The one-year forward rates, deferred t years, are estimated to be:

Year (t)	0	1	2	3	4
Forward Rate	4%	6%	8%	10%	12%

Calculate the spot rate for a zero-coupon bond maturing three years from now.

• 4%
• 5%
• 6%
• 7%
• 8%

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.