Annuity A pays 1 at the beginning of each year for three years. Annuity B pays 1 at the beginning of each year for four years. The Macaulay duration of Annuity A at the time of purchase is 0.93. Both annuities offer the same yield rate.

Calculate the Macaulay duration of Annuity B at the time of purchase.

• 1.240
• 1.369
• 1.500
• 1.930
• 1.965

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

Solution: B

The Macaulay duration of Annuity A is

[[math]] 0.93=\frac{0(1)+1( v)+2( v^{2})}{1+ v+ v^{2}}=\frac{ v+2 v^{2}}{1+ v+ v^{2}} [[/math]]

, which leads to the quadratic equation

[[math]] 1.07v^2 + 0.07v -0.93 = 0. [[/math]]

The unique positive solution is v = 0.9. The Macaulay duration of Annuity B is

[[math]] {\frac{0(1)+1( v)+2( v^{2})+3( v^{3})}{1+ v+ v^{2}+ v^{3}}}=1.369 [[/math]]

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