Math Problem Statement

A town wants to build a park for ​$1 comma 200 comma 000 and issues 600 bonds for ​$2 comma 000 ​each, with a term of 7 years and a coupon rate of 6​%. Consider the purchase of one such bond. Helpful video.opens in a new tab ​(a) What is the total return of this​ investment? ​(b) What is the average rate of​ return? Question content area bottom Part 1 ​(a) The total return is ​$    enter your response here. ​(Simplify your​ answer.) ​(b) The average rate of return is    enter your response here​%. ​(Simplify your answer. Round to two decimal places as​ needed.)

Solution

To solve this problem, we’ll calculate both the total return and the average rate of return for one bond.


Step 1: Total Return

  1. Bond Price: Each bond costs $2,000.
  2. Coupon Rate: 6% annually.
    • The annual interest earned = 2,000×0.06=1202,000 \times 0.06 = 120.
  3. Term: 7 years.
    • Total interest earned over 7 years = 120×7=840120 \times 7 = 840.
  4. Face Value: Assume the bond is redeemed at its face value of $2,000.
    • Total return = Total interest+Face value=840+2,000=2,840\text{Total interest} + \text{Face value} = 840 + 2,000 = 2,840.

The total return is $2,840.


Step 2: Average Rate of Return

The average rate of return can be calculated using the formula: Average Rate of Return=(Total ReturnInitial InvestmentInitial Investment×Number of Years)×100\text{Average Rate of Return} = \left( \frac{\text{Total Return} - \text{Initial Investment}}{\text{Initial Investment} \times \text{Number of Years}} \right) \times 100

  1. Initial Investment = $2,000.
  2. Total Return = $2,840.
  3. Number of Years = 7.

Average Rate of Return=(2,8402,0002,000×7)×100\text{Average Rate of Return} = \left( \frac{2,840 - 2,000}{2,000 \times 7} \right) \times 100

Average Rate of Return=(84014,000)×100=6%.\text{Average Rate of Return} = \left( \frac{840}{14,000} \right) \times 100 = 6\%.

The average rate of return is 6.00%.


Final Answers:

(a) The total return is $2,840.
(b) The average rate of return is 6.00%.


Would you like detailed explanations of the calculations or further help?
Here are 5 related questions to explore:

  1. What happens to the total return if the coupon rate is increased to 8%?
  2. How does the average rate of return compare for a 10-year bond with the same coupon rate?
  3. What would the total return be if the bond is sold at a discount (e.g., $1,800 face value)?
  4. How is the coupon rate determined in bond pricing?
  5. What is the present value of the bond's future cash flows at a 5% discount rate?

Tip: Understanding bond terms like face value, coupon rate, and maturity is key to evaluating investments effectively.

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Math Problem Analysis

Mathematical Concepts

Bond Valuation
Simple Interest
Rates of Return

Formulas

Total Interest = Bond Price × Coupon Rate × Term
Total Return = Total Interest + Face Value
Average Rate of Return = ((Total Return - Initial Investment) / (Initial Investment × Term)) × 100

Theorems

Principles of Simple Interest
Rate of Return Calculations

Suitable Grade Level

Grades 10-12