A 23-year coupon bond with face and redemption value of $1,000 has a coupon rate of 9% per annum payable semiannually and a yield to maturity of 14% per annum compounded semiannually. Algebraically find the price of the bond. Your final answer should be correct to 2 places after the decimal point.

To calculate the price of the bond, we need to use the formula for the present value of a coupon bond:

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

Where:
- $P$ is the price of the bond.
- $C$ is the coupon payment per period.
- $r$ is the yield to maturity (YTM) per period.
- $F$ is the face (or redemption) value of the bond.
- $n$ is the total number of periods.

### Step-by-Step Breakdown:

#### 1. Given Data:
- Face value $ F = 1000 $.
- Redemption value is the same as face value: $ F = 1000 $.
- Coupon rate = 9% per annum, payable semiannually. Therefore, the coupon payment $ C = \frac{9\% \times 1000}{2} = 45 $ dollars every 6 months.
- Yield to maturity = 14% per annum, compounded semiannually, so $ r = \frac{14\%}{2} = 0.07 $ per 6-month period.
- The bond is a 23-year bond, and since payments are made semiannually, the total number of periods $ n = 23 \times 2 = 46 $.

#### 2. Calculate the Price of the Bond:
The price of the bond is the sum of the present value of the coupon payments and the present value of the redemption value.

\[
P = 45 \times \left( 1 - \frac{1}{(1 + 0.07)^{46}} \right) \div 0.07 + \frac{1000}{(1 + 0.07)^{46}}
\]

Now, let’s calculate this step-by-step.

### Step 1: Calculate the present value of the coupon payments
\[
45 \times \left( 1 - \frac{1}{(1 + 0.07)^{46}} \right) \div 0.07
\]

### Step 2: Calculate the present value of the face value
\[
\frac{1000}{(1 + 0.07)^{46}}
\]

Let me calculate these values.The price of the bond is approximately $ 658.75 $ dollars.

Would you like further details or have any questions about this solution?

Here are some related questions you might find helpful:
1. How does the bond price change with different coupon rates?
2. What effect does increasing the yield to maturity have on bond prices?
3. Can you explain why bond prices and yields move inversely?
4. How would the price change if the bond paid coupons annually instead of semiannually?
5. What is the difference between coupon bonds and zero-coupon bonds in terms of pricing?

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

Learn how to algebraically calculate the price of a 23-year coupon bond with a face value of $1,000, a coupon rate of 9%, and a yield to maturity of 14% compounded semiannually. Detailed step-by-step breakdown included. Ideal for undergraduate finance students.

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