest,
\begin{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t},
\end{equation*}
Substituting the values P = $10,000, r = 0.0217, n = 2 (semiannually means twice per year), t = 5, we find the account will be worth,
\begin{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t}
\end{equation*}
\begin{equation*}
A(t) = \displaystyle $10,000 \left(1+\frac{0.0217}{2}\right)^{2*5}
\end{equation*}
\begin{equation*}
A(t) = $10,000(1.01085)^{10}
\end{equation*}
\begin{equation*}
A(t) = $11,239.53
\end{equation*}
The CD will be worth $11,239.53 after 5 years.
Checkpoint 4.7.4.
Denise deposits $3,500 in a CD bearing 2.23% interest compounded quarterly for 3 years. How much will Denise’s CD be worth after those 3 years?
Solution.
$3,741.46
Subsection 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.
Example 4.7.5.
Single Deposit to a Money Market Account
Marietta opens a money market account, and deposits $2,500.00 in the account. It bears 1.76% interest compounded monthly. If she makes no other transactions on the account, how much will be in the account after 4 years?
Solution.
This also uses the compound interest formula from Compound Interest,
\begin{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t},
\end{equation*}
Substituting the values P = $2,500, r = 0.0176, n = 12, t = 4, we find the account will be worth,
\begin{equation*}
A(t) = \displaystyle P \left(1+\frac{r}{n}\right)^{n*t}
\end{equation*}
\begin{equation*}
A(t) = \displaystyle $2,500 \left(1+\frac{0.0176}{12}\right)^{12*4}
\end{equation*}
\begin{equation*}
A(t) = $2,500(1.00146)^{48}
\end{equation*}
\begin{equation*}
A(t) = $2,682.20
\end{equation*}
The money market account will be worth $2,682.20 after 4 years.
Checkpoint 4.7.6.
Chuck opens a money market account, and deposits $8,500.00 in the account. It bears 1.83% interest compounded quarterly. If he leaves makes no other transactions on the account, how much will be in the account after 3 years?
Solution.
$8,978.57
Subsection 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.
Formula.
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{equation*}
ROI = \displaystyle \dfrac{FV - P}{P},
\end{equation*}
expressed as a percentage.
Checkpoint.
The length of time of the investment is not considered in ROI.
Example 4.7.7.
Determine the return on investment for the 5-year CD from Example 4.7.3. Round the percentage to two decimal places.
Solution.
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 FV = $11,239.53. Substituting and computing we find the return on investment.
\begin{equation*}
ROI = \displaystyle \dfrac{FV - P}{P},
\end{equation*}
\begin{equation*}
ROI = \displaystyle \dfrac{11,239.53 - 10,000}{10,000},
\end{equation*}
\begin{equation*}
ROI = \displaystyle \dfrac{1,239.53}{10,000},
\end{equation*}
\begin{equation*}
ROI = 0.123953
\end{equation*}
The Return on Investment, ROI is 12.40%.
Example 4.7.8.
Calculating Return on Investment
Determine the return on investment for the money market account from Example 4.7.5. Round the percentage to two decimal places.
Solution.
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 FV = $2,682.20. Substituting and computing we find the return on investment.
\begin{equation*}
ROI = \displaystyle \dfrac{FV - P}{P},
\end{equation*}
\begin{equation*}
ROI = \displaystyle \dfrac{2,682.20 - 2,500}{2,500},
\end{equation*}
\begin{equation*}
ROI = \displaystyle \dfrac{182.20}{2,500},
\end{equation*}
\begin{equation*}
ROI = 0.07288
\end{equation*}
The Return on Investment, ROI is 7.29%.
Checkpoint 4.7.9.
1. The amount of $13,000 is invested in a savings account. After 10 years the account has $15,250.00. Find the return on investment for this account.
2. The amount of $6,500 is deposited in a money market account. After 7 years, the account has $7,358.00. Find the return on investment for this account.
Solution.
1. 17.31%
2. 13.2%
Subsection 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.
Formula for the Future Value of an Ordinary Annuity.
The formula for the future value of an ordinary annuity is:
\begin{equation*}
FV = \frac{pmt\left(\left(1+\frac{r}{n}\right)^{nt}-1\right)}{\left(\frac{r}{n}\right)},
\end{equation*}
- FV
- 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.
- t
- is the number of years.
Checkpoint.
1. It is important to note that the number of deposits per year and the number of periods per year are the same.
2. Another form of annuity if 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 4.7.11.
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?
Solution.
