Section 4.5: Annuities
- Page ID
- 216983
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Annuities arise in many real-world situations, including retirement income plans, pension benefits, mortgage payments, and lottery prize distributions. In general, an annuity is a sequence of equal payments made at regular intervals over a specified period of time. Some annuities are created through financial contracts, while others occur naturally as part of everyday financial transactions. Because annuities involve predictable streams of cash flows, they are important applications of mathematics in finance and economics.
For example, many retirees receive regular income from Social Security or employer-sponsored pension plans. These programs typically provide fixed monthly payments for the remainder of the recipient's life, making them common examples of annuities. Another example occurs when lottery winners choose to receive their prize as a series of payments rather than a lump sum. Because the payments are made at regular intervals according to a predetermined schedule, they also constitute an annuity.
Although several types of annuities exist including fixed, variable, indexed, and immediate annuities, this text focuses on fixed annuities.
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.
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. Note that the number of payments made per year also equals the number of times the interest 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, what if you don't have $5,000 saved up to invest? Instead, let's invest $500 twice a year for 5 years. After 5 years, you will have deposited $5,000 equal to the principal in the previous example. However, investing $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. Now, we ask,
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 \($\text{1,003.05}\).
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 basically get the same result of $6,003.05 (off by one cent due to rounding error) when we use the table as we did by using the formula.
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.
- A periodic annuity: $500 twice per year for 5 years (for a total of $500 × 10 = $5,000).
When you invest $5,000 as a single lump sum, the entire amount remains in the account for all 5 years. Since interest is compounded semiannually, the investment earns interest 10 times, and each interest amount becomes part of the principal. This allows the account to earn interest on both the original investment and previously earned interest, resulting in the greatest possible growth.
With a periodic annuity, you invest $500 every 6 months. Although the total amount invested is still $5,000, each deposit remains in the account for a different length of time. The first deposit earns interest for the full 5 years, while later deposits have less time to grow. The final deposit is invested for only 6 months. As a result, the later deposits earn less interest, causing the annuity to accumulate less interest than the lump-sum investment.
In summary, a lump-sum investment benefits from having the entire amount invested for the full term, allowing it to earn the maximum amount of compound interest. A periodic annuity offers greater flexibility because deposits are made over time, but the money has less time to grow. As a result, the future value of the lump-sum investment is $7,401.22, while the future value of the periodic annuity is $6,003.05. In this example, the annuity accumulates $1,398.17 less than the lump-sum investment.
Sheldon 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 Sheldon have at the end of 8 years? How much interest did the account accumulate?
✅ Solution:
Start with the future value of an annuity 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 \($\text{6,247.32}\).
Anna & Tom 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 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
\[\begin{align} I &=\text{Future Value}-\text{(Total Number of Payments)} \cdot \text{(Periodic Payment)}\nonumber \\[8pt] I &=$\text{11,489.00}-\text{(12}\cdot \text{3.5)} \cdot ($\text{250})\nonumber \\[8pt] I&=$\text{11,489.00}-$\text{10,500} \nonumber \\[8pt] \end{align}\]
\[\boxed {I=$989}\nonumber \\[8pt]\]
The interest, which is the money saved on the wedding is \($989\).
Sandy 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 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 \($\text{1,465.50}\).
T.Q. 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. T.Q. 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 T.Q. fall short of his estimate?
✅ Solution:
Start with the future value of an annuity 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 \($\text{960.33}\).
Yes, Johnny will fall short of his estimate.
Refer to Example #4.55. Since T.Q. fell short of his $12,000 goal, what would the actual periodic payment T.Q. 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. In this example, we want to set up a sinking fund with future value \(A\) = $12,000. To find the required monthly payment, we solve for \(R\) in the formula for the future value of an annuity 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, the required periodic payment is \($\text{227.27}\).
Note: 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.
- Syed 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 earns no interest on his savings, then he will need to save $2,000 \(\div\) 36 = $55.56 per 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 interest was earned?
- 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 = $4,512.60
- Total contributions = $4,000
- I = $512.60
- 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 = $21,619.36
- Total contributions = $21,600
- I = $19.36
-
- R = $523.01
- 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.
-