Given A Five-Year 8% Coupon Bond With A Face Value Of 100

Given A Five Year 8 Coupon Bond With a Face Value Of 100

Given a five-year, 8% coupon bond with a face value of $1,000 and coupon payments made annually: a. What is the bond value if it is trading at the yield of 6%? b. What is the bond value if it is trading at the yield of 8%? c. What is the bond value if it is trading at the yield of 10%? d. Comment on the price and yield relation you observe. What are the percentage changes in value when the yield goes from 6% to 8% and when it goes from 8% to 10%?

Paper For Above instruction

The valuation of bonds is fundamental in understanding the relationship between bond prices and yields, which directly influences investment decisions and portfolio management. In this analysis, we consider a five-year, 8% coupon bond with a face value of $1,000, and examine its value at different market yields of 6%, 8%, and 10%. The bond pays annual coupons, thus providing a straightforward case to demonstrate bond pricing mechanics. Additionally, the bond's price sensitivity to changes in yield can be appreciated through percentage change calculations, which are vital for assessing interest rate risk.

Part A: Bond value at a 6% yield

Using the standard present value formula for bonds, the value of the bond (PV) is the sum of the present value of its coupons and face value, discounted at the market yield of 6%. The annual coupon payment (C) is 8% of $1,000, i.e., $80. The bond matures in 5 years.

PV = \(\sum_{t=1}^5 \frac{C}{(1 + y)^t} + \frac{F}{(1 + y)^5}\)

Where F = $1,000, y = 0.06 (6% yield).

Calculating:

PV = \(\frac{80}{1.06} + \frac{80}{1.06^2} + \frac{80}{1.06^3} + \frac{80}{1.06^4} + \frac{80}{1.06^5} + \frac{1000}{1.06^5}\)

Summing these, the bond value approximately is $1,055.65.

Part B: Bond value at an 8% yield

Repeating the calculation with y = 8%:

PV = \(\frac{80}{1.08} + \frac{80}{1.08^2} + \frac{80}{1.08^3} + \frac{80}{1.08^4} + \frac{80}{1.08^5} + \frac{1000}{1.08^5}\)

The resulting bond price is approximately $1,000.00, matching the face value, as expected for a bond trading at par when coupon rate equals yield.

Part C: Bond value at a 10% yield

At y=10%:

PV = \(\frac{80}{1.10} + \frac{80}{1.10^2} + \frac{80}{1.10^3} + \frac{80}{1.10^4} + \frac{80}{1.10^5} + \frac{1000}{1.10^5}\)

This sum yields approximately $954.05.

Part D: Price-Yield Relationship and Percentage Changes

From the above calculations, the bond's price is inversely related to the yield: as yield increases from 6% to 8%, the bond price decreases from approximately $1,055.65 to $1,000.00; then from 8% to 10%, it further declines to about $954.05. Quantitatively, the percentage change in bond price from 6% to 8% is about 5.2%, while from 8% to 10% it is approximately 4.6%. The larger percentage price decline when yield increases from 6% to 8% reflects the convex nature of bond prices and their sensitivity to yield changes. The change in price diminishes somewhat as yields rise, illustrating the nonlinear, convex relationship between bond prices and yields (Fabozzi, 2016). This demonstrates that bonds are more sensitive to yield decreases at lower levels and less sensitive when yields are higher.

References

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