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Methods of Savings

Distinguish various basic forms of savings plans.

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.

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.

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 Money from the CD cannot be withdrawn (without penalty) until the investment period is up.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.

5-Year CD

Try it.

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, $A=P{(1+\frac{r}{n})}^{\text{nt}}$, 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

\[\begin{array}{lll}A & = & P{(1+\frac{r}{n})}^{\text{nt}} \\ & = & \text{\$}10,000.00{(1+\frac{0.0217}{2})}^{2\times 5} \\ & = & \text{\$}10,000.00{(1.01085)}^{10} \\ & = & \text{\$}11,239.53\end{array}\]

The CD will be worth $11,219.53 after 5 years.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

Saving for a Car

Try it.

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?

Solution

Yaroslava has a goal and needs to know the payments to make to reach the goal. Her goal is $\text{FV}$ = $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.

\[\begin{array}{lll}\text{pmt} & = & \frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1} \\ & = & \frac{35,500\times (0.0425/12)}{{(1+0.0425/12)}^{12\times 3}-1} \\ & = & \frac{35,500\times (0.003541\overset{\bar}{6})}{{(1.003541\overset{\bar}{6})}^{36}-1} \\ & = & \frac{125.7291\overset{\bar}{6}}{0.13572901696} \\ & = & 926.325\end{array}\]

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

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

Formulas

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

$\text{ROI}=\frac{FV-P}{P}$

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

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

Practice (7)

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

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?

Cevabı açıkla.

To find the compound interest, use the formula from Compound Interest, $A=P{(1+\frac{r}{n})}^{nt}$, 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
\[\begin{array}{lll}A & = & P{(1+\frac{r}{n})}^{nt} \\ & = & \text{\$}4,520.00{(1+\frac{0.0145}{1})}^{1\times 3} \\ & = & \text{\$}4,520.00{(1.0145)}^{3} \\ & = & \text{\$}4,719.48\end{array}\]
Or, Violet will have $4,719.48 after 3 years.
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?

Cevabı açıkla.

This also uses the compound interest formula from Compound Interest, $A=P{(1+\frac{r}{n})}^{\text{nt}}$, 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
\[\begin{array}{lll}A & = & P{(1+\frac{r}{n})}^{\text{nt}} \\ & = & \text{\$}10,000.00{(1+\frac{0.0217}{2})}^{2\times 5} \\ & = & \text{\$}10,000.00{(1.01085)}^{10} \\ & = & \text{\$}11,239.53\end{array}\]
The CD will be worth $11,219.53 after 5 years.
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?

Cevabı açıkla.

This, once again, uses the compound interest formula from Compound Interest: $A=P{(1+\frac{r}{n})}^{\text{nt}}$, Substituting the values $P$ = $25,000, $r$ = 0.0176, $n$ = 12, and $t$ = 4, we find the account will be worth
\[\begin{array}{lll}A & = & P{(1+\frac{r}{n})}^{\text{nt}} \\ & = & \text{\$}2,500.00{(1+\frac{0.0176}{12})}^{12\times 4} \\ & = & \text{\$}2,500.00{(1.0014\overset{\bar}{6})}^{48} \\ & = & \text{\$}2,682.20\end{array}\]
The money market account will be worth $2,682.20 after 4 years.
1. Determine the return on investment for the 5-year CD from . Round the percentage to two decimal places.
2. Determine the return on investment for the money market account from . Round the percentage to two decimal places.
Cevabı açıkla.
1. 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.
  \[\begin{array}{lll}\text{ROI} & = & \frac{FV-P}{P} \\ & = & \frac{\text{\$}11,239.53-\text{\$}10,000}{\text{\$}10,000} \\ & = & \frac{\text{\$}1,239.53}{\text{\$}10,000} \\ & = & 0.123953\end{array}\]
  The ROI is 12.40%.
1. 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.
  \[\begin{array}{lll}\text{ROI} & = & \frac{FV-P}{P} \\ & = & \frac{\text{\$}2,682.20-\text{\$}2,500}{\text{\$}2,500} \\ & = & \frac{\text{\$}182.20}{\text{\$}2,500} \\ & = & 0.07288\end{array}\]
  The ROI is 7.29%.
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?

Cevabı açıkla.

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 $\text{pmt}$ = 250, $r$ = 0.0375, $n$ = 12, and $t$ = 8 into the formula, we find the future value of the account.
\[\begin{array}{lll}FV & = & \text{pmt}\times \frac{{(1+r/n)}^{n\times t}-1}{r/n} \\ & = & 250\times \frac{{(1+0.0375/12)}^{12\times 8}-1}{0.0375/12} \\ & = & 250\times \frac{{(1.003125)}^{96}-1}{0.003125} \\ & = & 250\times \frac{1.34922752406-1}{0.003125} \\ & = & 250\times \frac{0.34922752406}{0.003125} \\ & = & 250\times 111.752807699 \\ & = & 27,938.202\end{array}\]
The account, after 8 years, will contain $27,938.20.
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?

Cevabı açıkla.

To find the future value of the account, we use the ordinary annuity formula $FV=\text{pmt}\times \frac{{(1+r/n)}^{n\times t}-1}{r/n}$. 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
\[\begin{array}{lll}FV & = & \text{pmt}\times \frac{{(1+r/n)}^{n\times t}-1}{r/n} \\ & = & \text{\$}2,500\times \frac{{(1+0.034/1)}^{1\times 18}-1}{0.034/1} \\ & = & \text{\$}2,500\times \frac{{(1.034)}^{18}-1}{0.034} \\ & = & \text{\$}2,500\times \frac{1.82544897331-1}{0.034} \\ & = & \text{\$}2,500\times 24.2779109798 \\ & = & \text{\$}60,694.77\end{array}\]
After saving for 18 years, Rita and George will have $60,694.77 for Yusef’s college.
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?

Cevabı açıkla.

Yaroslava has a goal and needs to know the payments to make to reach the goal. Her goal is $\text{FV}$ = $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.
\[\begin{array}{lll}\text{pmt} & = & \frac{FV\times (r/n)}{{(1+r/n)}^{n\times t}-1} \\ & = & \frac{35,500\times (0.0425/12)}{{(1+0.0425/12)}^{12\times 3}-1} \\ & = & \frac{35,500\times (0.003541\overset{\bar}{6})}{{(1.003541\overset{\bar}{6})}^{36}-1} \\ & = & \frac{125.7291\overset{\bar}{6}}{0.13572901696} \\ & = & 926.325\end{array}\]
To reach her goal, Yaroslava would need to deposit $926.33 in her account each month.

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: Methods of Savings

Distinguish various basic forms of savings plans.
Compute return on investment for basic forms of savings plans.
Compute payment to reach a financial goal.
Determine the return on investment for the 5-year CD from
Determine the return on investment for the money market account from
The initial deposit in the CD was $10,000, so
The initial deposit in the money market was $2,500, so
Savings account

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.

Kendini dene.

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

Daha fazlası Arithmetic

Order of operations Fractions Percentages Powers and roots