What is a consol bond? What is the duration of a consol bond that sells at a yield to maturity of 8 percent? 10percent? 12percent? Would a consol trading at a yield to maturity of 10 percent have a greater duration than a 20-year zero-coupon bond trading at the same yield to maturity? Why?
The Correct Answer and Explanation is :
What is a Consol Bond?
A consol bond is a type of perpetuity bond that pays a fixed coupon indefinitely, without ever maturing. Instead of repaying the principal, the issuer pays interest payments forever. Its price can be calculated using the formula:
[
P = \frac{C}{r}
]
Where:
- ( P ): Price of the bond
- ( C ): Annual coupon payment
- ( r ): Yield to maturity (YTM) or discount rate
Duration of a Consol Bond
The duration of a consol bond is its Macauley duration, which measures the sensitivity of the bond’s price to changes in interest rates. For a perpetuity, the duration is calculated as:
[
D = \frac{1 + r}{r}
]
Where:
- ( r ): Yield to maturity (YTM)
Calculations:
- At 8% YTM (( r = 0.08 )):
[
D = \frac{1 + 0.08}{0.08} = \frac{1.08}{0.08} = 13.5 \text{ years}
] - At 10% YTM (( r = 0.10 )):
[
D = \frac{1 + 0.10}{0.10} = \frac{1.10}{0.10} = 11 \text{ years}
] - At 12% YTM (( r = 0.12 )):
[
D = \frac{1 + 0.12}{0.12} = \frac{1.12}{0.12} = 9.33 \text{ years}
]
Comparison with a 20-Year Zero-Coupon Bond
The duration of a zero-coupon bond equals its time to maturity, i.e., 20 years. A consol bond trading at a 10% YTM has a duration of 11 years, which is significantly less than the duration of the 20-year zero-coupon bond.
Explanation
The duration of a zero-coupon bond is equal to its maturity because all the cash flow occurs at the end. In contrast, the duration of a consol bond is determined by its infinite series of fixed coupon payments, making it less sensitive to changes in interest rates over long horizons. This reflects the time-weighted average of receiving cash flows, which is shorter for a consol bond than for a long-dated zero-coupon bond. Thus, the consol bond has less interest rate risk.