Revision as of 08:45, 18 November 2023 by Admin (Created page with "Mike buys a perpetuity-immediate with varying annual payments. During the first 5 years, the payment is constant and equal to 10. Beginning in year 6, the payments start to increase. For year 6 and all future years, the payment in that year is K% larger than the payment in the year immediately preceding that year, where K < 9.2. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. Calculate K. <ul class="mw-excansopts"><li>4.0</li...")
ABy Admin
Nov 18'23
Exercise
Mike buys a perpetuity-immediate with varying annual payments. During the first 5 years, the payment is constant and equal to 10. Beginning in year 6, the payments start to increase. For year 6 and all future years, the payment in that year is K% larger than the payment in the year immediately preceding that year, where K < 9.2. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50.
Calculate K.
- 4.0
- 4.2
- 4.4
- 4.6
- 4.8
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 18'23
Solution: A
[[math]]
\begin{align*}
167.50 &=10a_{\overline{5}|9.2\%}+10(1.092)^{-5}\sum_{t=1}^{c}\left[\frac{(1+k)}{1\cdot092}\right]^{t} \\
167.50 &=38.6955+6.44001{\frac{(1+k)/1.092}{1-(1+k)/1.092}} \\
(167.50-38.6955)[1-(1+k)/1.092] &= 6.4400](1+k)/1.092 \\
128.8045 &= 135.24451(1+k)/1.092 \\
1+k &= 0.0400 \\
k &= 0.0400 \Rightarrow K= 4.0\%
\end{align*}
[[/math]]
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.