Refer back to the Series EE savings bonds we discussed at the very beginning of the chapter.

Assuming you purchased a $50 face value bond, what is the exact rate of return you would earn if you held the bond for 20 years until it doubled in value?
Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

If you purchased a $50 face value bond in 2023 at the then-current interest rate of 2.10 percent per year, how much would the bond be worth in 2033?
Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

In 2033, instead of cashing the bond in for its then-current value, you decide to hold the bond until it doubles in face value in 2043. What rate of return will you earn over the last 10 years?
Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

### Problem 1: Rate of Return Over 20 Years

You're asked to find the exact rate of return if you purchase a $50 face value Series EE savings bond, and after 20 years, it doubles in value to $100. 

To calculate the rate of return ($ r $) over 20 years, we use the formula for compound interest:

\[
A = P(1 + r)^t
\]

Where:
- $ A $ is the future value ($100 in this case),
- $ P $ is the initial principal or face value ($50),
- $ r $ is the annual interest rate (what we need to find),
- $ t $ is the time in years (20 years).

Substituting the known values:

\[
100 = 50(1 + r)^{20}
\]

First, solve for $ 1 + r $:

\[
\frac{100}{50} = (1 + r)^{20}
\]

\[
2 = (1 + r)^{20}
\]

Now, take the 20th root of both sides to isolate $ r $:

\[
1 + r = 2^{\frac{1}{20}}
\]

Now calculate $ 2^{\frac{1}{20}} $:

\[
1 + r = 1.035264923
\]

\[
r = 1.035264923 - 1
\]

\[
r = 0.035264923
\]

So the rate of return is approximately:

\[
r = 3.53\% \text{ per year}
\]

---

### Problem 2: Value of the Bond in 2033

You purchased a $50 face value bond in 2023 at an interest rate of 2.10% per year. You want to find out how much it would be worth in 2033, after 10 years.

We again use the compound interest formula:

\[
A = P(1 + r)^t
\]

Where:
- $ P = 50 $,
- $ r = 0.021 $ (2.10% per year),
- $ t = 10 $.

Substituting the known values:

\[
A = 50(1 + 0.021)^{10}
\]

Now calculate $ (1.021)^{10} $:

\[
A = 50 \times 1.231144413
\]

\[
A = 61.56
\]

So, the bond will be worth approximately **$61.56** in 2033.

---

### Problem 3: Rate of Return Over the Last 10 Years (2033-2043)

In 2033, the bond is worth $61.56, and you hold it until 2043 when it doubles in value to $100. We need to calculate the rate of return over this last 10-year period.

We use the same formula for compound interest, but this time:
- $ A = 100 $,
- $ P = 61.56 $,
- $ t = 10 $,
- $ r $ is what we need to find.

Substitute the known values:

\[
100 = 61.56(1 + r)^{10}
\]

First, solve for $ 1 + r $:

\[
\frac{100}{61.56} = (1 + r)^{10}
\]

\[
1.624634078 = (1 + r)^{10}
\]

Take the 10th root of both sides:

\[
1 + r = 1.049461725
\]

\[
r = 1.049461725 - 1
\]

\[
r = 0.049461725
\]

So the rate of return over the last 10 years is approximately:

\[
r = 4.95\% \text{ per year}
\]

---

### Summary of Results:

1. **Rate of return over 20 years**: $ \mathbf{3.53\%} $
2. **Bond value in 2033**: $ \mathbf{61.56} $
3. **Rate of return over the last 10 years (2033-2043)**: $ \mathbf{4.95\%} $

---

Do you have any questions or need more details?

Here are 5 related questions you might explore:
1. How does the interest rate change if the bond doubles in a different period, like 25 years?
2. What is the effect of inflation on the real value of the bond over 20 years?
3. Can you calculate the value of a bond after 30 years at an interest rate of 2%?
4. How do different types of savings bonds (like Series I bonds) compare to Series EE bonds in terms of rate of return?
5. What is the impact of compounding frequency (e.g., monthly vs. yearly) on the bond's growth?

**Tip:** Compounding more frequently (monthly vs. yearly) slightly increases the future value of an investment.

Refer back to the Series EE savings bonds we discussed at the very beginning of the chapter. Assuming you purchased a $50 face value bond, what is the exact rate of return you would earn if you held the bond for 20 years until it doubled in value? Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

If you purchased a $50 face value bond in 2023 at the then-current interest rate of 2.10 percent per year, how much would the bond be worth in 2033? Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

In 2033, instead of cashing the bond in for its then-current value, you decide to hold the bond until it doubles in face value in 2043. What rate of return will you earn over the last 10 years? Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

Learn how to calculate the rate of return on a Series EE bond, its value after 10 years at 2.10% interest, and the return if held for another 10 years. Understand compound interest and exponential growth in financial scenarios.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation