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# Hull RMFI End of Chapter Question 4.16

#### ericbmoreira

##### New Member
I have tried searching the forums but I cannot figure out where to post a question like this nor if it has been posted before. This is on page 51 of the practice question set.

I'm confused with this answer. I thought the geometric mean would be, ((4th root (1.08*.92*1.12*.88))-1 = -.0052. Since the arithmetic mean is 0 it leads me to believe that the geometric would be -.0052 rather than -.0207.

Thank you.

Question 4.16

Good years are followed by equally bad years for a mutual fund. It earns +8%, -8%, +12%, -12% in successive years. What is the investor's overall return for the four years?

The investor's overall return is 1.08 x 0.92 x 1.12 x 0.88 - 1 = 0.0207 or -2.07% for the four years.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @ericbmoreira Thank you for searching, this question has not yet been posted. As you know, it's a Hull EOC question; we did not write it. I say that because ... unlike most of Hull's questions, this is not good, for the reason you cite. We haven't "vetted" the RM&FI (like we have the primary Hull OF&OD book).

That said, you are totally right here. First, such an imprecise question, as stated, would never be asked on the exam. Second, if the question did want the answer it would need to ask "What is the 5-year simple return?" or "What is the 5-year holding return?" ... however, this would be highly atypical, but it does appears the simple return over five years is -2.07%. Neither of these would be the default. To your point, the best guess is the CAGR or geometric return, for which I also get -0.52% an is verified by (1-0.52%)^5 = 0.974183, which is the terminal value. Even then, a proper question needs to request with: "What is the compound annual growth rate (aka, geometric mean)?" ... even there, I'd be careful to follow Hull's OFOD terminology and aka this a geometric mean, but primarily I would call it a compound annual growth rate (CAGR). I hope that's helpful, thank you for a keen observation!

#### ericbmoreira

##### New Member
Thanks again for your prompt reply. I have found a few other seemingly vague questions, and try to reverse engineer them if needed without spending too much time on them. I value all of the questions from your question sets though, because for me that is the best way to learn the content. I can also tell you that, in my opinion, there would be a much, much greater chance of failing without your content.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @ericbmoreira Thank you, that's nice to hear! As I looked subsequently at more of the EOC questions n RM&FI, I have realized there is a qualitative difference between the question's in Hull's RM&FI and his OF&OD. Perhaps it is because "The level of mathematical sophistication and the way material is presented [in RM&FI] have been managed carefully so that the book is accessible to as wide an audience as possible," or who knows, maybe even different authors wrote the RM&FI questions. The average OF&OD question seems to have a notch or two higher level of rigor. Thank you again!

#### Sixcarbs

##### Active Member
Subscriber
I initially did the geometric mean as well because it was fresh in my head from question 4.7.

I thought, ok, let me be slick here, but no good.

@David- Do you mean to say to the 4th power here not 5th?
"....is verified by (1-0.52%)^5"

And then 1-(1-.52%)^4 gets us the same -2.07% as the answer.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Sixcarbs Yes, I meant ^4 since there are only 4 returns, sorry. Looks like the CAGR should be (1.080 * 0.920 * 1.120 * 0.880)^(1/4) - 1 = -0.005218 = -0.5218%. Thanks,