A bond has a price of $110, a Macaulay Duration of 20.8 years, and a convexity of 160 years. If rates decrease from 8% to 7 ½%
per annum compounded semiannually then use calculus to find the approximate new price of the bond. Your final answer should be correct to 3 places after the decimal point.


To calculate the approximate new price of the bond using duration and convexity, we can use the following formula:

\[
\Delta P \approx - D \cdot P \cdot \Delta y + \frac{1}{2} \cdot C \cdot P \cdot (\Delta y)^2
\]

Where:
- $ \Delta P $ is the change in bond price.
- $ D $ is the Macaulay Duration.
- $ P $ is the initial price of the bond.
- $ C $ is the convexity.
- $ \Delta y $ is the change in yield.

We are given the following information:
- $ P = 110 $
- $ D = 20.8 $ years
- $ C = 160 $ years
- Initial yield $ y_0 = 8\% $ (or 0.08 per annum)
- New yield $ y_1 = 7.5\% $ (or 0.075 per annum)
- The yield change $ \Delta y = y_1 - y_0 = 0.075 - 0.08 = -0.005 $ per annum.

Since the rates are compounded semiannually, the change in yield on a semiannual basis is:
\[
\Delta y_{\text{semiannual}} = \frac{\Delta y}{2} = \frac{-0.005}{2} = -0.0025
\]

Now, substitute these values into the formula.

### Step 1: Calculate the first-order effect (using Duration)
\[
\Delta P_1 \approx - D \cdot P \cdot \Delta y_{\text{semiannual}} = - 20.8 \cdot 110 \cdot (-0.0025)
\]
\[
\Delta P_1 \approx 5.72
\]

### Step 2: Calculate the second-order effect (using Convexity)
\[
\Delta P_2 \approx \frac{1}{2} \cdot C \cdot P \cdot (\Delta y_{\text{semiannual}})^2 = \frac{1}{2} \cdot 160 \cdot 110 \cdot (-0.0025)^2
\]
\[
\Delta P_2 \approx 0.055
\]

### Step 3: Calculate the total change in price
\[
\Delta P \approx \Delta P_1 + \Delta P_2 = 5.72 + 0.055 = 5.775
\]

### Step 4: Calculate the new price
\[
P_{\text{new}} \approx P + \Delta P = 110 + 5.775 = 115.775
\]

Thus, the approximate new price of the bond is **$115.775** (to 3 decimal places).

Let me know if you would like more details or have questions! 

### 5 Related Questions:
1. How does Macaulay duration impact the sensitivity of bond prices to interest rate changes?
2. What is the difference between modified duration and Macaulay duration?
3. How does convexity influence bond price changes for large interest rate movements?
4. What assumptions does the duration-convexity approximation make about interest rate changes?
5. How can you hedge interest rate risk using duration and convexity?

### Tip:
Always use convexity to refine bond price estimates when dealing with larger changes in interest rates, as duration alone becomes less accurate in such cases.

A bond has a price of $110, a Macaulay Duration of 20.8 years, and a convexity of 160 years. If rates decrease from 8% to 7 ½% per annum compounded semiannually, use calculus to find the approximate new price of the bond. Your final answer should be correct to 3 places after the decimal point.

Learn how to calculate the new bond price when interest rates change using Macaulay Duration, Convexity, and calculus. This detailed example covers the duration-convexity approximation for semiannual compounding, suitable for finance students. Includes a step-by-step calculation of a bond's price with a change from 8% to 7.5% interest rate.

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