Revision as of 17:19, 20 November 2023 by Admin (Created page with "'''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 display="block"> \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> Mod...")
Exercise
Nov 20'23
Answer
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.
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