Section 4.5: Annuities
- Page ID
- 216983
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- Compute the future value of an annuity
In many financial situations, money is deposited or paid out in equal amounts at regular intervals. Examples include monthly deposits into a savings account, quarterly contributions to a retirement plan, or annual payments into an education fund. Situations like these are modeled using annuities.
An annuity is a sequence of equal payments made at equal time intervals, where each payment is deposited into an account that earns compound interest.
Common payment intervals include monthly, quarterly, or yearly deposits. In this chapter, unless otherwise noted, we assume that each payment is made at the end of the period. This type of annuity is called an ordinary annuity.
Why Annuities Are Important
Many long-term financial plans, such as, retirement savings, emergency funds, or education accounts, are built using regular contributions. Understanding how annuities work enables individuals to:
- Estimate how much their savings will grow over time
- Determine how much they need to save to reach a financial goal
- Make informed decisions about budgeting and long-term planning
In this section, we focus on the future value of annuities, which emphasizes the accumulation of savings.
Types of Common Annuities
There are several different annuities such as fixed annuities, variable annuities, indexed annuities, and immediate annuities. Later on in the text, we will focus on the simple fixed annuity. There are also deferred annuities employer‑sponsored plans like 403(b) and 401(k). Theses plans are the most common annuities that are offered within the workforce.
- 401(k) - A 401(k) is a retirement savings account offered by many employers. It allows you to save money from your paycheck before you spend it and before taxes are taken out. The name comes from a section of the U.S. tax code, Section 401(k).
- 403(b) - A 403(b) is a special type of retirement savings account for people who work in public schools, colleges and universities, nonprofit organizations, churches and religious groups, or some hospitals and public service organizations. It works a lot like a 401(k), but it’s designed for teachers and nonprofit workers.
- 457(b) - A 457(b) is a deferred compensation plan. That means you choose to set aside part of your paycheck into the plan, and you pay taxes later, when you withdraw the money in retirement. These plans are offered in public schools, state and local government agencies, and certain nonprofits. This makes them common among teachers, police, firefighters, city employees, and nonprofit staff. Unlike 401(k) and 403(b) plans, 457(b) plans do NOT charge a 10% early‑withdrawal penalty if you take money out before age 59½, as long as you’ve separated from your employer.
How a Traditional 401(k), 403(b), and 457(b) Works
- Money comes out of your paycheck automatically.
- You choose how much to contribute. For example, 5% of your paycheck.
- You never “see” that money in your checking account, so it’s easier to save.
- You usually don’t pay taxes on that money right away.
- Contributions are pre‑tax, meaning:
- You pay less in taxes today.
- The money grows in the account over many years.
- You pay taxes later, when you retire and withdraw it.
- Many employers may give you free money. They may match your contribution or something less to. For example, you contribute 5% of your paycheck, your employer matches 3% or even 5%. That means your employer adds money to your 401(k)/403(b)/457(b) for free, just because you saved. This is one of the biggest financial benefits an employer can offer.
- The money is typically invested in options such as stocks, bonds, or mutual funds, depending on the financial companies partnered with the employer. These investments have the potential to grow significantly over long periods of time.
Some major 401(k) providers for corporations are, Fidelity Investments, Vanguard, T. Rowe Price, Principal Financial Group, and Empower Retirement. Some other providers that are popular with small & mid‑size corporations are Guideline, ADP Retirement Services, Paychex Retirement Services, Human Interest, ShareBuilder 401k, and Charles Schwab Retirement Services. Some major 403(b) providers are Fidelity Investments, TIAA, Vanguard, and Empower Retirement. There aren't as many 403(b) providers because the plans have stricter eligibility rules. Lastly, some major providers for corporations are, Fidelity Investments, TIAA, Voya Financial, Empower Retirement, Nationwide, ICMA-RC, Lincoln Financial Group, Colebridge Financial, Principal Financial Group, and Metlife.
How a Traditional 401(k), a 403(b), and a 457(b) are Different
- Who Can Offer Them
- 401(k) is for for‑profit companies.
