What are the durations of a 10- and of a 30-year par bond if both yield 3% (semiannually compounded)?

To calculate the duration of a bond, we use the formula for Macaulay duration, which is:

D = (C * (1 + r) + T * M * r) / (M * r * (1 + r)^T)

where:

• D is the duration
• C is the annual coupon payment
• r is the yield to maturity (per period)
• T is the number of periods
• M is the face value of the bond

For a par bond, the coupon payment (C) is equal to the yield to maturity (r) times the face value (M). Since these are par bonds, the face value is 100.

Given that the yield is 3% semiannually compounded, the yield per period (r) is 3%/2 = 1.5%.

For a 10-year bond compounded semiannually, the number of periods (T) is 10*2 = 20.

For a 30-year bond compounded semiannually, the number of periods (T) is 30*2 = 60.

Now we can calculate the duration for each bond.

For the 10-year bond: D = (1.5 * (1 + 1.5) + 20 * 100 * 1.5) / (100 * 1.5 * (1 + 1.5)^20) = 10.21 years

For the 30-year bond: D = (1.5 * (1 + 1.5) + 60 * 100 * 1.5) / (100 * 1.5 * (1 + 1.5)^60) = 19.61 years

So, the duration of a 10-year par bond yielding 3% semiannually compounded is approximately 10.21 years, and the duration of a 30-year par bond yielding 3% semiannually compounded is approximately 19.61 years.