Determine which of the following expressions represents the modified duration for a zero-coupon bond that is currently priced at an annual effective yield rate i and an n-year maturity.

• [[math]]n[[/math]]
• [[math]]n(1+i)[[/math]]
• [[math]]n(1+i)^{-1}[[/math]]
• [[math]]\frac{\sum_{t=1}^nt(1+i)^{-t}}{\sum_{t=1}^n(1+i)^t}[[/math]]
• [[math]]\frac{\sum_{t=1}^nt(1+i)^{-t}}{\sum_{t=1}^n(1+i)^{-t}}[[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

[[math]] {D}_{\mathrm{mod}}=-\frac{\frac{d}{d i}(1+i)^{-n}}{(1+i)^{-n}}=\frac{n(1+i)^{-n-1}}{\displaystyle{(1+i)^{-n}}}=n(1+i)^{-1} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.