Revision as of 12:10, 20 November 2023 by Admin (Created page with "'''Solution: B''' The Macaulay duration of Annuity A is <math display = "block"> 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 display = "block"> 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 display = "block"> {\frac{0(1)+1( v)+2( v^{2})+3( v^{3})}{1+ v+ v^{2}+ v^{3}}}=1.369 </math> {{soacopyright | 2023 }}")
Exercise
Nov 20'23
Answer
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.