- 403(b) is for nonprofits, public schools, colleges, universities, and churches.
- 457(b) is for teachers, police, firefighters, city employees, and nonprofit staff.
- Investment Options
- 401(k) and 457(b) plans typically offer mutual funds and ETFs (Exchange Trades Funds; i.e, stocks and bonds).
- 403(b) plans historically focused on annuities, though many now include mutual funds.
- Employer Matching & Profit Sharing
- Matching is more common in 401(k)s; less common in 403(b)s; no matching in 457(b).
- Profit sharing is allowed in 401(k)s, but NOT in 403(b)s (because nonprofits have no profits to share).
- Additional Catch-Up Contributions (Catch‑up contributions are extra amounts you’re allowed to put into certain retirement accounts once you reach age 50 or older, beyond the standard annual IRS contribution limits. They are designed to help people who may be behind on retirement saving or who simply want to boost their savings in the years closer to retirement.)
- All plans allow age‑50+ catch-ups.
- 403(b) offers an extra catch-up for employees with 15+ years of service (up to $3,000/year).
- 473(b) offers a special “3‑year” pre‑retirement catch‑up
- ERISA Rules (Compliance & Oversight)
- Most 401(k)s are covered by ERISA (the Employee Retirement Income Security Act of 1974), a federal law that requires that sets standards for private‑sector retirement plans that must follow strict fiduciary rules. Its purpose is to protect employees’ retirement money.
- Some 403(b)s, especially governmental and church plans, are NOT fully subject to ERISA. It’s not guaranteed by any federal insurance program, and
it does not have ERISA’s strict fiduciary protections, meaning you may face higher fees or less oversight depending on your state and plan sponsor. - Governmental 457(b) plans are not ERISA‑covered (they follow state rules instead).
Why Students Should Care
Many students choose a career because they’re passionate about the work or attracted to a certain salary. But there’s another important factor to consider: the retirement benefits your future employer offers. Even though retirement may feel decades away, understanding plans like a 401(k), a 403(b), and a 457(b) matters because:
- You’ll likely be offered one in your first full‑time job
- Starting early allows your money to grow significantly over time
- Employer matching is essentially free money toward your future
- It helps you build long‑term financial stability
Time is the most powerful force in investing and young people have more of it than anyone else. Starting early can make an enormous difference in your financial future.
The future value of an annuity is the total amount accumulated in the account after the final payment has been made, including all interest earned. Because each payment is deposited at a different time, each payment earns a different amount of interest. Earlier payments earn interest over many periods, while later payments earn interest over fewer periods. Compound interest is interest calculated on both the initial principal (the original amount of money) and the accumulated interest from previous periods. In simple terms, you earn interest on your interest.
The future value of an annuity formula is as follows:
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
where:
- \(A\) = Future value (the accumulated amount)
- \(R\) = Regular periodic payment at the end of each period
- \(r\) = Annual interest rate (expressed as a decimal)
- \(n\) = Number of payments made per year
- \(t\) = Term of annuity (in years)
This formula accounts for the compound interest earned on each payment. As a note, the number of payments made per year also matches the number of times the amount each time is compounded per year.
Let's look back at a Section 4.3 example involving compound interest:
Suppose you invest $5,000 at an 8% interest rate for 5 years compounded semiannually. Find the future value.
Using the compound interest formula from Section 4.3, we have:
\[\begin{align} A &= $\text{5,000} \left(1 + \frac{0.08}{2}\right)^{2\cdot 5}\nonumber \\ A &= $\text{7,401.22}\nonumber \\ \end{align}\]
So, after 5 years, the future value is \($\text{7,401.22}\). The interest earned is \($\text{2,401.22}\).
Here, this is a one time lump sum of $5,000 invested over 5 years compounded twice a year. But, maybe you don't have $5,000 saved up to invest. So, what if we invested $5,000 a different way. Instead, let's invest $500 twice a year for 5 years. This is the same amount of $5,000; however, $500 every 6 months may be more manageable than a one time deposit of $5,000, especially for a student or anyone on a fixed income. So, here is the new question:
What amount will accumulate, if we deposit $500 semiannually for the next 5 years at an interest rate of 8%?
