For simplicity assume annual compounding and annual coupon payments for this question.

Exactly 2 years ago, an investor bought a newly issued 8.0%, 10-year, $1000-face value coupon bond from Effortless Fraud Inc. at a time when the yield to maturity on the company's bonds was 5.47%.

Part A: What did the investor pay for this bond at the time? (2 marks)

(Please answer to 2 decimal places)

Part B: Today, the yield to maturity on the bond of Effortless Fraud Inc. is 6.69%. What is this bond worth currently? (2 marks)

$

(Please answer to 2 decimal places)

To solve the problem, we need to calculate the price of the bond at two different points in time: when the bond was first purchased (Part A) and its value today (Part B). 

### **Part A: Price of the Bond When Purchased**
The price of a bond can be calculated using the formula:

\[
P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N}
\]

Where:
- $P$ is the price of the bond.
- $C$ is the annual coupon payment.
- $r$ is the yield to maturity (YTM) at the time of purchase.
- $F$ is the face value of the bond.
- $N$ is the number of years to maturity.

Given:
- Annual coupon rate = 8.0%
- Face value ($F$) = $1000
- Yield to maturity ($r$) = 5.47% or 0.0547
- Total maturity = 10 years
- Years remaining after 2 years = 8 years

The annual coupon payment ($C$) is calculated as:

\[
C = \text{Coupon Rate} \times F = 0.08 \times 1000 = 80 \text{ dollars}
\]

Now, calculate the price of the bond at the time of purchase:

\[
P = \sum_{t=1}^{10} \frac{80}{(1 + 0.0547)^t} + \frac{1000}{(1 + 0.0547)^{10}}
\]

This is the sum of the present value of all future coupon payments and the present value of the face value.

Let's calculate this step-by-step:

1. **Present Value of Coupon Payments:**

\[
PV_{\text{Coupons}} = \sum_{t=1}^{10} \frac{80}{(1 + 0.0547)^t}
\]

2. **Present Value of Face Value:**

\[
PV_{\text{Face Value}} = \frac{1000}{(1 + 0.0547)^{10}}
\]

I'll calculate this now.

\[
P_{\text{initial}} = \frac{80}{(1 + 0.0547)} + \frac{80}{(1 + 0.0547)^2} + \ldots + \frac{80}{(1 + 0.0547)^{10}} + \frac{1000}{(1 + 0.0547)^{10}}
\]

Calculating this sum:

\[
P_{\text{initial}} \approx 80 \times \frac{1 - (1.0547)^{-10}}{0.0547} + \frac{1000}{(1.0547)^{10}}
\]

\[
P_{\text{initial}} = 80 \times 7.6295 + 586.69 = 610.36 + 586.69 = 1197.05
\]

So the price paid initially:

\[
P_{\text{initial}} = 1197.05
\]

### **Part B: Price of the Bond Today**
Now, let's calculate the current price of the bond using the new yield to maturity of 6.69%:

Given:
- New yield to maturity ($r$) = 6.69% or 0.0669
- Remaining years ($N$) = 8 years

Again, using the formula for bond price:

\[
P_{\text{current}} = \sum_{t=1}^{8} \frac{80}{(1 + 0.0669)^t} + \frac{1000}{(1 + 0.0669)^{8}}
\]

Calculating the sum of the present values of the coupon payments and the face value:

\[
P_{\text{current}} = 80 \times \frac{1 - (1.0669)^{-8}}{0.0669} + \frac{1000}{(1.0669)^{8}}
\]

This sum evaluates to approximately:

\[
P_{\text{current}} \approx 80 \times 6.3928 + 607.83 \approx 511.42 + 606.98 = 1103.26
\]

---

### **Summary of Answers:**

1. **Part A:** The price paid for the bond initially: **$1,197.05**
2. **Part B:** The bond's current value: **$1,103.26**

Would you like more details on any part of this solution, or do you have any further questions?

### Related Questions:
1. How does an increase in yield to maturity affect the price of a bond?
2. What happens to the bond's price if the yield to maturity drops below the coupon rate?
3. How do you calculate the yield to maturity if the bond's price and coupon rate are known?
4. What is the significance of bond duration in price sensitivity?
5. How does bond convexity relate to the change in bond prices with interest rate changes?
6. What is the difference between current yield and yield to maturity?
7. How would a bond's price change if the yield to maturity increases after each year?
8. What factors can cause changes in a bond's yield to maturity?

**Tip:** The price of a bond is inversely related to its yield to maturity; as yields rise, bond prices fall, and vice versa.

Learn how to calculate the price of a bond and its current value using annual compounding and coupon payments. Solve for the initial price paid and the current bond value based on changing yield to maturity. Suitable for advanced undergraduate finance students.

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