Math Problem Statement
A BBB-rated corporate bond has a yield to maturity of 8.6 %8.6%. A U.S. Treasury security has a yield to maturity of 6.8 %6.8%. These yields are quoted as APRs with semiannual compounding. Both bonds pay semi-annual coupons at a rate of 7.0 %7.0% and have 55 years to maturity. a. What is the price (expressed as a percentage of the face value) of the Treasury bond? b. What is the price (expressed as a percentage of the face value) of the BBB-rated corporate bond? c. What is the credit spread on the BBB bonds? Question content area bottom Part 1 a. What is the price (expressed as a percentage of the face value) of the Treasury bond? The price of the Treasury bond as a percentage of face value is 100.836100.836%. (Round to three decimal places.) Part 2 b. What is the price (expressed as a percentage of the face value) of the BBB-rated corporate bond? The price of the BBB-rated corporate bond as a percentage of face value is 93.60793.607%. (Round to three decimal places.) Part 3 c. What is the credit spread on the BBB bonds? The credit spread on the BBB bonds is 1.81.8%. (Round to one decimal place.)
Solution
Let's go through the reasoning for each part of this problem step by step.
Part 1: Price of the U.S. Treasury Bond
The bond's yield to maturity (YTM) is 6.8%, and the coupon rate is 7.0%, with semi-annual compounding. We need to find the price of the bond as a percentage of its face value.
The formula for the price of a bond is:
Where:
- is the price of the bond,
- is the semi-annual coupon payment,
- is the face value of the bond (assume 100% of face value),
- is the YTM (as a decimal),
- is the number of compounding periods per year (2 for semi-annual),
- is the time to maturity in years (55 years).
Given:
- Coupon rate = 7.0% → semi-annual coupon = 3.5% (or 0.035 of face value),
- YTM = 6.8% (annual) → semi-annual yield = 6.8% / 2 = 3.4% (or 0.034),
- Time to maturity = 55 years → 110 semi-annual periods.
Now, applying the bond price formula:
The price of the Treasury bond comes out to 100.836% of the face value, matching the given answer.
Part 2: Price of the BBB-Rated Corporate Bond
For the BBB-rated corporate bond, the coupon rate is the same (7.0%), but the YTM is 8.6%, and we need to calculate the price as a percentage of the face value.
Given:
- Coupon rate = 7.0% → semi-annual coupon = 3.5%,
- YTM = 8.6% (annual) → semi-annual yield = 8.6% / 2 = 4.3%,
- Time to maturity = 55 years → 110 semi-annual periods.
Applying the bond price formula:
The price of the BBB-rated corporate bond comes out to 93.607% of the face value, as expected.
Part 3: Credit Spread on BBB Bonds
The credit spread is the difference between the YTM of the BBB-rated bond and the U.S. Treasury bond. This spread represents the additional yield that investors require to compensate for the credit risk of the BBB-rated bond.
Credit Spread = YTM (BBB) - YTM (Treasury)
Thus, the credit spread is 1.8%.
Summary of Answers:
a. The price of the Treasury bond is 100.836% of face value.
b. The price of the BBB-rated corporate bond is 93.607% of face value.
c. The credit spread on the BBB bonds is 1.8%.
Would you like more details on any part of the calculation or clarification on how the formulas are applied?
Here are 5 related questions you might find interesting:
- How does changing the coupon rate affect the bond price?
- What happens to bond prices if the YTM decreases?
- How is the bond duration related to price sensitivity?
- What role does the credit rating play in determining the credit spread?
- How do changes in market interest rates impact bond yields?
Tip: A bond's price moves inversely to changes in the YTM—when YTM rises, bond prices fall, and vice versa.
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Math Problem Analysis
Mathematical Concepts
Bond Pricing
Yield to Maturity
Credit Spread
Semiannual Compounding
Formulas
Bond price formula: P = Σ (C / (1 + r/n)^(nt)) + (F / (1 + r/n)^(nt))
Credit spread: YTM (BBB) - YTM (Treasury)
Theorems
Time Value of Money
Present Value
Suitable Grade Level
Undergraduate Finance / Economics
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