✅ Solution:
Start with the future value of an annuity interest formula.
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
We know that \(R=$500;~r=0.08;~t=5;~n=2\), so
\[\begin{align} A &=\frac{500\left[\left(1+\frac{0.08}{2}\right)^{2 \cdot 5}-1\right]}{\frac{0.08}{2}}\nonumber \\[8pt] A &=\frac{500\left[(1.04)^{10}-1\right]}{0.04}\nonumber \\[8pt] A &=\frac{500\left[0.48024428491834\right]}{0.04}\nonumber \\[8pt] A &= $\text{6,003.0535614796}\nonumber \\[8pt] \end{align}\]
\[\boxed {A \approx $\text{6,003.05}} \nonumber\\ \]
So, after 5 years, the future value is \($\text{6,003.05}\).
The interest earned is
\[\begin{align} I &=\text{Future Value}-\text{(Total Number of Payments)} \cdot \text{(Periodic Payment)}\nonumber \\[8pt] I &=$\text{6,003.05}-\text{(2}\cdot \text{5)} \cdot ($\text{500})\nonumber \\[8pt] I&=$\text{6,003.05}-$\text{5,000} \nonumber \\[8pt] \end{align}\]
\[\boxed {I=$\text{1,003.05}}\nonumber \\[8pt]\]
The interest earned is \($\text{1,003.05}\).
Let's focus on Example #4.5.1 and look at within a year-by-year process to further understand to see what is really happening. Here is a table showing the same calculations:
| Year | Beginning Balance |
Simple Interest (8% per year; 4% per period) |
Deposit (at end of period) |
Ending Balance after t Years (Beginning Balance + Interest + Deposit) |
|---|---|---|---|---|
| 0 | $0 | $0 | $0 | $0 |
| \(\frac{1}{2}\) | $0 | ($0)∙(0.08)∙\((\frac{1}{2})\) = $0 | $500 | $500 |
| 1 | $500 | ($500)∙(0.08)∙\((\frac{1}{2})\) = $20 | $500 | $1,020 |
| 1\(\frac{1}{2}\) | $1,020 | ($1,020)∙(0.08)∙\((\frac{1}{2})\) = $40.80 | $500 | $1,560.80 |
| 2 | $1,560.80 | ($1,560.80)∙(0.08)∙\((\frac{1}{2})\) = $62.43 | $500 | $2,123.23 |
| 2\(\frac{1}{2}\) | $2,123.23 | ($2,123.23)∙(0.08)∙\((\frac{1}{2})\) = $84.93 | $500 | $2,708.16 |
| 3 | $2,708.16 | ($2,708.16)∙(0.08)∙\((\frac{1}{2})\) = $108.33 | $500 | $3,316.49 |
| 3\(\frac{1}{2}\) | $3,316.49 | ($3,316.49)∙(0.08)∙\((\frac{1}{2})\) = $132.66 | $500 | $3,949.15 |
| 4 | $3,949.15 | ($3,949.15)∙(0.08)∙\((\frac{1}{2})\) = $157.97 | $500 | $4,607.12 |
| 4\(\frac{1}{2}\) | $4,607.12 | ($4,607.11)∙(0.08)∙\((\frac{1}{2})\) = $184.28 | $500 | $5,291.40 |
| 5 | $5,291.40 | ($5,291.40)∙(0.08)∙\((\frac{1}{2})\) = $211.66 | $500 | $6,003.06 |
We essentially get the same result of $6,003.05 (off by one cent due to round off error) when we use the table as we did by using the formula.
