Suppose GM Issues A Bond With Ten Years Until Maturity

Uppose That Gm Issues A Bond With Ten Years Until Maturity A Face Val

Uppose That GM issues a bond with ten years until maturity, a face value of $1000, and a coupon rate of 7% (annual payments). The yield to maturity on this bond when it was issued was 6%. A. What was the price of this bond when it was issued? B. Assuming the yield to maturity remains constant, what is the price of the bond immediately before it makes its first coupon payment? C. Assuming the yield to maturity remains constant, what is the price of the bond immediately after it makes its first coupon payment? (USE THE ATTACHED FILE FOR FORMULAS AND COMPLETING THE ASSIGNMENT)

Paper For Above instruction

The issuance and valuation of bonds are fundamental aspects of corporate finance, allowing companies like General Motors (GM) to raise capital through debt. Understanding the bond's price at issuance and how its value fluctuates over time, especially around coupon payments, provides insight into fixed-income securities and investor strategies. This paper explores the bond issued by GM with a ten-year maturity, emphasizing the calculation of its initial price, and subsequent prices just before and after the first coupon payment, under the assumption of a constant yield to maturity (YTM).

Introduction

Bonds are debt instruments that obligate the issuer to pay a fixed interest rate (coupon rate) to the bondholder periodically, with the promise to repay the face value at maturity. The valuation of bonds hinges on the present value of future cash flows, which include periodic coupon payments and the face value. When a bond is issued at a YTM different from its coupon rate, its price adjusts accordingly. Specifically, if the YTM is lower than the coupon rate, the bond will be issued at a premium; if higher, at a discount.

Calculating the Initial Price of the Bond

The bond's initial price at issuance depends on the present value of its future cash flows discounted at the bond's yield to maturity at that time, which is 6%. The bond pays an annual coupon of 7% of the face value, amounting to $70 annually (7% of $1000). The present value (PV) of the bond is calculated as the sum of the present value of the coupon payments and the present value of the face value:

P₀ = \(\sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n}\)

Where:

• C = Coupon payment = $70
• F = Face value = $1000
• r = Yield to maturity at issue = 6% or 0.06
• n = Number of periods = 10

Calculating the present value using these parameters gives:

P₀ = $70 \(\times\) \(\frac{1 - (1 + r)^{-n}}{r}\) + \(\frac{F}{(1 + r)^n}\)

Plugging in the numbers:

P₀ = $70 \(\times\) \(\frac{1 - (1 + 0.06)^{-10}}{0.06}\) + \(\frac{1000}{(1 + 0.06)^{10}}\)

Calculating each component:

• \(\frac{1 - (1 + 0.06)^{-10}}{0.06}\) ≈ 7.3601
• \(\frac{1000}{(1 + 0.06)^{10}}\) ≈ 558.39

Therefore:

P₀ ≈ $70 \(\times\) 7.3601 + 558.39 ≈ $515.21 + 558.39 ≈ $1073.60

This initial price indicates that the bond was issued at a slight premium, consistent with the coupon rate exceeding the YTM.

Price Immediately Before the First Coupon Payment

Assuming the same YTM of 6% remains constant, the bond's price immediately before the first coupon payment accounts for all remaining cash flows, which include nine annual coupon payments and the face value at the end of ten years. The time remaining is now 9 years, so the price is calculated similarly to the initial but with n=9:

P_before = $70 \(\times\) \(\frac{1 - (1 + 0.06)^{-9}}{0.06}\) + \(\frac{1000}{(1 + 0.06)^9}\)

Calculating these:

• \(\frac{1 - (1 + 0.06)^{-9}}{0.06}\) ≈ 7.5842
• \(\frac{1000}{(1 + 0.06)^9}\) ≈ 589.27

Thus:

P_before ≈ $70 \(\times\) 7.5842 + 589.27 ≈ $531.89 + 589.27 ≈ $1121.16

Price Immediately After the First Coupon Payment

Just after paying the first coupon, the bond's price is based on nine remaining payments plus the face value at maturity, discounted at the same YTM. The calculation is identical to the previous one, as the payment has just been made, and the next coupon payment is due in one year with 9 remaining years:

P_after = $70 \(\times\) \(\frac{1 - (1 + 0.06)^{-9}}{0.06}\) + \(\frac{1000}{(1 + 0.06)^9}\)

This is the same as the price before the coupon payment, approximately $1121.16.

Conclusion

The bond's initial price at issuance reflects the present value of its future cash flows, considering the market YTM at issuance. Over time, as each coupon payment is made, the observable bond price adjusts to reflect remaining cash flows. The calculations demonstrate the dynamic nature of bond valuation and how factors such as remaining maturity and prevailing YTM influence bond prices. Investors and issuers utilize these principles to assess the fair value of bonds and make informed decisions regarding buying, selling, or issuing debt instruments.

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