CFA CFA Level 1 Quants – Probability of returns

Quants – Probability of returns

  • This topic has 9 replies, 3 voices, and was last updated Apr-17 by Snippy.
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    • Snippy
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      I found this question while doing a practice exam and didn’t understand Schweser’s explanation of it.

      I would be very grateful if you can make it clear to me.

      Q. An investment has a mean return of 15% and a standard deviation of returns equal to 10%. If the distributions of returns is approximately normal, which of the following statements is least accurate? The probability of obtaining a return:

      A. less than 5% is about 16%
      B. greater than 35% is about 2.5%
      C. between 5% and 25% is about 95%

      Answer according to Schweser is C. Their explanation being: About 68% of all observations fall within +/- 1 standard deviation of the mean. Thus, about 68% of the values fall between 5 and 25.

      I know how to calculate probability of returns greater than a certain percentage which is why i know B is right. But the rest, i couldn’t figure.

    • lulu123
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      hmm you are right…. maybe Schweser is wrong on this one ?

      anybody else have any thoughts?!

    • lulu123
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      if the explanation was the 68% sentence, then maybe they made a typo in the C) (meant to write 68% lol) because clearly +/- TWO stdevs is 95%…

    • lulu123
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      OMG I FIGURED IT OUT…
      the question says “LEAST ACCURATE!!!!!!”
      so yes C is least accurate hahaha

    • Sophie Macon
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      OK @sidmenon. Just to check that you’re clear about how z tables are shown, i.e. it’s always showing the area below of a Z score (called Area A in the image below). The ‘middle’ of the distribution is 0 at the axis. It’s essential to get this understanding right.

      So, let’s find the Z-score for all 3 answers. Z = (X – mean)/ std deviation

      Z(A) = (5%- 15%)/10% = -1
      Z(B) = (35%-15%)/10% = +2
      Z(C) = we need to look at 25% here, as 5% is found in A already. So Z = (25%-15%)/10% = +1

      So for Z(B), given that the question ask if it’s greater than 35%, because of symmetry and the way the Z tables are shown (area to the ‘left’, as mentioned above), this is equivalent to looking at Z(-2), which is 2.28% (“about 2.5%”), so you’re right on this.

      For Z(A), it’s asking for less than 5%, so all you have to look for is Z(-1), it’s 15.87% based on Z-tables, which is also correct.

      For Z(C), it’s easy to derive based on the logic of symmetry and the way the Z-table is shown (i.e. area of the ‘left’), I have drawn these to illustrate how to calculate it below.

      So for 5%, you already know from answer A that the Z value is -1. For 25%, the Z value is +1. To get the area between 5% to 25%, you should take Z(+1) minus Z(-1). But note that the area on the left and the right of the shaded area (refer to last image below) is the same given the symmetry of a normal distribution. And since you know from A that the area below Z(-1) is 15.87%, it stands to hold that it’s the same on the right unshaded area. Therefore the answer here is 100% – 2*(15.87%) = 68.26%. So C is incorrect.

      Hope this is clear!

    • Snippy
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      @lulu123 When i said B is right, i meant it’s correct so i can eliminate that option.


      @Sophie
      Man, I’ve never understood anything better! You’ve explained in such great detail and put so much effort into this, i’m more than grateful. Thank you so much. You’re the boss 😀

    • Sophie Macon
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      You’re welcome @sidmenon! That’s just the way I understand distributions and it made sense to me drawing it out, plus I’m just a stats nerd. We’re all in to kick Quant’s ass, eh? 😀

    • lulu123
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      technically, u can go even faster without calculating/going to the z-tables

      Since you know that 1 stdev = 68%, 2 = 95%, 3=99.7%, and that the distribution is symmetrical, meaning that if its one stdev away from mean, then half of 68 is 34% –> 34% will lie within the stdev to one side only, leaving 50%-34% = 16% until the end of the curve

      A. less than 5% is about 16%

      mean is 15 so 5% is one stdev to the left
      as I said above, the area remaining after that 1 stdev will be 50% – 34% = 16%
      thus, this is correct

      B. greater than 35% is about 2.5%

      mean is 15 so 35% is 2 stdevs to the right
      2 stdevs is the 95% rule
      meaning that to the right hand-side there is 50%- (95/2) = 2.5%
      alternatively its (100% – 95% )/2 = 2.5%
      thus, this is also correct

      just think in terms of the curve, drawing it often helps visualise what is asked!

    • Sophie Macon
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      Nice one @lulu123, indeed that’s a quicker way to go, especially for such time-constrained exam! My lengthy one was to make sure the fundamentals are covered too. @sidmenon, makes sure you try @lulu123’s advice in the exam for quick elimination!

    • Snippy
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      @lulu123 I like your style 😀

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