So, let's compare the two different ways of investing the same total amount of money:
- A one‑time lump sum of $5,000 invested for 5 years compounded semiannually - (Compound Interest - Section 4.3)
- A periodic annuity: $500 twice per year for 5 years (for a total of $500 × 10 = $5,000) - (Annuities - Current Section 4.5)
When you invest $5,000 all at once, every dollar sits in the account for the entire 5‑year period. Because the interest is compounded twice a year (semiannually):
- The money earns interest 10 times
- Each interest amount becomes part of the principal
- The investment grows on itself (“interest on interest”)
This method gives the maximum possible growth, because the full $5,000 is working for the entire five years.
With the periodic annuity:
- You invest $500 every 6 months
- You still invest a total of $5,000, but…
- Each $500 deposit is invested for a different length of time
Specifically:
- The first $500 payment sits in the account all 5 years
- The second sits for 4.5 years
- The third for 4 years
- …
- The last payment is invested for only 6 months
- So only the earliest payments grow significantly.
- The later ones earn much less interest.
In short, a lump sum benefits from the longest possible time in the account whereas an annuity gives you flexibility, but your money does not grow as much, because it has less time to earn compound interest. (Lump sum FV = $7,401.22 vs. Periodic deposit FV = $6003.05. One would make $1,398.17 less under an annuity).
Notice that in the annuity, the final deposit makes no interest. Even though the last payment in an ordinary annuity does not earn interest, it is still included because the annuity definition requires equal payments. An ordinary annuity is defined as equal payments made at regular intervals at the end of each period. If you removed the last payment, the pattern would break. Thus, the payments would no longer be equal or consistent. The structure here matters more than the individual interest gained on any one payment. Even though the last payment does contribute to the total future value earning zero interest, it still increases the final balance. Think of it like this: If you put $500 into your account on the last day, your account still has $500 more than it did the day before. You don’t lose anything; you simply don’t gain interest on that particular payment because the period ends immediately afterward. In real world situations, the final payment always exists. Financial institutions don't say, “Your last deposit doesn’t earn interest, so don’t make it.” You would still make your last 401(k) deposit, last car payment, last insurance premium, or last contribution to a savings plan.
Think of an annuity like climbing stairs. Each payment is a step. Interest is the height added beneath earlier steps. Even though the last step doesn’t get a boost from interest,
you still have to take the step to reach the top. Skipping the final step changes the number of steps, not the height boost from interest.
Alex dreams of purchasing a vacation cabin in eight years. To build a down payment, he deposits $600 quarterly into an investment account earning 7% annual interest, compounded quarterly. How much will Alex have at the end of 8 years? How much interest did the account accumulate?
✅ Solution:
Start with the future value of an annuity interest formula.
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
We know that \(R=$600;~r=0.07~t=8;~n=4\), so
\[\begin{align} A &=\frac{600\left[\left(1+\frac{0.07}{4}\right)^{4 \cdot 8}-1\right]}{\frac{0.07}{4}}\nonumber \\[8pt] A &=\frac{600\left[(1.0175)^{32}-1\right]}{0.0175}\nonumber \\[8pt] A &=\frac{600\left[0.74221349218035\right]}{0.0175}\nonumber \\[8pt] A &= $\text{25,447.319731898}\nonumber \\[8pt] \end{align}\]
\[\boxed {A \approx $\text{25,447.32}} \nonumber\\ \]
So, after 8 years, the future value is \($\text{25,447.32}\).
The interest earned is
\[\begin{align} I &=\text{Future Value}-\text{(Total Number of Payments)} \cdot \text{(Periodic Payment)}\nonumber \\[8pt] I &=$\text{25,447.32}-\text{(4}\cdot \text{8)} \cdot ($\text{600})\nonumber \\[8pt] I&=$\text{25,447.32}-$\text{19,200} \nonumber \\[8pt] \end{align}\]
\[\boxed {I=$\text{6,247.32}}\nonumber \\[8pt]\]
The interest accumulated is \($\text{6,247.32}\).
Tara & Ben are planning their wedding and want to avoid wedding loans. They decide to save for three and a half years by depositing $250 at the end of each month into an account earning 5.2% interest, compounded monthly. How much will they have at the end of the term? How much money did they save in the process?
