Question

You have been investing $165 a month for the last 12 years. today, your investment account is worth $60,508.29. what is your average rate of return on your investments? 13.77 percent 13.80 percent 13.84 percent 13.89 percent 14.03 percent

Answer 1

Given:
Future value, F=60508.29
Monthly payment, A = 165
Compounding period = month
Number of periods, n = 12*12=144
interest per period = i   [ to be found ]

We have the relationship
F=A((1+i)^n-1)/i
but there is no explicit formula for i for given F, A and n.
We need to solve a non-linear equation for the value of i, the monthly interest rate.
One of the ways is to solve it by fixed iteration, i.e. 
1. using the given relation, express i in terms of other parameters.
2. select an initial value of i
3. evaluate i according the equation in step 1 until the value is stable.

Here we will use the relationship to express
i=((60508.29*i)/165+1)^(1/144)-1  [ notice that i is on both sides of = sign ]
using an initial value of i=0.01 (about 1% per month).
Successively, we get
i=((60508.29*0.01)/165+1)^(1/144)-1=0.01075571
i=((60508.29*0.01075571)/165+1)^(1/144)-1=0.011160681, similarly
i=0.0113685
i=0.0114728
i=0.0115246
i=0.0115502
i=0.0115628
i=0.0115690
i=0.0115720
Assuming the above has stablilized, and the APR is 12 time the above value, namely
Annual percentage rate = 0.01157205998210142*12=0.13886=13.89%