SOLUTION (Investment Strategies - first order DDS):
The DDS will be:
a(n+1) = (1+r) a(n) + 100, a(0) = 100.
The solution for this DDS is:
a(n) = (100 r + 100)/r (1 + r)^n - 100/r
This solution can be found in any text on discrete dynamical systems. It can be built from a conjectured form: a(n) = c (1+r)^n + b, where c and be are constants to be determined from the nohomogeneous term 100 and the given initial condition a(0) = 100.
After 12 years at interest rate r we shall have t(r) dollars in our account where:
t(r) = (100 r + 100)/r (1 + r)^12 - 100/r - 9000 r^2,
the last term being the brokerage fee for obtaining interest rate r.
Input :=
t[r_] = (100 r + 100)/r (1 + r)^12 -
100/r - 90000 r^2;
We plot this amount as a function of r.
Input :=
Plot[t[r],{r,.00001,.2}]
Output =
-Graphics-
Thus, if we go for the interest rate of 1.44% we shall be able to attain $1,400 after 12 years with the penalty and interest in effect. We note that we could obtain $1,400 after 12 years with another interest rate - higher though. This is due to the fact that while we earn more interest we shall have to pay a higher brokerage fee.
And this is our fee.
Input :=
fee = 90000 r^2/.sol[[1]]
Output =
513.607