✅ Solution:
Start with the future value of an annuity interest formula.
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
We know that \(R=$250;~r=0.052~t=3.5;~n=12\), so
\[\begin{align} A &=\frac{250\left[\left(1+\frac{0.052}{12}\right)^{12 \cdot 3.5}-1\right]}{\frac{0.052}{12}}\nonumber \\[8pt] A &=\frac{250\left[(1.0043333333333)^{42}-1\right]}{0.0043333333333}\nonumber \\[8pt] A &=\frac{250\left[0.1991426008935\right]}{0.0043333333333}\nonumber \\[8pt] A &= $\text{11,488.996205482}\nonumber \\[8pt] \end{align}\]
\[\boxed {A \approx $\text{11,489.00}} \nonumber\\ \]
So, after 3.5 years, the future value is \($\text{11,489.00}\).
The interest earned is
\[\begin{align} I &=\text{Future Value}-\text{(Total Number of Payments)} \cdot \text{(Periodic Payment)}\nonumber \\[8pt] I &=$\text{11,489.00}-\text{(12}\cdot \text{3.5)} \cdot ($\text{250})\nonumber \\[8pt] I&=$\text{11,489.00}-$\text{10,500} \nonumber \\[8pt] \end{align}\]
\[\boxed {I=$999}\nonumber \\[8pt]\]
The interest, which is the money saved on the wedding is \($999\).
Stefani G. wants to take a special vacation in five years. She plans to save money regularly so she doesn’t have to use credit cards. Her bank offers a savings account that earns 6% annual interest, compounded monthly. She decides to deposit $150 at the end of every month. What is the future value? How much interest was made?
✅ Solution:
Start with the future value of an annuity interest formula.
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
We know that \(R=$150;~r=0.06~t=5;~n=12\), so
\[\begin{align} A &=\frac{150\left[\left(1+\frac{0.06}{12}\right)^{12 \cdot 5}-1\right]}{\frac{0.06}{12}}\nonumber \\[8pt] A &=\frac{150\left[(1.005)^{60}-1\right]}{0.005}\nonumber \\[8pt] A &=\frac{150\left[0.34885015254932\right]}{0.005}\nonumber \\[8pt] A &= $\text{10,465.504576479}\nonumber \\[8pt] \end{align}\]
\[\boxed {A \approx $\text{10,465.50}} \nonumber\\ \]
So, after 5 years, the future value is \($\text{10,465.50}\).
The interest earned is
\[\begin{align} I &=\text{Future Value}-\text{(Total Number of Payments)} \cdot \text{(Periodic Payment)}\nonumber \\[8pt] I &=$\text{10,465.50}-\text{(12}\cdot \text{5)} \cdot ($\text{150})\nonumber \\[8pt] I&=$\text{10,465.50}-$\text{9,000} \nonumber \\[8pt] \end{align}\]
\[\boxed {I=$\text{1,465.50}}\nonumber \\[8pt]\]
The interest earned is \($\text{1,465.50}\).
Johnny wants to buy a reliable used car in four years. Based on current prices, he estimates he will need about $12,000 to comfortably purchase a car and cover registration, taxes, and insurance. Johnny decides to set up a savings plan where he contributes $200 at the end of each month into a savings account that earns 4.8% annual interest, compounded monthly. What is the future value? How much interest was made? Will Johnny fall short of his estimate?
✅ Solution:
Start with the future value of an annuity interest formula.
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
We know that \(R=$200;~r=0.048~t=4;~n=12\), so
\[\begin{align} A &=\frac{200\left[\left(1+\frac{0.048}{12}\right)^{12 \cdot 4}-1\right]}{\frac{0.048}{12}}\nonumber \\[8pt] A &=\frac{200\left[(1.004)^{48}-1\right]}{0.004}\nonumber \\[8pt] A &=\frac{200\left[0.2112065613749\right]}{0.004}\nonumber \\[8pt] A &= $\text{10,560.328068746}\nonumber \\[8pt] \end{align}\]
\[\boxed {A \approx $\text{10,560.33}} \nonumber\\ \]
So, after 4 years, the future value is \($\text{10,560.33}\).
