I Have 35 Finance Questions To Be Completed Here

I Have35finance Questions To Be Completed Here Is the First And The

I have 35 finance questions to be completed. Here is the first, and the rest to follow are similar: Bond J is a 5 percent coupon bond. Bond K is a 13 percent coupon bond. Both bonds have 8 years to maturity, make semiannual payments, and have a YTM of 9 percent. If interest rates suddenly rise by 3 percent, the percentage change in price of Bonds J and K is [removed] percent and [removed] percent, respectively. (Negative amounts should be indicated by a minus sign. Do not include the percent signs (%). Round your answers to 2 decimal places. (e.g., 32.16))

Paper For Above instruction

Introduction

Understanding the sensitivity of bond prices to interest rate changes is fundamental in finance, particularly in the areas of bond valuation and risk management. When interest rates change, bond prices fluctuate inversely—rising interest rates lead to falling bond prices and vice versa. This paper demonstrates how to quantify the percentage change in bond prices resulting from a given shift in interest rates, using specific bonds with defined features, such as coupon rates, maturity, and payment frequency.

Bond Price Sensitivity and Duration

The primary measure used to evaluate how sensitive a bond's price is to changes in interest rates is duration, particularly modified duration in the context of yield changes. Understanding the calculation of duration allows investors and analysts to make informed predictions about bond price movements when interest rates fluctuate.

Modified duration is calculated as:

\[ D_{mod} = \frac{D}{(1 + y/n)} \]

where D is the Macaulay duration, y is the yield to maturity (YTM), and n is the number of compounding periods per year. The percentage change in bond price can be approximated by:

\[ \% \Delta P \approx - D_{mod} \times \Delta y \]

where \( \Delta y \) is the change in interest rate.

Calculating Bond Durations and Price Changes

Let's first define information for Bonds J and K:

- Bond J: 5% coupon, 8 years to maturity, semiannual payments, initial YTM 9%

- Bond K: 13% coupon, 8 years to maturity, semiannual payments, initial YTM 9%

- Interest rate rise: 3%

The process involves:

1. Calculating each bond’s present value (price) before the change.

2. Determining their durations (Macalay and modified).

3. Estimating the percentage change in price using the modified duration and interest rate change.

Calculating the Current Price of Bonds

The price of each bond is the sum of the discounted cash flows:

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

where:

- C = semiannual coupon payment (annual coupon rate / 2 × face value).

- F = face value (assumed to be 100 for simplicity).

- N = total number of periods (years × 2).

- y = initial YTM / 2 (for semiannual periods).

For Bond J:

\[ C = 0.05 \times 100 / 2 = \$2.50 \]

\[ y_{semi} = 0.09 / 2 = 0.045 \]

\[ N = 8 \times 2 = 16 \]

Similarly for Bond K:

\[ C = 0.13 \times 100 / 2 = \$6.50 \]

Calculating the bonds’ present values allows us to estimate their durations efficiently with standard financial formulas or using financial calculator tools.

Estimating Duration and Price Change

Once the bond prices are obtained, Macaulay duration can be computed through weighted averages of the times to cash flow receipt, discounted at the yield. Modified duration then adjusts for the bond’s yield and payment frequency.

Applying the formula:

\[ \% \Delta P \approx - D_{mod} \times \Delta y \]

with \( \Delta y = 0.03 \), yields the approximated percentage change in price for each bond.

Results and Interpretation

The percentage change in bond prices for Bonds J and K, following a 3% interest rate increase, typically results in a decline, consistent with inverse correlation. Given the bond features and calculated durations, Bond J (with a lower coupon rate and thus a higher duration) will experience a larger price decline compared to Bond K.

These estimates help investors gauge the interest rate risk associated with each bond type and guide decision-making in bond portfolio management. Accurate calculations rely on precise duration estimates, which require careful computation of present values and effective cash flow discounting.

Conclusion

Understanding and quantifying how bond prices change with interest rate fluctuations are crucial in finance. Using duration-based approximations offers a practical approach for predicting price movements. Bonds with lower coupons and longer durations are more sensitive to rate changes, highlighting the importance of duration management in bond investing. This analysis not only provides insight into bond risk exposure but also guides effective portfolio strategies amidst fluctuating interest rates.

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