Question 1193715: Topics In Contemporary Math
21: Simple Interest
4) To fix the damage caused by the Springfield Monorail, the city sold $1,000,000 in 20-year bonds that paid a semi-annual coupon with an annual rate of 6.5%. What is the value of each coupon payment? How much interest is paid in total until the bonds are mature? What is the total cost of fixing the damage?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! ### Given Information:
- **Principal**: \( \$1,000,000 \)
- **Annual Coupon Rate**: \( 6.5\% \)
- **Bond Term**: \( 20 \, \text{years} \)
- **Coupons**: Paid semi-annually.
---
### Step 1: Value of Each Coupon Payment
The value of each coupon payment is calculated as:
\[
\text{Coupon Payment} = \text{Principal} \times \frac{\text{Annual Coupon Rate}}{\text{Number of Coupons Per Year}}
\]
Substituting the values:
\[
\text{Coupon Payment} = 1,000,000 \times \frac{6.5}{2} \% = 1,000,000 \times 0.0325 = 32,500
\]
The value of each coupon payment is **\$32,500**.
---
### Step 2: Total Interest Paid Over 20 Years
The total number of coupon payments over 20 years is:
\[
\text{Total Payments} = 20 \, \text{years} \times 2 \, \text{payments/year} = 40 \, \text{payments}.
\]
The total interest paid is:
\[
\text{Total Interest} = \text{Coupon Payment} \times \text{Total Payments}
\]
\[
\text{Total Interest} = 32,500 \times 40 = 1,300,000
\]
The total interest paid is **\$1,300,000**.
---
### Step 3: Total Cost of Fixing the Damage
The total cost includes the original principal plus all interest payments:
\[
\text{Total Cost} = \text{Principal} + \text{Total Interest}
\]
\[
\text{Total Cost} = 1,000,000 + 1,300,000 = 2,300,000
\]
The total cost of fixing the damage is **\$2,300,000**.
---
### Final Answers:
1. **Value of Each Coupon Payment**: **\$32,500**
2. **Total Interest Paid**: **\$1,300,000**
3. **Total Cost of Fixing the Damage**: **\$2,300,000**
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