A bond has a modified duration of 8 and a price of 112,955 calculated using an annual effective interest rate of 6.4%. [math]E_{MAC}[/math] is the estimated price of this bond at an interest rate of 7.0% using the first-order Macaulay approximation. [math]E_{MOD}[/math] is the estimated price of this bond at an interest rate of 7.0% using the first-order modified approximation.
Calculate [math]E_{MAC}-E_{MOD}[/math].
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
Solution: E
Modified duration = (Macaulay duration)/(1 + i) and so Macaulay duration = 8(1.064) = 8.512.
[math]E_{MAC}=1\,12,955{\left({\frac{1.064}{1.07}}\right)}^{8.512}=107,676, \quad E_{MOD}=1\;12,955[1-(0.07-0.064)(8)]=107,533.[/math]
Then, [math]E_{M A C} - E_{MOD} = 107, 676 -107,533 = 143[/math]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.