I am currently working on this problem and I am having a hard time finding the right number.
A mutual fund advertises that average annual compound rate of returns for various periods ending Dec 31 2005 are as follows: 10 yrs = 13%, 5 yrs = 17%, 2 yrs = 15% and 1 yr = 22%. a), find the average annual compound interest rate of return for the period Jan 1 1996 to Dec 31 2000. b), find the annual rate of return for calendar year 2004.
a), My claim is that from 1996 to 2000 there are 5 conversion periods with 17%, so the number is
$$\sqrt[5]{1.17}-1 \approx 3.199%$$
but the answer is supposedly approximately 9.1%.
b), In the 2004 period, there is only one conversion period, so the return interest should be the same as the claimed interest rate which is 22%.
But the answer is supposedly 8.4%
Can some one help?
Note that the stated returns are for periods ending in 2005 so the 1 year return is for 2005 only and not 2004. Similarly, the 5 year is for the period of 2001-2005 and not 1996-2000. Please be careful and read the dates in the question or else you will be fooled here.
It may help to consider the following interpretation of the data given in annual returns:
Return in 2005: 22%
Return from 2004 to 2005: 15%
Return from 2001 to 2005: 17%
Return from 1996 to 2005: 13%
For a which is asking for the returns from 1996 to 2000: Note that the returns stated are as of the end of 2005 which means the 5 year return given would be 2001-2005 and so to get the returns for 1996-2000 you would have to use the 10 year return and 5 year return to determine the new average. In this case, the 10 year average is 13% and the 5 year is 17% which means that the first 5 years had to be low enough to bring the average down by 4% which would be the simpler way to see a ~9% average return though if you had to you could compute the following:
$\frac{(1.13)^{10}}{(1.17)^5}^{1/5} \approx 1.0913675213675213675213675213675$ which is 9.1%
For b which is asking for the return in 2004:
Same as a in a way where the return is the year before the 1 year return. There is the one year of 22% and the 2 year annual return of 15% where to find the other return, one could just take $\frac{1.15^2}{1.22}-1 \approx 1.0840163934426229508196721311475$ which is 8.4%. You are ignoring the dates in the question. The 1 year return would be just 2005 and the 2 year would be 2004 and 2005 combined. If you want 2004, you have to deduce this which is the part you are missing.