MLK Bank has an asset portfolio that consists of $160 million of 15-year, 8.5 percent annual coupon, $1,000 bonds that sell at par.

a-1. What will be the bonds’ new prices if market yields change immediately by ± 0.10 percent?
a-2. What will be the new prices if market yields change immediately by ± 2.00 percent?
b-1. The duration of these bonds is 9.0101 years. What are the predicted bond prices in each of the four cases using the duration rule?
b-2. What is the amount of error between the duration prediction and the actual market values?

Let's break down each part of the problem step by step to solve it effectively.

### Part a-1: Bond Price with ± 0.10% Change in Market Yields
The bond in question is a 15-year, 8.5% annual coupon bond priced at par (i.e., \$1,000), and its price will change as the market yield changes by ±0.10%.

We will use the bond price formula:

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

Where:
- $ P $ = price of the bond
- $ C $ = annual coupon payment = 8.5% of \$1,000 = \$85
- $ F $ = face value of the bond = \$1,000
- $ y $ = yield to maturity (current yield is 8.5%, and it changes by ±0.10%)
- $ n $ = number of years to maturity = 15 years

We will calculate the price for:
1. Yield = 8.4% (8.5% - 0.10%)
2. Yield = 8.6% (8.5% + 0.10%)

### Part a-2: Bond Price with ± 2.00% Change in Market Yields
We will follow the same process as part a-1 but with a ±2.00% change in yield, i.e., the yields will be:
1. Yield = 6.5% (8.5% - 2.00%)
2. Yield = 10.5% (8.5% + 2.00%)

### Part b-1: Predicted Bond Prices Using Duration Rule
The **duration rule** estimates the percentage change in bond price due to a change in yield:

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

Where:
- $ D $ = bond duration = 9.0101 years
- $ \Delta y $ = change in yield
- $ y $ = initial yield = 8.5%

For each yield change (± 0.10% and ± 2.00%), we will calculate the predicted price change and then use it to estimate the new price.

### Part b-2: Amount of Error Between Duration Prediction and Actual Prices
Finally, we will compare the actual prices calculated in parts a-1 and a-2 to the predicted prices from part b-1, and find the error as:

\[
\text{Error} = \left| \text{Actual Price} - \text{Predicted Price} \right|
\]

Let's now proceed to calculate each part.### Part a-1: Bond Prices with ± 0.10% Change in Market Yields
- At 8.4% yield: \$1,008.35
- At 8.6% yield: \$991.75

### Part a-2: Bond Prices with ± 2.00% Change in Market Yields
- At 6.5% yield: \$1,188.05
- At 10.5% yield: \$852.12

Next, let's calculate the predicted bond prices using the **duration rule** for these yield changes.It seems like I can’t do more advanced data analysis right now. Please try again later. 

If you'd like, I can explain the duration rule further or guide you through the calculations manually. Let me know how you'd like to proceed!

Here are some related questions that can help you understand the concepts further:

1. What is the significance of bond duration in price prediction?
2. How does a bond's coupon rate affect its price sensitivity to yield changes?
3. Why do bonds with longer maturities generally have higher durations?
4. What role does the yield curve play in bond pricing?
5. How does convexity provide a more accurate prediction than duration alone?

**Tip:** Duration estimates price changes under small yield shifts, but for larger shifts, convexity improves accuracy.

Learn how to calculate bond prices with changes in market yields and predict bond price shifts using the duration rule. This example includes detailed steps for market yield shifts of ±0.10% and ±2.00%, comparing actual and predicted prices with error analysis. Suitable for college-level finance students.

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