**All other things equal (ytm = 10%), which of the following has the longest duration?**

**Options**

- A) a 30-year bond with a 10% coupon
- B) a 20-year bond with a 9% coupon
- C) a 20-year bond with a 7% coupon
- D) a 10-year zero-coupon bond

**Correct Answer:** A) a 30-year bond with a 10% coupon

**Answer Explanation:**

The concept of duration is an essential component in understanding bond investments. It measures the sensitivity of a bond’s price to changes in interest rates. Duration is usually measured in years and provides investors with valuable information about a bond’s price risk.

In this scenario, we are given four different bonds with various maturities and coupon rates, and we need to determine which one has the longest duration while assuming that all other factors, including the yield to maturity (YTM), are equal at 10%.

**Let’s break down the options and explain why the answer is correct and why the other options are not correct in detail:**

**A) 30-year bond with a 10% coupon:**

This bond has a maturity of 30 years and a coupon rate of 10%, which means it pays an annual interest of 10% of its face value. Since the YTM is also 10%, this bond is trading at par, which means its price is equal to its face value. When YTM and coupon rate are the same, the bond’s price equals its face value.

Now, to calculate the duration, we need to consider the weighted average time to receive the bond’s cash flows (coupon payments and principal repayment). In this case, since it’s a 30-year bond, the bondholder will receive 30 coupon payments and the face value of the bond at the end of 30 years.

To calculate the duration, you’ll assign weights to each cash flow based on the present value of those cash flows. In this scenario, all the cash flows are equally weighted because the YTM is equal to the coupon rate, making the bond trade at par.

The duration of this bond will be close to its maturity, which is 30 years. Therefore, option A has the longest duration among the given choices.

**B) 20-year bond with a 9% coupon:**

This bond has a maturity of 20 years and a coupon rate of 9%. The YTM is given as 10%, which means it’s trading at a discount because the coupon rate is lower than the YTM. When the coupon rate is less than the YTM, the bond’s price is lower than its face value.

To calculate the duration of this bond, you’ll again need to consider the weighted average time to receive its cash flows. In this case, you have 20 coupon payments and the face value of the bond at the end of 20 years.

Because it’s trading at a discount, the bond’s price is less than its face value, and the principal repayment at maturity will have a higher present value than the coupon payments. Therefore, the weighted average time will be less than 20 years. The duration of this bond will be less than 20 years.

**C) 20-year bond with a 7% coupon:**

This bond is similar to option B, with a maturity of 20 years. However, it has a lower coupon rate of 7%. Given that the YTM is 10%, this bond is also trading at a discount, just like option B. As mentioned earlier, when the coupon rate is lower than the YTM, the bond’s price is less than its face value.

To calculate the duration of this bond, you’ll again consider the weighted average time to receive its cash flows. In this case, you have 20 coupon payments and the face value of the bond at the end of 20 years.

Since it’s trading at a discount, the principal repayment at maturity will have a higher present value than the coupon payments. Therefore, the weighted average time will be less than 20 years, just like option B.

**D) 10-year zero-coupon bond:**

A zero-coupon bond does not make regular coupon payments. Instead, it is sold at a discount to its face value, and the investor receives the face value at maturity. In this case, we have a 10-year zero-coupon bond, and the YTM is given as 10%.

The duration of a zero-coupon bond is equal to its time to maturity. In this case, the duration is 10 years, which is the same as the bond’s maturity.

**In summary:**

- Option A (30-year bond with a 10% coupon) has the longest duration because it is trading at par, and its duration is close to its maturity of 30 years.
- Options B (20-year bond with a 9% coupon) and C (20-year bond with a 7% coupon) have shorter durations than option A because they are trading at discounts, resulting in higher present values for the principal repayment at maturity, which reduces the weighted average time.
- Option D (10-year zero-coupon bond) has a duration equal to its maturity of 10 years because it does not make coupon payments.

In conclusion, when comparing bond durations, it’s essential to consider not only the time to maturity but also the relationship between the coupon rate and the YTM, as well as the bond’s price relative to its face value. In this scenario, option A has the longest duration because it’s trading at par, and its maturity is the longest among the given options.