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.)

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:

\[
P = \sum \left( \frac{C}{(1 + r/n)^{nt}} \right) + \frac{F}{(1 + r/n)^{nt}}
\]

Where:
- \(P\) is the price of the bond,
- \(C\) is the semi-annual coupon payment,
- \(F\) is the face value of the bond (assume 100% of face value),
- \(r\) is the YTM (as a decimal),
- \(n\) is the number of compounding periods per year (2 for semi-annual),
- \(t\) 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:

\[
P = \sum_{i=1}^{110} \frac{3.5}{(1 + 0.034)^{i}} + \frac{100}{(1 + 0.034)^{110}}
\]

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:

\[
P = \sum_{i=1}^{110} \frac{3.5}{(1 + 0.043)^{i}} + \frac{100}{(1 + 0.043)^{110}}
\]

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)

\[
Credit\ Spread = 8.6\% - 6.8\% = 1.8\%
\]

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:
1. How does changing the coupon rate affect the bond price?
2. What happens to bond prices if the YTM decreases?
3. How is the bond duration related to price sensitivity?
4. What role does the credit rating play in determining the credit spread?
5. 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.

A BBB-rated corporate bond has a yield to maturity of 8.6%. A U.S. Treasury security has a yield to maturity of 6.8%. These yields are quoted as APRs with semiannual compounding. Both bonds pay semi-annual coupons at a rate of 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?

Learn how to calculate the price of a U.S. Treasury bond and a BBB-rated corporate bond using semiannual compounding, and determine the credit spread between the two. This problem uses concepts like yield to maturity, bond pricing formulas, and credit spread, suitable for undergraduate finance or economics students.

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