Suppose a​ 10-year, $ 1 comma 000 bond with an 8.7 % coupon rate and​ semi-annual coupons is trading for a price of ​$1 comma 035.25.
a. What is the​ bond's yield to maturity​ (expressed as an APR with​ semi-annual compounding)?
b. If the​ bond's yield to maturity changes to 9.3 % ​APR, what will the​ bond's price​ be?

Let's solve this problem step-by-step.

### Given Information
- **Bond Face Value**: $1,000
- **Coupon Rate**: 8.7% (semi-annual coupons, so each coupon is 4.35% of the face value)
- **Time to Maturity**: 10 years (so there are $10 \times 2 = 20$ semi-annual periods)
- **Current Price**: $1,035.25

### Part (a): Calculate the Yield to Maturity (YTM)

The yield to maturity (YTM) is the interest rate that equates the present value of all future cash flows (coupons and face value) to the bond's current price. With semi-annual compounding, the cash flows for each period are:

1. **Coupon Payment per Period** = $ \frac{8.7\%}{2} \times 1000 = 43.50$
2. **Face Value at Maturity** = $1,000

The bond price equation, with YTM as the unknown, is:

\[
1,035.25 = \sum_{t=1}^{20} \frac{43.50}{(1 + r)^t} + \frac{1000}{(1 + r)^{20}}
\]

Here, $ r $ is the semi-annual YTM rate. The annual YTM (expressed as APR with semi-annual compounding) is $ 2r $.

This equation needs to be solved for $ r $ using iterative or numerical methods, as it cannot be solved algebraically.

I'll compute the YTM.

The bond's yield to maturity (YTM), expressed as an APR with semi-annual compounding, is approximately **8.18%**.

### Part (b): Calculate the Bond's Price if YTM Changes to 9.3% APR

If the YTM changes to 9.3% APR, we need to find the new price of the bond. Since this is with semi-annual compounding, the semi-annual YTM becomes:

\[
\text{New Semi-Annual YTM} = \frac{9.3\%}{2} = 4.65\%
\]

We can use this rate to discount each cash flow to determine the bond's new price.

Let's calculate it.If the bond's yield to maturity changes to 9.3% APR, the bond's price will be approximately **$961.48**.

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Would you like more details on any of the calculations, or do you have further questions?

### Related Questions to Consider:
1. How does the coupon rate affect the bond’s price sensitivity to interest rate changes?
2. What is the difference between yield to maturity and current yield?
3. How would the bond's price change if the coupon rate were higher or lower?
4. What factors typically influence changes in a bond’s yield to maturity?
5. How does the compounding frequency impact the effective yield of a bond?

### Tip:
When interest rates rise, bond prices generally fall, and vice versa. This inverse relationship can be useful in portfolio diversification strategies.

Suppose a 10-year, $1,000 bond with an 8.7% coupon rate and semi-annual coupons is trading for a price of $1,035.25. a. What is the bond's yield to maturity (expressed as an APR with semi-annual compounding)? b. If the bond's yield to maturity changes to 9.3% APR, what will the bond's price be?

Learn how to calculate a bond's yield to maturity (YTM) and determine its price change when YTM fluctuates. This detailed guide walks through a 10-year, $1,000 bond with an 8.7% coupon rate, showing the effect of a YTM change to 9.3%. Key topics include present value, compounding, and bond valuation. Ideal for finance students and professionals.

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