N this study, frick (1990) clarified the distinction between statistical linear models and analysis of patterns in time (apt). Compared to the linear models approach (which includes particular mathematical models), apt is a straightforward way to create temporal maps and analyze them for recurring patterns of events. Moreover, in apt probabilities of temporal patterns are estimated by relative frequency and duration of those events

You wish to retire in 13 years, at which time you want to have accumulated enough money to receive an annual annuity of $23,000 for 18 years after retirement. during the period before retirement you can earn 9 percent annually, while after retirement you can earn 11 percent on your money. what annual contributions to the retirement fund will allow you to receive the $23,000 annuity? use appendix c and appendix d for an approximate answer, but calculate your final answer using the formula and financial calculator methods.