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{equation*}
FV = \frac{pmt\left(\left(1+\frac{r}{n}\right)^{nt}-1\right)}{\left(\frac{r}{n}\right)},
\end{equation*}
\begin{equation*}
FV = \frac{250\left(\left(1+\frac{0.0375}{12}\right)^{12*8}-1\right)}{\left(\frac{0.0375}{12}\right)},
\end{equation*}
\begin{equation*}
FV = \dfrac{250((1.003125)^{96}-1)}{0.003125}
\end{equation*}
\begin{equation*}
FV = 27.938.202
\end{equation*}
The account, after 8 years will contain $27,938.20.
Checkpoint 4.7.12.
Kelly invests $525 every third month (quarterly payments), at the end of the compounding period, into an account bearing 3.89% interest compounded quarterly. How much will be in the account after 15 years?
Solution.
$42,499.63
Setting Savings Account Interest Rates.
There are a number of factors that contribute to the amount 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 they lend out money. Finally, interest rates are impacted by the Federal Reserve Bank. When the Fed raises interest rates, so do banks.
The Federal Reserve Chairperson.
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. Currently, the Fed Chair is Jerome Powell, who has served since 2018.
Example 4.7.13.
Future Value of an Ordinary Annuity: 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?
Solution.
To find the future value of the account, we use the ordinary annuity formula;
\begin{equation*}
FV=\frac{pmt\left(\left(1+\frac{r}{n}\right)^{nt}-1\right)}{\left(\frac{r}{n}\right)},
\end{equation*}
The payment pmt = $2,500, the interest rate r = 0.034 (as a decimal) the number of compunding periods n = 1, the number of years, t = 18. We find the future value of the account by substituting these values and computing,
\begin{equation*}
FV=\frac{pmt\left(\left(1+\frac{r}{n}\right)^{nt}-1\right)}{\left(\frac{r}{n}\right)},
\end{equation*}
\begin{equation*}
FV=\frac{2500\left(\left(1+\frac{0.034}{1}\right)^{1*18}-1\right)}{\left(\frac{0.034}{1}\right)},
\end{equation*}
\begin{equation*}
FV=\dfrac{2,500((1.034)^{18}-1)}{0.034}
\end{equation*}
\begin{equation*}
FV = $60,694.77
\end{equation*}
After saving for 18 years, Rita and George will have $60,694.77 for Yusef’s college.
Checkpoint 4.7.14.
Bemnet saves $280 per month in a savings account bearing 3.11% interest compounded monthly. After 20 years, how much does Bemnet have in the account?
Solution.
$93,037.59
Subsection 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.
Formula for the Amount that Needs to be Deposited per period pmt, of an ordinary annuity.
The formula for the amount that needs to be deposited per period, pmt, of an ordinary annuity to reach a specified goal, FV is:
\begin{equation*}
pmt = \frac{FV\left(\frac{r}{n}\right)}{\left(\left(1+\frac{r}{n}\right)^{nt}-1\right)},
\end{equation*}
- pmt
- is the payment to be calculated for.
- FV
- is the future value of the annuity.
- r
- is the annual interest rate (in decimal form)
- n
- is the number of compounding periods per year.
- t
-
is the number of years.With this formula, it is possible to plan the amount to be saved
Example 4.7.15.
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 willl 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?
Solution.
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{equation*}
pmt = \frac{FV\left(\frac{r}{n}\right)}{\left(\left(1+\frac{r}{n}\right)^{nt}-1\right)},
\end{equation*}
\begin{equation*}
pmt = \frac{35,500\left(\frac{0.0425}{12}\right)}{\left(\left(1+\frac{0.0425}{12}\right)^{12*3}-1\right)},
\end{equation*}
\begin{equation*}
pmt = \frac{35,500 * 0.0035416}{1.0035416^{36}-1},
\end{equation*}
\begin{equation*}
pmt = \frac{125.72916}{0.13572901696}
\end{equation*}
\begin{equation*}
pmt = 926.325
\end{equation*}
To reach her goal, Yaroslava would need to deposit $926.33 in her account each month.
Note 4.7.16.
This has been rounded up, so that the deposits don’t fall short of the goal. However, some round off using the standard rounding rules: if the last digit is 1, 2, 3, or 4, the number is rounded down; if the last digit is 5, 6, 7, 8, or 9 the number is rounded up.
Checkpoint 4.7.17.
Chione decides to put new siding on her house. She finds that it will cost about $27,800. She decides to begin saving for the purchase so that she doesn’t take on debt to side the house. How much would Chione need to deposit every 6 months in an ordinary annuity that yields 5.16% compounded semi-annually for 5 years?
Solution.
$2,250.60
Exercises Exercises
1.
