Graham is the beneficiary of an annuity due. At an annual effective interest rate of 5%, the present value of payments is 123,000 and the modified duration is [math]D_{MOD}[/math].

Tyler uses the first-order Macaulay approximation to estimate the present value of Graham’s annuity due at an annual effective interest rate was 5.4%. Tyler estimates the present value to be 121,212.

Calculate [math]D_{MOD}[/math], the modified duration of Graham’s annuity at 5%.

• 3.67
• 3.75
• 3.85
• 3.95
• 4.04

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

Solution: A

[[math]] 121,212=123,000\bigg(\frac{1.05}{1.054}\bigg)^{D_{MAC}}\Rightarrow D_{MAC}=\frac{\ln(121,212/123,000)}{\ln(1.05/1.054)}=3.851 [[/math]]

Then [math]D_{MOD} = 3.8512 /1.05 = 3.67[/math]

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