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.