One bond has a coupon rate of 6.6%, another a coupon rate of 8.3%. Both bonds pay interest annually, have 15-year maturities, and sell at a yield to maturity of 8.0%.

If their yields to maturity next year are still 8.0%, what is the rate of return on each bond?

To calculate the rate of return on each bond, we need to consider both the interest (coupon) payments received and the change in the bond's price over the year. Since the yield to maturity (YTM) remains the same at 8.0% after one year, the bond prices will not change over the year. The rate of return will only reflect the coupon payment relative to the bond's price at the start of the year.

### Step-by-Step Solution:

1. **Determine the bond prices today** using the given YTM of 8.0%. Since both bonds have a 15-year maturity and pay interest annually, we'll use the formula for the present value of a bond:

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

   where:
   - $ P $ is the bond price,
   - $ C $ is the annual coupon payment,
   - $ YTM $ is the yield to maturity (8.0%),
   - $ n $ is the number of years to maturity (15),
   - $ F $ is the face value of the bond (assumed to be $1000 for simplicity).

2. **Calculate the price of each bond**:
   - **Bond 1**: Coupon rate = 6.6%, so annual coupon payment $ C = 6.6\% \times 1000 = 66 $.
   - **Bond 2**: Coupon rate = 8.3%, so annual coupon payment $ C = 8.3\% \times 1000 = 83 $.

3. **Price of Bond 1** (6.6% coupon):
   \[
   P_1 = \sum_{t=1}^{15} \frac{66}{(1 + 0.08)^t} + \frac{1000}{(1 + 0.08)^{15}}
   \]

4. **Price of Bond 2** (8.3% coupon):
   \[
   P_2 = \sum_{t=1}^{15} \frac{83}{(1 + 0.08)^t} + \frac{1000}{(1 + 0.08)^{15}}
   \]

5. **Find the prices numerically** using the bond pricing formula to get precise bond prices.

6. **Calculate the rate of return** for each bond over the one-year period. If the YTM remains at 8.0%, the bond prices one year later can be calculated similarly, but now with 14 years remaining to maturity.

7. **Rate of Return Formula**:
   \[
   \text{Rate of Return} = \frac{\text{Coupon Payment} + (\text{Price}_{\text{End}} - \text{Price}_{\text{Start}})}{\text{Price}_{\text{Start}}}
   \]

   Since the price at the end and start of the year will remain the same if YTM remains unchanged:
   \[
   \text{Rate of Return} = \frac{\text{Coupon Payment}}{\text{Price}_{\text{Start}}}
   \]

### Let's calculate this in detail:

1. **Bond 1 (6.6% coupon rate)**:
   \[
   P_1 = \sum_{t=1}^{15} \frac{66}{(1 + 0.08)^t} + \frac{1000}{(1 + 0.08)^{15}}
   \]
2. **Bond 2 (8.3% coupon rate)**:
   \[
   P_2 = \sum_{t=1}^{15} \frac{83}{(1 + 0.08)^t} + \frac{1000}{(1 + 0.08)^{15}}
   \]

I'll calculate these values and the final rates of return now.Here are the calculations:

1. **Bond 1 (6.6% coupon rate):**
   - Price: $880.17
   - Rate of Return: $ \approx 7.50\% $

2. **Bond 2 (8.3% coupon rate):**
   - Price: $1025.68
   - Rate of Return: $ \approx 8.09\% $

### Explanation:
- The rate of return for Bond 1 is lower than its coupon rate because it sells at a discount.
- The rate of return for Bond 2 is slightly higher than the YTM due to its premium pricing.

Do you want more details or have any questions about this calculation?

### Further Questions:
1. How would the rates of return change if the yields to maturity increased to 9% next year?
2. What happens to the bond prices if the yield to maturity drops to 7%?
3. Why is the rate of return different from the coupon rate for these bonds?
4. How does the duration of a bond affect its sensitivity to changes in interest rates?
5. What is the impact of semi-annual coupon payments on bond pricing and returns?

### Tip:
The rate of return on a bond depends on both the coupon payments and the price changes. If the market interest rates (YTM) remain constant, the bond's return comes primarily from the coupon payments.

One bond has a coupon rate of 6.6%, another a coupon rate of 8.3%. Both bonds pay interest annually, have 15-year maturities, and sell at a yield to maturity of 8.0%. If their yields to maturity next year are still 8.0%, what is the rate of return on each bond?

Calculate the rate of return for two bonds with coupon rates of 6.6% and 8.3%, 15-year maturities, and a yield to maturity of 8.0%. This solution includes step-by-step bond pricing and rate of return calculation, using key formulas like present value of a bond and rate of return. Suitable for finance students and professionals.

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