A 18 year bond at YTM 6%, coupon rate of 9.215% face value of $1000, that was $1350.95. Called price $1120, called at 5 years at 12% premium over par for a realized yield of 3.675%? If you bought one of these bonds today, calculate the expected rate of return

To calculate the expected rate of return on the bond, we need to consider two possible scenarios:

1. **The bond is held until maturity** (18 years from now).
2. **The bond is called** after 5 years at a 12% premium over par.

### 1. **Rate of Return if Held Until Maturity**

If the bond is held until maturity:

- **Coupon Rate (C):** 9.215% of face value = 0.09215 * $1,000 = $92.15 per year.
- **Face Value (FV):** $1,000
- **Purchase Price (P):** $1,350.95
- **Years to Maturity (n):** 18 years

The expected yield (YTM) can be approximated using the formula:

\[
\text{YTM} \approx \frac{C + \frac{FV - P}{n}}{\frac{FV + P}{2}}
\]

Substituting the values:

\[
\text{YTM} \approx \frac{92.15 + \frac{1000 - 1350.95}{18}}{\frac{1000 + 1350.95}{2}}
\]

\[
\text{YTM} \approx \frac{92.15 + \frac{-350.95}{18}}{1175.475}
\]

\[
\text{YTM} \approx \frac{92.15 - 19.497}{1175.475} 
\]

\[
\text{YTM} \approx \frac{72.653}{1175.475} \approx 0.0618 \text{ or } 6.18\%
\]

The YTM if held to maturity is approximately **6.18%**.

### 2. **Rate of Return if Called in 5 Years**

If the bond is called after 5 years:

- **Call Price (CP):** $1,120 (since 12% premium over par of $1,000)
- **Years to Call (n):** 5 years

The yield to call (YTC) can be calculated using:

\[
\text{YTC} \approx \frac{C + \frac{CP - P}{n}}{\frac{CP + P}{2}}
\]

Substituting the values:

\[
\text{YTC} \approx \frac{92.15 + \frac{1120 - 1350.95}{5}}{\frac{1120 + 1350.95}{2}}
\]

\[
\text{YTC} \approx \frac{92.15 + \frac{-230.95}{5}}{\1235.475}
\]

\[
\text{YTC} \approx \frac{92.15 - 46.19}{1235.475} 
\]

\[
\text{YTC} \approx \frac{45.96}{1235.475} \approx 0.0372 \text{ or } 3.72\%
\]

The yield to call in 5 years is approximately **3.72%**.

### **Expected Rate of Return**
Given that the bond could be called, you would typically use the lower of the YTM and YTC as the expected rate of return, considering the worst-case scenario for an investor.

**Expected Rate of Return:** **3.72%**

### Next Steps

Would you like more details or have any questions?

Here are some additional questions to consider:

1. How does the coupon rate affect the bond’s price?
2. What is the impact of the bond being called on the expected rate of return?
3. How is the realized yield different from the yield to maturity?
4. What are the risks of buying a callable bond?
5. How would changes in market interest rates affect the bond’s price?

**Tip:** Callable bonds often offer higher coupon rates to compensate for the risk that the issuer might redeem them before maturity, potentially affecting your expected returns.

Learn how to calculate the expected rate of return on a callable bond with a detailed step-by-step solution. This problem covers bond valuation concepts, including Yield to Maturity (YTM) and Yield to Call (YTC). Suitable for financial professionals and investors.

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