The duration of a ten-year, 8 percent coupon bond when the interest rate is 10 percent is 7.76 years. What happens to the price of the bond if the interest rate increases to 12 percent?

		


To determine the price change of the bond when the interest rate increases to 12%, we can use the following bond pricing formula:

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

Where:
- \( P \) = price of the bond
- \( C \) = annual coupon payment = \( 0.08 \times F \)
- \( F \) = face value of the bond (usually 100 or 1000)
- \( r \) = market interest rate (expressed as a decimal)
- \( n \) = number of years to maturity

### Given:
- Coupon rate = 8% (so coupon payment \( C = 0.08 \times F \))
- Initial interest rate = 10%
- Duration of bond at 10% = 7.76 years
- New interest rate = 12%
- Maturity = 10 years

#### Step 1: Calculate price at 10% interest rate (Initial Price)
We can approximate the price using the bond duration formula:

\[
\frac{\Delta P}{P} = -D \times \frac{\Delta r}{1 + r}
\]

Where:
- \( \Delta P \) = percentage change in bond price
- \( D \) = bond duration = 7.76 years
- \( \Delta r \) = change in interest rate (from 10% to 12%, so \( \Delta r = 0.02 \))
- \( r \) = initial interest rate (10% = 0.10)

Now, calculate the percentage change in price:

\[
\frac{\Delta P}{P} = -7.76 \times \frac{0.02}{1 + 0.10} = -7.76 \times 0.01818 = -0.141
\]

This indicates a 14.1% price decrease when the interest rate rises from 10% to 12%.

#### Step 2: Conclusion
The price of the bond will decrease by approximately 14.1% when the interest rate increases to 12%. If the initial price at 10% interest rate is \( P_0 \), then the new price \( P_1 \) can be approximated as:

\[
P_1 = P_0 \times (1 - 0.141)
\]

If you know the actual price of the bond at a 10% interest rate, you can multiply it by this factor to get the new price at a 12% interest rate.

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Would you like a deeper dive into any of these steps or additional clarifications?

Here are 5 follow-up questions that expand this topic:
1. How is bond duration used to measure interest rate risk?
2. What is the relationship between bond prices and interest rates?
3. How does coupon rate affect the duration of a bond?
4. What factors influence a bond's sensitivity to interest rate changes?
5. Can the bond price increase if the interest rate falls below the coupon rate?

**Tip**: Bond prices and interest rates move inversely—when interest rates rise, bond prices fall, and vice versa.

The duration of a ten-year, 8 percent coupon bond when the interest rate is 10 percent is 7.76 years. What happens to the price of the bond if the interest rate increases to 12 percent?

Learn how bond prices are affected by changes in interest rates with this example of a ten-year, 8% coupon bond. Using bond pricing and duration formulas, we calculate a 14.1% price decrease when the interest rate increases from 10% to 12%. Suitable for university-level finance and economics students.

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