A 30 year zero coupon bond yields 4.5% annually. Price it per $100 of face to the nearest cent in your head, and give its DV01.

A 30 year zero coupon bond yields 4.5% annually. Price it per $100 of face to the nearest cent in your head, and give its DV01.

Approach: Take logs of the discount factor with a three-term expansion, exponentiate, then use the fact that a zero's Macaulay duration equals its maturity to reach the modified duration.

26.70. The price of the zero coupon bond is 100 divided by 1.045 to the power 30. Take logs: ln(1.045) = 0.045 - 0.0010125 + 0.0000304 = 0.0440179, so 30 times that is 1.32054 and the growth factor is e to that power, which is 3.7453. The price is 100/3.7453 = 26.700 per 100 of face. The rule of 72 gives the sanity check, since 72/4.5 = 16 years to double, so 30 years is just under two doublings and the price should land a little above 25. The Macaulay duration of a zero equals its maturity, so the modified duration is 30/1.045 = 28.71 and the DV01 is 26.700 * 28.71 * 0.0001, which is $0.0767 per $100 of face.

Follow-up: Yields rise to 5.5%. What is the new price by duration alone, and how much does the true price differ because of convexity?

Key concepts: zero coupon bond, modified duration, DV01, rule of 72.