A $1,000 par value bond with four years left to maturity pays an interest payment semiannually with a 5 percent coupon rate and is priced to have a 4.5 percent yield to maturity.



If interest rates unexpectedly rise by 0.3 percent, by how much will the bond’s price change?

To find how much the bond's price changes due to an unexpected increase in interest rates, we can use the bond price formula. However, for simplicity, we can approach this using **duration and convexity**—two measures used to estimate the sensitivity of a bond's price to changes in interest rates.

Here’s a step-by-step breakdown:

### 1. **Determine the bond’s price before the rate change**  
We can calculate the bond price initially using the semiannual bond price formula:

\[
P = \sum_{t=1}^{n} \frac{C}{(1 + r/2)^{2t}} + \frac{F}{(1 + r/2)^{2n}}
\]

Where:
- $C = \frac{0.05 \times 1000}{2} = 25$ (semiannual coupon payment),
- $F = 1000$ (face value),
- $r = 4.5\%$ (yield to maturity),
- $n = 4 \times 2 = 8$ (since 4 years means 8 semiannual periods).

We’ll compute the price using this formula for the initial yield.

### 2. **Calculate the bond’s price after the rate increase**  
Now, calculate the bond price again using a new yield $r = 4.5\% + 0.3\% = 4.8\%$.

### 3. **Find the difference in bond price**  
The change in bond price is simply the difference between the price before and after the rate increase.

Let’s calculate that now.The bond's initial price before the rate increase was approximately \$1,018.12. After the interest rate unexpectedly rose by 0.3%, the new price dropped to approximately \$1,007.20. 

Thus, the bond's price decreased by approximately \$10.92.

Would you like more details, or do you have any questions about this calculation?

Here are five related questions:
1. What is the relationship between bond prices and interest rates?
2. How does duration measure a bond's sensitivity to interest rate changes?
3. What is convexity, and how does it affect bond price sensitivity?
4. Can you explain how yield to maturity (YTM) is calculated for bonds?
5. What are the risks involved in bond investments due to fluctuating interest rates?

**Tip**: Bonds with longer maturities and lower coupon rates are more sensitive to interest rate changes.

A $1,000 par value bond with four years left to maturity pays an interest payment semiannually with a 5 percent coupon rate and is priced to have a 4.5 percent yield to maturity. If interest rates unexpectedly rise by 0.3 percent, by how much will the bond’s price change?

Explore how a bond's price changes when interest rates rise unexpectedly by 0.3%. This detailed solution covers bond pricing using duration and convexity, and includes key formulas for calculating the bond's price before and after the interest rate change. Suitable for college-level finance and mathematics students.

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