Practice problem regarding The Expected Future value and Standard Deviation.

CFA Level I Quant
#1

Hi,

I don’t understand the explanation given in the CFA Institute book. I wonder if someone can help solving this problem please?

An exchange rate has a given expected future value and standard deviation.

A. Assuming that the exchange rate is normally distributed, what are the probabilities that the exchange rate will be at least 2 or 3 standard deviations away from its mean?

B. Assume that you do not know the distribution of exchange rates. Use Chebyshev’s inequality (that at least 1 − 1/k2 proportion of the observations will be within k standard deviations of the mean for any positive integer k greater than 1) to calculate the maximum probabilities that the exchange rate will be at least 2 or 3 standard deviations away from its mean.

(Institute 597)

Institute, CFA. 2016 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. CFA Institute, 07/2015. VitalBook file.

#2

Help us understand your confusion.

A. The normal distribution has values of 95 for two standard deviations and 99.7 for 3 standard deviations. So turning those around, you will generally fine 5% of values outside of two standard deviatiosn and .3% of values outside three stnadard deviations.

B. The application of Chebyshev’s inquality for 2 and 3 standard deviations is:

1 - 1/2^2 = 3/4 (so 3/4 of the results should be within 2 standard deviations)

1 - 1/3^2 = 8/9 (and 8/9 of the results should be within 3 standard deviations).

But you want the probability that it will be outside those ranges so you need to subtract by 1:

1/4 of the results should be outside 2 standard devations and

1/9 of the results should be outside 3 standard deviations.

Does this help?