Complicated Retirement Problem Revisited-growing perpetuity

[jonah]

While viewing a GoogleBooks preview of Shaum’s Outlines in Mathematics of Finance by P. Zima and R. Brown, I came across another application (a financial one at that) of infinite geometric series. This in turn recalled to my mind condition b (i.e. growing geometric perpetuity) of annuity_man’s complicated retirement problem.

Nick and Sharon would like to retire when they are both 50 years old. Nick and Sharon are both 26 years old. Nick is currently employed, but Sharon is trying to become a lawyer. Nick currently has $30,000 in an investment account, which earns him 6% per year compounded semi-annually. Nick would like to contribute to their joint retirement fund every two months until Sharon finishes her studies when they’re both 30. Nick’s contributions would be made at the end of each two month period. From that point Sharon would like to make all of the contributions to their retirement fund. She would also contribute every two months. These contributions would be made at the end of each two month period. During their retirement, they would like to withdraw $15,000 per month at the beginning of each month. They would like the amount they receive to increase by ½% each month. Nick is expected to live until he is 85. Sharon expects to live until she is 90. However, after Nick’s death she’s content with receiving only 50% of what she received while Nick was alive. These payments would be made at the beginning of each month and would continue to grow by ½% each month. During retirement they expect to earn 4% compounded quarterly.

b) Nick would like to establish a scholarship for students. He would like the scholarship to pay $5,000 the first year it was paid. He would like the first scholarship to be paid 5 years after he retires. He would also like the scholarship payment to grow by 3% per year and the payments under the scholarship to go on in perpetuity. To fund both the scholarship and their retirement what will be the amount of the contributions they have to make?

The effective rate of 4% compounded quarterly is 4.060401%. If the scholarship is to earn at this effective rate, then the discounted value, B, of the future scholarship at the end of 5 years after Nick’s retirement, or at the end of the 55th year, is given by
\(\displaystyle B = 5,000 + 5,000\left( {1.03} \right)\left( {1.04060401} \right)^{ - 1} + 5,000\left( {1.03} \right)^2 \left( {1.04060401} \right)^{ - 2} + \cdot \cdot \cdot\)
The expression on the right-hand side is the sum of an infinite geometric progression whose first term is
\(\displaystyle b = 5,000\) and whose common ratio, the absolute value of which is less than 1, is
\(\displaystyle u = \left( {1.03} \right)\left( {1.04060401} \right)^{ - 1}\)
Applying the formula for such a convergent infinite series, we have
\(\displaystyle B = \frac{b}{{1 - u}} \approx \$ 490,665.3285...\)
The discounted value of B at the end of age 50 is
\(\displaystyle B\left( {1.04060401} \right)^{ - 5} \approx \$ 402,122.0568...\)
Adding this discounted value of B on the right-hand side of the three assumptions (interpretations) that I came up with on my third post at http://www.freemathhelp.com/forum/viewtopic.php?f=17&t=29218&start=0, we get:

Assumption A
\(\displaystyle \begin{gathered} 30,000\left( {1 + \tfrac{{.06}}{2}} \right)^{24 \times 2} + R \cdot s_{\left. {\overline {\, {24 \times 6} \,}}\! \right| i_6 } \hfill \\ = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}} + \tfrac{1}{2} \cdot 15,000\left( {1 + g} \right)^{419} \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot \left( {1 + i_{12} } \right)^{ - 420} \hfill \\ + B\left( {1.04060401} \right)^{ - 5} \hfill \\ \Leftrightarrow R \approx \$ 32,725.6565... \hfill \\ \end{gathered}\)

Assumption B
\(\displaystyle \begin{gathered} 30,000\left( {1 + \tfrac{{.06}}{2}} \right)^{24 \times 2} + R \cdot s_{\left. {\overline {\, {24 \times 6} \,}}\! \right| i_6 } \hfill \\ = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}} + 7,500\left( {1 + g} \right)^{420} \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot \left( {1 + i_{12} } \right)^{ - 420} \hfill \\ + B\left( {1.04060401} \right)^{ - 5} \hfill \\ \Leftrightarrow R \approx \$ 32,740.66051... \hfill \\ \end{gathered}\)

Assumption C
\(\displaystyle \begin{gathered} 30,000\left( {1 + \tfrac{{.06}}{2}} \right)^{24 \times 2} + R \cdot s_{\left. {\overline {\, {24 \times 6} \,}}\! \right| i_6 } \hfill \\ = 15,000 \cdot \frac{{r^{420} - 1}}{{r - 1}} + 3,750\left( {1 + g} \right)^{420} \cdot \frac{{r^{60} - 1}}{{r - 1}} \cdot \left( {1 + i_{12} } \right)^{ - 420} \hfill \\ + B\left( {1.04060401} \right)^{ - 5} \hfill \\ \Leftrightarrow R \approx \$ 31,232.75749... \hfill \\ \end{gathered}\)

Note: Whatever happened to annuity_man? Haven't heard a feedback from him since the deluge of opinion on his little problem.

 

[Denis]

Jonah, that same problem was posted a few times over the past years, including here I'm sure.

Quite a few solutions were also posted...perhaps the moderators know how to find them ?

 

[jonah]

I'm sure there was one but I wasn't aware of it. I did a preliminary scan on all the threads on the Finance/Business Math sub-forum several weeks ago and I'm pretty sure I didn't ran across one that dealt with such a problem. Engaging the forum's search and advanced search feature (using the key words infinite series) landed me back to annuity_man's problem and 101 other threads (mostly calculus related).