7) By how much will the price of a 30-year zero-coupon bond change if, due to a permanent change in interest rates, the price of a 3-year zero-coupon bond decreased by 4%? Assume the short- and long-term interest rates are the same. If your answer depends on other parameters not given in this question (e.g., current interest rate, bond prices, etc.) write your answer as a function of these parameters and specify what these parameters are.
7) By how much will the price of a 30-year zero-coupon bond change if, due to a permanent change in interest rates, the price of a 3-year zero-coupon bond decreased by 4%? Assume the short- and long-term interest rates are the same. If your answer depends on other parameters not given in this question (e.g., current interest rate, bond prices, etc.) write your answer as a function of these parameters and specify what these parameters are.
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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Transcribed Image Text:### Bond Price Sensitivity to Interest Rate Changes
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**7) By how much will the price of a 30-year zero-coupon bond change if, due to a permanent change in interest rates, the price of a 3-year zero-coupon bond decreased by 4%? Assume the short- and long-term interest rates are the same. If your answer depends on other parameters not given in this question (e.g., current interest rate, bond prices, etc.) write your answer as a function of these parameters and specify what these parameters are.**
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### Explanation:
In this question, we analyze the relationship between the price sensitivity of bonds to changes in interest rates.
**Key Points:**
- **Zero-Coupon Bond:** A bond that pays no coupon and is sold at a discount. It returns the face value at maturity.
- **Interest Rate Impact:** Changes in interest rates inversely affect the price of bonds.
- **Price Sensitivity:** Longer-term bonds are generally more sensitive to interest rate changes.
In order to solve this, one must understand the concept of duration and its role in bond price volatility due to interest rate fluctuations. Here, the question instructs us to consider equal short- and long-term interest rates and to formulate the answer based on provided and non-provided parameters.
### Detailed Steps:
1. **Interpret the Impact:** The given information states a 4% decrease in the price of a 3-year zero-coupon bond.
2. **Calculate Duration:** Bonds' sensitivity to interest rate changes is often measured by "duration."
#### Mathematical Approach:
1. **Price Formula for Zero-Coupon Bonds:**
- \( P = \frac{F}{(1 + r)^n} \)
Where:
- \( P \) = Price of the bond
- \( F \) = Face value of the bond
- \( r \) = Interest rate
- \( n \) = Term to maturity
2. **Determine the Factor:**
- Since the prices change inversely relative to the power of \( n \), the price change for longer maturities will be more pronounced.
3. **Utilize Provided Information:**
- A 4% decrease for a 3-year bond sets the baseline.
4. **Calculate 30-Year Bond Sensitivity:**
- Given longer time to maturity (30 years), expect a larger percentage decrease if calculated similarly.
Formulate the answer in
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