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### Part (a) Price of a 2-Year Zero-Coupon Bond

To determine the price of a 2-year zero-coupon bond with a face value of €1,000, we need to use the **spot rate** for a 2-year bond. The spot rate is the interest rate for a specific maturity, allowing us to discount the bond's face value back to the present value.

The spot rate for a 2-year zero-coupon bond is not directly given. However, we will derive it in part (b). First, let's walk through the steps necessary to compute the bond price once we have the 2-year spot rate:

\[
\text{Price} = \frac{\text{Face Value}}{(1 + r_2)^2}
\]
Where:
- Face Value = €1,000
- \(r_2\) is the 2-year spot rate.

We will return to this after deriving the spot rates.

---

### Part (b) Deriving the Zero-Coupon Yield Rates (Spot Rates)

We can derive the spot rates from the bond prices and the coupon payments provided. Let's calculate the spot rates for years 1 to 4.

**Bond A (1-year bond):**
- This is a zero-coupon bond, so the spot rate for year 1 is the same as the yield to maturity.
- Spot rate \( r_1 = 2.00\% \).

**Bond B (2-year bond):**
- The coupon is 10% of the face value, so it pays €100 annually.
- The bond price equation is:

\[
P_B = \frac{100}{(1 + r_1)} + \frac{100 + 1000}{(1 + r_2)^2}
\]
Where:
- \(P_B\) is the price of bond B, which we can compute from the yield-to-maturity (YTM) given.
- \(P_B = \frac{100}{1.02} + \frac{1100}{(1 + r_2)^2}\).

Next, solve for \(r_2\), the 2-year spot rate.

**Bond C (3-year bond):**
- Coupon rate = 6%, so it pays €60 annually.
- The bond price equation is:

\[
P_C = \frac{60}{(1 + r_1)} + \frac{60}{(1 + r_2)^2} + \frac{60 + 1000}{(1 + r_3)^3}
\]
Where \(r_3\) is the 3-year spot rate.

**Bond D (4-year bond):**
- Coupon rate = 12%, so it pays €120 annually.
- The bond price equation is:

\[
P_D = \frac{120}{(1 + r_1)} + \frac{120}{(1 + r_2)^2} + \frac{120}{(1 + r_3)^3} + \frac{120 + 1000}{(1 + r_4)^4}
\]
Where \(r_4\) is the 4-year spot rate.

Let me calculate the spot rates \(r_2\), \(r_3\), and \(r_4\).The real and feasible solution for the spot rates from the calculations are:

- \( r_2 = 0.1525 \) or 15.25%
- \( r_3 = 0.1275 \) or 12.75%
- \( r_4 = 0.2192 \) or 21.92%

These are the spot rates for years 2, 3, and 4, respectively. Now, using \( r_2 = 15.25\% \), we can calculate the price of the 2-year zero-coupon bond in part (a).

### Part (a) Price of the 2-Year Zero-Coupon Bond:
\[
\text{Price} = \frac{1000}{(1 + r_2)^2} = \frac{1000}{(1 + 0.1525)^2}
\]
Let's calculate this now.The price of the 2-year zero-coupon bond with a face value of €1,000 is approximately **€752.87**.

### Summary:
1. The price of the 2-year zero-coupon bond is €752.87.
2. The derived zero-coupon yield (spot rates) are:
   - \( r_1 = 2.00\% \) (given)
   - \( r_2 = 15.25\% \)
   - \( r_3 = 12.75\% \)
   - \( r_4 = 21.92\% \)

Let me know if you need any clarifications or further details!

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### Related Questions:
1. How is the yield to maturity (YTM) different from spot rates?
2. What factors influence the price of a bond in financial markets?
3. How can bond prices be calculated for different coupon rates?
4. What is the relationship between bond prices and interest rates?
5. How can zero-coupon bonds be used in financial strategies?

**Tip:** Remember that higher interest rates (spot rates) lower the bond price because future payments are discounted more heavily!

Given information about four different, default-free bonds, each with a face value of €1,000:
(a) What is the price of a 2-year zero-coupon bond with a face value of €1,000?
(b) Derive the zero-coupon yield rates (spot rates) for years 1 to 4.

Learn how to calculate the price of a 2-year zero-coupon bond and derive the zero-coupon yield rates (spot rates) for a set of bonds with different maturities. This problem covers key finance concepts including bond pricing, spot rates, and the time value of money. Suitable for undergraduate finance students.

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