Final answer:
The annual rate of return on the investment is -2.875%, which was calculated using the formula for compound interest rearranged to solve for the annual interest rate (r). The negative value indicates the investment's value has decreased over the 12-year period.
Step-by-step explanation:
To calculate the annual rate of return on an investment, we need to look at how much the value of the investment has changed over time. In this case, the initial investment was $264,500 and it is now valued at $204,000 after a 12-year period. To find the annual rate of return, we use the formula for compound interest, which is A = P(1 + r)^t, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), and t is the time the money is invested for in years.
We rearrange the formula to solve for r as follows:
204,000 = 264,500(1 + r)^12
0.77143 = (1 + r)^12
Now we take the 12th root of both sides to isolate (1 + r):
(1 + r) = 0.77143^(1/12)
1 + r = 0.97125
Subtracting 1 from both sides gives us:
r = -0.02875 or -2.875%
Thus, the annual rate of return is -2.875%, indicating that the investment has decreased in value over the 12-year period.