Quants – Probability of returns

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

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%…

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

anybody else have any thoughts?!

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!

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!

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!

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? 😀