A 27-year coupon bond with a face and redemption value of $1,000 has a coupon rate of 20% per annum payable semiannually and a yield to maturity of 12% per annum compounded semiannually. Algebraically find the price of the bond. 

To find the price of the bond algebraically, we can use the present value formula for bonds. The price of the bond is the sum of the present value of the coupon payments and the present value of the redemption value. The formula is as follows:

\[
P = C \times \left( 1 - (1 + r)^{-n} \right) \div r + \frac{F}{(1 + r)^n}
\]

Where:
- $P$ is the price of the bond.
- $C$ is the semiannual coupon payment.
- $F$ is the face value (redemption value) of the bond.
- $r$ is the semiannual yield (which is half of the annual yield).
- $n$ is the total number of semiannual periods.

### Step 1: Calculate the semiannual coupon payment
Since the coupon rate is 20% per annum, and the bond pays interest semiannually, the semiannual coupon rate is half of that:

\[
\text{Semiannual coupon rate} = \frac{20\%}{2} = 10\%
\]

Thus, the semiannual coupon payment is:

\[
C = 10\% \times 1000 = 100
\]

### Step 2: Calculate the semiannual yield
The yield to maturity is 12% per annum compounded semiannually, so the semiannual yield is:

\[
r = \frac{12\%}{2} = 6\% = 0.06
\]

### Step 3: Calculate the total number of semiannual periods
The bond has 27 years to maturity, and since interest is paid semiannually, the total number of periods is:

\[
n = 27 \times 2 = 54 \text{ semiannual periods}
\]

### Step 4: Apply the bond pricing formula
Now, plug all these values into the bond pricing formula:

\[
P = 100 \times \left( 1 - (1 + 0.06)^{-54} \right) \div 0.06 + \frac{1000}{(1 + 0.06)^{54}}
\]

Let’s break this down:

1. First, calculate the present value of the coupon payments:

\[
100 \times \left( 1 - (1.06)^{-54} \right) \div 0.06
\]

2. Then, calculate the present value of the redemption value:

\[
\frac{1000}{(1.06)^{54}}
\]

Once we calculate these two components, their sum will give the price of the bond.

Let's solve these.The price of the bond is approximately **$1,638**.

Would you like more details or have any questions?

Here are 5 related questions that can extend your understanding:
1. How would the bond price change if the yield to maturity increased to 15%?
2. What happens to the bond price as the number of years to maturity decreases?
3. How does the frequency of coupon payments affect the bond price?
4. What is the difference between coupon bonds and zero-coupon bonds in terms of pricing?
5. How would the bond price be affected if the coupon rate was lower, say 10%?

**Tip**: A bond's price decreases when the yield to maturity is higher than the coupon rate because future cash flows are discounted at a higher rate.

A 27-year coupon bond with a face and redemption value of $1,000 has a coupon rate of 20% per annum payable semiannually and a yield to maturity of 12% per annum compounded semiannually. Algebraically find the price of the bond.

Learn how to algebraically find the price of a 27-year coupon bond with a 20% coupon rate and a yield to maturity of 12%. This solution covers key concepts in bond pricing, including the present value formula, semiannual coupon payments, and yields. Ideal for undergraduate finance or economics students.

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