You set up a savings plan for retirement in 35 years. You will deposit $250 each month for 35 years. The account will earn an average of 6.5% APR compounded monthly.
- How much will you have in your retirement plan in 35 years?
- How much interest did you earn.
- What percent of the balance is interest?
2.
You set up a savings plan for retirement in 40 years. You will deposit $75 each week for 40 years.The account will earn an average of 8.5% APR compounded weekly.
- How much will you have in your retirement plan in 40 years?
- How much interest did you earn.
- What percent of the balance is interest?
3.
You set up a savings plan for retirement in 30 years. You will deposit $750 each quarter for 30 years. The account will earn an average of 7.75% APR compounded quarterly.
- How much will you have in your retirement plan in 30 years?
- How much interest did you earn?
- What percent of the balance is interest?
4.
You set up a savings plan for retirement in 25 years. You will deposit $20 per day for 25 years. The account will earn an average of 2.35% APR compounded daily.
- How much will you have in your retirement plan in 25 years?
- How much interest did you earn?
- What percent of the final balance is interest?
5.
Suppose you invest $130 a month for 5 years into an account earning 9% APR compounded monthly. After 5 years, you leave the money, without making additional deposits, in the account for another 25 years.
- How much will you have in the end?
- How much interest did you earn?
- What percent of balance is interest?
6.
Suppose you invest $200 per month for 10 years into an account earning 5% APR compounded monthly. You then leave the money, without making additional deposits, in the account for another 20 years.
- How much will you have after the first 10 years?
- How much will you have after the additional 20 years?
- How much total interest did you earn?
- What percent of the final balance is interest?
7.
Suppose you have 30 months in which to save $3,500 for a cruise for your family. If you can earn an APR of 3.8%, compounded monthly, how much should you deposit each month?
8.
You wish to have $3,000 in 2 years to buy a fancy new stereo system. How much should you deposit each quarter into an account paying 6.5% APR compounded quarterly?
9.
Jamie has determined they need to have $450,000 for retirement in 30 years. Their account earns 6% APR. How much would Jamie need to deposit in the account each month?
10.
Lashonda already knows that she wants $500,000 when she retires. If she sets up a saving plan for 40 years in an account paying 10% APR, compounded quarterly, how much should she deposit each quarter?
11.
Jose’ inherits $55,000 and decides to put it in the bank for the next 25 years to save for his retirement. He will earn an average of 5.6% APR compounded monthly for the next 25 years. His partner deposits $375 a month in a separate savings plan that earns 5.6% APR compounded monthly for the next 25 years.
- How much will each have at the end of 25 years?
- How much interest did each person earn?
- What percent of balance is interest for each person?
12.
Akiko inherits $45,000 and decides to put it in the bank for the next 30 years to save for her retirement. She will earn an average of 7.8% APR compounded monthly for the next 30 years. Her spouse deposits $200 a month in a separate savings plan that earns 7.8% APR compounded monthly for the next 30 years.
- How much will each have at the end of 30 years?
- How much interest did each person earn?
- What percent of balance is interest for each person?
13.
Sylvin makes an initial deposit of $1000 into a savings account and then adds $100 each month for 10 years into an account pays 4.5% APR compounded monthly.
- What will be her final balance?
- How much interest did she earn?
- What percent of the final balance is interest?
14.
Elena makes an initial deposit of $5000 into a savings account and then adds $1000 each year for 20 years into an account pays 2.35% APR compounded annually.
- What will be her final balance?
- How much interest did she earn?
- What percent of her final balance is interest?
15.
Vanessa just turned 40 years old. Her plan is to save $100 per month until retirement at age 65. Suppose she deposits that $100 each month into a savings account that earns 4% APR compounded monthly.
- What will her balance be when she turns 65 years old?
- If she started saving when she turned 25 years old instead, what would her balance be?
16.
Chris wants to start saving money for retirement. Suppose he deposits $1000 every year into a savings account that pays 5% APR compounded annually.
- How much will Chris have saved in 20 years?
- How much will Chris have saved in 40 years?
17.
Fareshta and Ahmad want to save to help send their child to college. Their plan is to put aside $50 every week. Suppose they deposit that money into an account that pays 3.5% APR compounded weekly.
- How much money will be in the account in 18 years? (assume 52 weeks in a year)
- What minimum initial lump sum deposit would they need to make today to have the same balance in 18 years if they weren’t putting aside the $50 per week?
18.
Elisa decides to cancel her cable TV and to deposit the $100 she will save each month into an account that pays 4.5% APR compounded monthly.
- How much will she have in the account in 10 years?
- What minimum initial lump sum deposit would she need to make today to have the same balance in 10 years without saving the $100 per month?
