A company's liabilities are 20,000 today and 100,000 five years from today. The Macaulay duration of the company's liabilities with respect to the market annual effective yield rate is 3.70 years.

Calculate the modified duration of the company's liabilities, in years.

• 3.26
• 3.31
• 3.52
• 3.65
• 4

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

Solution: B

Let [math]i[/math] represent the effective market annual yield rate and [math]v=\frac{1}{1+i}[/math]. The Macaulay duration is 3.70 years, which is equal to the present-value-weighted times of the liabilities. Therefore, we have

[[math]] \begin{aligned} & 3.70=\frac{20,000(0)+100,000 v^5(5)}{20,000+100,000 v^5}=\frac{25 v^5}{1+5 v^5} \\ & 3.70+18.5 v^5=25 v^5 \\ & 3.70=6.5 v^5 \\ & v=0.89342 \\ & 1+i=1.11929 \end{aligned} . [[/math]]

Modified duration equals Macaulay duration divided by [math](1+i)[/math], so the modified duration is [math]\frac{3.70}{1.11929}=3.30567[/math] years.

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