The interest earned is
\[\begin{align} I &=\text{Future Value}-\text{(Total Number of Payments)} \cdot \text{(Periodic Payment)}\nonumber \\[8pt] I &=$\text{10,560.33}-\text{(12}\cdot \text{4)} \cdot ($\text{200})\nonumber \\[8pt] I&=$\text{10,560.33}-$\text{9,600} \nonumber \\[8pt] \end{align}\]
\[\boxed {I=$\text{960.33}}\nonumber \\[8pt]\]
The interest earned is \($\text{960.32}\).
Yes, Johnny will fall short of his estimate.
Refer to Example #4.55. Since Johnny fell short of his $12,000 goal, what would the actual periodic payment Johnny needs to make in order to make that goal?
✅ Solution:
To actually know the exact payment for a given future value, you would use what is called a sinking fund. A sinking fund is a long‑term savings plan in which regular, equal deposits are made into an account in order to accumulate a specific amount of money in the future. So if we start off with the future value of an annuity interest formula, solve in terms of the periodic payment, \(R\).
The future value of an annuity interest formula:
\[A=\frac{R\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}\nonumber \]
Solving for \(R\),
\[R=\frac{A\frac{r}{n}}{\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}\nonumber \]
We know that \(A=$\text{12,000};~r=0.048~t=4;~n=12\), so
\[\begin{align} R &=\frac{$\text{12,000}\frac{0.048}{12}}{\left[\left(1+\frac{0.048}{12}\right)^{12 \cdot 4}-1\right]} \nonumber \\[8pt] R &=\frac{48}{\left[\left(1.004\right)^{48}-1\right]} \nonumber \nonumber \\[8pt] R &=\frac{48}{0.21120656137492} \nonumber \\[8pt] R &=\$\text{227.26566678387} \nonumber \\[8pt] \end{align}\]
\[\boxed {R \approx $\text{227.27}} \nonumber \\[8pt] \]
So, after 4 years, the periodic payment is \($\text{227.27}\).
Thus, there will be 48 scheduled payments of \($\text{227.27}\). This translates to \($\text{10,908.96}\) which means \($\text{12,000}-$\text{10,908.96}=$\text{1,101.04}\) of interest made on the annuity.
- A student wants to deposit $200 quarterly into an investment account earning 5% annual interest, compounded quarterly for 5 years.
- What is the future value?
- How much of the final amount is the student's own contributions?
- How much interest was earned?
- Leo wants a high‑performance laptop for his graphic design classes. He knows he will need about $2,000 in three years. He notes that if he divides $2,000 by 36 months it is worst case $55.56 a month. He hopes that if he deposits $50 at the end of each month into an account earning 6% annually, compounded monthly, he would have the $2,000 needed for purchase. Will Leo have the $2,000?
- Sad Sally’s mother thought at one point to deposit $100 at the end of each month into a Wells Fargo savings at a rate of 0.01% compounded monthly for 18 years. After she did the calculations, she realized not to do so.
- What would the future value be in 18 years?
- How much of the final amount is her own contributions?
- How much little interest was made?
- A 25-year old decides that her goal is to retire at age 60 with at least $1,000,000 in savings. The company investment annuity offers 7.25% annual returns, compounded monthly.
- What amount will she need to invest each month?
- Compare the amount earned from interest to the amount she deposited.
- Answers
-
- a) A = $5,640.74; b) Total contributions = $4,000; c) I = $1,640.74.
- A = $1,966.81; Leo comes up only $33.19 short. So, we suggest to Leo to make an extra $33.19 payment on the final payment of $50 to get exactly $2,000.
- a) A = $21,619.36; b) Total contributions = $21,600; c) I = $19.36.
- a) R = $523.01; b) I = $780,335.80; Total amount in deposits made = $523.01 x 420 = $219,664.20; Investment is more than tripled over the 40 years.