To solve for $y$, which represents the semi-annual yield, let's break down the equation step-by-step. Afterward, we'll double $y$ to express the yield to maturity as an APR with semi-annual compounding.

The equation given is:

$1038.00 = \frac{46.00}{y} \left[ 1 - \frac{1}{(1 + y)^{20}} \right] + \frac{1000}{(1 + y)^{20}}$

Here's the plan:

1. Isolate $y$: Rearranging this equation directly can be complex due to the exponent and terms involving $y$ in both the denominator and base of the exponent. Generally, equations like this are solved numerically.
2. Use Numerical Methods: We can apply numerical methods, such as the Newton-Raphson method or a financial calculator, to approximate $y$.

Step 1: Rearrange for a Numerical Solution

Let's rewrite the formula with $y$ explicitly on one side to set up for a numerical method:

$1038.00 = \frac{46.00}{y} \left[ 1 - (1 + y)^{-20} \right] + 1000 \cdot (1 + y)^{-20}$

This equation does not lend itself to algebraic solutions because of its complexity, so the next step involves using an iterative approach (like Newton-Raphson or bisection) or a financial calculator. Let’s outline a few details for implementing it numerically if solving manually.

1. Initial Guess: Begin with a reasonable guess for $y$.
2. Iterate: Use a calculator or software to solve for $y$ such that both sides of the equation match closely.

Step 2: Yield to Maturity (APR)

Once you find the semi-annual yield $y$, double it to get the yield to maturity (APR), as follows:

$\text{APR} = 2 \times y$

Would you like a more detailed step-by-step numerical approximation? Or do you need help with another part of this problem?

---

Further Practice Questions

1. What numerical method can we use to solve equations of this form, and how does it work?
2. How would the equation change if the compounding period were quarterly instead of semi-annually?
3. How can you verify the solution for $y$ once obtained through a numerical method?
4. Why is doubling the semi-annual yield an appropriate way to express APR?
5. How does increasing the face value from 1000 to 2000 affect the yield to maturity?

---

Tip:

For complex bond valuation and yield calculations, a financial calculator or spreadsheet software can save significant time and ensure precision.