Revision as of 01:21, 20 November 2023 by Admin (Created page with "The present value of cash flows at an effective interest rate of i is given by the following function: <math display = "block"> P\left(i\right)=100+500\left(1+i\right)^{-3}-1000\left(1+i\right)^{-4},{}~{\mathrm{for}}\,i>0. </math> Determine which of the following expressions represents the modified duration of these cash flows. <ul class="mw-excansopts"><li><math display = "block">\frac{1500(1+i)^{-4}-4000(1+i)^{-5}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}</math></li><li><...")
Nov 20'23
Exercise
The present value of cash flows at an effective interest rate of i is given by the following function:
[[math]]
P\left(i\right)=100+500\left(1+i\right)^{-3}-1000\left(1+i\right)^{-4},{}~{\mathrm{for}}\,i\gt0.
[[/math]]
Determine which of the following expressions represents the modified duration of these cash flows.
- [[math]]\frac{1500(1+i)^{-4}-4000(1+i)^{-5}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
- [[math]]\frac{1500(1+i)^{-3}-4000(1+i)^{-4}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
- [[math]]\frac{-1500(1+i)^{-4}+4000(1+i)^{-5}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
- [[math]]\frac{-1500(1+i)^{-3}+4000(1+i)^{-4}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
- [[math]]\frac{-6000(1+i)^{-5}+20,000(1+i)^{-6}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
Nov 20'23
Solution: A
[[math]]
-\frac{P^{\prime}\left(i\right)}{P(i)}=\frac{1500\left(1+i\right)^{-4}-4000\left(1+i\right)^{-5}}{100+500\left(1+i\right)^{-4}-1000\left(1+i\right)^{-4}}
[[/math]]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.