You own investment A and 10 bonds of bond B. The total value of your holdings is 10,364.59. BB has a coupon rate of 11.26% par value of $1000 YTM of 13.20% 14 years until maturity and semi annual coupons with the next coupon expected in six months. Investment A is expected to pay $253 per year for 11 years, has an expected return of X percent, and is expected to make its first payment later today. What is X?

To find $ X $, the expected return for Investment A, we can break down the problem by first calculating the value of Bond B and then determining $ X $ for Investment A based on the total portfolio value.

### Step 1: Calculate the Present Value of Bond B

Each bond (Bond B) has:
- **Par Value** = $1000
- **Coupon Rate** = 11.26%
- **Yield to Maturity (YTM)** = 13.20%
- **Maturity** = 14 years
- **Semi-annual coupons** (twice per year, so each period is 6 months)
- **Number of Periods** = 14 years * 2 = 28

**Semi-annual Coupon Payment**:
\[
\text{Coupon Payment} = \frac{\text{Coupon Rate} \times \text{Par Value}}{2} = \frac{0.1126 \times 1000}{2} = 56.30
\]

The present value of each bond can be calculated as the sum of the present values of the coupon payments and the face value discounted at the bond's semi-annual YTM rate:

1. **Discount Rate per Period (Semi-annual YTM)**:
   \[
   \text{Discount Rate} = \frac{\text{YTM}}{2} = \frac{0.1320}{2} = 0.066
   \]

2. **Present Value of Coupon Payments**:
   \[
   \text{PV of Coupons} = \sum_{t=1}^{28} \frac{56.30}{(1 + 0.066)^t} = 613.10 \, \text{(approximated)}
   \]

3. **Present Value of Face Value**:
   \[
   \text{PV of Face Value} = \frac{1000}{(1 + 0.066)^{28}} = 235.89 \, \text{(approximated)}
   \]

   Total present value of each bond is approximately:
   \[
   \text{PV of Bond B} = 613.10 + 235.89 = 848.99
   \]

   Since there are 10 bonds:
   \[
   \text{Total Value of Bond B Holdings} = 10 \times 848.99 = 8489.90
   \]

### Step 2: Calculate the Present Value of Investment A

Since the total portfolio value is $10,364.59, we can find the value of Investment A:
\[
\text{Value of Investment A} = 10,364.59 - 8,489.90 = 1,874.69
\]

Investment A is expected to pay $253 per year for 11 years. We are given the value of these cash flows and need to find the expected return rate $ X $.

Using the formula for the present value of an annuity:
\[
1,874.69 = \frac{253 \times (1 - (1 + X)^{-11})}{X}
\]

Solving this equation for $ X $ involves some iteration, but the approximate solution is:
\[
X \approx 10.47\%
\]

### Answer:
The expected return $ X $ for Investment A is approximately **10.47%**.

Would you like a more detailed breakdown or have any questions?

### Related Questions
1. How would the expected return change if Investment A’s payments started a year from today instead of immediately?
2. How is the value of a bond affected by changes in interest rates and time to maturity?
3. What is the impact of semi-annual versus annual compounding on bond yields?
4. How would a higher YTM for Bond B influence the portfolio value?
5. What formula is used to find the present value of an annuity?

#### Tip:
When calculating bond prices, remember that longer time to maturity amplifies the effect of interest rate changes on the bond's value.

This solution demonstrates how to find the expected return on an investment given the total value of holdings in bonds and an annuity. It includes detailed bond valuation calculations, present value of annuities, and expected return determination. Suitable for undergraduate finance students.

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