Standard deviation

CFA Level I Quant
#1

Hello everyone!

I’m having trouble understanding this question, can someone please explain? Thank you in advance.

The forecast rate of return from an investment has the following probability distribution: Rate of Return Probability 15% 0.250 20% 0.500 24% 0.125 26% 0.125 The standard deviation of the rate of return is closest to:

  • A) 0.13%.
  • B) 2.50%.
  • C) 3.57%.
#2

Expected Return = 0.25 * 15 + 0.5 * 20 + 0.125 * 24 + 0.125 * 26= 20% (0.2)

variance = 0.25^2 * (0.15-0.2)^2 + 0.5^2 * (0.2 - 0.2)^2 + 0.125^2 * (0.20 - 0.24)^2 + 0.125^2 * (0.20-0.26)^2= 0.0002375

std dev = sqrt(variance) = 0.015411035007422 ==> 1.5%

this does not match any of the answers shown … so not sure

#3

First determine the Expected return = (0.25 * 0.15) + (0.5 * 0.20) + (0.125 * 0.24) + (0.125 * 0.26) = 0.2

To calculate variance of the returns you multiply the probability of each outcome by the deviation between return and expected return squared.

0.25 * (0.15 - 0.2)^2 + 0.5 * (0.2 - 0.2)^2 + 0.125 * (0.24 - 0.2)^2 + 0.125 * (0.26 - 0.2)^2 = 0.0006 + 0 + 0.0002 + 0.0005 = 0.0013

To calculate standard deviation take the square root of variance (0.0013) = 0.00354 or 3.54% so the answer is C. The reason I am slightly of with my calculation is because of rounding errors.

Hope this helps!

#4

I’m getting 3.57%. Is that what the given solution is?

I get this from:

* Getting the population mean of .20 for the returns (just a weighted average).

* Summing the squares of the differences of the samples fromthe mean * the frequency of each to get the population variance, which turns out to be .001275.

* Taking the square root of the population variance to get the population standard deviation, which is .035707.

This is shown (kinda) in Section 7.3 of Reading 7.

Let me know if this is right.

#5

you are right. I multiplied by square of the weight … hence my error.

#6

Thanks!

#7

You’re correct. I just did it and got the same result.

#8

Just a tip if you want to do this quickly on the calculator (given you understand the calculations of course).

Using the TI BA II Plus: Open the DATA worksheet, put in the returns as X values, probabilities as Y values. Open the adjacent STAT worksheet, select 1-V (one variable) and press down arrow and check up on the population standard deviation. Just checked if it worked for your question and it did, but it doesn’t accept decimal values (12.5 in this case), so I rounded one to 13 other to 12, answer is 3.56% (0,01% off due to the rounding).